text:
\begin{document}

\maketitle

\abstract{
We study the differential and Riemannian geometry of
  algebras $A$ endowed with an
  action of a triangular
  Hopf algebra $H$ and noncomutativity  compatible with the associated braiding.
The modules of one forms and of braided derivations are modules in a
symmetric ribbon category of $H$-modules $A$-bimodules, whose internal morphisms correspond to tensor fields.
Different approaches to curvature and torsion
  are proven to be equivalent by extending the Cartan calculus
  to left (right) $A$-module connections. The Cartan structure
  equations and the Bianchi identities
  are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary braided symmetric
  pseudo-Riemannian metrics is proven}.

\tableofcontents

\section{Introduction}
Noncommutative Riemannian geometry is an active and interdisciplinary
research field. One one hand it is studied using Connes' approach
and the spectral geometry of the Laplacian (see
\cite{Khalkhali} for a recent review). On the other hand it is studied  algebraically,
starting with a differential calculus on a noncommutative algebra, a
notion of Riemannian metric and addressing the problem of a
Levi-Civita connection.

The interest in the field is also due to
gravity on noncommutative spacetime possibly capturing aspects of a
quantum gravity theory.

Given a noncommutative algebra $A$ and an associated differential
calculus $(\dd,\Omega(A))$, the notion of Levi-Civita connection
relies on the possibility of imposing the metric compatibility
condition $\nabla \g=0$ where $\g$ is a (properly defined symmetric)
element in $\Omega(A)\otimes_A\Omega(A)$.  This implies lifting the connection on
the module $\Omega(A)$ of one forms to the module
$\Omega(A)\otimes_A\Omega(A)$.

The problem of defining connections on tensor products modules is nontrivial. In
the literature it is usually overcome constraining the connection to
be a bimodule connection \cite{D-VM96}. For approaches to
Levi-Civita connections along
these lines see \cite{Majid},
\cite{BGL1}, \cite{BGL2}, \cite{Weber} and references therein. This method leads to constrain the possible
metrics $\g$, typically requiring $\g$ to be in the center of the
bimodule $\Omega(A)\otimes_A\Omega(A)$ ($a\g=\g a$ for all $a\in
A$).
For a selected class of noncommutative algebras on the other hand it is possible to
overcome this contraint on  the metric.
In case of $\mathbb{R}^n$ with Moyal-Weyl noncommutativity the Levi-Civita
connection of an arbitray symmetric metric was constructed in
\cite{NCG1} using a
noncommutative Koszul formula (see also \cite[\S 3.4, \S 8.5]{book}). A similar result holds  on
the noncommutative torus  \cite{Rosenberg}.
These results and those in \cite{Arnlind} rely on the existence
of (undeformed) derivations of the noncommutative algebra $A$
generating the $A$-module of vector fields (dual to that of
one forms). A generalization via local charts  in the deformation
quantization context is in \cite{AC}.

In this paper we consider arbitrary pseudo-Riemmannian metrics $\g$
and avoid the requirement of preferred derivations. We study noncommutative algebras $A$  endowed with a representation of a
triangular Hopf algebra $H$, i.e., $H$-module algebras $A$. If the noncommutativity is compatible with the braiding given
by the triangular structure these algebras are called braided
commutative or quasi commutative. Examples include the noncommutative
torus, the Connes-Landi  spheres \cite{CLandi} and more generally
the noncommutative algebras obtained from Drinfeld twists of
commutative ones. Indeed, this paper is inpired by \cite{NCG2} where the
noncommutative differential geometry associated with Drinfeld twists
was pioneered. The present setting is however
independent form twisting commutative geometries. For
example it applies whenever $A$ is a cotriangular Hopf algebra.

There is a canonical differential calculus on braided commutative
$H$-module algebras $A$ \cite{Weber},
and one does not have to consider
bimodule connections in order to lift connections to tensor product
modules \cite{AS}. We show that in this case the differential and the Riemannian geometry on
finitely generated projective modules can be developped with arbitrary right
$A$-module connections on the $A$-bimodule of one forms $\Omega(A)$.
This is achieved by considering operators acting on $A$-bimodules from
the left and from the right.
For example, the covariant derivative is a composition of a left
connection $\std$ (acting from the right) and an
inner derivative (acting from the left). It is this combination that
leads to the braided Cartan relation for the covariant derivative
$[\stDu,\ii_v]=\ii_{[u,v]}$  allowing to prove the equivalence between
different formulations of the curvature and the torsion tensors.
The relation between (left) connections on vector fields and the
dual (right) connections on one forms further leads to the Cartan
structure equations for curvature and torsion and to the associated Bianchi identities.
Upon considering a pseudo-Riemannian metric, this noncommutative
differential geometry is used to provide a Koszul formula for  metric
compatible torsion free connections, leading to
uniqueness and existence of the Levi-Civita connection.

The algebraic structure underlying this study is that of the
categories of $H$-modules and of relative $H$-modules $A$-bimodules
(or $H$-equivariant $A$-bimodules). Following \cite{AS} and the
sharpened results in \cite{BSS1}, \cite{BSS2}  we recall and develop
the different structures of modules and module maps we need in
noncommutative Riemannian geometry. This clarifies the constructions
and the different general properties needed in the progress of the paper.
For example, left (right) connections are linear maps but are not morphism in the
category ${}^H\MMM$ of $H$-modules, in categorical terms they are internal morphisms. Left connections have different $H$-action form right connections; they are different internal morphisms in
${}^H\MMM$.  The covariant derivative of a left connection on the
other hand is a left $A$-linear map and an internal morphism in the
category $\catm$ of relative $H$-modules $A$-bimodules. Its torsion and
curvature are yet internal morphisms (left $A$-linear maps
transforming under the $H$-adjoint action) in the
subcategory of braided symmetric $H$-modules $A$-bimodules
$\cat$. In particular, we study internal morphisms associated to tensor
products with  finitely generated and projective modules and their duals.
This allows, as in classical differential geometry,  to understand noncommutative tensors fields in all their
different forms, as elements of $A$-bimodules (sections), or as various left (right)
$A$-module maps. The underlying category in this richest case is the ribbon category
of relative $H$-modules $A$-bimodules
finitely generated and projective as $A$-modules (more specifically, it
is a  compact closed category since $H$ is triangular).

This underlying categorical context is presented in
Section 2, togheter with the examples where $A$ is a cotriangular Hopf
algebra and where
it is the Drinfeld twist deformation of the algebra of smooth
functions on a manifold $M$.
In Section 3 the differential and Cartan calculus canonically
constructed in \cite{Weber} is revisited.
In Section 4 we study right
connections and left connections since both are relevant for
uderstanding curvature and torsion. The Cartan formula for covariant
derivatives $[\stDu,\ii_v]=\ii_{[u,v]}$ is established and implies that the
curvature tensor, defined as the square of the connection, can be
equivalently defined via the commutator of covariant derivatives along
vector fields. Similarly, two different definitions of torsion are
shown to be equivalent.
In Section 5 the relation between connections and curvatures
on modules and dual modules is studied, this leads to the Cartan
structure equations for curvature and torsion and to the associated
Bianchi identities.
In the last section existence and uniqueness of the Levi-Civita
connection is proven for any braided symmetric metric.

\section{Hopf algebras, braidings and
  representations}\label{section2}
 We work in the category of $\Bbbk$-modules, with $\Bbbk$ a fixed
 commutative field of charachteristic $\not=2$
or the ring of formal power series in a variable $\hbar$ over such field;
much of what follows holds for a commutative unital ring. The tensor product over $\Bbbk$ is denoted
$\otimes$. Algebras over $\Bbbk$ are assumed associative and unital. Hopf algebras are
assumed with invertible antipode.

In Section \ref{SCMC} we study right (left) $\Bbbk$-linear maps
and $A$-linear maps, hence introducing biclosed (left and right closed)
monoidal categories and, when the Hopf algebra $H$ is triangular, braided biclosed monoidal categories.
In Section \ref{fgp} we continue the study of right (left) $A$-linear
maps considering, for an arbitrary Hopf algebra $H$, the case of
a finitely generated and projective $A$-module, this is the same as a rigid module.
In Section \ref{sect:ribbon} we study rigid $H$-modules with
$H$ triangular, thus obtaining a ribbon category of
relative $H$-modules $A$-bimodules.
The last section is devoted to the two examples of the category of
bicovariant bimodules of a cotriangular Hopf algebra and of
that of noncommutative vector bundles obtained via Drinfeld twist
deformation.

\subsection{Closed monoidal categories and symmetric braidings}\label{SCMC}
We start recalling basic Hopf algebra notions, the  category of
$H$-modules for an arbitrary Hopf algebra $H$ and also for
a triangular Hopf algebra $H$, this is standard material covered in
textbooks (see eg.  \cite{Kassel,Ma95}). In this simple context we
introduce  $\Bbbk$-linear
maps that are not invariant under the $H$-action, they come with two
different $H$-actions structuring them as $\Bbbk$-linear maps acting
from the right or from the left.
Their categorical interpretation, as internal-hom morphisms, is also discussed. They structure the monoidal category of $H$-modules
${}^H\MMM$ as a biclosed monoidal category.
If $H$ is a triangular Hopf algebra then $({}^H\MMM, \otimes, {}_\Bbbk\hom,\hom_\Bbbk)$ is furthermore a
braided symmetric biclosed monoidal category and there is a
tensor product $\otimes_\RR$ of internal morphisms.

Given an $H$-module algebra $A$ we then study relative $H$-modules
$A$-bimodules ($H$-equivariant $A$-bimodules)
and the associated right $A$-linear maps and
left $A$-linear maps (internal-hom morphisms). This is the biclosed
monoidal category $(\catm, \otimes_A,{}_A\hom, \hom_A)$. If $H$ is triangular and the product in $A$
is compatible with the braiding, restricting to modules where the
$A$-bimodule structure is compatible with the braiding (braided
symmetric $A$-bimodules)
we have the biclosed  monoidal subcategory  $(\cat  \otimes_A,{}_A\hom, \hom_A)$, which is braided symmetric.

This chain of results holds as well when we consider a graded algebra
instead of $A$, and graded modules.

\subsubsection{Modules over a Hopf Algebra}
Let $H$ be a Hopf algebra $(H,\mu,\eta,\Delta,\varepsilon, S)$ over $\Bbbk$.
We denote by $^{H}\MMM^{}$ the category of left $H$-modules, where objects
 in ${}^{H}\MMM^{}$ are $\Bbbk$-modules $V$ with a left
$H$-action $\ra : H\otimes {V}\to {V}$, while morphisms in ${}^H\MMM$
are $\Bbbk$-module maps $f: V\to W$ that are $H$-equivariant,  i.e.,
\begin{equation}\label{eqn:Hequivariance}
 \xi\ra f(v) =f(\xi\ra v) ~,
\end{equation}
for all $\xi\in H$ and $v\in {V}$; we write $f\in \Hom^{}_{{}^{\,H}\!\MMM}(V,W)$.
In this paper $H$-modules will be always left $H$-modules and will be
simply called $H$-modules.

Since $H$ is a  bialgebra ${}^H\MMM$ is a (strict)  monoidal
category.
Given two $H$-modules $V$ and $W$
their tensor product $V\otimes W$
is an $H$-module  with $H$-action
\begin{equation}
\ra: H \otimes {V} \otimes {W} \longrightarrow  {V} \otimes {W} ~,
\qquad h \otimes v \otimes w \longmapsto h\ra (v\otimes w):=(h_{(1)} \ra v) \otimes (h_{(2)} \ra w)~,
\end{equation}
where we have used the Sweedler notation $\Delta(h) = h_{(1)}\otimes h_{(2)}$ (with summation understood)
for the coproduct of $H$.
The tensor product of two morphisms  in ${}^{H}\MMM^{}$, $f: V \to V^\prime,~ g: W\to
W^\prime$ is the morphism in ${}^{H}\MMM^{}$ defined by
$f\otimes g:
V\otimes W\to V'\otimes W', v\otimes w\mapsto f(v)\otimes
g(w)$. The tensor product functor $\otimes$ is associative.
The unit object in ${}^H\MMM$ is $\Bbbk$ with left $H$-action given by the counit of $H$,
$\ra : H\otimes \Bbbk\to \Bbbk\,,~h\otimes \lambda \mapsto \epsilon(h)\,\lambda$.

Since the bialgebra $H$ is  a Hopf algebra,  ${}^H\MMM$ is a closed monoidal category.
For any $V,W$ in ${}^H\MMM$,  we denote by
$\hom_\Bbbk(V,W)$ in ${}^H\MMM$ the $\Bbbk$-module
${\rm Hom}_{\Bbbk}(V,W)$ of $\Bbbk$-linear maps  $L : V\to W$
 equipped with the  adjoint $H$-action
\begin{equation}\label{adjact}
\RA_{\,} : H \otimes^{} {\hom_\Bbbk}^{}\big({V}, {W}\big) \longrightarrow
 {\hom_\Bbbk}^{}\big({V}, {W}\big)~, ~~
h \otimes L \longmapsto h\RA L:=h_{(1)}\!\!\: \ra\,\circ \,L \,\circ\, S(h_{(2)})\ra ~\!,
\end{equation}
i.e., $(h\RA L)(v)= h_{(1)} \ra (L(S(h_{(2)})\ra v))$. Given morphisms $f^\op: V\to V', g: W\to W'$ (where $f^\op: V\to V'$ is
just $f: V'\to V$ thought as a morphisms in the opposite category
$({}^H\MMM)^\op$) we have the morphism

\begin{equation}\label{eqn:internalhommorph}
{\hom_\Bbbk}^{}(f^\op,g): {\hom_\Bbbk}^{}\big({V},{W}\big) \longrightarrow
{\hom_\Bbbk}^{}\big({V}^\prime,{W}^\prime\, \big)~, \qquad L \longmapsto g \circ L \circ f~.
\end{equation}
This way we have defined the so-called internal-hom functor
$
{\hom_\Bbbk}: \big({}^H_{}\MMM^{}\big)^\op \times {}^H_{}\MMM^{} \longrightarrow {}^H_{}\MMM^{}~.
$
This is compatible with the tensor product functor $\otimes$, indeed since any
$H$-equivariant map $f:V\otimes W\to Z$ can be considered as an
$H$-equivariant map $\zeta(f): V\to \hom(W,Z)$ via $\zeta(f)(v):=f(v,
\mbox{-})$, we have that the functor $\mbox{-}\otimes W$ is left adjoint to $\hom_\Bbbk(W,\mbox{-})$,
thus $({}^H\MMM, \otimes, \hom_\Bbbk)$ is a
closed  monoidal category.

The submodule $\hom_\Bbbk\!\!\:(V,W)^H\subset \hom_\Bbbk(V,W)$ of
$H$-invariant elements, i.e., $h\ra L=\varepsilon(h)L$ coincides with
that of $H$-equivariant maps $V\to W$ and is hence indentified with $\Hom_{{\!\:}^{H\!\!\:}\MMM}(V,W)$.
\\

Let $V,W$ be modules in ${}^H\MMM$, we can define another
$H$-adjoint action,
$\RA^\cop$,
on the $\Bbbk$-module ${\rm Hom}_{\Bbbk}(V,W)$ of $\Bbbk$-linear maps
$V\to W$. We denote
by ${}_\Bbbk\hom(V,W)$ the $\Bbbk$-module ${\rm Hom}_{\Bbbk}(V,W)$
with $H$-action $\RA^\cop$ defined by
\begin{equation}\label{copadjact}
\RA_{\,}^\cop : H \otimes^{} {{\,}_\Bbbk\hom}^{}\big({V}, {W}\big) \longrightarrow
 {{}_\Bbbk\hom}^{}\big({V}, {W}\big)~\!, ~~
\xi_{}\otimes \tilde L \longmapsto \xi\RA^\cop \tilde L:=\xi_{(2)}
\ra \, \circ \,\tilde L \,\circ\, S^{-1}(\xi_{(1)}) \ra ~\!,
\end{equation}
$$
~~~~~(h\ra^\cop\tilde L)(v)=\xi_{(2)}
\ra (\tilde L(S^{-1}(\xi_{(1)}) \ra v))~\!.
$$
This gives the monoidal structure  $({}^H\MMM, \otimes,
{}_\Bbbk\hom)$, with the functor  $V\otimes \mbox{-}$ that is  left
adjoint to ${}_\Bbbk\hom(V,\mbox{-})$ (via $\zeta(f): W\to {}_\Bbbk\hom(V,Z)$,
$\zeta(f)(w):=f(\mbox{-},w)$,  for any  $f\in
\Hom^{}_{{}^{\,H}\!\MMM}(V\otimes W,Z)$).

 While $\Bbbk$-linear maps
$L\in \hom_\Bbbk(V,W)$ naturally act from the left, indeed the $\RA$ adjoint
action satisfies, for all $\xi \in H$, $v\in V$, $\xi\ra
(L(v))=(\xi_{(1)}\RA L)(\xi_{(2)}\ra v)$, $\Bbbk$-linear maps
$\tilde L\in {}_\Bbbk\hom(V,W)$ naturally act from the right, indeed the $\RA^\cop$ adjoint
action satisfies,
\begin{equation}\label{copadjact2}
\xi\ra
(\tilde L(v))=(\xi_{(2)}\RA^\cop \tilde L)(\xi_{(1)}\ra v)~,
\end{equation}
 that, evaluating
$\tilde L$ on $v$ from  the right, reads $\xi\ra((v)(\tilde L))=(\xi_{(1)}\ra
v) (\xi_{(2)}\RA^\cop \tilde L)$.

Summarizing, associated with the Hopf algebra $H$, we have the biclosed monoidal category $$({}^H\MMM, \otimes, \hom_\Bbbk, {}_\Bbbk\hom)~.$$

\subsubsection{Modules over a triangular Hopf Algebra}
Let now $H$ be a triangular Hopf
algebra with universal
$\mathcal{R}$-matrix $\mathcal{R}\in H\otimes H$ .
We recall that it satisfies
$$
\Delta^{\cop}(\xi)
=\mathcal{R}\Delta(\xi)\mathcal{R}^{-1}
\text{ for all }\xi\in H,
$$
$$
    (\Delta\otimes\mathrm{id})(\mathcal{R})
    =\mathcal{R}_{13}\mathcal{R}_{23}
   ~,~~
    (\mathrm{id}\otimes\Delta)(\mathcal{R})
    =\mathcal{R}_{13}\mathcal{R}_{12}
$$
and the triangularity condition $\mathcal{R}_{21}=\mathcal{R}^{-1}$.
Because of the triangular structure the  monoidal category ${}^H\MMM$
is braided symmetric (also called symmetric):
the braiding $\tau^{}:=\tau(\RR)$ is the natural isomorphism $\tau: \otimes^{} \Rightarrow \otimes^\op$
with components defined by,
\begin{equation}
\tau^{}_{V,W}: V \otimes^{} W \longrightarrow W \otimes^{} V ~, \qquad v \otimes^{} w
\longmapsto

\big(\bar R^{\alpha} \ra w\big) \otimes^{} \big(\bar R_{\alpha} \ra
v\big)

\end{equation}
where we used the notation $\mathcal{R}=R^\alpha\otimes R_\alpha$,
$\mathcal{R}^{-1}=\bar R^\alpha\otimes \bar R_\alpha$.
${}^H\MMM$ is braided symmetric because for
all $V, W$, $\tau^{}_{W,V}\circ \tau^{}_{V,W}=\id_{V\otimes W}$, hence
$\tau$ provides a representation of the permutation group.
With slight abuse of notation we  shall frequently write $\tau$ for $\tau^{}_{V,W}$.

The functors $\,\mbox{-}\otimes W$ and $W\otimes \mbox{-}$ in this
triangular case are naturally isomorphic via the
braiding; correspondingly, the two internal-hom functors $\hom_\Bbbk$
and  ${}_\Bbbk\hom$ are naturally isomorphic via the family of isomorphisms,
for all $V,W\in {}^H\MMM$,

\begin{equation}\label{LefttoRight}
{\cal D}_{V,W}:\hom_\Bbbk(V,W) \longrightarrow {}_\Bbbk\hom(V,W)~,~~
L\mapsto {\cal D}^{}_{V,W}(L):=
(\oR^\al\ra L) \circ \oR_\al\ra
\end{equation}
with inverse given by (cf. \cite[\S 3.2]{AS})
\begin{equation}\label{RighttoLeft}
~{\cal D}^{-1}_{V,W}: {}_\Bbbk\hom(V,W)\longrightarrow  \hom_\Bbbk(V,W)~, ~~\tilde L\mapsto
{\cal D}^{-1}_{V,W}(\tilde L)=(R^\al\RA^\cop \tilde L)\circ R_{\al\,}\ra~.
\end{equation}
Here we just prove $H$-equivariance. For all $\xi\in H$,
the equality
$h\ra {\cal D}^{-1}_{V,W}(\tilde L)=~{\cal
  D}^{-1}_{V,W}(h \ra^\cop\tilde L)$
 is equivalent to
$\xi_{(1)}\RA((R^\al\RA^\cop\tilde L)\circ
R_\al\ra)\circ \xi_{(2)}\ra=(R^\al\RA^\cop(\xi_{(1)}\RA^\cop\tilde
L))\circ R_\al\ra\circ\xi_{(2)}\ra$. Using the action properties of
$\RA^\cop$ and $\ra$
and quasi-cocommutativity, $\Delta^{\cop}(\xi)
=\mathcal{R}\Delta(\xi)\mathcal{R}^{-1}$, both members of this
equation are shown to equal
$\xi\ra\circ ((R^\al\RA^\cop\tilde L)\circ
R_\al\ra)$.
\\

Summarizing, when the Hopf algebra $H$ is triangular
 $$({}^H\MMM, \otimes, \hom_\Bbbk, {}_\Bbbk\hom)$$
is a braided symmetric biclosed  monoidal category.
In a braided closed monoidal category we can evaluate, compose and
consider tensor products not just of morphisms but also of internal
morphisms. For quasitriangular Hopf algebras and hence for triangular
Hopf algebras, internal morphisms  evaluation and composition is the usual evaluation of
$\Bbbk$-linear maps on  $\Bbbk$-modules. The tensor
product of internal morphisms in ${}^H\MMM$ on the other hand
differs from that of the category of $\Bbbk$-modules
(cf. \cite[Corollary 9.3.16]{Ma95}).

Given $\Bbbk$-linear maps $L \in\hom_\Bbbk(V, W)$, $L' \in \hom_\Bbbk(V', W')$ the tensor product $L\otimes_{\mathcal
  R} L' $ is the $\Bbbk$-linear map
\begin{equation}\label{otimesR}
L\otimes_\RR L' := (L\circ \bar R^\alpha\ra\,)\otimes (\bar R_\alpha\RA
L')\in \hom_\Bbbk(V\otimes V', W\otimes{W'})~,
\end{equation}
i.e., for all $v\in V,w\in W,~ (L\otimes_\RR L')(v\otimes w)=L(\oR^\al\ra
v)\otimes (\oR_\al\RA L')(w)$. It is this associative tensor product that is
compatible with the $H$-module structure:
$\xi \ra
(L\otimes_\RR L')=\xi_{(1)}\ra L\otimes_\RR  \xi_{(2)}\ra L'$.
From the definition it follows that
\begin{equation}\label{POQPcircQ}
L\otimes_\RR L'= (L\otimes \id)\circ (
\oR^\alpha\ra\;\otimes \,\oR_\alpha\RA L')
=(L\otimes_\RR \id)\circ (\id\otimes_\RR L')~.
\end{equation}
While
$L\otimes_\RR\id=L\otimes \id$, we have $
\id\otimes_\RR L'=
\oR^\alpha\ra\;\otimes \,\oR_\alpha\RA L'=\tau\circ (L'\otimes \id ) \circ
\tau^{-1}~.$

Similarly, it can be proven that given $\Bbbk$-linear maps $\tilde L \in{}_\Bbbk\hom(V, W)$,
$\tilde L'\in {}_\Bbbk\hom(V', W')$ we have the corresponding tensor
product $\tilde L\,\tilde\otimes_\RR \,\tilde L' $
\begin{equation}\label{tildeotimesR}
\tilde L\,\tilde\otimes_\RR\, \tilde L' :=  (\bar R^\alpha\RA^\cop
\tilde L)\otimes (\tilde L'\circ \bar R_\alpha\ra\,)\in
{}_\Bbbk\hom(V\otimes  V',W\otimes W')~,
\end{equation}
i.e., for all $v\in V$, $v'\in V'$, $(\tilde L\,\tilde\otimes_\RR \,\tilde
L') (v\otimes v')= (\oR^{\alpha}\RA^\cop\tilde L)(v) \otimes \tilde L'
(R_\alpha\ra v')$. This tensor product is associative and compatible
with the $H$-adjoint action $\RA^\cop$, indeed we have
$\xi\RA^\cop (\tilde L\,\tilde\otimes_\RR \tilde L') =(\xi_{(1)}\RA^\cop
\tilde L)\,\tilde\otimes_\RR (\xi_{(2)}\RA^\cop \tilde L')$.
\\

\subsubsection{Algebras and bimodules}\label{ALGBIMOD}
Let $H$ be a Hopf algebra. A left $H$-module algebra $A$ is an algebra with a compatible
$H$-module structure,
$$
\xi\rhd(a b)=(\xi_{(1)}\rhd a)(\xi_{(2)}\rhd b) ~,~~
\xi\rhd 1_{A}=\epsilon(\xi)1_{A}
$$
for all $\xi\in H$ and $a,b\in {A}$.
We denote by $^{H}_{\AA}\MMM^{}_\AA$ the category of
relative $H$-modules $A$-bimodules (or $H$-equivariant $A$-bimodules).
An object $V$ in $^{H}_{\AA}\MMM^{}_\AA$ is an
$A$-bimodule with a compatible $H$-module structure, i.e., $H\otimes V\to
V$, $\xi\otimes v\mapsto \xi \ra v\,;~  \xi \ra av =(\xi_{(1)}\ra a)(\xi_{(2)}\ra v)$
and similarly for the right $A$-module structure.
Morphisms in
$^{H}_{\AA}\MMM^{}_\AA$ are $H$-equivariant maps that are also
$A$-bimodule morphisms.

If $V,W$ are modules in $\catm$ also $\hom_\Bbbk(V,W)$ and
${}_\Bbbk\hom(V,W)$ are modules in $\catm$  with $H$-action as in
\eqref{adjact} and \eqref{copadjact} respectively, and with  $A$-bimodule
structure given via the left $A$-module structures of $V$
and $W$ for  $\hom_\Bbbk(V,W)$ and the right $A$-module structures of
$V$ and $W$  for ${}_\Bbbk\hom(V,W)$; for all $a\in A, v\in V, L\in
\hom_\Bbbk(V,W), \tilde L\in {}_\Bbbk\hom(V,W)$,
\begin{equation}
\label{aLvLav}
(aL)(v)=a(L(v))~,~~(La)(v)=L(av)~,
\end{equation}
\\[-2.6em]\begin{equation}\label{AhomAA}
(\tilde L a)(v)=\tilde
L(v)\:\!a~,\,\,\,\,~~(a\tilde L)(v)=\tilde L(va)~.
\end{equation}
Let  $\hom_A(V,W)\subset \hom_\Bbbk(V,W)$ be the  submodule in $\catm$ of right $A$-linear maps: for all $a\in A$, $L(va)=L(v)a$,
and let
${}_A\hom(V,W)\subset {}_\Bbbk\hom(V,W)$ be the submodule in $\catm$
of left $A$-linear maps: for all $a\in A$, $\tilde L(av)=\tilde a\tilde L(v)$.

Associated with $\hom_A(V,W)\subset \hom_\Bbbk(V,W)$  and ${}_A\hom(V,W)\subset {}_\Bbbk\hom(V,W)$  we have the  functors $\hom_A: (\catm)^\op\times \catm\to \catm$ and
${}_A\hom: (\catm)^\op\times \catm\to \catm$, with the action on morphisms
$(f^\op,g)$ in $(\catm)^\op\times \catm$ as in
\eqref{eqn:internalhommorph}.
The functor $\,\mbox{-}\otimes_A W$ is left adjoint to
$\hom_A(W,\mbox{-})$ \cite{BSS1},
and similarly,
 $V\otimes_A \mbox{-}$ is left
adjoint to ${}_A\hom(V,\mbox{-})$,
 thus $$(\catm, \otimes_A, \hom_A,{}_A\hom)$$ is a
biclosed  monoidal category.
\\

When $H$ is triangular we consider $\AA$ to be braided
symmetric (also called symmetric or quasi commutative), for all $a,b\in \AA$,
$$ab=(\bar R^\alpha\ra b)(\bar R_\alpha \ra a)~.$$
Similarly, $V$ in $^{H}_{\AA}\MMM^{}_\AA$  is braided symmetric if $$a\:\!v =
(\oR^{\al}\ra v )  (\oR_{\al}\ra a)~.$$

We denote by $\cat$ the
full subcategory of braided symmetric modules in $^{H}_{\AA}\MMM^{}_\AA$.
Let $V,W$ be modules in $\cat$, the modules $V\oA W$, $\hom_A(V,W)$
and ${}_A\hom(V,W)$ are also in $\cat$,
for example it is easy to see that for all $a\in A$, $L\in \hom_A(V,W)$,  $\tilde L\in
{}_A\hom(V,W)$, $La=(\oR^\al \ra a)(\oR_\al \RA L)$ and $\tilde La=(\oR^\al \ra
a)(\oR_\al \RA \tilde L)$.
Extending to the biclosed case the results of \cite{BSS1} we have that
$$(\cat, \otimes_\AA, \hom_A, {}_A\hom)~$$ is a full closed monoidal
subcateogry of $(\catm, \otimes_A, \hom_A. {}_A\hom)$ which  is braided
symmetric.

The braiding is induced from that in ${}^H\MMM$ and the  isomorphisms in \eqref{LefttoRight} restrict to
isomorphisms
\begin{equation}\label{Dcallr}
{\cal D}_{V,W}: {}_A\hom(V,W)\to\hom_A(V,W)
\end{equation}
 in $\cat$,
thus proving that ${}_A\hom$ and $\hom_A$ are naturally isomorphic
functors (cf. \cite[\S 5.6]{AS}).
\\

Since $(\cat, \otimes_\AA, \hom_A)$ is a braided closed monoidal
category we have the tensor product of internal morphisms, that with slight abuse of notation we
still denote $\otimes_\RR$. Indeed, similarly to the braiding, this can be
seen as induced from the tensor product of internal morphisms  in
$({}^H\MMM, \otimes, \hom_\Bbbk)$. Let $L\in\hom_A(V,W)\subset \hom_\Bbbk(V,W)$ and $L'\in
\hom_A(V',W')\subset  \hom_\Bbbk(V',W')$, then
$L\otimes_\RR L'\in
\hom_\Bbbk(V\otimes V',W\otimes W')$, as defined in \eqref{otimesR}
is trivially right $A$-linear and
induces a well defined right $A$-linear map
in $\hom_A(V\otimes_A V',W\otimes_A W')$ that we still denote
$L\otimes_\RR L'$  (cf. \cite[Theorem 5.16]{AS}). Associativity
is straightforward. Moreover, for each quadruple
$V,W,V',W'$ of modules in $\cat$ the map
$${\otimes_\RR}^{}_{{V,W,V',W'}}:
\hom_A(V,W)\otimes_A\hom_A(V',W')\to\hom_A(V\otimes_A
V,W\otimes_A W')~,~~L\otimes_AL'\mapsto L\otimes_\RR L'$$ is a morphism in $\cat$
(cf. \cite[\S 5.6]{BSS1}, \cite[Proposition 9.3.13]{Ma95}).
\\

Similarly, in $(\cat, \otimes_\AA, {}_A\hom)$ the tensor product of
internal morphisms is denoted $\tilde\otimes_\RR$ and can be seen as induced from \eqref{tildeotimesR}. Here too for each quadruple
$V,W,V',W'$ of modules in $\cat$ we have that
$${\tilde\otimes_\RR}{}^{\phantom{J_i}}_{{V,W,V',W'}}:
{}_A\hom(V,W)\otimes_{\!\!\:A}{}_{\!\:A}\hom(V',W')\to{}_A\hom(V\otimes_A
V,W\otimes_A W')~,~~\tilde L\otimes_A\tilde L'\mapsto \tilde
L_{\,}\tilde\otimes_\RR\tilde L'$$ is a morphism in $\cat$.

\subsubsection{Graded algebras and bimodules}
The results of the previous subsection
can be extended to the case of $\mathbb{Z}$-graded modules
$V=\bigoplus_{n\in \mathbb{Z}}V^n$.
We consider $H$ to be
$\mathbb{Z}$-graded and nontrivial only in degree zero. Let $\OMM$ be
a graded algebra and an $H$-module algebra, it
is graded braided symmetric if
$$\theta\wedge\theta'=(-1)^{|\theta|_{}|\theta'|}(\bar R^\alpha\ra \theta')\wedge (\bar R_\alpha
\ra \theta)~,$$ where $\wedge$ denotes the product in $\OMM$ and
$\theta,\theta'$ are arbitrary elements in $\OMM$ of  homogeneous degree $|\theta|$ and $|\theta'|$.
Correspondingly,  $\CatM$ denotes the category of $\mathbb{Z}$-graded
modules that are relative
$H$-modules $\OMM$-bimodules (with  grade compatible $\OMM$-module
actions)  and that are graded braided
symmetric:
$V=\bigoplus_{n\in \mathbb{Z}}V^n$ is in  $\CatM$ if
$$\theta\,v =
(-1)^{|\theta|_{}|v|}(\oR^{\al}\ra v )  (\oR_{\al}\ra \theta)~,$$ where
$|v|$ is the degree of the homogeneous element $v\in V$.

The category $\CatM$ is monoidal with tensor product $\otimes_\OMM$. It is also
closed, indeed first observe that for each $V,W$ in $\CatM$, we have that ${}_\Bbbk\hom(V,W)$
and $\hom_\Bbbk(V,W)$ are naturally graded $H$-comodules. Then define
$$
{}_\OMM\hom(V,W)
$$
to be the graded $H$-submodule of ${}_\Bbbk\hom(V,W)$ spanned by
{\sl{graded}} left $\OMM$-linear maps; these are maps
 $\tilde L\in  {}_\Bbbk\hom(V,W)$ of homogenous degree $|\tilde L|$
 (i.e. $\tilde L:V^n\to
 W^{n+|\tilde L|}$, $n\in \mathbb{Z}$) such that
\begin{equation}\label{gradedOMlinear}
\tilde L(\theta v)=(-1)^{|\tilde L||\theta|}\theta \:\!\tilde
 L(v)~.
\end{equation}
Similarly, $\hom_\OMM(V,W)$ is the module of right $\OMM$-linear
 maps, $L(v\theta)=L(v)\theta$.
The modules $ {}_\OMM\hom(V,W)$  and $\hom_\OMM(V,W)$ are  in $\CatM$; their $\OMM$-bimodule
structure reads, for all $L\in \hom_\OMM(V,W)$, and for all $\tilde L\in {}_\OMM\hom(V,W)$,  $\theta\in \OMM$, $v\in V$,
respectively of homogenous degree
$|\tilde L|$, $|\theta|$ and $|v|$,

\begin{equation}\label{bimodofOM}
\begin{split}&(\theta L)(v)=\theta (L(v))~,~~~~~~~~~~~~~(L\theta)(v)=L(\theta v)~,\\[.4em]
&(\tilde L \theta)(v)=(-1)^{|\theta||v|}\tilde
L(v)\!\:\theta~,~~~~(\theta\tilde L)(v)=(-1)^{|\theta|(|\tilde L|+|v|)}\tilde L(v\theta)~.
\end{split}
\end{equation}
This defines the functors $\hom_\OMM: (\CatM)^\op\times \CatM\to \CatM$ and
${}_\OMM\hom: (\CatM)^\op\times \CatM\to \CatM$, where their action on
morphisms
$(f^\op,g)$ in $(\CatM)^\op\times \CatM$ is as in \eqref{eqn:internalhommorph}.

Similarly to the ungraded case we have the braided symmetric biclosed monoidal
category
$$(\CatM,\otimes_\OMM, \hom_\OMM, {}_\OMM\hom)$$
with tensor product of internal morphisms respectively denoted
$\otimes_\RR$ and $\tilde\otimes_\RR$.

\subsection{Finitely generated projective modules and their duals}\label{fgp}
In this section $H$ is a Hopf algebra  (no
triangularity structure is assumed).  We study finitely generated and
projective left (right) $A$-modules in $\catm$ and their duals. It is known that finitely
generated and projective $\Bbbk$-modules are rigid modules. Similarly, finitely generated
and projective right (left) $A$-modules in $\catm$ are rigid
modules in $\catm$.  We continue the study of internal morphisms  proving key canonical isomorphisms
for internal morphisms in $\catm$ arising from tensor products with a rigid
module (cf.~{\it 2$\,$} in Theorem \ref{Thmfgp123}).

Among the various equivalent definitions of finitely generated
projective module (see e.g.~the monograph
\cite{LamBook}) we use the convenient characterization in terms of a pair of dual bases
\begin{lemma}  ({\it Dual Basis Lemma}).
 Let $A$ be an algebra.
A left $A$-module $\Gamma$ is
finitely generated and projective if and only if there exists a family of elements
$\lbrace s_i\in \Gamma:i=1,\dots,n\rbrace$ and left $A$-linear maps
$\lbrace {}^{*\!}s^i \in\stG
:={_A}\Hom(\Gamma,A):i=1,\dots,n \rbrace$
with $n\in\mathbb{N}$, such that for any $s\in \Gamma$ we have
(sum over repeated indices understood)
\begin{equation}\label{astsisi}
 s= {}^{*\!}s^i(s)\, s_i~.
\end{equation}
A right $A$-module $\Sigma$  is
finitely generated and projective if and only if there exists a family of elements
$\lbrace \sigma^i\in \Sigma :i=1,\dots,n\rbrace$ and right $A$-linear maps
$\lbrace \sigma_i^*\in \Sigma^*:=\Hom_A(\Sigma,A):i=1,\dots,n \rbrace$
with $n\in\mathbb{N}$, such that for any $\sigma\in \Sigma$ we have
\begin{equation}
 \sigma=\sigma^i\, \sigma^*_i(\sigma)~.
\end{equation}
\end{lemma}
The set $\lbrace s_i, {}^{*\!}s^i:i=1,\dots,n \rbrace$ is loosely
referred to a ``pair of dual bases'' for the left $A$-module $\Gamma$,
even though $\lbrace s_i\rbrace$ is just a generating set of $\Gamma$
and not necessarily a basis. Similarly $\lbrace \sigma^i,
{\sigma}{}^*_i:i=1,\dots,n \rbrace$ is a pair of dual bases for the right $A$-module $\Sigma$.
\\

The dual $\stG :={_A}\Hom(\Gamma,A)$ of a finitely generated and projective left $A$-module $\Gamma$ is a finitely
generated and projective right $A$-module, with right $A$-action
 as in \eqref{AhomAA}. Moreover, the dual $({\stG})^{*_{}}
 :=\Hom_A(\stG,A)$  of the dual is a left $A$-module canonically
identified with the original module $\Gamma$. Similarly, we have the
left $A$-module
$\Sigma^*:=\Hom_A(\Sigma,A)$ dual to the right $A$-module $\Sigma$ and the canonical
identification ${}^*(\Sigma^*):= {}_A\Hom(\Sigma^*,A)\simeq
\Sigma$. We state these properties for left $A$-modules (the proof can be easily derived from e.g. ~\cite[\S 2B]{LamBook}).
\begin{proposition}\label{propo:fgpmodprop}
 Let $\Gamma$ be a finitely generated and projective left
 ${A}$-module. Denote by
$\lbrace s_i,\sts^i:i=1,\dots,n \rbrace$ a pair of dual bases. For any $s\in \Gamma$, let
$\iota (s) \in (\stG)^*:= \Hom_A(\stG,A)$ be defined by
$\iota(s)(\sts) := \sts(s)$, for all $\sts \in \stG$. We have
\begin{enumerate}
 \item[{\it 1.}$\phantom i\!\!$] $\lbrace \sts^i,\iota (s_i):i=1,\dots,n\rbrace$ is a pair of dual bases for $\stG$,
 \item[{\it 2.}$\phantom i\!\!$] $\stG$ is a finitely generated and projective right $A$-module,
 \item[{\it 3.}$\phantom i\!\!$] The canonical map $\iota: \Gamma\to ({\stG})^{*_{}}\,,~s\mapsto
  \iota({s})$ is an isomorphism of left $A$-modules.
\end{enumerate}
\end{proposition}
If we consider modules in $\catm$, Proposition
§\ref{propo:fgpmodprop} holds in $\catm$. Let  $\Gamma$ be a module in
$\catm$ that is finitely generated and projective as left $A$-module, then
\begin{enumerate}
\item[{\it  1$'\!.$}] $\stG:={}_A\hom(\Gamma, A)$
is in $\catm$ and  is finitely generated and
projective as right $A$-module,
\item[{\it 2$\:\!'\!.$}]
$({\stG})^{*_{}}
 :=\hom_A(\stG,A)$
is in $\catm$ and is finitely
generated and projective as  left $A$-module,
\item[{\it 3$\!\:'\!.$}] The canonical
map $\iota : \Gamma\to ({\stG})^{*_{}}\,,~s\mapsto \iota(s)$ is an isomorphism
in $\catm$.
\end{enumerate}

We further recall that for finitely generated and projective left
$A$-modules $\Gamma$ and $A$-bimodules $W$ there are
isomorphisms ${}_A\Hom(\Gamma,W)\simeq
{}\stG\otimes_A W$ and $\Hom_A(\stG,W)\simeq W\otimes_A\Gamma$. If $\Gamma$ and $W$ are in $\catm$
then so are these isomorphisms.
\begin{proposition}\label{propo:projhomiso}
Let $\Gamma$ be a finitely generated and projective left $A$-module, so
that $\stG$ is a finitely generated and projective right $A$-module. Let $W$ be an $A$-bimodule.
Then there exist  right $A$-module and left $A$-module isomorphisms
(evaluation maps)
\begin{equation}\label{jmathdef}
\begin{split}
\flat : {}^*\Gamma\otimes_AW\, &\to\, {}_A\Hom(\Gamma,W)\\
\sts\otimes_A w\,\;&\mapsto\, (\sts\otimes\omega)^\flat(s):=
\sts(s)\,\omega~~~~
\end{split}
\begin{split}
\imath :  W\otimes_A \Gamma\,& \to\, \Hom_A(\stG,W)\\
w\otimes_A s\;\,&\mapsto\,\imath(w\otimes_A s)\,(\sts):= w\:\sts(s)\;.~~
\end{split}
\end{equation}
If in addition $\Gamma$ and $W$ are modules in $\catm$
then
$$\flat: {}^*\Gamma\otimes_AW \to {}_A\hom(\Gamma,W)~~,~~~\imath :  W\otimes_A \Gamma\, \to\, \hom_A(\stG,W)$$
are module isomorphisms in $\catm$.
\end{proposition}

\noindent
{\it Sketch of the proof. } Right $A$-linearity of $\flat$ is
immediate. Let $\sharp:=\flat^{-1}$ be the inverse of $\flat$; it is  given by $\tilde L\mapsto
{\tilde L}^\sharp= \sum_{i=1}^n\sts^i\otimes_A\tilde L(s_i)$. Left
$A$-linearity is also immediate. $H$-equivariance reads
$\big(h\ra^\cop (\sts\otimes_A\omega)^\flat\big)(s)=\big(h\ra
(\sts\otimes_A\omega)\big)^\flat(s)$, and follows from \eqref{Hppair}.
Similarly for $\imath$. $~~~~~~~~~~~~~~~~${$\square$}
\\

A pairing between modules $\Gamma$, $\Sigma$ in $\catm$ is a
morphism $\Gamma\otimes_A\Sigma\to A$. We denote by
\begin{equation}\label{ppair}
\le~,~\re:\Gamma\otimes_A\stG\rightarrow A~,~~s\otimes_A \sts\mapsto \le
s,\sts\re:=\sts(s)=\iota(s)(\sts)
\end{equation}
the pairing due to the evaluation of ${}^*\Gamma={}_A\hom(\Gamma,A)$
on $\Gamma$ or equivalently of $\Gamma\simeq
({}^*\Gamma)^*=\hom_A({}^*\Gamma,A)$ on ${}^*\Gamma$.
It is well defined on the balanced tensor product $\otimes_A$ because of the left
$A$-module structure of $\stG$. It is easily seen to be left and right $A$-linear,
indeed the notation $\le~,~\re$ conveniently takes into account the $A$-bimodule structures of
$\Gamma$ and
$\stG$, as well as that $\stG$ are left $A$-linear maps while
$\Gamma\simeq({\stG})^*$ are right $A$-linear maps,
for all $a\in A, s\in \Gamma, \sts\in \stG$, $\langle a
s,\sts\rangle=a\langle  s,\sts\rangle$,
$\langle
s a,\sts\rangle=\langle  s,a\:\! \sts\rangle$,
$
\langle s,\sts a\rangle=\langle  s,\sts\rangle a
$.
Furthermore, $H$-equivariance
\begin{equation}\label{Hppair}
h\ra\le s,\sts\re=\le h_{(1)} \ra s, h_{(2)}\ra^\cop \sts\re~
\end{equation}
is due to the $H$-module structure of $\stG$, cf. \eqref{copadjact2}.

We extend the pairing $\le~,~\re : \Gamma\otimes_A\stG\to
A$ to the morphisms in $\catm$
\begin{equation}\label{convimath}\le ~\,,\,~~\re: \Gamma\otimes_A\otimes \stG\otimes_A W\to
W~,~~s\otimes_A \sts\otimes_A w\mapsto \le s,\sts\otimes_A w\re:=\le
s,\sts\re w
\end{equation}
\begin{equation}\label{convjmath}\le ~~\,\,,~\re: W\otimes_A \Gamma\otimes_A\otimes \stG \to
W~,~~w\otimes_A s\otimes_A \sts\mapsto \le w\otimes_A s,\sts\re:=w\le
s,\sts\re
\end{equation}
so that the internal morphisms $(\sts\otimes\omega)^\flat$  and  $\imath(w\otimes_A s)$ of
Proposition \ref{propo:projhomiso} are respectively simply denoted as
$\le~\,,\sts\otimes\omega\re$ and
$\le w\otimes_A s,~\re$.

\begin{definition}\label{defdualcat}
Given modules $\Gamma\in \catm$, $\Sigma\in \catm$ we say that $\Gamma$
has right dual $\Sigma$, or equivalently that $\Sigma$ has left dual $\Gamma$, if we have maps
$\ev:\Gamma\otimes_A\Sigma\rightarrow A$ (evaluation map) and
$\coev:A\rightarrow \Sigma\otimes_A\Gamma$ (coevaluation map) in $\catm$
such that the compositions
\begin{equation}\label{coerecoev}
\begin{split}
\end{split}
\begin{split}\Gamma\simeq \Gamma\otimes_A
A
\xrightarrow{\id_\Gamma\otimes_{\!A\,}\coev\,}\,
\Gamma\otimes_A\Sigma\otimes_A\Gamma\xrightarrow{\ev\otimes_{\!A\,}\id_\Gamma\,}\Gamma\,,\\[.4em]
\Sigma\simeq
A\otimes_A\Sigma
\xrightarrow{\coev\otimes_{\!A\,}\id_{\Sigma}}
\Sigma\otimes_A\Gamma\otimes_A\Sigma\xrightarrow{\id_{\Sigma\,}\otimes_{\!A\,}
\ev}\Sigma\,,
\end{split}
\end{equation}
are respectively the identity maps
$\id_\Gamma$ and $\id_{\Sigma}$.
If $\Gamma$ has a right dual we say that it is right rigid.
If $\Sigma$ has a left dual we similarly  say that it is left rigid.
\end{definition}
We will also denote  by $\ev^{}_\Gamma$ and $\coev^{}_\Gamma$ the
evaluation and coevaluation maps of the right rigid module $\Gamma$,
and  frequently use the notations $\le~,~\re:\Gamma\otimes_A\Sigma\to
A$ and $\le~,~\re_\Gamma$
for the evaluation map.

Left (right)  duals are unique up to
isomorphisms, in this sense we can simply speak of {\sl the} left (right) dual
of a module, and we say that the module is rigid. In order to prove uniqueness
up to isomorphism,  let $\Gamma$ also admit right dual $\Sigma
\check{\phantom i\!\!}$ with
maps $\ev\check{\phantom s\!\!}: \Gamma\otimes_A \Sigma\check{\phantom i\!\!}\to A$, $\coev\check{\phantom s\!\!}: A\to
\Sigma\check{\phantom i\!\!}\otimes_A\Gamma$. Then it is easy to see that $\varphi:=(\id_\Sigma\otimes_A \ev \check{\phantom s\!})\circ
(\coev\otimes_A \id_{\Sigma\check{\phantom i\!\!}}) :\Sigma\check{\phantom i\!\!}\to \Sigma$ has inverse
 $\varphi^{-1}:=(\id_{\Sigma \check{\phantom i\!\!}}\otimes_A \ev)\circ
(\coev\check{\phantom s\!\!}\otimes_A \id_{\Sigma}) :\Sigma\to \Sigma \check{\phantom i\!\!}$ and that
\begin{equation}\label{ev'coev'}
\ev\check{\phantom s\!\!}=\ev\circ(\id_\Gamma\otimes_A\varphi)~,~~
\coev\check{\phantom s\!\!}=(\varphi^{-1}\otimes_A\id_\Gamma)\circ\coev~.
\end{equation}
This implies that if there exists a coevaluation map, it
is uniquely determined by the pairing $\ev: \Gamma\otimes_A \Sigma\to
A$.
A pairing $\le~,~\re:\Gamma\otimes_A \Sigma\to A$  is
{\it exact} if there exits a map $\coev :A\to
\Sigma\otimes_A\Gamma$ fulfilling the conditions of Definition
\ref{defdualcat}.

The pairing in \eqref{ppair} is exact if and only if $\Gamma$ in
$\catm$ is finitely generated and projective as left $A$-module.

\begin{theorem}\label{Thmfgp123} Let $\Gamma$ in $\catm$ and  $\stG:={}_A\hom(\Gamma,A)$.\begin{enumerate}
\item  $\Gamma$ is finitely generated and projective as left
  $A$-module in $\catm$
if and only if  $\Gamma$ is right rigid.
\item If $\Gamma$ is right rigid,
 for all $V,W\in \catm$,
 \begin{equation}\label{iso1e2}
{}_A\hom(\Gamma\otimes_A V,W)\simeq {}_A\hom(V,{}^*\Gamma\otimes_A
  W)~~,~~~~\hom_A(V\otimes_A\stG,W)\simeq
  \hom_A(V,W\otimes_A\Gamma)~~
\end{equation}
  are isomorphisms in $\catm$.
\item Let $\Upsilon$ be also right rigid
  in $\catm$ and  ${}^*\Upsilon:={}_A\hom(\Upsilon,A)$, then so is
  $\Gamma\otimes_A\Upsilon$, and we
further have the isomorphism in $\catm$, $\stU\otimes_A \stG\simeq
{}^*(\Gamma\otimes_A \Upsilon)$.
\end{enumerate}
\begin{proof}
 {\it 1.} Let  $\Gamma$ be finitely generated and projective as  left
  $A$-module in $\catm$.
We show that $\Gamma$ satisfies Definition \ref{defdualcat} with $\Sigma=\stG$.
The evaluation map is given by the pairing in \eqref{ppair}.
The coevaluation map is the $A$-bilinear map that associates to the
unit $1_A\in A$ the element $\id_\Gamma^{~\sharp}\in \stG\otimes_A\Gamma$,
where $\sharp=\flat^{-1}$ is as in Proposition \ref{propo:projhomiso} with
$W=\Gamma$ and $\id_\Gamma\in\Hom_\catm(\Gamma,\Gamma)\subset {}_A\hom(\Gamma,\Gamma)$. This map is well defined because $\id_\Gamma^{~\sharp}$ is central in
$\stG\otimes_A\Gamma$; indeed  for all $a\in A$,
the equality $a \,\id_\Gamma^{~\sharp}=\id_\Gamma^{~\sharp}\, a$ is
equivalent  to its image under the $A$-bilinear map $\flat$, that
reads $a_{\,}\id_\Gamma=\id_{\Gamma\,} a$, which holds
true due to the $A$-bimodule structure \eqref{AhomAA} of ${}_A\hom(\Gamma,\Gamma)$.
Explicitly, using a pair of dual bases, we have
 $\id_\Gamma^{~\sharp}=\sts^i\otimes_A s_i$, and
\begin{equation}\label{expcoev}
\coev: A\rightarrow \stG\otimes_A\Gamma~,~~  a\mapsto a_{\,}\sts^i\otimes_A s_i=\sts^i\otimes_A s_{i\,}a~.
\end{equation}
Similarly, $H$-equivariance of $\coev$, which  is equivalent to,
for all $h\in H$, $h\ra \coev(1_A)=\varepsilon(h)
\coev(1_A)$, follows from that of $\flat$ and
$\id_\Gamma$ (this latter reading
$h\ra^\cop \id_\Gamma=\varepsilon(h) \id_\Gamma$).

The coherence conditions \eqref{coerecoev} follow using \eqref{expcoev}
and recalling that $\lbrace s_i, {}^{*\!}s^i:i=1,\dots,n \rbrace$ and $\lbrace  {}^{*\!}s^i, \iota( s_i):i=1,\dots,n \rbrace$ are
a pair of dual bases for $\Gamma$ and $\stG$, respectively
(cf. Proposition \ref{propo:fgpmodprop}).

Vice versa, let $\Gamma$ be right rigid.
Since
$\coev(1_A)\in \Sigma\otimes_A\Gamma$, we have
$\coev(1_A)=\sigma^i\otimes_A s_i$ (finite sum understood)
$\sigma^i\in \Sigma$, $s_i\in \Gamma$. The condition that the first
composition in \eqref{coerecoev} equals $\id_\Gamma$
reads,
for all $s\in \Gamma$,
$\le s,\sigma^i\re s_i=s$. Setting $\sts^i=\le~\,,\sigma^i\re\in\stG$
we see that $\Gamma$ is a finitely
generated and projective left $A$-module.
\\
\\
{\it 2.}
It follows from point  {\it 1} that if $\Gamma$
is right rigid we can consider $\stG$ to be its right dual, with
coevaluation map  $\coev : A\to \stG\otimes_A \Gamma$. For all $\tilde L\in {}_A\hom(\Gamma\otimes_A V,W)$, define $${\tilde
  L}^\sharp:=(\id_{\stG}\otimes_A\tilde L)\circ (\coev\otimes_A\id_V):
V\to \stG\otimes_A W~,~~v\mapsto {\tilde L}^\sharp(v)=\sts^i\otimes_A\tilde
L(s_i\otimes_A v)~.$$ The map ${\tilde L}^\sharp$ is  left $A$-linear
because so is  $\tilde L$ and because $\coev$ is
$A$-bilinear.
Furthermore, the map $\sharp: {}_A\hom(\Gamma\otimes_A V,W)\to
{}_A\hom(V,{}^*\Gamma\otimes_AW)$ is $H$-equivariant since so is
$\coev$. It is invertible with inverse $\flat:
{}_A\hom(V,{}^*\Gamma\otimes_A W)\to  {}_A\hom(\Gamma\otimes_A V,W)$
given by, for all $\tilde P\in {}_A\hom(V,{}^*\Gamma\otimes_A W)$,
${\tilde P}^\flat:=(\ev\otimes_A\id_W)\circ (\id_\Gamma\otimes_A
\tilde P) : \Gamma\otimes_A V\to W$,
$\,s\otimes_A v\mapsto {\tilde P}^\flat(s\otimes v)=\le s, {\tilde
  P}(v)\re$, (recall definition \eqref{convimath}).  Indeed we have
\begin{equation*}
\begin{split}
{\tilde L}^{\sharp^{\mbox{\scriptsize$\,{}\flat$}}} \!(s\otimes_A v)&=\le s, {\tilde
  L}^\sharp(v)\re=\le s,\sts^i\otimes_A\tilde L(s_i\otimes_Av)\re=\le
s, \sts^i\re \tilde L(s_i\otimes_A v)=\tilde L(\le s,\sts^i\re
s_i\otimes_A v)\\ &=\tilde L(s\otimes_A v)
\end{split}
\end{equation*}
where in the third equality we used left $A$-linearity of $\tilde L$.
We similarly have
$$
{{\tilde P}^{\flat^{\mbox{\scriptsize$\,{}\sharp$}}}} (v)=
\sts^i\otimes_A\tilde P^\flat (s_i\otimes_A v)=\sts^i\otimes_A\le
s_i,\tilde P(v)\re=\tilde P(v)$$
where we used, for all $\sts\in \stG, w\in W$,
$\sts^i\otimes_A\le s_i,\sts\otimes_A w\re=
\sts^i\otimes_A\le s_i,\sts\re w=
\sts^i\le s_i,\sts\re\otimes_A w=\sts\otimes_A w$.

The second isomorphism in \eqref{iso1e2} is similarly proven.
In this case we define $\sharp: \hom_A(V\otimes_A\stG,W)\to
\hom_A(V,W\otimes_A\Gamma)$, $L\mapsto L^\sharp$ with
$L^\sharp(v):=L(v\otimes \sts^i)\otimes s_i$. Right $A$-linearity of
$L^\sharp$ follows from \eqref{expcoev}. $H$-equivariance  of $\sharp$ follows
from that of $\coev$. The inverse of $\sharp$ is
 $\flat: \hom_A(V,W\otimes_A\Gamma)\to \hom_A(V\otimes_A\stG,W)$,
 $P\mapsto P^\flat$
given by $P^\flat(v\otimes_A\sts):=\le P(v),\sts\re$.
\\
\\
{\it 3.} $\Gamma\otimes_A\Upsilon$ is finitely generated and
projective if so are $\Gamma$ and $\Upsilon$. For example, denoting by
$\{u_j,\stu^j : j=1,\ldots,m\}$ a dual basis of $\Upsilon$, then a dual
basis of  $\Gamma\otimes_A\Upsilon$
is given by the elements $t_{ij}=s_i\otimes_A u_j\in
\Gamma\otimes_A\Upsilon$, and the elements $\stt^{ij}\in
{}^*\!(\Gamma\otimes_A\Upsilon)$ defined by, for all $s\in \Gamma,
u\in \Upsilon$,  $\stt^{ij}(s\otimes_A
u) :=\le s\le u,\stu^j\re,\sts^i\re$. Indeed, $\stt^{ij}(s\otimes_A
u)_{\,}t_{ij}=\le s\le u,\stu^j\re,\sts^i\re_{\,} s_i\otimes_A u_j= s\le
u,\stu^j\re\otimes_A u_j=s\otimes_A\le
u,\stu^j\re u_j=s\otimes_A u$.

In order to prove the isomorphism $\stU\otimes_A\stG\simeq \stGU$,
define
$\varphi^{}_{\Gamma,\Upsilon}:\stU\otimes_A\stG \to \stGU$,
$$
\varphi^{}_{\Gamma,\Upsilon}:=(\id^{}_{\stGU}\otimes_A\ev^{}_\Gamma)\circ
(\id^{}_{\stGU\otimes_A\Gamma}\otimes_A\ev^{}_\Upsilon\otimes_A\id^{}_{\stG})\circ(\coev^{}_{\Gamma\otimes_A\Upsilon}\otimes_A\id^{}_{\stU\otimes_A\stG})\,,
$$
$\stu\otimes_A\sts\mapsto
\varphi^{}_{\Gamma,\Upsilon}(\stu\otimes_A\sts)=\stt^{ij}\le\le t_{ij},\stu\re,\sts\re
$, (recall definition \eqref{convjmath}).
The map $\varphi^{}_{\Gamma,\Upsilon} $ is easily seen to be in
$\catm$; it is invertible
with inverse $\varphi^{-1}_{\Gamma,\Upsilon}:\stGU\to\stU\otimes_A\stG $,
$$
\varphi^{-1}_{\Gamma,\Upsilon}=(\id^{}_{\stU\otimes_A\stG}\otimes_A\ev^{}_{\Gamma\otimes_A\Upsilon})\circ
(\id^{}_{\stU}\otimes_A\coev^{}_{\Gamma}\otimes_A\id^{}_{\Upsilon\otimes_A\stGU})\circ(\coev^{}_{\Upsilon}\otimes_A\id^{}_{\stGU})\,,
$$
$\stt\mapsto
\varphi^{-1}_{\Gamma,\Upsilon}(\stt)=\stu^j\otimes_A\sts^i\le s_i\otimes_Au_j,\stt\re$.
Indeed we have
\begin{equation*}
\begin{split}
\varphi^{-1}_{\Gamma,\Upsilon}\circ\varphi^{}_{\Gamma,\Upsilon\,}(\stu\otimes_A\sts)&\,=\,\varphi^{-1}_{\Gamma,\Upsilon}(\stt^{i'j'}\le\le
t_{i'j'},\stu\re,\sts\re)
=\stu^j\otimes_A\sts^i
\le s_i\otimes_A u_j,\stt^{i'j'}\re \le\le t_{i'j'},\stu\re,\sts\re
\\[.4em] &\,=\,\stu^j\otimes_A\sts^i\le\le_{\,}\le s_i\otimes_A
u_j,\stt^{i'j'}\re_{\,} t_{i'j'\,},\stu\re, \sts\re
=
\stu^j\otimes_A\sts^i\le\le
s_i\otimes_A u_j,\stu\re,\sts\re
\\[.4em] &
\,=\,\stu^j\otimes_A\sts^i\le s_i,\le u_j,\stu\re\sts\re
=\stu^j\otimes_A\le
u_j,\stu\re\sts

=\stu\otimes_A\sts~,
\end{split}
\end{equation*}
where in the first line we used right $A$-linearity of
$\varphi^{-1}_{\Gamma,\Upsilon} $, in the second line left
$A$-linearity of $\ev^{}_\Gamma$ and $\ev^{}_\Upsilon$, in the third
middle $A$-linearity of $\ev^{}_\Gamma$.
Similarly one shows that
$\varphi^{}_{\Gamma,\Upsilon}\circ\varphi^{-1}_{\Gamma,\Upsilon\,}=\id^{}_{\stGU}$.
\end{proof}\end{theorem}

\begin{remark}\label{rigidhom}
The first isomorphism in \eqref{iso1e2} is a natural isomorphism
between the functors ${}_A\hom\circ (\Gamma\!\otimes_{\!A} \times \:\id)$
and ${}_A\hom\circ (\id\,\times{}\stG\otimes_{\!A})$ from
$ (\catm)^\op\times \catm$ to $\catm$. Similarly for the second.
Restricting  \eqref{iso1e2} to
$H$-equivariant internal morphisms and recalling that these are
identified with the morphisms in
$\catm$  we obtain the natural isomorphisms
$$\Hom_{\,\catm\!}(\Gamma\otimes_A V,W)\simeq \Hom_{\,\catm\!}(V,{}^*\Gamma\otimes_A
  W)~,~~\Hom_{\,\catm\!}(V\otimes_A\stG,W)\simeq
  \Hom_{\,\catm\!}(V,W\otimes_A\Gamma)~,$$
showing that $\Gamma\otimes_A$ is left adjoint to $\stG\otimes_A $ and
that $\otimes_A\stG$ is left adjoint to $\otimes_A\Gamma$.
\end{remark}
\begin{remark}\label{newevcoev}
Due to the isomorphism  $\stU\otimes_A \stG\simeq
{}^*(\Gamma\otimes_A \Upsilon)$, in the following the right dual of
$\Gamma\otimes_A\Upsilon$ will be considered to be $\stU\otimes_A \stG$ with evaluation
and coevaluation maps given as in \eqref{newevcoev}. Explicitly these read
\begin{equation*}
\begin{split}
\le~~,~~\re &:  \Gamma\otimes_A \Upsilon\otimes_A \stU\otimes_A
\stG\rightarrow A~,~~ s\otimes_A u \otimes_A \stu\otimes_A\sts \mapsto
\le s\otimes_A u, \stu\otimes_A\sts\re=\le s\le u, \stu\re,\sts\re_{}~,\\[.4em]
\coev&: A\to  \stU\otimes_A\stG\otimes_A \Gamma\otimes_A
\Upsilon~,~~a\mapsto a\,\stu^j\otimes_A\sts^i\otimes_As_i\otimes_A u_j~.
\end{split}
\end{equation*}
with $\{s_i,\sts^i:i=1,\dots,n \}$ and $\{u_j,\stu^j :j=1,\dots,m \}$
dual basis
of $\Gamma$ and $\Upsilon$, respectively.
\end{remark}

\begin{remark}\label{Thmright}
A right version of Theorem \ref{Thmfgp123} holds as well. In
particular, modules in $\catm$ are  finitely generated and projective
as right $A$-modules if and only if they are left rigid. If $\Sigma$ and $\Lambda$ are left rigid, so
is $\Sigma\otimes_A\Lambda$ and $\Lambda^*\otimes_A\Sigma^*\simeq (\Sigma\otimes_A \Lambda)^*$. We hence consider the left dual of
$\Sigma\otimes_A\Lambda$ to be $\Lambda^*\otimes_A \Sigma^*$ with
evaluation and coevaluation canonically induced via the isomorphism $\Lambda^*\otimes_A\Sigma^*\simeq (\Sigma\otimes_A \Lambda)^*$.
Explicitly these read
\begin{equation*}
\begin{split}
\le~,~\re &:  \Lambda^*\otimes_A \Sigma^*\otimes_A \Sigma\otimes_A
\Lambda\rightarrow A~\!,\,~ \lambda^*\otimes_A \sigma^* \otimes_A \sigma\otimes_A\lambda \mapsto
\le \lambda^*\otimes_A \sigma^*, \sigma\otimes_A\lambda\re=\le
\lambda^*, \le \sigma^*, \sigma\re\lambda\re\:,\\[.4em]
\coev&: A\to  \Sigma\otimes_A\Lambda\otimes_A \Lambda^*\otimes_A
\Sigma^*~\!,\,~a\mapsto
a\,\sigma^i\otimes_A\lambda^j\otimes_A\lambda^*_j\otimes_A \sigma^*_i\;,
\end{split}
\end{equation*}
with $\{\sigma^i, \sigma^*_i : i=1,\ldots n\}$ and
$\{\lambda^j,\lambda^*_j : j=1,\ldots m\}$ dual basis of
$\Sigma$, $\Lambda$, respectively.
\end{remark}

\begin{remark}
Definition \ref{defdualcat} holds in a generic monoidal category. For
example also when the tensor product is topological. In this case
rigid modules are projective and topologically finitely generated
(i.e., there exists        a finite number of
elements  that span a dense subset of the module). This
is the case of Example \ref{ex:bundleformal}.
\end{remark}

The right dual (or adjoint) of a morphism $f:
\Gamma\to \Upsilon$ of left rigid modules in $\catm$ is
the morphism ${}^*f:\stU\to \stG$ defined by $\le s, {}^{*\!}f(\stu)\re
= \le f(s),\stu\re$ and more explicitly ${}^{*\!}f(\stu)=\sts^i\,\le
f(s_i),\stu\re$
(using a pair of dual bases for $\Gamma$). Duals of internal morphisms
will be studied in Section \ref{dualinternalmorph}.

\subsection{Ribbon categories\label{sect:ribbon}}
If the Hopf algebra $H$ has a triangular structure $\RR$
we can consider finitely generated (left or right) $A$-modules in the braided category $\cat$.
Let $\Gamma$ in $\cat$ be finitely generated and
projective as left $A$-module, hence it is right rigid in $\cat$,
with right dual ${}^*\Gamma={}_A\hom(\Gamma,A)$ that is finitely
generated and projective as right $A$-module  in $\cat$.
All the results in section \ref{fgp} concerning  modules in $\catm$
holds true in the full subcategory $\cat$.

In a braided category, right rigid modules $\Gamma$ are also left rigid
(cf. e.g. \cite[Chapter XIV.3]{Kassel}), with the module
${}^*\Gamma$ that is also left dual to $\Gamma$ with evaluation and
coevaluation maps obtained via the braiding $\tau^{}_{\stG,\Gamma}$,
\begin{equation}\label{tevcoev}
\begin{split}
\le~,~\re':=\langle~,~\rangle\circ \tau^{}_{\stG,\Gamma}&:
\stG\otimes_A\Gamma\to A~,\\[.4em]
\coev':=
\tau^{}_{\stG,\Gamma}\circ\coev&: A\to \Gamma\otimes_A\stG~.
\end{split}
\end{equation}
These evaluation and coevaluation
maps show that
the  finitely generated and projective right $A$-module $\stG$ is also finitely generated and projective as
a left $A$-module.
We can then speak of finitely generated and projective $A$-modules in
$\cat$, there is no need to specify if they are left or right
$A$-modules.

\begin{remark}
The induced maps $\le~,~\re'\circ({\cal D}\oA\id^{}_\Gamma)$ and
$(\id^{}_\Gamma\oA {\cal D}^{-1})\circ\coev'$ under the isomorphism
${\cal D}^{-1}:\stG\longrightarrow\Gamma^*$, where ${\cal D}:={\cal
  D}_{\Gamma,A}$, cf. \eqref{Dcallr},
structure $\Gamma^*=\hom_A(\Gamma,A)$ as left dual to
$\Gamma$, with
the evaluation map   $\le~,~\re'\circ({\cal D}\oA\id^{}_\Gamma)$ that is the canonical evaluation of
$\Gamma^*=\hom_A(\Gamma,A)$ on $\Gamma$; indeed, for all $s^*\in \Gamma^*,s\in \Gamma$, $$\le \sts,{\cal D}s^*\re'=\le\oR^\al\ra s,\oR_\al
\ra^\cop{\cal D}s^*\re=(\oR_\al\ra^\cop{\cal D}s^*)\circ\oR^\al\ra(s)={\cal
  D}^{-1}({\cal D}s^*)(s)=s^*(s)~.$$ In other terms, if $\{s_i,\sts^i:
i=1\ldots, n\}$ is a dual basis of the left
$A$-module $\Gamma$, we then have that $\{\oR^\al\ra s_i,{\cal
  D}^{-1}(\oR_\al\ra^\cop\sts^i): \al=1,\ldots,
m,\, i=1,\ldots n\}$  is a  dual basis of  $\Gamma$ as right $A$-module.
\end{remark}

\begin{remark}The exact pairing $\le~,~\re':
\stG\oA\Gamma\to A$ induces the isomorphisms in $\cat$ (cf. Proposition \ref{propo:projhomiso})
\begin{equation}\label{jpip}
\begin{split}
\le~\,,\,~~\re'&: \Gamma\oA W \longrightarrow {}_A\hom(\stG, W)~,~~
s\otimes w\mapsto \le ~\,, s\otimes_A w\re':=\le ~\,, s\re' \otimes_A w~,
\\[.4em]
\le~~~,~\re'&: W\oA \Gamma \longrightarrow \hom_A(\Gamma^*, W)~,~~s\otimes
w\mapsto \le w\otimes_A s, ~\, \re':=w\oA\le s,\, ~\re' ~.
\end{split}
\end{equation}
\end{remark}

  We denote by $\catfp$ the full subcategory in $\cat$ of finitely generated and
  projective modules.
  Every module has a left and a right dual and therefore $\catfp$
  is a rigid category.
Recall that a ribbon category
  ${\cal C}$ is a braided monoidal category $({\cal C},\otimes,\tau)$  where each object has a left dual and
  with a family of isomorphisms $\theta_\Gamma:\Gamma\to\Gamma$ for
  each object $\Gamma$ in ${\cal C}$ such that  $\theta^{}_{\Gamma\otimes\Upsilon}=\tau^{}_{\Upsilon,\Gamma}\circ\tau^{}_{\Gamma, \Upsilon}\circ{}\theta_\Gamma\otimes\theta_\Upsilon$
  and $(\theta_\Gamma)^*=\theta_{\Gamma^*}$ for all $\Gamma,
  \Upsilon$ objects in ${\cal C}$. We then immediately conclude that $\catfp$
  is a ribbon category with $\theta_V=\id_V$.
\\

For later use we notice that the
dual of the braiding isomorphism equals the braiding on the dual modules, i.e.,  in the
notations of Remark \ref{newevcoev},
\begin{equation}\label{taudual}
{}^{*\!}(\tau^{}_{\Gamma,\Upsilon})=\tau^{}_{\stG,\stU}~.
\end{equation}
This follows form  $\le\tau^{}_{\Gamma,\Upsilon}(s\otimes
u),\tau^{}_{\stU,\stG}(\stu\otimes_A\sts)\re=\le s,\oR^\al \ra
\sts\re\,\oR_\al\ra\le u,\sts \re=\le s\otimes_A u,\stu\otimes_A\sts\re$
and triangularity of the universal $\RR$-matrix.

\subsection{Examples}
We discuss the ribbon category of bicovariant bimodules of a
triangular Hopf algebra (quantum group),  the one of equivariant vector
bundles on a manifold and (adapting the treatment in \cite[\S 6]{BSS1}
to the ribbon category context)  its
noncommutative Drinfeld twist deformation.
\begin{example}\label{cotriangularK}
Let $K$ be a finite dimensional Hopf algebra, and let $U$ be the
$\Bbbk$-dual Hopf algebra. Left (right) $K$-coactions correspond to right
(left) $U$-actions on modules. In particular, since  $K$ is a $K$-bicomodule
algebra via the coproduct, then it is a $U$-bimodule algebra.
It follows that the braided monoidal category of $K$-bicovariant
bimodules \cite{Schauenburg} is equivalent to that of relative $U$-bimodules
$K$-bimodules. Now, $U$-bimodules are equivalently left
${U}\otimes U^\op$-modules (where $U^\op$ is the Hopf algebra with opposite
product and ${U}\otimes U^\op$ the tensor product Hopf algebra),
hence we have the braided monoidal category
${}^{U\otimes U^\op}_{}{}^{}_{\!\!\!\!\!\!\!K}\MMM^{}_K$ of finitely generated
$K$-modules. These are free modules and hence this is a rigid
category. If $U$ is triangular with $\RR$-matrix $\mathscr{R}$, then $K$ is
cotriangular, for all
$k,k'\in K$, $k'k=\mathscr{R}(k^{}_{(1)}\otimes k'_{(1)}) k^{}_{(2)}k'_{(2)} \mathscr{R}^{-1}(k^{}_{(3)}\otimes k'_{(3)})$;
furthermore $U^\op$ is triangular with  $\RR$-matrix $\mathscr{R}^{-1}$, and
$U\otimes U^\op$ is also triangular with $\RR$-matrix
${\RR}=(\id\otimes \rm{flip}\otimes\id)(\mathscr{R}\otimes
\mathscr{R}^{-1})$. Then, the cotriangularity property of $K$ is just
the braided commutativity property
of $K$ with respect to the $\RR$ matrix of  $U\otimes U^\op$. Henceforth,
${}^{{U\otimes U^\op}}_{}{}^{}_{\!\!\!\!\!\!\!K}\MMM^\sym_K$ is a
ribbon category with braided commutative Hopf algebra $K$.

Similarly, we can consider $K$ a cotriangular quantum group, (for
example of the $A,B,C,D$ series given via  multiparametric $\RR$
matrices \cite{res}) with dually paired triangular Hopf algebra  $U$.
As before, $K$ is braided commutative and
${}^{{U\otimes U^\op}}_{}{}^{}_{\!\!\!\!\!\!\!K}\MMM^\sym_K$ is the
ribbon category of finitely generated free modules over $K$.
\end{example}
$\phantom{.}$\\[-1.6em]

Ribbon categories are easily obtained via Drinfeld twists
of ribbon categories. Recall that a Drinfeld twist is an
invertible element
$\FF\in H\otimes H$ satisfying the cocycle and normalization properties:
\begin{equation}\label{twist}
(\FF\otimes
\id)(\Delta\otimes\id)\FF=(\id\otimes\FF)(\id\otimes\Delta)\FF~,~~(\varepsilon\otimes\id
\FF)=(\FF\otimes\id)=1_H~.
\end{equation}
Given a twist we deform the triangular Hopf algebra
$H$ with universal $\RR$-marix $\RR$ in the triangular Hopf algebra $H^\FF$
that as algebra is the same as $H$, while it has coproduct $\Delta^\FF(\xi)=\FF\Delta(\xi)\FF^{-1}$ and
universal $\RR$-matrix $\RR^\FF=\FF_{21}\RR\FF^{-1}$. We deform the algebra
$A$ in the braided commutative algebra $A^\FF$ that equals $A$ as
$\Bbbk$-module, while it has
multiplication,  for all $a,b\in A$,
$a\cdot^{{_{^\FF}}} b:=(\of^\al\ra a) (\of_\al\ra b)$, where $\FF^{-1}=\of^\al\otimes\of_\al$. The
normalization conditions imply that the unit of
$A$ is also the unit of $A^\FF$.
We further associate to any module $\Gamma$ in $\catfp$ a new module
$\Gamma^\FF$ in $\catFF$ that has the same $\Bbbk$-module structure.
The $H^\FF$-module structure is given by the $H$-action (indeed as
algebras $H^\FF$ equals $H$) and the left and right $A^\FF$-actions are
given by, for all $a\in A^\FF$, $s\in \Gamma$,
$a\dotF s:=(\of^\al\ra a)(\of_\al \ra s)
$
and $s
\dotF a=(\of^\al\ra s)(\of_\al \ra a)$.
Due to these assignments the ribbon category $\catfp$ is
equivalent to the ribbon category $\catfpFF$. We refer to Appendix
\ref{catequivF} for a proof.

 \begin{example}\label{ex:bundle}{\it $U\gg$-equivariant
  $C^\infty(M)$-bimodules from $G$-equivariant vector bundles on $M$.}
Let $G$ be a Lie group and $M$ a $G$-manifold with right $G$-action
$r: M\times G\to M\,,~(m,g)\mapsto m\:\!g$.  We assume manifolds to be
finite dimensional and second countable. In this case every finite
rank vector bundle has a finite open covering of the base trivializing the bundle.

Recall that a $G$-equivariant vector bundle $E\to M$ is a vector
bundle $E\to M$ where $E$ and $M$ are right $G$-manifolds, the
$G$-actions on $E$ and $M$ are compatible with the projection $E\to M$ and $G$ acts via linear isomorphisms on the fibers of $E$.

The right $G$-action on $M$ pulls back to a left $G$-action
$G\times C^\infty(M)\to C^\infty(M)$ on the algebra $C^\infty(M)$
of smooth complex valued functions on $M$.
Using the $G$-action on $E$  we also have a left $G$-action on the
$C^\infty(M)$-module of sections $\Gamma(E)$ (for all $m\in M,  s\in
\Gamma(E)$, $(g\ra s)(m)=s(mg)g^{-1}$). With this action $\Gamma(E)$
is a relative $G$-module $C^\infty(M)$-module (or $G$-equivariant
$C^\infty(M)$-module), i.e., for all $a\in
C^\infty(M), s\in  \Gamma(E) , g\in G, g\ra as=(g\ra a) (g\ra s)$.
Since $E\to M$ is trivialized by a finite open covering of $M$ it can
be shown that $\Gamma(E)$ is finitely generated and projective as a $C^\infty(M)$-module.
This construction leads to a functor $\Gamma: G$-{\sl VecBun}$_M\to
{}^G_{C^\infty(M)}\MMM^{\sym,\fp}_{C^\infty(M)} $ from the category of
$G$-equivarint vector bundles over $M$ to that of
$G$-equivariant $C^\infty(M)$-modules that are
finitely generated and
projective over $C^\infty(M)$. (If $G$ is the trivial group the
functor is an equivalence proving Serre-Swan theorem in the smooth
context).
Both categories are symmetric rigid monoidal, hence are ribbon
categories and the functor is braided monoidal.

Associated with the $G$-action on $M$ we have a (smooth) action of the
Lie algebra $\gg=$Lie$\,G$ on $C^\infty(M)$ via derivations.
It canonically lifts to an action of the universal enveloping
algebra $U\gg$ on $A=C^\infty(M)$ that therefore  becomes a
$U\gg$-module algebra. Similarly, $\Gamma(E)$ is a relative
$U\gg$-module $A$-bimodule.  (Technically, $C^\infty(M)$ and the
finitely generated projective module $\Gamma(E)$ are nuclear
Fr{\'e}chet spaces with respect to the usual
$C^\infty$-topology. Furthermore, $C^\infty(M)$ is a unital Fr{\'e}chet algebra
with the usual pointwise product $\mu:= \mathrm{diag}_M^\ast :
C^\infty(M)\,\widehat\otimes\, C^\infty(M) \to C^\infty(M) $ given by pull-back of
the diagonal map $\mathrm{diag}_M : M\to M\times M$.
Here $C^\infty(M)\,\widehat\otimes\, C^\infty(M)\simeq C^\infty(M\times M)$
denotes the completed tensor product. Therefore $C^\infty(M)$ is an algebra object in the category of $U\gg$-Fr{\'e}chet spaces).
It follows that we have a braided monoidal functor {\sl ULie}$\,\circ \Gamma$ from the
category $G$-{\sl VecBun}$_M$
to the category of relative $U\gg$-modules $A$-bimodules. The functor
is valued in the sub-category
 ${}^{U\gg}_A\MMM_{A\,\geom}^{\sym,\fp}$
of  $U\gg$-modules $A$-bimodules, where objects are
the modules of sections $\Gamma(E)$ of $G$-equivariant vector bundles
$E\to M$ and morphisms are $U\gg$-equivariant $A$-bimodule morphisms.
This is a ribbon category, indeed it is a symmetric rigid category (where the dual $A$-bimodule
$\Gamma(E)^*=\hom_A(\Gamma(E), A)$ carries the  adjoint
$U\gg$-action).
\end{example}

\begin{example}\label{ex:bundleformal} {\it Formal braided equivariant vector bundles.}
Associated with the ring $\mathbb{C}[[\hbar]]$ of formal power
series in $\hbar$ with coefficients in $\mathbb{C}$ we have the
 formal power series extension
of the $\mathbb{C}$-modules $U\gg$, $A$ and $\Gamma(E)$, denoted as
usual $U\gg[[\hbar]]$, $A[[\hbar]]$ and $\Gamma(E)[[\hbar]]$.
The tensor product in the definition of the
${\mathbb{C}}[[\hbar]]$-Hopf algebra  $U\gg[[\hbar]]$ is the
completed one in the $\hbar$-adic topology (cf. e.g. \cite[Chapter XVI]{Kassel}), so that
 $U\gg[[\hbar]]\,\widehat\otimes\, U\gg[[\hbar]]\simeq (U\gg\otimes
U\gg)[[\hbar]]$ (and the tensor product in the definition of the
$\mathbb{C}[[\hbar]]$-algebra $A[[\hbar]]$ is the
completed one in the Fr\'echet and $\hbar$-adic topologies, so that
$A[[\hbar]]\,\widehat \otimes\, A[[\hbar]]\simeq C^\infty(M\times M)[[\hbar]]$).
The $A[[\hbar]]$-bimodule $\Gamma(E)[[\hbar]]$ is topologically  finitely
generated and projective and we denote by
${}^{U\gg[[\hbar]]}_{A[[\hbar]]}\MMM_{A[[\hbar]]\,\geom}^{\sym,\fp}$ the associated
ribbon category.
It has been obtained via the change of base ring $\mathbb{C}\to\mathbb{C}[[\hbar]]$
that induces the braided monoidal functor $\mathbb{C}[[\hbar]]\widehat\otimes:
{}^{U\gg}_A\MMM_{A\,\geom}^{\sym,\fp}
\to{}^{U\gg[[\hbar]]}_{A[[\hbar]]}\MMM_{A[[\hbar]]\,\geom}^{\sym,
  \fp}$ that to
$\Gamma(E)$ associates the extension
$\mathbb{C}[[\hbar]]\!\;\widehat\otimes\!\;\Gamma(E)\simeq \Gamma(E)[[\hbar]]$
and extends morphisms by $\mathbb{C}[[\hbar]]$-linearity.

By considering a twist $\FF$ of $U\gg[[\hbar]]$  we obtain the braided commutative rigid
monoidal category ${}^{U\gg^\FF}_{A^\FF}\MMM_{A^\FF\,\geom}^{\sym,
      \fp}$ (cf.
Proposition \ref{prop:twist}). This provides a deformation quantization of $G$-equivariant
    bundles on $A=C^\infty(M)$.
\end{example}

A further example, as shown in the next section, is provided by the
braided monoidal category of the modules of
covariant and contravariant tensor fields canonically associated with
the differential calculus on a braided commutative $H$-module algebra $A$.

\section{Differential and Cartan Calculus}\label{sectCC}

In \cite{Weber} it is shown that given a triangular Hopf algebra $H$
and an $H$-module algebra $\AA$ that is braided symmetric (also called
symmetric or quasi commutative) there is a canonical construction of a
differential calculus on $\AA$ and of a Cartan calculus.
We revisit the construction in the notations and conventions
that stem from Section \ref{section2}, and that will be used
throughout the paper.

Let  $\Der_\RR(\AA)$ be the $\Bbbk$-module  of
braided derivations. These are $\Bbbk$-linear maps $u\in
\hom_\Bbbk(\AA,\AA)$ that satisfy the braided Leibniz rule, for all $a,b\in A$,
\begin{equation}\label{braidedleibniz}
u(ab)=u(a)b\,+\,\oR^\alpha\ra a \,(\oR_\alpha \RA u)(b)~.
\end{equation}
This braided Leibniz rule, due to the triangular structure $\RR$ of $H$,
implies the identity $u(ab)=u(\oR^\al\ra b\,\oR_\al\ra a)$ showing
compatibility with the braided symmetry of the product in $A$.

The braided commutator
$$[~,~]: \Der_\RR(\AA)\otimes \Der_\RR(\AA)\to \Der_\RR(\AA)~,~~u\otimes v\mapsto
u v-{\oR}^\al\ra v\,{\oR}_\al\ra u$$
(where composition of operators is understood) closes in
$\Der_\RR(\AA)$, is an $H$-equivariant map and structures $\Der_\RR(\AA)$ as a braided Lie
 algebra, i.e.,
\begin{equation}\label{brLie}
[u,v]=-[{\oR}^\al \ra v,{\oR}_\al \ra u]~,~~
[u,[v,z]]=[[u,v],z]+[{\oR}^\al \ra v,[{\oR}_\al\ra u,z]]~.
\end{equation}
Braided derivations are furthermore a module in ${}^{H}_\AA\MMM_\AA^\sym$
by defining, for all $a,b\in A$,
\begin{equation}\label{aub}
(au)(b)=a\,u(b)~~,~~~ua=(\oR^\al\ra a)\oR_\al\ra u~;

\end{equation}
$a u$ is again a braided derivation because of the braided symmetric property of $\AA$.

From now on we set $$\Vect(A):=\Der_\RR(A)$$ and call it the bimodule
of (braided) vector fields.
\\

We assume that $\Vect(A)$ is finitely generated and projective as left
$A$-module. While the existence of the differential and Cartan calculus is
indipendent from this assumption,  if $\Vect(A)$ is finitely generated
and projective all modules are in $\catfp$ rather than in $\cat$.
The right dual module  $\Omega(A):={}^*\Vect(A)={}_\AA
\hom(\Vect(A), A)$ of all left $A$-linear maps $\Vect(A)\to  A$ is by definition
the module of 1-forms. It  follows that it is finitely generated and projective as a
right $A$-module.
The evaluation and coevaluation maps are morhisms in $\catfp$ denoted as
in Section \ref{fgp},
\begin{equation}
\begin{split}
\langle~,~\rangle&:\Vect(A) \otimes_\AA \Omega(\AA)\to
\AA~,~~v\otimes_A\omega\mapsto \le v,\omega\re\\[.4em]
\coev&:A\to \Omega(\AA)\otimes_A \Vect(A)~,~~a\mapsto
\omega^i\otimes_A e_i
\end{split}
\end{equation}
where $\{ e_i\in \Vect(A), \omega^i\in \Omega(A) :
i=i,\ldots, n\}$ is a pair of dual bases for $\Vect(A)$.

 The contraction operator is defined by
\begin{equation}\label{contractionoperator}
\ii: \Vect(A)\to \hom_A(\Omega(A),A) ~,~~\ii_u(\omega):=\langle
u,\omega\rangle~
\end{equation}
and is just the isomorphism $\Vect(A)\simeq ({}^*\Vect(A))^*$ of
Proposition \ref{propo:fgpmodprop}.
Up to this canonical identification  $\{ e_i, \omega^i :
i=i,\ldots n\}$ is also a pair of dual bases for $\Omega(A)$.

Notice that since $\VV$ is a module in $\catfp$, the module $\Omega(A)$ is also
left dual to $\VV$ via $\le~,~\re\circ \tau_{\Omega(A),\VV}$ and
$\tau_{\Omega(A),\VV}\circ\coev$ (cf. Section \ref{sect:ribbon}).
 \\

Associated with the bimodules $\Vect(A)$
and $\Omega(\AA)$ in ${}^{H}_\AA\MMM_\AA^{\sym, \fp}$ we have the
module in ${}^{H}_\AA\MMM_\AA^{\sym, \fp}$ of
covariant and contravariant tensor fields
$\Tau^{\bullet,\bullet}=\oplus_{p,q} \Tau^{p,q}$, where
$\Tau^{p,0}=\Omega(A)^{\otimes_A p}$,
  $\Tau^{0,q}=\Vect(A)^{\otimes_A q}$
and
$\Tau^{p,q}=\Tau^{p,0}\otimes_\AA \Tau^{0,q}$. This is a graded
algebra with product that on elements of homogeneous degree is defined by
$$
\otimes_A : \Tau^{p,q}\otimes \Tau^{p',q'}\to
\Tau^{p+p',q+q'}~,~~\theta\otimes_A\nu\otimes\theta'\otimes_A\nu'\mapsto
\theta\otimes_A\oR^\al \ra \theta'\otimes_A
\oR_\al\ra\nu\otimes_A\nu'~,
$$
here $\theta\in \Tau^{p,0}$, $\nu\in \Tau^{0,q}$,  $\theta'\in
\Tau^{p',0}$, $\nu'\in \Tau^{0,q'}$.
By construction $\Tau^{\bullet,\bullet}$ is a graded
$H$-module algebra (it is not braided symmetric since for example
$\theta\otimes_A\theta'\not= \pm\, \oR^\al\ra\theta'\otimes_A
\oR_\al\ra\theta$).
\begin{proposition}\label{onionprop}
The module $\Tau^{0,r}=\Vect(A)^{\otimes_A r}$ in $\catfp$ is left dual to
$\Tau^{r,0}=\Omega(A)^{\otimes_A r}$, with
evaluation and coevaluation maps
\begin{equation*}

\begin{split}
\le~,~\re &
: \TT^{0,r} \otimes_\AA\TT^{r,0}\;\to\;
A~,~~

\le v_r\otimes_A\ldots v_2\otimes_A
v_1,\omega_1\otimes_A\omega_2\otimes_A\ldots \omega_r\re=\\[.2em]
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=
\le v_r,\ldots\le v_2,\le v_1,\omega_1\re\omega_2\re\ldots\omega_r\re \\[.2em]

\coev &: A \;\to\;\TT^{r,0}\otimes_\AA  \TT^{0,r} ~,~~\coev(a)=a_{\,}
\omega^{i_1}\otimes\omega^{i_2}\ldots\otimes_A\omega^{i_r}\otimes_A
e_{i_r}\otimes_A \ldots e_{i_2}\otimes_A e_{i_1}~.
\end{split}
\end{equation*}
\begin{proof}
The proposition follows from $\Omega(A)={}^*\Vect(A)$ and by iteration
from Remark \ref{newevcoev}, or
equivalently from $\Vect(A)\simeq\Omega(A)^*$ and Remark \ref{Thmright}.
\end{proof}
\end{proposition}
Recalling that  $\TT^{p,q}=\TT^{r,0}\otimes_A \TT^{p-r,q}$ (if
$p\geq r$) we
immediately extend the pairing to the evaluation of $\TT^{0,r}$ on
$\TT^{p,q}$. This is the morphism in $\catfp$ defined to be trivial if
$r>p$ and otherwise given by
\begin{equation}\label{evTT}
\le~,~\;\re
: \TT^{0,r} \otimes_\AA\TT^{p,q}\;\to\;
\TT^{p-r,q}~,~~\le\nu,\theta\otimes_A\tau\re:=\le\nu,\theta\re\tau
\end{equation}
for all $\nu\in \Tau^{0,r}$, $\theta\in \Tau^{r,0}$, $\tau\in
\Tau^{p-r,q}$.
In particular for $r=1$,  we obtain the extension of the contraction
operator to $\ii:\Vect(A)\to \hom_A(\TT^{p,q},\TT^{p-1,q})$. Hence the
evaluation \eqref{evTT} is just the iteration of the contraction operator
$r$-times:
$\le v_r\otimes_A\ldots v_2\otimes_A
v_1,\tau\re=
\ii_{v_r}\circ \ldots \ii_{v_2}\circ\ii_{v_1}(\tau)$.
\\

The graded
$H$-module algebra  of exterior forms is by definition $\OM:=\bigoplus_{n\in \mathbb{N\,}}\Omega^{
  n}(A)$. Here $\Omega^0_\RR(A)=A,~\Omega^1_\RR(A)=\Omega(A),$ and
$\Omega^n(A)$ is the module of completely braided antisymmetric tensors in
$\TT^{n,0}$, for example
$
\Omega^2_\RR(A)=\Omega(A)\wedge \Omega(A)\subset \Omega
_\RR (A)\otimes_\AA \Omega _\RR (A)$ is the image of the projector
\begin{equation}\label{PAproj}
P_A=\frac{1}{2}(\id^{\otimes 2}-\tau_\RR): \Omega^{\otimes 2}(A)\to
\Omega^{\otimes 2}(A)~
\end{equation}
that is a morphism in $\catfp$.
The wedge product of one forms is defined by
\begin{equation}\label{wedgeonforms}
\omega\wedge \omega':=\omega\otimes_\AA
\omega'-{\oR}^\al\ra \omega'\otimes_\AA {\oR}_\al\ra\omega~,
\end{equation}
and is a braided antisymmetric 2-form: $\omega\wedge \omega'=- \oR^\al\ra \omega'\wedge  {\oR}_\al\ra\omega$.
Similarly for forms of higher degree, indeed the construction is as in the classical
case since the braiding $\tau_\RR$ provides a representation of the
premutation group. In particular  $\OM$ is equivalently the quotient
of the tensor algebra $\TT^{\bullet, 0}$ by the two-sided ideal
generated by $\ker \,P_A$.
The exterior algebra $\OM$ is generated in degree 1 and is a graded braided symmetric
$H$-module algebra: for all $\theta\in \Omega^p(A)$, $\theta'\in
\Omega^{p'}(A)$, $\theta\wedge\theta'= (-1)^{pp'}\oR^\al\ra\theta'\wedge
\oR_\al\ra\theta$.

This same construction applied to $\Vect(A)$ gives the graded braided antisymmetric
$H$-module algebra $\Vect^\bullet(A):=\bigoplus_{n\in \mathbb{N\,}}\Vect^{
  n}(A)$ of polyvector fields.
The duality of Proposition \ref{onionprop} then implies the duality
between the graded modules $\OM$ and $\Vect^\bullet(A)$.
For example, let  $P_A=\frac{1}{2}(\id^{\otimes 2}-\tau_\RR): \Vect^{\otimes 2}(A)\to
\Vect^{2}(A)\subset \Vect^{\otimes 2}(A)$, since $P_A$ is the
adjoint of the projector in \eqref{PAproj}, cf. \eqref{taudual},
we conclude that  $\Vect^2(A)$ is dual to $\Omega^2(A)$ as
modules in $\catfp$.
\\

It is not difficult (using induction on the degree of forms)  to see
that the contraction operator $\ii:
\VV\otimes_A\Omega^{\otimes_A p}\to\Omega^{\otimes_A p-1}$ restricts to
$\ii:\VV\otimes_A\Omega^{p}\to\Omega^{p-1}$, giving, for all $u\in \Vect(A)$, the (graded) braided inner derivation
$\ii_u:\OM\to\OMmu(A)$,
$$\ii_u(\theta\wedge\theta')=\ii_u(\theta)\wedge\theta'+(-1)^{|\theta|}(\oR^\al\ra
\theta)\wedge
\ii_{\oR_\al\ra u}(\theta')~,$$
where $|\theta|\in \mathbb{N}$ is the degree of the homogeneous form
$\theta$.
 This shows that the algebra
of polyvector fields $\Vect^\bullet(A)$ is the algebra generated by
degree $-1$ braided derivations of the algebra $\OM$.

Applying a second inner derivative in the above formula
 we
obtain that on exterior forms
\begin{equation}\label{ii}
\ii_u\circ \ii_v+\ii_{\oR^\al\ra v}\circ \ii_{\oR_\al \ra u}=0~,
\end{equation}
in accordance with the equivalent definition of $\Vect^\bullet(A)$ as the
quotient of the tensor algebra $\TT^{0,\bullet}$ via the  two-sided ideal
generated by $\ker P_A$.
\\

We now study the action of the braided Lie algebra of vector fields
$\Vect(A)$ on tensor fields, i.e.,  the Lie derivative. We define $\LLL:\VV\otimes A\to A$, $\LLL_u(a):=u(a)$

and  $\LLL:\VV\otimes \VV\to \VV$, $\LLL_u(v):=[u,v]$.
Since $\xi\ra (\LLL_u(a))=\LLL_{\xi_{(1)}\ra u}(\xi_{(2)}\ra a)$ and
$\xi\ra (\LLL_u(v))=\LLL_{\xi_{(1)}\ra u}(\xi_{(2)}\ra v)$, $\LLL$ is $H$-equivariant

and we can extend the action of $\Vect(A)$ to the tensor
algebra $\Tau_\RR^{0,\bullet}$
by requiring $\LLL_u$ to be a braided derivation:
$$\LLL_u(\nu\otimes_A\nu')=\LLL_u(\nu)\otimes
\nu'+\oR^\al\ra \nu\otimes_A\LLL_{\oR_\al\ra u}(\nu')~$$
for all $\nu,\nu'\in  \Tau_\RR^{0,\bullet}$. This implies the
commutativity property $\LL_u\circ
\tau=\tau\circ\LL_u$ between the  braiding and the Lie derivative
operators.
The Lie derivative on contravariant tensor fields
is canonically defined by duality, for all $\nu\in \Tau_\RR^{0,r}$ and
$\theta\in  \Tau_\RR^{r,0}$,
\begin{equation}\label{LonOm} \LLL_u\langle \nu,\theta\rangle=\langle\LLL_u\nu,\theta\rangle
+\langle \oR^\al\ra\nu,\LLL_{\oR_\al\ra u}\theta\rangle
\end{equation}
i.e., $\langle \nu,\LLL_{u}\theta\rangle:=\LLL_{\oR^\al \ra u}\langle \oR_\al\ra
\nu,\theta\rangle-\langle\LLL_{\oR^\al\ra u}\oR_\al \ra\nu,\theta\rangle
$.
It follows that vector fields acts on the tensor algebra
$\Tau^{\bullet,\bullet}$ as braided derivations.
On tensor fields  $\Tau^{\bullet,\bullet}$ we have
\begin{equation}\label{LLL}
\LLL_u\circ \LLL_v-\LLL_{\oR^\al \ra v}\circ \LLL_{\oR_\al \ra
  u}=\LLL_{[u,v]}~;
\end{equation}
this follows from the braided Jacoby identity in \eqref{brLie}, the
braided Leibniz rule and \eqref{LonOm}. Equation \eqref{LLL} shows
that the Lie derivative $\LLL:\VV\otimes\Tau^{\bullet,\bullet}\to  \Tau^{\bullet,\bullet}$
is an action of the braided Lie algebra of derivations $\VV$ on  $\Tau^{\bullet,\bullet}$.

Vector fields  also act as braided derivations on the exterior algebras
$\OM$ and
$\Vect^{\bullet\!}(A):=\oplus_{n\in \mathbb{N}}\Vect^n(A)$.
It is immediate to compute, for $\omega\in \Omega(A)$,
\begin{equation}\label{Lii}
(\LLL_u\circ \ii_v-\ii_{\oR^\al \ra v}\circ \LLL_{\oR_\al \ra
  u})\omega=\ii_{[u,v]}\omega~
\end{equation}
since both left hand side and right hand side are braided derivations,
this relation extends to arbitrary exterior forms.
\\

The exterior derivative $\dd: A\to \Omega(A)$ is defined by
$$\langle u, da\rangle=\LLL_ua~,$$ for all $u\in \Vect(A)$.
The definition is well given because both  $\le\,~,\dd a\re:\Vect(A)\to A$
and $\hat a: \Vect(A)\to A$, $u
\mapsto \LLL_u(a)=u(a)$ are left $A$-linear and transform under the
corresponding $H$-adjoint action, i.e. are internal homomorphism in
${}_A\hom(\Vect(A),A)$  (this follows from
$\LLL: \Vect(A)\to \Vect(A)$, $u\mapsto \LLL_u=u$ as well as $\langle~,~\rangle$ being morphisms in $\cat$).
The undeformed Leibniz rule $\dd(ab)=\dd(a)b+a\dd(b)$ is immediate from
$\le u,\dd (ab)\re=\LLL_u(ab)=\LLL_u(a)b+\oR^\al\ra a\,\LLL_{\oR_\al \ra
  u}b=\le u,(\dd a)b\re+(\oR^\al\ra a)\le\oR_\al\ra u,\dd b\re
=\le u,(\dd a)b\re+\le u , a\dd b\re$, where we used \eqref{aub} and
that the pairing $\le~,~\re$ is well defined on the balanced tensor
product $\VV\otimes_A\Omega(A)$. The $H$-equivariance of  $\dd: A\to \Omega(A)$ is also immediate.
In terms of a pair of dual bases $\{e_i, \omega^i : i=1,\ldots n\}$,
we have $\dd a=\omega^i\le
e_i,\dd a\re=\omega^i\LLL_{e_i}(a)=\omega^i\, e_i(a)$.
\\

On one forms the exterior
derivative is defined by $\dd: \Omega(A)\to \Omega ^2 (A)$, such that for all $u,v\in \Vect(A)$,
\begin{equation}\label{donOm}
\frac{1}{2}\le u\wedge v, \dd \omega\re=-\LLL_u\le v,\omega\re+\le
[u,v],\omega\re+\LLL_{{\oR^\al}\ra v}\le\oR_\al\ra u, \omega\re~.
\end{equation}
This definition is well given because it can be shown that both the right hand side and the
left hand side define left $A$-linear maps from
$\Vect^{2}(A)\to A$ that transform under the corresponding $H$-adjoint action,
i.e., are both internal homomorphisms in
${}_{A}\hom(\Vect^{2}(A), A)$.
Acting with $\xi\in H$ on \eqref{donOm} and recalling that $\LLL,
\wedge, [~,~]$ are H-equivariant maps and  that $\Delta^\op(\xi)=\RR\Delta(\xi)\RR^{-1}$
it immediately follows that $\dd\Omega(A)\to \Omega(A)^{ 2}$ is $H$-equivariant.
 It is also straighforward to see that $\dd^2a=0$.
The exterior derivative is extended to the whole exterior algebra
$\OM$ as a graded derivation (equivalently, since
$\dd$ is $H$-equivariant, as a graded braided derivation). This
defines the graded differential algebra $(\OM,\wedge,\dd)$
with $(\OM,\wedge)$ a braided graded algebra object in
$\cat$, and $\dd$ that squares to zero and is an $H$-equivariant
derivation of $(\OM, \wedge)$.
\\

Using induction on the form degree we have that the Lie derivative
commutes with the exterior derivative, indeed, for all $a\in A$,
$\omega\in\Omega(A)$, $z\in \Vect(A)$,
$\LLL_z\dd a=\dd\LLL_za$ and
$\LLL_z\dd \omega =\dd\LLL_z\omega$ as can be seen by applying the Lie
derivative to \eqref{donOm}. Similarly, for all $v\in \Vect(A)$,
$\LLL_v=\ii_v\circ \dd+\dd\circ \ii_v$ trivially holds on $A$. It also holds on
$\Omega(A)$, indeed, for all $u\in \Vect(A)$ we have
$$\le u,\LLL_v\omega\re=
\LLL_{\oR^\al \ra v}\le\oR_\al\ra
u,\omega\re-\le[{\oR^\al \ra v}\,,{\oR_\al \ra u}]\,,\omega\re=
\LLL_{\oR^\al \ra v}\le\oR_\al\ra
u,\omega\re+\le[u,v],\omega\re~;$$
now rewrite the last term using \eqref{donOm} and use $\frac{1}{2}\le u\wedge v, \dd w\re=\le u\otimes_\RR v, \dd \omega\re=\ii_u\circ \ii_v\circ \dd\omega
$ and $\LLL_u\le v,\omega\re=\ii_u\circ \dd\circ \ii_v\omega$ in order to obtain $\le
u,\LLL_v\omega\re=\ii_u\circ (\ii_v\circ \dd+\dd\circ \ii_v)\omega$. Since $\LLL_v$ and
$\ii_v\circ \dd+\dd\circ \ii_v$ are both braided derivatives this implies their
equality as derivations of $\OM$.
\\

The equations $\LLL_z\circ \dd =\dd\circ \LLL_z$, $\LLL_v=\ii_v\circ \dd+\dd\circ \ii_v $ and  \eqref{ii}, \eqref{LLL} (restricted to exterior forms),
\eqref{Lii}, $\dd^2=0$,  constitute the Cartan calculus of the
exterior, Lie and inner derivatives that  has
recently been studied in \cite{Weber} (it
was considered in the case of
noncommutative geometries arising from Drinfeld twists of commutative
geometries in \cite{proc}). Notice that the derivation of these
equations holds true also if $\VV$ is not finitely generated and
projective over $A$ (indeed the contraction operator in
\eqref{contractionoperator} need not be an isomorphism). We summarize
these equations in the following theorem,

\begin{theorem}[Braided Cartan calculus]
Let ${A}$ be a braided symmetric left $H$-module algebra
and consider the associated braided differential algebra
$(\OM,\wedge,\mathrm{d})$.
The exterior derivative, the Lie derivative and inner derivative along vector fields
$u,v\in \Vect(A)$ are graded braided derivations of
$\OM$ (respectively of degree $1,0,
-1$) that
 satisfy
\begin{equation*}
\begin{split}
        [\LLL_u,\LLL_v]
        =&\,\LLL_{[u,v]},~~\\
        [\LLL_u,\ii_v]
        =&\,\ii_{[u,v]},\\
        [\LLL_u,\mathrm{d}]=&\,0\,,
\end{split}
\begin{split}
        [\ii_u,\ii_v]=&\,0\,,\\
        [\ii_u,\mathrm{d}]
        =&\,\LLL_u,\\
        [\mathrm{d},\mathrm{d}]=&\,0\,,
\end{split}
\end{equation*}
where
$[L,L']=L\circ L'-(-1)^{|L||L'|}\oR^\al\ra L'\circ \oR_\al\ra L$
is the graded braided commutator of $\Bbbk$-linear maps $L,L'$  on
$\OM$ of
degree $|L|$ and $|L'|$.
\end{theorem}
\begin{example}{\it Braided derivations of a triangular Hopf algebra $K$
  from bicovariant differential calculus \`a la Woronowicz.}
Let $A=K$ be a cotriangular quantum group (Hopf algebra) with dually
paired triangular Hopf algebra $U$ and $\RR$-matrix $\mathscr{R}$,
cf. Example \ref{cotriangularK}.
Let $\Delta$ and
$\varepsilon$ be the comultiplication and counit in $U$, and let $$g:=\{\chi\in U, \Delta\chi=\chi\otimes 1+\mathscr{R}^\al\otimes\mathscr{R}_\al\ra\chi\}~.$$
Due to the triangular structure, $g$ is a
$\Bbbk$-submodule of  $\ker\varepsilon$, the bracket
$$[\chi,\chi']=\chi\chi'-(\mathscr{R}^\al\ra\chi') (\mathscr{R}_\al\ra\chi)$$
is braided antisymmetric and it satisfies the braided Jacoby identity,
cf. \eqref{brLie}. Thus $g$
is a quantum Lie algebra \`a la  Woronowicz \cite{Wor}. It defines the first order
bicovariant differential calculus on the quantum group $K$ with
triangular structure $\mathscr{R}$.

Under  the canonical injection $g\simeq g\otimes 1\subset U\otimes U^\op$, and setting
$X=\chi\otimes 1$, we have $\Delta_{U\otimes U^\op}X=X\otimes
1+\oR^\al\otimes\oR_\al\ra X$, where
$\RR=\mathscr{R}\otimes\mathscr{R}^{-1}$ (cf. Example
\ref{cotriangularK}). Since $K$ is a $U\otimes U^\op$-module we have
a canonical action of $X$ on $K$; it reads $X\ra a=\chi\ra
a=a^{}_{(1)}  <\!\chi,a^{}_{(2)}\!>$. Therefore the action of $g\simeq g\otimes 1$ on $K$ is via braided
derivations: $X\ra (ab)=(X\ra a) b +(\oR^\al\ra a)(\oR_\al\ra b)$,
for all $a,b\in K$.
 These left invariant vector fields freely generate the
$K$-module of vector fields $\Vect(K):=K\otimes g$, it is the
bicovariant bimodule of vector fields dual to that of one forms. Recalling from Example
\ref{cotriangularK} that $K$ is braided commutative under $\RR$,
we obtain that the  braided derivations of $K$ are given by the
quantum Lie algebra $g$ via $\Vect(K):=K\otimes g$.
\end{example}
\begin{example}
Let $G$ be a Lie group and $M$ a $G$-manifold as in Example
\eqref{ex:bundle}. Recall the Drinfeld twist deformation of
vector bundles on $M$ of Example \eqref{ex:bundle}.
Drinfeld twist deformations of the differential and Cartan
calculus on $M$  have
been considered in \cite{NCG2}  and in more generality in
\cite{Weber}. They exemplify the constructions presented in
this section.
\end{example}

\section{Right connections, left connections, curvature and torsion}
\label{sec:connections}
We study connections on modules $\Gamma\in \cat$ considered as right
$A$-modules or as left $A$-modules  and their
extensions to $\Cat$, where
 as usual $H$ is a triangular Hopf algebra, $A$ a braided
  commutative $H$-module algebra and
  $\bigl(\OM,\wedge,\dd\bigr)$ the associated
  braided differential graded algebra (differential calculus)
  constructed in Section \ref{sectCC}.
Covariant derivatives along vector fields are introduced and their
braided bracket with inner derivatives is an inner derivative
generalizing the braided Cartan relation
$[\LLL_u,\ii_v]=\ii_{[u,v]}$ to $[\stDu,\ii_v]=\ii_{[u,v]}$. This Cartan relation implies that the
curvature tensor, defined as the square of the connection, can be
equivalently defined via the commutator of covariant derivatives along
vector fields. This extends to the braided commutative geometry
setting the usual two equivalent definitions of curvature. Similarly for
the torsion tensor.

Right (left) connections on modules in $\cat$ are
 more general connections than the bimodule connections usually considered in the
literature \cite{D-VM96}. As shown in the last subsection, using the braided commutativity property of $A$
and of $\Gamma$, they however can be summed to give connections on
tensor product modules.

\subsection{Connections and Cartan formula}
  Let $H$ be a triangular Hopf algebra, $A$ be a braided
  commutative $H$-module algebra and
  $\bigl(\OM,\wedge,\dd\bigr)$ the associated
  braided differential graded algebra (differential calculus)
  constructed in Section \ref{sectCC}.
\begin{definition}\label{defi:connection}
A right connection on a module $\Gamma$ in $\cat$
is a $\Bbbk$-linear map
\begin{equation} \dst : \Gamma \rightarrow
\Gamma \otimes_A \Omega(A)
\end{equation}
in $\hom_\Bbbk(\Gamma,\Gamma\oA \Omega(A))$,
which satisfies the Leibniz rule, for all
$s\in\Gamma$, $a\in A$,
\begin{equation}
\label{prop_Delta}
\dst(s a) =
\dst(s) \:\!a+ s\otimes_A\dd a ~.
\end{equation}
A left connection on $\Gamma$ is a $\Bbbk$-linear map
\begin{equation} \std : \Gamma \rightarrow
\Omega(A)\otimes_A\Gamma
\end{equation}
in ${}_\Bbbk\hom(\Gamma,\Omega(A)\oA\Gamma)$,
which satisfies the Leibniz rule,
\begin{equation}
\std(a s) =
\dd a\otimes_A s+ a\std(s)~.
\end{equation}
\end{definition}

We denote by $\Con_A(\Gamma)$ and ${}_A\Con_A(\Gamma)$ the set of all
right, respectively left connections. Notice that in the definition of
$\Con_A(\Gamma)$
(respectively ${}_A\Con(\Gamma)$),  no compatibility condition with the left
(right) $A$-module structure of $\Gamma$ is required.

Given any connection $\dst \in \Con_A(\Gamma)$ and any right $A$-linear map
$L\in\hom_A(\Gamma,\Gamma\otimes_A\Omega(A))$, the sum $\dst+ L$ is a
connection  in $\Con_A(\Gamma)$.
The  action $\dst\mapsto \dst +L$ is free and transitive and
hence $\Con_A(\Gamma)$ is an affine space over the module $\hom_A(\Gamma,\Gamma\otimes_A\Omega(A))$ in $\cat$.

Similarly, the difference $\std-\std'$ of two left
connections is a left $A$-linear map, and ${}_A\Con(\Gamma)$ is an
affine space over the module ${}_A\hom(\Gamma,\Gamma\otimes_A\Omega(A))$ in $\cat$.
Notice that left connections, as left $A$-linear maps, acts from the
right, cf. \eqref{copadjact} and \eqref{copadjact2}. Their properties
become more intuitive by evaluating them on
the right of elements $s\in \Gamma$, hence writing  $(s)\std$ rather than $\std(s)$.
\\

We can act with the
$H$-adjoint action $\RA$ defined in \eqref{adjact} on $\dst\in \Con_A(\Gamma)\subset \hom_\Bbbk(\Gamma,\Gamma\otimes_A\Omega(A))$, for all $\xi\in H$,
\begin{equation}
\label{eqn:conadjoint}
 \xi\RA\dst := \xi_1\ra\,\circ \dst \circ S(\xi_2)\ra\,~.
\end{equation}
This $\Bbbk$-linear map $\xi\RA\dst\in
\hom_\Bbbk(\Gamma,\Gamma\otimes_A\Omega(A))$ is easily seen to satisfy
(cf. \cite[\S 6.2]{AS}),
for all $s\in \Gamma$ and $a\in A$,
$ (\xi\RA\dst\,) (sa)=
(\xi\RA \dst\,) (s)\;\! a\, +\,s\otimes_A  \varepsilon(\xi)\dif a~.$
In particular we see that if $\varepsilon(\xi)=0$ then
$\xi\RA \dst \in\hom_A(\Gamma,\Gamma\otimes_A \Omega(A))$, while if
$\varepsilon(\xi)=1$ then $\xi\RA\dst\in\Con_A(\Gamma)$.
Similarly for $\std\in{}_A\Con(\Gamma)$.
Using this action and the braided commutativity of the $A$-bimodule $\Gamma$,  a right connection
$\dst$ on $\Gamma$ is shown to be also a braided left connection,
cf. \cite[Prop. 6.8]{AS}, and similarly
a left connection $\std$ on $\Gamma$ is also a braided right
connection, for all $a\in A, s\in \Gamma$,
\begin{equation}\label{braidedconn}
\begin{split}
~~~~~~~~~~~~~
\end{split}
\begin{split}
\dst(a s)&=(\oR^\al\ra a) (\oR_\al\ra
\dst)(s)+\oR^\al\ra s \otimes_A \oR_\al\ra \dif a~,\\
\std(s a)&= (\oR^\al\RA^\cop
\dst)(s) (\oR_\al\ra a)+\oR^\al\ra \dif a\otimes_A\oR_\al\ra s ~.
\end{split}
\end{equation}
If $\dst$ is $H$-equivariant we have $\dst(a s)= a
\dst(s)+\oR^\al\ra s\otimes_A \oR_\al\ra\dif a
$, and thus recover the notion of bimodule connection
studied in \cite{D-VM96}.
\\

The $H$-module algebra homomorphism (injection) $A\rightarrow \OM$
allows to associate to modules in $\cat$ modules in $\Cat$ via the
change of base ring
$\Gamma\to \OM\otimes_A\Gamma$.
Indeed, given $\Gamma$ in $\cat$, the bimodule in $\cat$
\begin{equation}\label{defOMM}
\OM\otimes_A \Gamma
\end{equation}
is naturally a left $\OM$-module
by defining, for all $\theta, \eta \in \OM$, $s\in \Gamma$, $\eta \wedge (\theta \otimes_\AA s):=
(\eta \wedge \theta) \otimes_\AA s$.
Since $\OM$ is a graded braided commutative
$H$-module algebra, $\OM\otimes_A \Gamma=\bigoplus_{n\in\mathbb{N}\,}
\Omega^{n\!}(A)\otimes_A\Gamma$  becomes a graded braided
commutative $\OM$-bimodule by defining
$(\theta \otimes_\AA s)\wedge\eta:=
\theta\wedge \oR^\al\ra \eta \,\otimes_\AA \oR_\al \ra s$. Hence
$\OM\otimes_A \Gamma$
is a module in $\Cat$. Furthermore, the equality
$(\OM\otimes_A \Gamma)\otimes_\OM (\OM\otimes_A \Upsilon)=\OM\otimes_A
(\Gamma\otimes_A\Upsilon)$, for any $\Gamma,\Upsilon$ in $\cat$, implies that the
association $\Gamma\to \OM\otimes_A\Gamma$
defines a strict monoidal functor from $\cat$ to $\Cat$, and
hence from $\catfp$ to $\Catfp$.

More in general,  given $W$ in $\Cat$  (and hence in $\cat$)
and $\Gamma$ in $\cat$,  we have
$W\otimes_A \Gamma$ in $\cat$ and we structure it as a module in $\Cat$ with the obvious left
$\OM$-module structure and with  right $\OM$-module structure
determined by
requiring  $W\otimes_A\Gamma$ to be braided graded symmetric. Notice that
if $\Sigma \in \cat$, the associativity $(W\otimes_A\Gamma)\otimes_A\Sigma =
W\otimes_A(\Gamma\otimes_A\Sigma)$ in $\cat$ lifts to $\Cat$.

Analogously, we define the module $ \Gamma\otimes_A \OM$ in
$\Cat$. The  $\OM$-bimodule actions read, for
all $s\in \Gamma$, $\theta, \eta\in \OM$,
$(s\otimes_\AA \theta)\wedge \eta:=s\otimes_A (\theta\wedge
\eta)$, and  $\eta\wedge (s\otimes_A\theta ):=(\oR^\al \ra s)
\otimes_A(\oR_\al\ra\eta)\wedge \theta$.
\\

Every left $A$-linear map $\tilde L\in {}_A\hom(\Gamma,\Omega^{n}\otimes_A
\Gamma)$, uniquely extends to a graded left $\OM$-linear  map
in  ${}_\OM\hom(\OM\otimes_A\Gamma,\Omega^{\bullet +n}(A)\otimes_A
\Gamma)$, that we still denote $\tilde L$ and with degree $|
\tilde L|=n$; it is given by
$\tilde L(\theta\otimes_A s)=(-1)^{|\tilde
  L||\theta|}\theta\wedge\tilde L(s)$ for all $\theta\in \OM$ of
homogeneous degree $|\theta|$ and $s\in
\Gamma$. This provides an isomorphims
${}_A\hom(\Gamma,\Omega\otimes_A
\Gamma)\simeq {}_\OM\hom(\OM\otimes_A\Gamma,\OMPu\otimes_A
\Gamma$) of modules in $\cat$. Similarly,
$\hom_A(\Gamma,\Gamma\otimes_A
\Omega)\simeq \hom_\OM(\Gamma\otimes_\OM\OM,\Gamma\otimes_A
\OMPu)$ as modules in $\cat$.

Correspondingly, right and left connections on $\Gamma$ uniquely extend as $\Bbbk$-linear operators of
degree one on the modules $\Gamma\otimes_\AA \OM$ and $\OM\otimes_\AA \Gamma$,
\begin{equation}\label{dOMdef}
\!\!\!\!\!\!\!\!\!\!\!\Dst: \Gamma\otimes_\AA \OM\longrightarrow  \Gamma\otimes_\AA \Omega^{\bullet+1\!}(A)~~,~~~
\stD: \OM\otimes_\AA \Gamma\longrightarrow  \Omega^{\bullet+1\!}(A) \otimes_\AA\Gamma~,
\end{equation}
by defining, for all $s\in \Gamma_\AA$, $\theta\in \Omega^k(A)$,
\begin{equation}\label{defofdnabla}
~~~~~\Dst(s\otimes_\AA
\theta)=\dst(s)\wedge\theta+s\otimes_\AA \dd\theta~~,~~~
\stD(\theta\otimes_\AA
s)=\dd\theta\otimes_\AA s+(-1)^k\theta \wedge {\std}( s)~.
\end{equation}
More in general, for any $h\in H$, we extend $h\RA \dst$ and
$h\RA^{\cop}\std$ by defining
\begin{equation} \label{hDD}
\begin{split}
\end{split}
\begin{split}\dd^{\phantom{J_J}}_{_{}h_{}\RA}\ddst(s\otimes_\AA
\theta)&=(h\RA\dst)(s)\wedge\theta+s\otimes_\AA
\varepsilon(h)\dd\theta~,~~\\[.2em]
\dd^{\phantom{J_J}}_{_{}h_{}\RA^\cop}\stdd(\theta\otimes_\AA
s)&=\varepsilon(h)\dd\theta\otimes_\AA s+(-1)^k\theta \wedge
{h\RA^\cop \std}( s)~.
\end{split}
\end{equation}
These definitions are  well given because they are  independent from the
representative chosen for the balanced tensor product and because they are compatible
with the $H$-module action: for all $h'\in H$,
$h'\RA \Dst=\dd^{\phantom{J_i}}_{h'\RA\!}\ddst$,
$h'\RA^\cop \stD=\dd^{\phantom{J_J}}_{_{}h'_{}\RA^\cop}\stdd$,
and similarly for $h'$ acting on $\dd^{\phantom{J_J}}_{_{}h_{}\RA}\ddst$ and $\dd^{\phantom{J_J}}_{_{}h_{}\RA^\cop}\stdd$.
This can be proven observing that  definitions \eqref{defofdnabla} also read (the second
expression holding on  $\Omega^k(A)\otimes_\AA \Gamma$),
$$\Dst=\dst\otimes_\RR \id+\id\otimes_\RR \dd
=\dst\otimes \id+\id\otimes \dd\,,~~
\stD=\dd\tilde\otimes_\RR \id+(-1)^k\id\tilde\otimes_\RR \std
=\dd\otimes \id+(-1)^k\id\otimes \std\,,$$
(and similarly for $\dd^{\phantom{J_J}}_{_{}h_{}\RA}\ddst$ and $\dd^{\phantom{J_J}}_{_{}h_{}\RA^\cop}\stdd$).
These $\Bbbk$-linear maps on $\Gamma\otimes_\AA \OM$ and $\OM\otimes_\AA \Gamma$
are induced from corresponding $\Bbbk$-linear maps on $\Gamma\otimes \OM$ and
$\OM\otimes \Gamma$, that are constructed using sums of  the canonical
tensor products $\otimes_\RR$ and $\tilde\otimes_\RR$ of $\Bbbk$-linear maps
transforming under the adjoint $H$-actions $\RA$ and $\RA^\cop$
respectively, cf. \eqref{otimesR} and \eqref{tildeotimesR}.  These
tensor producs reduce to the usual one $\otimes$  because
$\dst\in \Con_A(\Gamma)\subset \hom_\Bbbk(\Gamma,\Gamma\otimes_A\Omega)$,
that acts from the left, appears only on the left, while $\std\in _{\;A\!\!\!}\Con(\Gamma)\subset
_{\;\Bbbk\!\!}\hom(\Gamma,\Omega\otimes_A\Gamma)$, that acts from the right,
appears only on the right, and because
of $H$-equivariance of the identity $\id$ and of the exterior
derivative $\dd$ (cf. again \eqref{otimesR} and \eqref{tildeotimesR}).

It is easy to show that $\stD$ and $\Dst$ satisfy the Leibniz rule, for all $\varsigma\in
\Gamma\otimes_\AA\Omega^k(A)$, $\vartheta\in\OM$
and for all $\sigma\in
\OM\otimes_\AA\Gamma$ and
$\theta\in \Omega^k(A)$,
\begin{equation}\label{dleibOM}
\Dst(\varsigma\wedge \vartheta)= \Dst_{}\varsigma\wedge
\vartheta+(-1)^k\varsigma\wedge \dd\vartheta~~,~~~
\stD(\theta\wedge \sigma)= \dd\theta\wedge
\sigma+(-1)^k\theta\wedge \stD\sigma~;
\end{equation}
indeed because of linearity it is enough to consider
$\varsigma=s\otimes_\RR \eta$ with  $\eta\in \Omega^k(A)$,
and similarly for $\sigma$.

Vice versa  any $\Bbbk$-linear map $\Dst$ and $\stD$ as in \eqref{dOMdef} that satisfies the
Leibniz rule \eqref{dleibOM} is uniquely defined by a connection
$\dst$ and $\std$, respectively. In view of this bijection we also call  $\Dst$ and
$\stD$ connections.
\\

We canonically extend the inner derivative $\ii:\Vect(A)\to
\hom_A(\OM,\OMmu(A))$ to $\ii:\Vect(A)\to
\hom_A(\OM\otimes_A \Gamma,\OMmu\oA \Gamma)$ by defining, for all $u\in
\Vect(A)$,
\begin{equation}\label{defcontraction}
\ii_u: \OM \otimes_\AA\Gamma\to
\OMmu(A)\otimes_\AA\Gamma~,~~\theta\otimes_A s\mapsto\ii_u(\theta\otimes_\AA s)=\ii_u(\theta)\otimes_\AA s~.
\end{equation}

The covariant derivative of a left connection
 along a vector field $u\in
\Vect(A)$ is the linear operator of zero degree $\stD{}_{\mbox{$\!{}^{}_u$}}: \OM\otimes_\AA\Gamma\to \OM\otimes_\AA\Gamma$ defined by
\begin{equation}\label{defCD}
\stD{}_{\mbox{$\!{}^{}_u$}}:=\ii_u\circ \stD+\stD
\circ \ii_u~,
\end{equation}
in particular on $\Gamma$ we have $\stD{}_{\mbox{$\!{}^{}_u$}}=\ii_u\circ \std$
 that as usual we
denote by $\std{}_u$. Notice however that $\std{}_u$ is the composition of a
$\Bbbk$-linear map $\std$ acting from the right and a $\Bbbk$-linear
map $\ii_u$ acting from the left. From \eqref{defofdnabla} we have
that the covariant derivative satisfies, for all $u,v\in\VV,~a\in A,
\;s,t \in\Gamma$,
\begin{equation}\label{PROPDu}
\begin{split}
\std{}_{u+v} s&=\std{}_u s+\std{}_v s~,\\ \std{}_{au} s&=a\std{}_{u}s~,\\
 \std{}_u (s+t)&=\std{}_u (s)+\std{}_u (t)~,

\end{split}
\end{equation}
and
\begin{equation}\label{LEIBDu}
{}~~~~~~~~~~~~~~~~~~
\std{}_u(a s)
=\LLL_u(a) s+ (\oR^\al\ra a)\std{}_{\oR^\al\ra u} s~.
\end{equation}

\begin{remark} \label{remcd}
We also term covariant derivative a map $\std^\cd:\Vect(A)\times \Gamma\to \Gamma$ that satisfies
\eqref{PROPDu} and \eqref{LEIBDu}; equivalently, since \eqref{PROPDu} and
\eqref{LEIBDu} imply $\Bbbk$-bilinearity, a left $A$-linear map
$\std^{\cd}:\Vect(A)\otimes\Gamma\to \Gamma$ satisfying the Leibniz
rule  $\std^\cd(u\otimes a s)
=\LLL_u(a) s+ (\oR^\al\ra a)\std^\cd(\oR^\al\ra u\otimes
s)$. This latter more elegantly reads
\begin{equation}\label{leibnablacd}
\std^\cd(u\otimes a s)
=\LLL_u(a) s+ \std^\cd(ua\otimes s)~.
\end{equation}
Since $\std\in
{}_A\Con(\Gamma)\subset {}_\Bbbk\hom(\Gamma,\Omega(A)\oA\Gamma)$, we also require $\std^\cd\in{}_\Bbbk\hom(\VV\otimes
\Gamma,\Gamma)$. A covariant derivative is therefore a map
 $\std^\cd\in{}_A\hom(\VV\otimes
\Gamma,\Gamma)$ satisfying the Leibniz rule \eqref{leibnablacd}.
Notice that ${}_A\hom(\VV\otimes
\Gamma,\Gamma)$ is a module in $\catm$, not in $\cat$, indeed $\VV\otimes
\Gamma$ is  in $\catm$ not in $\cat$.

Let  $\pi: \VV\otimes \Omega(A)\to \VV\otimes_A \Omega(A)$ be the
canonical projection associated with the unit $\eta:
\Bbbk\to A$ (the $\Bbbk$-linear map $1_\Bbbk\mapsto
\eta(1_\Bbbk)=1_A$). To any connection  $\std\in
{}_A\Con(\Gamma)$ we associate a covariant derivative  via the map
 $$\std\longmapsto
\std^\flat:=(\le~,~\re\circ \pi\otimes_A \id_\Gamma)\circ
(\id_{\VV}\otimes\std)~,~~ \std^\flat(u\otimes s)=\ii_u\circ\std(s)~.$$
If the module $\VV$ is finitely generated and projective, this map is a bijection with inverse
$\std^\cd\mapsto (\std^\cd)^\sharp:=(\id_\Omega\otimes_A\std^\cd)\circ
(\coev\circ \eta \otimes \id_\Gamma)$,
$(\std^\cd)^\sharp(s)=\omega^i\otimes_A\std^\cd(e_i\otimes
s)$; here as usual $\Bbbk\otimes \Gamma\simeq\Gamma$ and $\{e_i,\omega^i :
i=1\ldots, n\}$
with $\coev(1_A)=\omega^i\otimes_A e_i$ is a dual basis of $\VV$.

It is easy to see that covariant derivatives form an affine
space over the module ${}_A\hom(\Vect(A)\oA\Gamma,\Gamma)$ in $\cat$.
The bijection $\flat=\sharp^{-1}$ is compatible with
the affine structures of the space of connections ${}_A\Con(\Gamma)$
and the affine space of covariant derivatives; thus it lifts to an
isomorphism of affine spaces over the isomorphic modules
${}_A\hom(\Gamma,\Omega\oA\Gamma)$ and
${}_A\hom(\Vect(A)\oA\Gamma,\Gamma)$ in $\cat$ (cf. Theorem \ref{Thmfgp123}).
\end{remark}

\begin{lemma}
The covariant derivative $\stDu$ along a vector field $u\in \VV$ of a left connection satisfies the braided Leibniz rule, for all
$\theta\in \OM$, $\sigma\in\OM\otimes_\AA\Gamma$
\begin{equation}
\stDu (\theta\wedge\sigma)=\LLL_u(\theta)\wedge\sigma+(\oR^\al\ra\theta)\wedge \stDalu(\sigma)~.
\end{equation}
\begin{proof} Because of linearity it is enough to consider a form of
  homogeneous degree $\theta\in \Omega^k(A)$. We apply the
  definition and use the Leibniz rule for $\stD$ and
  for  $\ii_u$ thus obtaining
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! \stDu\!(\theta\wedge\,\sigma)=\nn\\[.2em]
&=&\!\!\!(\ii_u\circ \stD+\stD
\circ \ii_u)(\theta\wedge\sigma)\\[.2em]
&=&\!\!\!\ii_u\big(\dd\theta\wedge\sigma+(-1)^k\theta\wedge
\stD\sigma\big)+
\stD\big(\ii_u(\theta)\wedge\sigma+(-1)^k(\oR^\al\ra\theta)\wedge
\ii_{\oR_\al\ra u}\sigma\big)\\[.2em]
&=&\!\!\!\ii_u(\dd\theta)\wedge\sigma-(-1)^k
(\oR^\al\ra\dd\theta)\wedge \ii_{\oR_\al\ra u}\sigma+(-1)^{k\,}\ii_u(\theta)\wedge
 \stD\sigma+(\oR^\al\ra\theta)\wedge \ii_{\oR_\al\ra u} \stD\sigma
\\[.2em]
&&\!\!\!+\,\dd(\ii_u\theta)\wedge\sigma-(-1)^k\:\!\ii^{}_u(\theta)\wedge
\stD\sigma+(-1)^k\dd(\oR^\al\ra\theta)\wedge
\ii_{\oR_\al\ra u }\sigma+(\oR^\al\ra\theta)\wedge
\stD\ii_{\oR_\al\ra u}\sigma
\\[.2em]
&=&\!\!\!
\LLL_u(\theta)\wedge\sigma+(\oR^\al\ra\theta)\wedge
\std{}_{\oR_\al\ra u}(\sigma)
\end{eqnarray*}
where we  used the identity
$\dd(\oR^\al\ra\theta)\otimes\oR_\al=\oR^\al\ra
(\dd\theta)\otimes\oR_\al$ due to  $H$-equivariance of the exterior derivative.
\end{proof}
\end{lemma}

\begin{theorem}\label{THM44}
For all vector fields $u, v\in \Vect(A)$ the covariant derivative
$\stDu: \OM\otimes_A\Gamma\to \OM\otimes_A\Gamma$ and the inner derivative
$\ii_v: \OM\otimes_A\Gamma\to\OMmu \otimes_A\Gamma$, satisfy the
braided Cartan relation
\begin{equation}
\stDu\circ \ii_v -\ii_{\oR^\al\ra v} \circ \stDalu=\ii_{[u,v]}~,\label{cartanCalculuscovder}
\end{equation}
equivalently,
\begin{equation}\label{CRID}
 \ii_u \circ \stDv -\stDalpv\circ \ii_{\oR_\al\ra u} =\ii_{[u,v]}~.
\end{equation}
 \begin{proof}
Because of linearity, in order to  prove the first relation it is enough to
evaluate it on  $\theta\otimes s$, with
$\theta\in \OM$ and $s\in \Gamma$.
We then compute
\begin{eqnarray*}

\big(\stDu\circ \ii_v -\ii^{}_{\oR^\al\ra v} &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\circ\, \stDalu{}_\big)
(\theta \otimes_\AA s)\,=\\[.4em]
&=&\!\!
\stDu
\big(\ii_v\theta\otimes_\AA s\big)
-\ii_{\oR^\al\ra v}\big(\LLL_{\oR_\al\ra u}\theta\otimes_\AA s+
(\oR^\beta\ra\theta)\otimes_\AA \std{}_{\oR_\be\oR_\al\ra u}  s\big)
\\[.4em]
&=&\!\!
\LLL_u\circ  \ii_v(\theta)\otimes_\AA s
+\oR^\al\ra (\ii_v\theta)\otimes_\AA  \std{}_{\!\!^{\phantom{.}}_{\oR_\al\ra u}}  s\\[.4em]
&&\!\!-\ii_{\oR^\al\ra v}\circ \LLL_{\oR_\al\ra u}(\theta)\otimes_\AA s
-\ii_{\oR^\al\ra v}(\oR^\be\ra \theta)\otimes_\AA
\std{}_{\oR_\be\oR_\al\ra u} s
\\[.4em]
&=&\!\!
\ii_{[u,v]}(\theta \otimes_\AA s)~;

\end{eqnarray*}
where in the last passage the second and fourth term cancel because
$\oR^\al\ra(\ii_v\theta) \otimes\oR_\al
=\oR^\al\ra \le v,\theta\re \otimes \oR_\al=
\le \oR^\al_{{(1)}} \ra v,\oR^\al_{{(2)}}\ra\theta\re
\otimes\oR_\al
=\le \oR^\al\ra
          v , \oR^\be\ra\theta\re\otimes\oR_\be\oR_\al=
          \ii_{\oR^\al\ra v}(\oR^\be\ra\theta)\otimes
\oR_\be\oR_\al~.
$

The equivalent Cartan relation \eqref{CRID} is obtained by observing that
both the left hand side and the right hand side in \eqref{cartanCalculuscovder} are $\Bbbk$-linear expressions
in $u$ and in $v$,  by considering the
substitution $u\otimes v \to \oR^\al\ra v \otimes \oR_\al\ra u$,  and by recalling the braided antisymmetry of the
braided commutator (cf. the first identity in \eqref{brLie}).
\end{proof}
\end{theorem}

The braided Cartan relation \eqref{cartanCalculuscovder} equivalently
reads $[\stDu,\ii_v]=\ii_{[u,v]}$ where the braided bracket, despite the
connection $\stD$ is not $H$-equivariant, braids nontrivially only
the vector fields $u$ and $v$, as in the Cartan relation $[\LLL_u,\ii_v]=\ii_{[u,v]}$.
\subsection{Curvature}

The curvatures of the connections $\dst\in \Con_A(\Gamma)$ and $\std\in
{}_A\Con(\Gamma)$ are respectively defined by
\begin{equation}~~~~

\Dst^{\!\!\!\!\!2}=\Dst\circ \Dst~\,,\,~~

{\stD}^{\!\!\!\!\!2}=\stD\circ \stD~.~~~
\end{equation}
These are respectively right and left $\OM$-linear maps,
$$

\Dst^{\!\!\!\!\!2}\;\in\hom_{\OM}(\Gamma\otimes_\AA \OM, \Gamma\otimes_\AA \OMP2\,)~,~~

{\stD}^{\!\!\!\!\!2}\;
\in
{}_{\OM}\hom(\OM\otimes_A\Gamma, \OMP2\otimes_A\Gamma)~.$$
We easily prove for example the second relation. Trivially
${\stD}^{\!\!\!\!\!2}\; $ transforms under the  $H$-adjoint action
$\RA^\cop$, since so does $\stD$. We are left to prove
left $A$-linearity, for all $\theta\in \Omega^k(A),
\varsigma\in\OM\otimes_A \Gamma $,
\begin{align}
~~~~~~~~~~~~~~\stD\circ \stD
(\theta\wedge\varsigma )&\,=\,\stD\big(\dd\theta\wedge
\varsigma+(-1)^{|\theta|}\theta\wedge
\stD\varsigma\big)\nonumber\\
&\,=\,(-1)^{|\theta+1|}\dd\theta\wedge
\stD\varsigma+
(-1)^{|\theta|}\dd\theta\wedge
\stD\varsigma+\theta\wedge \stD\circ
\stD\varsigma\\
&\,=\,\theta\wedge \stD\circ \std{}_{}\varsigma~.\nonumber
\end{align}
Recalling that $\Omega(A)={}_A\hom(\VV,A)$ we have a morphism
\begin{equation}\label{oVVG}
{}_A\hom(\Gamma,\Omega^2(A)\otimes_A\Gamma)\longrightarrow {}_A\hom(\Vect^2(A)\otimes_A\Gamma,\Gamma)
\end{equation}
(that becomes an isomorphism
if the $A$-module $\Vect(A)$ is  finitely generated and projective in
$\cat$, cf. Theorem \ref{Thmfgp123} and Remark \ref{newevcoev}).
In $ {}_A\hom(\Vect^2(A)\otimes_A\Gamma,\Gamma)$ we have a second definition of curvature of a
left connection $\std\in  {}_A\Con(\Gamma)$. As in \cite{NCG2}
we define the curvature $R\stdd$ to be the  $\Bbbk$-linear map
$R\stdd:\Vect(A)\otimes\Vect(A)\otimes\Gamma\to \Gamma$, $u\otimes
v\otimes s\mapsto R\stdd(u,v,s)$,
\begin{equation}\label{curvatureuvs}
R\stdd(u,v,s):=(\std{}_u\circ
\std{}_v-\std{}_{\oR^\al\ra v}\circ \std{}_{\oR_\al\ra u}-\std{}_{[u,v]})(s)~.
\end{equation}
We now show that it is a left $A$-linear map
$\Vect^2(A)\otimes_A\Gamma\to \Gamma$ and relate it to the
curvature ${\stD}^{\!\!\!\!\!2}\,\,$.
To this end we extend the evaluation $\le~,~\re:
\TT^{0,r}\otimes_A\TT^{p,q}\to \TT^{p-r,q}$, of $\TT^{0,r}$ on $\TT^{p,q}$,  defined in \eqref{evTT},
to the evaluation of
$\TT^{0,r}$ on $\TT^{p,q}\otimes_A\Gamma$,
\begin{equation}
\le~,~~\re
: \TT^{0,r} \otimes_\AA\,\TT^{p,q}\otimes_A\Gamma\;\longrightarrow\;
\TT^{p-r,q}\otimes_A\Gamma~,~~\nu\otimes_A\,\tau\otimes_A s\mapsto \le\nu,\tau\otimes_A s\re:=\le\nu,\tau\re s~,
\label{oppopairing}
\end{equation}
that is a morphism in $\cat$. Similarly, from the duality between
$\Vect^r(A)$ and $\Omega^r(A)$ (submodules of  $\TT^{0,r}$ and
$\TT^{r,0}$) we have the evaluation
of $\Vect^r(A)$ on $\Omega^r(A)\otimes_A\Gamma$, that we still denote
$\le~,~~\re
: \Vect^r(A)  \otimes_\AA\,\Omega^r(A)\otimes_A\Gamma\;\rightarrow\;\Gamma$.

\begin{theorem}\label{RD2}
Let $\VV$ be finitely generated and projective as $A$-module in $\cat$
and  let $\Gamma$ in $\cat$.  Consider a left connection $\std\in
{}_A\Con(\Gamma)$.
i)
The curvature $R\stdd$
defined in
\eqref{curvatureuvs} satisfies, for all $u,v\in \Vect(A)$ and $s\in \Gamma$,
\begin{equation}
R\stdd(u,v,s)=
-\,\ii_ u\circ \ii_v\circ {\stD}^{\!\!\!\!\!2}\;(s)~.
\end{equation}
ii) The curvature $R\stdd\in {}_A\hom(\Vect^2(A)\otimes_A\Gamma,\Gamma)$.
\begin{proof} Part i):
\begin{equation*}\begin{aligned}
~~~~~~~~~~~~~\,\ii_ u\circ \ii_v\circ {\stD}^{\!\!\!\!\!2}\;(s)&=\,\ii_ u\big(
                                                    \ii_{v}\circ
                                                    {\stD}(\std
                                                    s)\big)=\ii_u \big(\stDv(\std
                                                    s)-\stD(\std{}_v
                                                    s)\big)\\[.2em]
&=\ii_{[u,v]}(\std s)+\stDalpv (\ii_{\oR_\al\ra u}\std
  s)-\std{}_u(\std{}_v s)\\[.2em]
&=-(\std{}_u\circ
\std{}_vs-\std{}_{\oR^\al\ra v}\circ \std{}_{\oR_\al\ra
  u}s-\std{}_{[u,v]}s)=-R\stdd(u,v,s)
\end{aligned}
\end{equation*}
where in the second equality we added and subtracted $\stD\circ
\ii_v$, in the third we used Theorem \ref{THM44}.
\\

\noindent Part ii):
Another expression for the curvature is
\begin{equation}\begin{split}
R\stdd(u,v,s)&=-\,\ii_ u\circ \ii_v\circ {\stD}^{\!\!\!\!\!2}\;(s)=
-\,\ii_ u\le v\,,{\stD}^{\!\!\!\!\!2}\;s\re=-\le u\,,\le v, {\stD}^{\!\!\!\!\!2}\;s\re\re=
-\le
  u\otimes_A v, {\stD}^{\!\!\!\!\!2}\;s\re\\
&=-\frac{1}{2}\le u\wedge
  v, {\stD}^{\!\!\!\!\!2}\; s\re~,
\end{split}
\end{equation}
where in the last equality we used that $\Vect(A)\otimes_A\Vect(A)$ is
the direct sum of braided antisymmetric plus braided symmetric vector fields,
and that these latter have vanishing pairing with 2-forms. Since ${\stD}^{\!\!\!\!\!2}_{}\in
{}_{A}\hom(\Gamma, \Omega^2(A)\otimes_A\Gamma)$
and $\le~\,,\,~\re :
\Vect^2(A)\otimes_A\Omega^2(A)\otimes_A\Gamma\longrightarrow
\Gamma$ we see that $R\stdd$ is well defined
as  a map $\Vect^2(A)\otimes_A\Gamma\to \Gamma$. In other terms we can write
$$
R\stdd(u,v,s)=\frac{1}{2}R\stdd(u\wedge v\otimes_As)~.
$$
Moreover, $\le~,~\re$ is left $A$-linear because it is a morphism in
$\cat$, hence also $R\stdd$ is left $A$-linear: for every $a\in A,\; u,v\in
\Vect(A)$ and $s\in \Gamma$, $R\stdd(au,v,s)=a R\stdd(u,v,s)$. Finally, we
have $R\stdd\in {}_A\hom({\Vect^2(A)\otimes_A\Gamma},\Gamma)$ because $R\stdd$ transforms according to the $H$-adjoint action
$\RA^\cop$, indeed, for all $h\in H$,
\begin{equation*}
\begin{split}
h\ra \big(R\stdd(u,v,s)\big)&=
-\frac{1}{2}\le h_{(1)}\ra u\,\wedge h_{(2)}\ra v , (h_{(4)}\RA^\cop
{\stD}^{\!\!\!\!\!2}\;)(h_{(3)}\ra s) \re\\[.2em]
&=
(h_{(4)}\RA^\cop R\stdd)(h_{(1)}\ra u,h_{(2)}\ra
v,h_{(3)}\ra s)~,
\end{split}
\end{equation*}
i.e., $h\ra \big(R\stdd(u\wedge v\otimes_A s)\big)=(h_{(2)}\RA^\cop
R\stdd)(h_{(1)}\ra (u\wedge v\otimes_A s))$.

\end{proof}
\end{theorem}

An analogous theorem holds for right connections; it involves
considering the $A$-bimodule of forms $\hom_A(\VV,A)$. This is obtained via the isomophism
${\cal D}_{\VV, A}: \Omega (A)\!= \!{}_A\hom(\VV,A)\longrightarrow
\hom_A(\VV,A)$, cf. \eqref{Dcallr}.
Equivalently, one can use the evaluation map
\begin{equation}\label{VVtildeOm}
\langle~,~\rangle':=
\langle~,~\rangle\circ \tau^{}_{\Omega(A),\Vect(A)}:
\Omega(A)\otimes_A\Vect(A)\to A~.
\end{equation}

\subsection{Torsion}
In this subsection we set $\Gamma=\Vect(A)$ with $\VV$ finitely
generated and projective as $A$-module  in $\cat$. As usual
$\Omega(A)={}^*\Vect(A)={}_A\hom(\VV,A)$ is the right dual module. Consider the canonical element
$I:=\coev(1_A)\in \Omega(A)\otimes\Vect(A)$.
Notice that $I$ is invariant under the $H$-action: for all $h\in H,
h\ra I=\varepsilon (h)I$ (indeed, so is $1_A$, and $\coev: A\to \Omega(A)\otimes_A\Vect(A)$ is a
morphism in $\cat$).

Given a left
connection $\std$, we define the associated torsion 2-form with values in vector
fields to be the tensor field
$$
\stD(I)\in \Omega^2(A)\otimes_A \Vect(A)~.
$$
We also define the torsion $T\stdd$ as the
$\Bbbk$-linear map
$T\stdd:\Vect(A)\otimes\Vect(A)\to\Vect(A)$, $u\otimes
v\mapsto T\stdd(u,v)$,
\begin{equation}\label{Torsion-uv}
T\stdd(u,v):=\std{}_uv-\std{}_{\oR^\al\ra v}\:\!{\oR_\al\ra u}\,-\:\!{[u,v]}~.
\end{equation}
We now show that it is a left $A$-linear map
$\Vect^2(A)\to  \Vect(A)$ and relate it to the
torsion 2-form $\stD(I)$.
\begin{theorem}\label{Tasd}
Consider a left connection $\std\in
{}_A\Con(\Vect(A))$.
For all $u,v\in \Vect(A)$ we have
\begin{equation}
T\stdd(u,v)=
-\,\ii_ u\circ \ii_v\circ {\stD}\,(I)~,
\end{equation}
hence $T\stdd\in  {}_A\hom(\Vect^2(A),\Vect(A))$.
\begin{proof}
The first of the coherence conditions \eqref{coerecoev}
between the evaluation and coevaluation maps
$\le~,~\re:\Vect(A)\otimes_A\Omega(A)\to A$ and $\coev:A\to
\Omega(A)\otimes\Vect(A)$ equivalently reads, for all $v\in \Vect(A)$, $\ii_v\:\!I=v$.
We then recall the definition of the covariant derivative in equation \eqref{defCD}, and compute, for all $u,v\in \Vect(A)$,
\begin{align}{}
~~~~~~~~~~~~~~~~~~~~~~~~~~\ii_ u\circ \ii_v\circ {\stD}\,(I)&=
\ii_ u\circ \stDv\,(I)- \ii_u\circ {\stD}\circ  \ii_v \,(I)\nonumber \\
&=\ii_{[u,v]}(I)+\stDalpv \circ \ii_{\oR_\al\ra u} (I)- \ii_u\circ
  {\stD}(v)\nonumber\\
& =-\big({[u,v]}+\stDalpv (\oR_\al\ra u) - \std{}_uv\big)\nonumber\\
&=-T\stdd(u,v)
\end{align}
where in the second line we used the Cartan identity \eqref{CRID}.
\\

The equality
$$T\stdd(u,v)=-\ii_ u\circ \ii_v\circ{\stD}\,(I)=-\ii_ u\le v,
{\stD}\,I\re=-\le u\otimes_A v,{\stD}\,I\re=-\frac{1}{2}\le
u\wedge v,{\stD}\,I\re~$$
shows that $T\stdd$ is a well defined map $\Vect^2(A)\to
\Vect(A)$; therefore we can write
$$T\stdd(u,v)=\frac{1}{2}T\stdd(u\wedge v)~.$$
Left $A$-linearity of $T\stdd$
immediately follows from left $A$-linearity of $\le~,~\re$. In order to show that $T\stdd\in
{}_A\hom(\Vect^2(A),\Vect(A))$ it remains
to prove that
$T\stdd$ transforms according to the $H$-adjoint action
$\RA^\cop$, indeed, recalling the $H$-coinvariance of the canonical element
$I$ we have, for all $h\in H$,
\begin{align*}h\/\ra \big(T\stdd(u,v)\big)&=\,
-\frac{1}{2}\le h_{(1)}\ra u\wedge   h_{(2)}\ra v , (h_{(4)}\RA^\cop
{\stD})(h_{(3)}\ra I) \re\\ &=\,
-\frac{1}{2}\le h_{(1)}\ra u\wedge   h_{(2)}\ra v , (h_{(3)}\RA^\cop
{\stD})(I) \re=
(h_{(3)}\ra T\stdd)(h_{(1)}\ra u,h_{(2)}\ra v)
\end{align*}
i.e., $h\ra \big(T\stdd(u\wedge v)\big)=(h_{(2)}\RA^\cop
T\stdd)(h_{(1)}\ra (u\wedge v))$.

\end{proof}
\end{theorem}
Notice that recalling the second definition in \eqref{hDD} we also have
$h \RA^\cop T\stdd=
T^{\phantom{J\!}}_{h{\tiny{{\,\RA^\cop}}}}\stdd$.
\\

An analogous result holds for right connections. In this case we consider
 $\Vect(A)$ right dual to $\Omega(A)$ with evaluation map as in
 \eqref{VVtildeOm} and coevalution map $
\coev' :=
\tau^{}_{\Omega(A),\Vect(A)}\circ\coev: A\to
\Vect(A)\otimes_A\Omega(A)$, cf. equation \eqref{tevcoev}.
 We further consider the canonical element
$I'=\coev'(1_A)\in \Vect(A)\otimes_\AA \Omega(A)$.
The torsion of a right connection $\dst$ is then defined by
$\Dst(I')\in \Vect(A)\otimes_A\Omega^{2}(A)$.

\subsection{Sum of connections}
Let $\Gamma$ and $\widehat\Gamma$ be two modules in $\cat$ with
left (right) connections. The sum of these left (right) connections
is the left (right) connection on the tensor product module
$\Gamma\otimes_A\widehat\Gamma$. We recall this construction and
present corollaries that will be used in Section 6 in order to study
connections compatible with a metric structure on $A$, these are
connections on the tensor product $\Omega(A)\otimes_\AA\Omega(A)$.

\begin{theorem}[Sum of connections] \label{defsumofD} Given connections $\dst \in \Con_A(\Gamma)$, $\widehat\dst \in
\Con_A(\widehat\Gamma)$ and  $\std \in {}_A\Con(\Gamma)$, $\widehat\std \in
{}_A\Con(\widehat\Gamma)$ on the modules $\Gamma$ and $\widehat\Gamma$
in $\cat$ their sums, respectively
defined by
\begin{align}
\begin{split}
~~~~~~~~~~~~~~~~~
\end{split}
\begin{split}
\dst\oplus_\RR\widehat\dst :\,\Gamma\otimes_A\widehat\Gamma&\,\longrightarrow\, \Gamma\otimes_A\widehat\Gamma\otimes_\AA\Omega(A)\\[.2em]
 s\otimes_\AA\hat s&\,\longmapsto\, \tau_{23}\circ(\dst(s)\otimes_A\hat
                                 s)+(\oR^\al\ra s)
                                 \otimes_A(\oR_\al\RA\widehat\dst)(\hat
                                 s)~,\nn\\[1em]
\std\,\tilde\oplus_\RR\widehat\std
  :\,\Gamma\otimes_A\widehat\Gamma&\,\longrightarrow\, \Omega(A)\otimes_\AA \Gamma\otimes_A\widehat\Gamma\nn\\
 s\otimes_\AA\hat s&\,\longmapsto\,  (\oR^\al\RA^\cop\std)(s)
                                 \otimes_A(\oR_\al\ra \hat
                                 s)+\tau_{12}\circ(s\otimes_A\widehat\std(\hat
                                 s))~,
\end{split}
\end{align}
where
$\tau_{23}:\Gamma\otimes_A\Omega(A)\otimes_A\widehat\Gamma\rightarrow
\Gamma\otimes_A\widehat\Gamma\otimes_A\Omega(A)$ and
$\tau_{12}:\Gamma\otimes_A\Omega(A)\otimes_A\widehat\Gamma\rightarrow
\Omega(A)\oA\Gamma\otimes_A\widehat\Gamma$ are the braiding
isomorphisms, are well defined connections
$\dst\oplus_\RR\widehat\dst \in
{}_A\Con(\Gamma\otimes_A\widehat\Gamma)$ and $\std \,\tilde\oplus_\RR
\widehat\std\in {}_A\Con(\Gamma\otimes_A\widehat\Gamma)$.
\begin{proof} We divide the proof is in three steps, see \cite[Theorem 6.9]{AS}
  for right connections.\\
{\it i)} The definitions are independent from the
representative chosen in $\Gamma\otimes\widehat\Gamma$ for the balanced tensor product
$\Gamma\otimes_A\widehat\Gamma$ (e.g. $s a\otimes \hat s$ versus
$s\otimes a\hat s$).\\
{\it ii)}  The sums  $\dst\oplus_\RR\widehat\dst $ and $\std \,\tilde\oplus_\RR
\widehat\std $ transform according to the  $H$-adjoint actions $\ra$
and $\ra^\cop$ so that  $$\dst\oplus_\RR\widehat\dst \in \hom
_\Bbbk(\Gamma\otimes_A\widehat\Gamma,\Gamma\otimes_A\widehat\Gamma\otimes_A\Omega(A))~,~~
\std\,\tilde\oplus_\RR\widehat\std\in {}_\Bbbk\hom(\Gamma\otimes_A\widehat\Gamma,\Omega(A)\otimes_A\Gamma\otimes_A\widehat\Gamma)~.$$
This holds because these maps are induced on the
quotient $\Gamma\otimes_A\widehat\Gamma$ from
the maps
\begin{equation}\label{sums}
\begin{split}
\tau_{23}\circ\pi\circ ( \dst\otimes_\RR\id_{\widehat\Gamma})
+\pi\circ(\id_\Gamma\otimes_\RR\widehat\dst)
&:
\Gamma\otimes\widehat\Gamma\longrightarrow\Gamma\otimes_A\widehat\Gamma\otimes_A\Omega(A)~,
\\[.2em]
\pi\circ (\std\,\tilde\otimes_\RR\,\id_{\widehat\Gamma})+\tau_{12}\circ\pi\circ
(\id_{\Gamma}\,\tilde\otimes_\RR\widehat\std)
&:
\Gamma\otimes\widehat\Gamma\longrightarrow\Omega(A)\otimes_A\Gamma\otimes_A\widehat\Gamma~,
\end{split}
\end{equation}
were $\pi$ is the projection from the tensor product  over $\otimes$ to
the balanced tensor product $\otimes_A$.
The maps in \eqref{sums}  are well defined in $\hom
_\Bbbk(\Gamma\otimes\widehat\Gamma,\Gamma\otimes_A\widehat\Gamma\otimes_A\Omega(A))$ and
$_\Bbbk\hom(\Gamma\otimes\widehat\Gamma,\Omega(A)\otimes_A\Gamma\otimes_A\widehat\Gamma)$
because they are sums of compositions of the
$H$-equivariant braiding $\tau$, the $H$-equivariant projections
$\pi$
and the canonical tensor products  $\otimes_\RR$ and
$\tilde\otimes_\RR$ for internal morphisms
$\dst$, $\widehat \dst$ and  $\std$, $\widehat\std$, respectively in the
categories $({}^H\MMM,\otimes, \hom_\Bbbk)$ and
$({}^H\MMM,\otimes,{}_\Bbbk\hom)$, cf. \eqref{otimesR} and
\eqref{tildeotimesR}.
\\
{\it iii)} Due to the braided left (right) property \eqref{braidedconn} the map
 $\dst\oplus_\RR\widehat\dst $ ($\std \,\tilde\oplus_\RR
\widehat\std $) satisfies the Leibniz rule showing that it is a
right (left) connection on
$\Gamma\otimes_A\widehat\Gamma$.
\end{proof}
\end{theorem}
From point {\it ii)}  in the above proof it follows that for all $h\in H$,
 $h\ra(\dst\oplus_\RR\widehat\dst)= h\ra\dst\,\oplus_\RR\, h\ra
 \widehat\dst$  and $h\ra^\cop(\std \,\tilde\oplus_\RR\widehat\std)=
 h\ra^\cop\std \;\tilde\oplus_\RR\, h\ra^\cop\widehat\std$.
Furthermore, it is not difficult to prove associativity of
the sum $\oplus_\RR$ of right connections and of the sum
$\tilde\oplus_\RR$ of left connections. This implies the
following corollary.

\begin{corollary}\label{corsumconn}Let $\bigoplus_{n\in {\mathbb{N}}}\Gamma^{\otimes
    n}$  be the tensor algebra generated by
  $\Gamma$ in $\cat$.
Connections $\dst$ and $\std$ on $\Gamma$
  uniquely lifts to connections on
$\bigoplus_{n\in {\mathbb{N}}}\Gamma^{\otimes
    n}$ that we still denote
  $\dst$ and $\std$ and that are given by the braided Leibniz rules
  of Theorem \ref{defsumofD}.
In particular, connections on $\Vect(A)$ lift to
connections on $\TT^{0,\bullet}$ and similarly, connections on
$\Omega(A)$ lift to connections on $\TT^{\bullet,0}$.
\end{corollary}

Associated with  the braided Leibniz rule for connections on
 $\TT^{0,\bullet}$ and on $\TT^{\bullet,0}$
we have the braided Leibniz rule for covariant
derivatives on $\TT^{0,\bullet}$ and on $\TT^{\bullet,0}$. We shall later
use the following corollary.
\begin{corollary}\label{Leibcovderu}
For any connection $\std\in\Con(\Vect(A))$ and vector field  $u\in
\Vect(A)$ the covariant derivative
$\std{}_u:\Vect(A)\to \Vect(A)$ lifts to the covariant derivative
$\std{}_u:\TT^{0,\bullet}\to \TT^{0,\bullet}$ defined via the braided Leibniz
rule
\begin{equation*}
\begin{split}
\std{}_u(v\otimes_A z)&=({\oR}^{\al\!}\ra^\cop\std)_u v \oA \oR{}_{\al\!}\ra\!
z\;+\; \oR^{\al\!} \ra\!\!\: v\otimes_A\std{}_{\oR_\al \ra u} z\\[.4em]
&={\oR}^\al\!\ra\!\!\; (\std{}_{\oR_\beta\ra u}\oR_\gamma\ra v)
\oA \oR_\al\oR^{\beta\!\!\;}\oR^{\gamma\!} \ra \! z
\;+\; \oR^{\al\!} \ra\!\!\; v\otimes_A\std{}_{\oR_\al \ra u} z

\end{split}
\end{equation*}
where $({\oR}^\al\ra^\cop\std)_u:=\ii_u\circ ({\oR}^\al\ra^\cop\std)$.
\begin{proof}
The first addend in the first equality is straighforward; the second
addend follows by considering the identity,

for all
$\theta\in\Omega(A)$, $z'\in \TT^{0,\bullet}$,

\begin{equation*}
\begin{split}
\ii_u\circ\tau_{12}(v\otimes_A\theta
\oA z')&\,=\,\le u,{\oR}^\beta\!\!\:\ra\theta\re \oR_\beta\!\!\:\ra v \oA z'
\,=\,{\oR}^\gamma \oR_\beta\!\!\: \ra\!\!\: v_{\,} \oR_{\gamma\!\!\:}\ra\!\!\: \le u, \oR^\beta\RA\theta\re\oA z'\\[.4em]
&=\oR^{\al}\!\!\:
\ra v\oA\ii_{\oR_{\al}\ra u}(\theta\oA z')
\end{split}
\end{equation*}
that is due to quasitriangularity of the $\RR$-matrix.
The second equality follows from triangularity of the $\RR$-matrix and
the identities
\begin{equation*}
\begin{split}
R_\al\ra\le\oR_\beta \ra u\,,\std\oR_\gamma\ra v\re\oA
R^\al\oR^\beta\oR^\gamma \ra z&
\,=\,
\le R_{\al_{(1)\!}}\oR_\beta \ra u\,,R_{\al_{(2)\!}}\ra (\std\oR_\gamma\ra v)\re\oA
R^\al\oR^\beta\oR^\gamma z\\[.2em]
&\,=\,\le u \,,R_\delta\ra(\std\:\!\oR_\gamma\ra v)\re\oA R^\delta\oR^\gamma
\ra z
\end{split}
\end{equation*}
and
$
{R_{\delta}}\ra(\std \:\!\oR_\gamma\ra v) \otimes_A R^\delta \oR ^\gamma\ra z\,=\,
{R_{\delta}}\ra(\std \;\!S^{-1\!}(R_\gamma)\ra v)
\otimes_A R^\delta R ^\gamma\ra z
\,=\,
(R_\al\ra^\cop\std\;\!)v\oA R^\al\ra
z\:\!,
$
both due to the quasitriangular structure.
\end{proof}
\end{corollary}

\section{Duality and Cartan structure equations}
In this section we fix $\Gamma$  to be a finitely generated and
projective $A$-module in $\cat$, i.e., to be a module in $\catfp$.  We
study the duals of left or right $A$-linear (more generally graded
$\OM$-linear) maps and the duals of connections.
This leads to the relation between the
curvature on a module $\Gamma$ and the associated
curvature on the dual module $\stG$ and similarly for the torsion. These are
the  Cartan structure equations for curvature and torsion in
braided noncommutative geometry in a global coordinate independent formalism.
Globally defined curvature and torsion coefficients, with respect to a
pair of dual bases for
the finitely generated and projective module of vector fields, are introduced and the
Bianchi identities proven.

\subsection{Connections on dual modules}\label{dualinternalmorph}
Let $\Gamma$ be in $\catfp$, the dual module $\stG={}_A\hom(\Gamma, A)$ is  in $\catfp$.
The associated modules
$\OM\otimes_\AA\Gamma$ and $\stG\otimes_\AA \OM$, defined in
\eqref{defOMM} and in the subsequent paragraphs,  are modules in
$\Catfp$, indeed $\stG\otimes_\AA \OM$ is right dual to
$\OM\otimes_\AA\Gamma$ in $\Cat$ with evaluation and coevaluation maps
that are the $\OM$-linear extensions of the ones of $\Gamma$, \begin{equation}\label{pairOMG}\!\!\!
\begin{split}
\le~,~\re&: (\OM\otimes_\AA
\Gamma)\,\otimes_\OM (\stG\otimes_\AA \OM)\rightarrow
\OM\;,~~\le\theta\otimes_A s,\sts\otimes_A\eta\re=\theta\wedge
\le s,\sts\re\wedge\eta \\[.4em]
\coev&: \OM \longrightarrow (\stG\otimes_A \OM)\otimes_{\OM}(\OM\otimes_A
\Gamma)\;,~~\coev(\theta)=\theta\; \sts^i\otimes_\OM s_i~.
\end{split}
\end{equation}
We study the duals of  $\Bbbk$-linear maps, of  morphisms in ${}^H\MMM$
($\Bbbk$-linear and $H$-equivariant maps) and of  internal morphisms
in $\Cat$ (graded left $\OM$-linear maps or right $\OM$-linear maps).
\begin{definition}\label{def-dualL} The dual  (or adjoint)  of a $\Bbbk$-linear
map
$\tilde L :
\OM\otimes_\AA\Gamma\to
\Omega^{\bullet+|\tilde L|}\!(A)\otimes_A \Gamma$, of degree $|\tilde
L|$,
 is the right $\OM$-linear map ${}^{*\!}\tilde L: \stG\otimes_\AA \OM\to\stG\otimes\Omega^{\bullet+|\tilde L|}\!(A)$, of degree $|{}^{*\!}\tilde L|=|\tilde L|$, defined by
\begin{equation}\label{LL'}
\le\sigma,{}^{*\!}\tilde L (\stsigma)\re=(-1)^{|\tilde  L||\sigma|}
\le\tilde L(\sigma), \stsigma\re
~,
\end{equation}
for all $\sigma \in
\OM\otimes_\AA \Gamma$  of homogeneous form
degree $|\sigma|$ and  $\stsigma\in \stG\otimes_\AA \OM$.

Vice versa, the dual of a
 $\Bbbk$-linear map $L:\stG\otimes_\AA \OM\to \stG\otimes_\AA
\Omega^{\bullet+|L|}\!(A)$ of degree $|L|$
is the graded left  $\OM$-linear map $L^*:  \OM\otimes_A\Gamma\to
\Omega^{\bullet+|\stL|}\!(A)\otimes_A\Gamma$ of degree $|L^*|=|L|$ defined by
\begin{equation}\label{L'L}
\le L^*(\sigma), \stsigma\re=(-1)^{|L||\sigma|}\le\sigma,L(\stsigma)\re~,
\end{equation}
for all $\sigma \in
\OM\otimes_\AA \Gamma$  of homogeneous form
degree $|\sigma|$ and  $\stsigma\in \stG\otimes_\AA \OM$.
\end{definition}
Notice that while ${}^*\tilde L$ is a right $\OM$-linear map, $L^*$ is
a graded left $\OM$-linear map, c.f. \eqref{gradedOMlinear}.

 By $\Bbbk$-linearity the map  ${}^{*\!}{\scriptstyle(\,~)} : \tilde L\mapsto
{}^{*\!}\tilde L$, defined in \eqref{LL'}  when $\tilde L$ has homogeneus
degree, extends to arbitrary $\Bbbk$-module maps $\tilde L :
\OM\otimes_\AA\Gamma\to
\OM\otimes_A \Gamma$. Similarly we have the $\Bbbk$-linear map
${\scriptstyle(\,~)}{}^{*} :  L\mapsto L^*$ defined on arbitrary $\Bbbk$-module maps $L :
\stG\otimes_\AA\OM\to\stG\otimes_A \OM$.
\begin{proposition}\label{dualfunc}
The $\Bbbk$-linear and grade preserving maps
${\,}^{*\!}{\scriptstyle(\,~)}$ and ${\scriptstyle(\,~)}{}^{\!*}$
restrict to $H$-equivariant maps
\begin{equation}\label{LtL}
{}^{*\!}{\scriptstyle(\,~)}: {}_\Bbbk\hom(\OM\otimes_\AA\Gamma , \OM\otimes_A \Gamma)
\,\longrightarrow \,{}\hom_\OM(\stG\otimes_\AA
 \OM , \stG\otimes\OM)~,~ ~\tilde L\mapsto
{}^*\tilde L~,~
\end{equation}
\begin{equation}\label{LtLp}
{}{\scriptstyle(\,~)}^{\!*}{}:
{}\hom_\Bbbk(\stG\otimes_\AA
 \OM , \stG\otimes\OM)
\,\longrightarrow \,
{}_\OM\hom(\OM\otimes_\AA\Gamma , \OM\otimes_A \Gamma) ~,~ ~ L\mapsto L^*~.~
\end{equation}
The map in \eqref{LtL} further restricts to an isomorphism in  $\Catfp$,
\begin{equation}\label{LtL'}
{}^{*\!}{\scriptstyle(\,~)}: {}_{\OM}\hom(\OM\otimes_\AA\Gamma , \OM\otimes_A \Gamma)
\,\longrightarrow \,{}\hom_\OM(\stG\otimes_\AA
 \OM , \stG\otimes\OM)~,

\end{equation}
with inverse
$
{\scriptstyle(\,~)}^{\!*}: {}\hom_\OM(\stG\otimes_\AA
 \OM , \stG\otimes\OM)
\,\longrightarrow
{}_{\OM}\hom(\OM\otimes_\AA\Gamma , \OM\otimes_A \Gamma)
$,
that is the restriction of the map in \eqref{LtLp}.

\begin{proof}
We show that the map ${}^{*\!}{\scriptstyle(\,~)}$ in
\eqref{LtL} is
$H$-equivariant: for all $h\in H$, $h\ra{}^*\tilde L={}^*(h\ra^\cop \tilde
L)$. By $\Bbbk$-linearity it is enough to prove $H$-equivariance on
elements $\tilde L$ of homogenous degree $|\tilde L|$. This is indeed
the case because for all $\sigma \in \OM\otimes_\AA \Gamma$  of
homogeneous form degree $|\sigma|$ and
all $\stsigma\in \stG\otimes_\AA \OM$, we have
\begin{equation}\label{equi*}
\begin{split}
\le \sigma, (h\ra {}^*\tilde L)(\stsigma)\re&=\le\sigma,h_{(1)}\ra
({}^*\tilde L (S(h_{(2)})\ra\stsigma)\re=h_{(2)}\ra\le
S^{-1}(h_{(1)})\ra\sigma, {}^*\tilde L (S(h_{(3)})\ra\stsigma)\re\\[.4em]
&=(-1)^{|\tilde L||\sigma|}h_{(2)}\ra\le \tilde
L(S^{-1}(h_{(1)})\ra\sigma),S(h_{(3)})\ra\stsigma\re\\[.4em]
&=
(-1)^{|\tilde L||\sigma|}\le h_{(2)}\ra \tilde
L(S^{-1}(h_{(1)})\ra\sigma),\stsigma\re\\[.4em]
&=(-1)^{|\tilde
  L||\sigma|}\le (h\ra^\cop \tilde L)(\sigma),\stsigma\re\\[.4em]
&=\le\sigma , {}^*(h\ra^\cop \tilde
L)(\stsigma)\re~.
\end{split}
\end{equation}
$H$-equivariance of ${\scriptstyle(\,~)}^{\!*}$ is proven substituting
in \eqref{equi*}
${}^*\tilde L$ and $\tilde L$ with $L$ and $L^*$, respectively.
\\

The restricted map
${}^{*\!}{\scriptstyle(\,~)}$ in \eqref{LtL'} is a morphism  in
$\Cat$ since for all internal morphisms
$\tilde L\in
{}_{\OM}\hom(\OM\otimes_\AA\Gamma , \Omega^{\bullet+|\tilde L|}\!(A)\otimes_A \Gamma)$,
$L\in \hom_\OM(\stG\otimes_\AA \OM, \stG\otimes_\AA
\Omega^{\bullet+|\tilde L|}\!(A))$ and
forms  $\theta\in
\OM$ of homogeneous degree $|\theta|$,
we have ${}^*(\theta\tilde L)=\theta\,{}^{*\!}\tilde L$
and
${}^*(\tilde L\theta)={}^{*\!}\tilde L\:\!\theta$.
We prove for example the first relation, for all
$\sigma \in \OM\otimes_\AA \Gamma$  of
homogeneous form degree $|\sigma|$ and  $\stsigma\in \stG\otimes_\AA
\OM$, we have
\begin{equation*}
\begin{split}
\le\sigma,{}^{*\!\:\!}(\theta\tilde L)(\stsigma)\re
&=(-1)^{|\theta\tilde  L||\sigma|}\le(\theta\tilde L)(\sigma),\stsigma\re
=(-1)^{|\theta||\tilde L|} (-1)^{|\theta||\sigma|}(-1)^{|\theta\tilde L||\sigma|}
     \le \tilde L(\sigma \theta),\stsigma\re\\[.2em]
&= (-1)^{|\tilde L||\theta\sigma|}
     \le\tilde L(\sigma \theta),\stsigma\re
=\le\sigma ,\theta \,{}^{*\!}\tilde L(\stsigma)\re
=\le\sigma ,(\theta\, {}^{*\!}\tilde L)(\stsigma)\re
\end{split}
\end{equation*}
where we used the definition \eqref{LL'},   the bimodule structure of internal
morphisms in $\Cat$ given in  \eqref{bimodofOM}, then the definition \eqref{L'L} and
again   \eqref{bimodofOM}.

Finally, the morphism ${}^{*\!}{\scriptstyle(\,~)}$ in \eqref{LtL'} is
an isomorphism, with inverse ${\scriptstyle(\,~)}^{\!*}$, since  for all $L$ and $\tilde L$,
$({}^*\tilde L )^*=\tilde L$
and ${}^*_{} (L^*)=L$. This immediately follows from  the definitions
\eqref{LL'} and \eqref{L'L}.
\end{proof}
\end{proposition}
We have explicit expressions for the isomorphisms
${}^{*\!}{\scriptstyle(\,~)}$ and ${\scriptstyle(\,~)}^{\!*}$ in
$\Cat$   in terms of a
dual basis of $\Gamma$, i.e., of the coevaluation map
$\coev(1_{\OM})=\sts^i\otimes_As_i$.
For all internal morphisms
$\tilde L\in
{}_{\OM}\hom(\OM\otimes_\AA\Gamma , \Omega^{\bullet+|\tilde L|}\!(A)\otimes_A \Gamma)$,
$L\in \hom_\OM(\stG\otimes_\AA \OM, \stG\otimes_\AA
\Omega^{\bullet+|\tilde L|}\!(A))$ and
 $\sigma \in \OM\otimes_\AA \Gamma$  of
homogeneous form degree $|\sigma|$ and  $\stsigma\in \stG\otimes_\AA
\OM$, we have
\begin{equation}\label{*Lexplicit}{}^{*\!}\tilde L(\stsigma)=\sts^i\le \tilde
L(s_i),\stsigma\re~,~~L^*(\sigma)=(-1)^{|L||\sigma|}\le \sigma,
L(\sts^i)\re s_i~.
\end{equation}
This is due to the identities
$
\le\sigma, \sts^i\le\tilde L(s_i),\stsigma\re\re=
\le\sigma, \sts^i\re\le\tilde
L(s_i),\stsigma\re=(-1)^{|\tilde L||\sigma|}\le\tilde L(\le\sigma,\sts^i\re
s_i),\stsigma\re=(-1)^{|\tilde L||\sigma|}\le\tilde
L(\sigma),\stsigma\re
=\le \sigma, {}^*\tilde L(\stsigma)\re~,
$
where we first used
right $\OM$-linearity of the pairing, then its left $\OM$-linearity
and that of $\tilde L$. The expression for $L^*$ is similarly proven.

\begin{definition}\label{Defdualconn}
Let $\std\in {}_A\Con(\Gamma)$. The dual of the left connection $\stD$ is the $\Bbbk$-linear map
${}^{*\!}(\stD): \stG \otimes_\AA \OM\longrightarrow
\stG\otimes_A\Omega^{\bullet+1\!}(A) $, defined by
\begin{equation}\label{dualstconn}
\dd\le\sigma,\stsigma\re=\le \stD\sigma,\stsigma\re+(-1)^{|\sigma|}\le\sigma,{}^{*\!}(\stD)\stsigma\re~,
\end{equation}
for all $\sigma \in  \Gamma\otimes_A\OM$ of homogeneous form degree $|\sigma|$ and $\stsigma\in \stG\otimes_\AA \OM$.
\label{dualconnst}
\end{definition}
From the definition it follows  that ${}^{*\!}(\stD)$ is a right connection.
Indeed
for all $\sigma$ and $\stsigma$ as above and
$\vartheta\in\OM$, we have the identity
$\le \sigma , {}^{*\!}(\stD)(\stsigma\wedge \vartheta)\re=
\le
\sigma, \,{}^{*\!}(\stD)\stsigma\wedge \vartheta\re+
(-1)^{|\stsigma|}\le
\sigma, \stsigma\wedge\dd\vartheta\re$.
\\

Vice versa, given a right connection $\dst \in {}_A\Con(\stG)$,
equation \eqref{dualstconn}, rewritten as
\begin{equation}\label{eqdualconnst}
\dd\le\sigma,\stsigma\re=\le (\Dst)^{*\,}\sigma,\stsigma\re+(-1)^{|\sigma|}\le\sigma,\Dst\,\stsigma\re~,
\end{equation}
{\sl defines} a left connection
$(\Dst)^*$.
Obviously, the dual of a dual connection
is the initial connection.

If $\sigma=s\in \Gamma\subset \Gamma\otimes_A\OM$ and similarly, if
$\sigma'=s'\in\Gamma'\subset\OM\otimes_A \Gamma'$,
equations \eqref{dualstconn} and \eqref{eqdualconnst} read
\begin{equation}\label{dddual}
\dd\le s, \sts\re=\le \std s, \sts\re+\le s, {}^*\std  \;\sts\re~~,~~~\dd\le
s,\sts\re=\le \dst^* s,\sts\re +\le s,\dst\; \sts\re~
\end{equation}
and define ${}^*\std\in\Con_A(\stG)$ in terms of $\std\in
{}_A\Con(\Gamma)$, and $\dst^*\in {}_A\Con(\stG)$ in terms of
$\dst\in\Con_{A}(\stG)$.
Since the extensions $\stD$ and $\Dst$ of the
connections $\std$ and $\dst$ are uniquely determined by the Leibniz
rule, we have  ${}^*(\stD)=\Dstst$ and  $(\Dst)^*=\Dst_{\!\!\!\!\!\!{\phantom{I^I}}^*}$.

Using a dual basis we have the explicit expressions, for all $s\in
\Gamma$, $\sts\in \stG$,
$$
{}^*\std\;\sts=\sts^i\otimes_A\dd\le
s^i,\sts\re-\sts^i\otimes_A\le\std s^i,\sts\re~,~~~\dst^*s=\dd\le
s,\sts^i\re\otimes_A s_i-\le s,\dst\,\,\sts^i\re\otimes_A s_i~.
$$
For example, applying the first expression on $s\in \Gamma$, and
using

$\le s,\sts^i\re\le\std s_i,\sts\re=\le\std(\le s,\sts^i\re s_i),\sts\re
$ $-$ $(\dd\le s,\sts^i\re)\le s_i,\sts\re =\le \std s,\sts\re- (\dd\le s,\sts^i\re)\le s_i,\sts\re $,
we  obtain the first expression in \eqref{dddual}.
\\

The difference of  two connections $\std, \std' \in {}_A\Con(\Gamma)$ is a left
$A$-linear map, the difference of their duals is the right $A$-linear
map
${}^{*}\:\!\std - {}^{*}\:\!\std'=-{}^{\,*\!\!\:}(\std - \std')$
where on the right hand side we used the restriciton to $\catfp$ of the isomorphism
${}^{*\!}{\scriptstyle(\,~)}$ of Proposition \ref{dualfunc}.
Since also $-^{\,*\!}{\scriptstyle(\,~)}$ is an isomorphism in $\catfp$  we immediately have
\begin{corollary}\label{dualconnaffineiso}
${}^{*\!}{\scriptstyle(\,~)}:{}_A\Con(\Gamma)\rightarrow
\Con_A(\stG)$ with  $\,-{}^{\,*\!}{\scriptstyle(\,~)}:
{}_A\hom(\Gamma,\Omega(A)\otimes_A \Gamma )\rightarrow
\hom_A(\stG,\Gamma\otimes_A\Omega(A))$
 is an isomorphism of affine spaces over modules in $\catfp$.
\end{corollary}

There is a unique way of inducing connections on
duals of tensor product modules, indeed,  the sum of dual connections is the
dual of the sum of connections.
\begin{proposition}\label{staroplus} Consider the connections $\std \in {}_A\Con(\Gamma)$, $\widehat\std \in
{}_A\Con(\widehat\Gamma)$ and
 $\dst \in \Con_A(\Gamma)$, $\widehat\dst \in
\Con_A(\widehat\Gamma)$ on the modules $\Gamma$, $\widehat\Gamma$
in $\catfp$. Let ${}^*\std \in \Con_A({}^*\Gamma)$, ${}^*\widehat\std \in
\Con_A({}^*\widehat\Gamma)$ and $\dst^* \in {}_A\Con(\Gamma^*)$, $\widehat\dst^* \in
{}_A\Con(\widehat\Gamma^*)$
 be the connections on the dual modules.
We have $${}^*(\std \:\tilde\oplus_\RR \:\!\widehat\std\:\!)={}^*\widehat\std\:\!\oplus_\RR\!
{}^*\std~,~~
(\widehat\dst\:\!\oplus_\RR\!\!\:\std\;\!)^*=\dst^* \,\tilde\oplus_\RR \widehat\dst^*~,~~
$$ as connections in
${}\Con_A({}^*\widehat\Gamma\otimes_A{}^*\Gamma)$ and in
${}_A\Con(\Gamma^*\otimes_A \widehat\Gamma^*)$, respectively.
\end{proposition}

We leave the proof of this proposition to the reader. It follows from
triangularity of the $\RR$-matrix, including the compatibility
\eqref{taudual}  of the braiding with the
dual braiding. The second equality follows form the first recalling
that the dual of a dual connection is the initial connection.

\subsection{Cartan structure equations for curvature and torsion}\label{CRcurv}
According to Corollary \ref{dualconnaffineiso} we have
\begin{lemma}\label{D'2D2'}Let $\Gamma$ be a module in $\catfp$. The
dual of the curvature 2-form of a left connection $\std\in
{}_A\Con(\Gamma)$ is minus the curvature two form of the dual
connection ${}^*\std\in  \Con_A(\stG)$, i.e.,
${{}^*({\stD}^{\!\!\!\!\!2\,})}=-\,{\Dstst}^{\!\!\!\!\!\!\!2\;\:\;}$
Similarly, for a right connection $\dst\in \Con_A(\Gamma)$,
${({\Dst}^{\!\!\!\!\!2}\,)^*} =-\,\Dst_{\!\!\!\!\!\!{\phantom{I^I}}^*}^{\!\!\!\!\:\!\!2\:}$.
\begin{proof}
Use twice \eqref{dualstconn} and  ${}^*(\stD)=\Dstst$ to obtain,
for all $\sigma \in \OM\otimes_A  \Gamma$ of homogeneous form degree
$|\sigma|$ and $\stsigma\in \stG\otimes_\AA \OM$,
\begin{equation}\label{RDRD'}
0=\dd^2\le\sigma,\stsigma\re=\dd\big(\le
\stD\sigma,\stsigma\re+(-1)^{|\sigma|}\le\sigma,
\Dstst\stsigma\re\big)=\le
{\stD}^{\!\!\!\!\!2}\,\,\sigma\,,\stsigma\re+\le\sigma,
{\Dstst}^{\!\!\!\!\!\!2\:\:\,}\stsigma\re~.
\end{equation}
By definition, the dual of the curvature 2-form satisfies, cf. \eqref{LL'},
$\le \sigma,
{}^*({\stD}^{\!\!\!\!\!2}\,\,)\stsigma\re=\le
   {\stD}^{\!\!\!\!\!2}\,\,\sigma\,,\stsigma\re$, hence
${{}^*({\stD}^{\!\!\!\!\!2\,})}=-\,{\Dstst}^{\!\!\!\!\!\!\!2\;\:\;}$. The
second equality,
${({\Dst}^{\!\!\!\!\!2}\,)^*}
=-\,\Dst_{\!\!\!\!\!\!{\phantom{I^I}}^*}^{\!\!\!\!\:\!\!2\:}$,
then follows  setting $\std=\dst^*$ and recalling that the double dual of
a connection (curvature) is the original connection (curvature).
\end{proof}
\end{lemma}

Let  $\Gamma$ and $\Vect(A)$ to be in $\catfp$. Their duals
$\stG$ and $\Omega(A)$ are also in $\catfp$. According to Remark
\ref{newevcoev} the right dual of $\Vect(A)\otimes_\AA \Gamma$
is $\stG\otimes_\AA\Omega(A)$. Similarly (cf. also Proposition
\ref{onionprop}) the right dual of
$\Tau^{0,r}\otimes_\AA \Gamma$ is $\stG\otimes_\AA\Tau^{r,0}$, $r\in
\mathbb{N}$. In the next theorem we use the associated exact pairing
$\le~,~\re
: \TT^{0,r} \otimes_A\Gamma\otimes_\AA\,\stG\otimes_A\TT^{r,0}\longrightarrow A$
with $r=2$, and also the exact pairing in $\Catfp$
defined in \eqref{pairOMG}.
\begin{theorem}[Second Cartan structure equation] \label{CartanII}
Let $\Gamma$ and $\VV$ be modules in $\catfp$ and let $\std\in {}_A\Con(\Gamma)$. For all
$u,v\in\Vect(A)$, $s\in \Gamma$, $\sts\in\stG$ we have
\begin{equation*}
\le R\stdd(u,v,s), \sts\re=\le u\otimes_A v\otimes_A s,
{\Dstst}^{\!\!\:\!\!\!\!\!\!2}\,\:\:\sts\re
\end{equation*}
or, equivalently, $\le R\stdd(u, v,s), \sts\re=\frac{1}{2}\le u\wedge
v\otimes_A s,
{\Dstst}^{\!\!\:\!\!\!\!\!\!2}\,\:\: \sts\re$.
\begin{proof}
First notice that for all $\vartheta\in \OM\subset \TT^{\bullet,0}(A), s\in
\Gamma, \sts\in\stG$, considering the pairing of
$\OM\otimes_A \Gamma$ with $\stG\otimes_A\OM$ defined in equation
\eqref{pairOMG} we have
\begin{equation}\label{pairpairs}
\begin{split}
\le \ii_u\circ \ii_v (\vartheta\otimes_A s),\sts\re&=\le \le u\otimes_A v,
\vartheta\re s,\sts\re= \le u\otimes_Av,
\vartheta\re\le s,\sts\re\\[.4em] &=
\le u\otimes_Av,
\vartheta \le s,\sts\re\re=
\le u\otimes_Av,
\le\vartheta\otimes_A s,\sts\re\re\\[.4em] &=
 \ii_u\circ \ii_v \le \vartheta\otimes_A s,\sts\re
\end{split}
\end{equation}
where in the second line we first used right $A$-linearity of
$\le~,~\re: \Vect^{\otimes 2}\otimes_A\Omega^{\otimes 2}\to A$ and
then left $\OM$-linearity of the pairing in \eqref{pairOMG}.
Then, recalling also  Theorem \ref{RD2} and  Lemma \ref{D'2D2'}, we have
\begin{equation}
\begin{split}\le R\stdd(u,v,s),\sts\re&=
-\le\,\ii_ u\circ \ii_v\circ {\stD}^{\!\!\!\!\!2}\;(s),\sts\re=
-\,\ii_ u \circ \ii_v \le  {\stD}^{\!\!\!\!\!2}\;(s),\sts\re
=\,\ii_ u \circ \ii_v  \le s,
{\Dstst}^{\!\!\:\!\!\!\!\!\!2}\,\:\: \sts\re\\[.4em]
&=\le u\otimes_A v,\le s,
{\Dstst}^{\!\!\:\!\!\!\!\!\!2}\,\:\: \sts\re\re=
\le u\otimes_A v\otimes_A s,
{\Dstst}^{\!\!\:\!\!\!\!\!\!2}\,\:\: \sts\re
\end{split}\end{equation}
where in the last equality we used right $\OM$-linearity of the pairing
in \eqref{pairOMG}. The equivalent expression  $\le R\stdd(u, v,s), \sts\re=\frac{1}{2}\le u\wedge
v\otimes_A s,
{\Dstst}^{\!\!\:\!\!\!\!\!\!2}\,\:\: \sts\re$ trivially follows from
$u\wedge v=u\otimes_A v-\oR^\al\ra v\/\otimes_A \oR_\al\ra u$.
\end{proof}
\end{theorem}

\begin{theorem}[First Cartan structure equation]\label{CartanI}
Let $\VV$ be in $\catfp$ and $\std\in {}_A\Con(\VV)$. For all
$u,v\in\Vect(A)$, $\theta\in \Omega(A)$, we have
\begin{equation*}
\le T\stdd(u,v), \theta\re=-\le u\otimes_A v,
(\dd+\wedge\circ {}^{\!*}\std\;\!)\,\theta\re
\end{equation*}
or, equivalently,
$\le T\stdd(u,v), \theta\re=-\frac{1}{2}\le u \wedge v,
(\dd+\wedge\circ {}^{\!*}\std\;\!)\theta\re$.
\begin{proof}
\begin{equation*}
\begin{split}
\le T\stdd(u,v), \theta\re
&=-\le\,\ii_ u\circ \ii_v\circ {\stD}(I),\theta\re
=-\,\ii_ u \circ \ii_v \le  {\stD}(I),\theta\re
=-\,\ii_ u \circ \ii_v (\dd \le I,\theta\re+\le I,{}^{*\!\;\!}\std\,\theta\re)\\[.4em]
&=-\,\ii_ u \circ \ii_v (\dd\theta+\wedge\circ  {}^{*\!\;\!}\std\,\theta)
=-\le u\otimes_A v , \dd\theta+\wedge\circ {}^{*\!\;\!}\std\,\theta\re
\end{split}\end{equation*}
where we used Theorem \ref{Tasd}, then \eqref{pairpairs} with
$\Gamma=\Vect(A)$, next the Definition \ref{Defdualconn}
of dual connection with $\stG=\Omega(A)$, and in the second line
$\OM$-bilinearity of the pairing
$\le~,~\re:\OM\otimes_A \Vect(A)\otimes_A\Omega(A)\otimes_A\OM\to
\OM$, so that $\le I,\theta\re=\le\coev(1_{\OM}),\theta\re=
\le\omega^i\otimes_Ae_i ,\theta\re=\theta$ and $\le
I,\omega\otimes_A\eta\re=\le\omega^i\otimes_A
e_i,\omega\otimes_A\eta\re=\omega^i\wedge\le
e_i,\omega\re\wedge\eta=\omega\wedge \eta$ for all $\omega\in
\Omega(A), \eta\in \OM$.
The equivalent expression $\le T\stdd(u,v), \theta\re=-\frac{1}{2}\le u \wedge v,
(\wedge\circ {}^{\!*}\std+\dd)\theta\re$ trivially follows from
$u\wedge v=u\otimes v-\oR^\al\ra v\/\otimes\/ \oR_\al\ra u$.\end{proof}
\end{theorem}

In the proof we have shown that for all $\theta\in \Omega(A)$, $\le  {\stD}(I),\theta\re=(\dd +
\wedge\circ {}^{*\!\;\!}\std\;\!)\theta$.
This defines the torsion
\begin{equation}\label{tordualCart}
(\dd+\wedge\circ {\!\!\:}^*\std\;\!):
  \Omega(A)\to \Omega^2(A)
\end{equation} of an arbitrary right connection on one forms ${}^*\std\in
\Con_A(\Omega(A))$ and shows that it is an internal morphism in
  $\hom_A(\Omega,\Omega^2(A))$.
As for the curvature ${\Dstst}^{\!\!\:\!\!\!\!\!\!2}~~\!$,

there is a unique lift to the torsion
$(\dd\circ \wedge\:\!+\:\!\wedge\circ\Dstst)\in
\hom_\OM(\Omega(A)\otimes_A\OM,\OMP2{\,})$.

\begin{remark}In Theorems \ref{CartanII} and \ref{CartanI} we use the pairing
$\le~,~\re
: \TT^{0,2} \otimes_A\Gamma\otimes_\AA\,\stG\otimes_A\TT^{2,0}\;\to
A$. The pairing between 2-vector fields and 2-forms $\le~,~\re^{}_{\!\wedge}
: \Vect^2(A) \otimes_A\Gamma\otimes_\AA\,\stG\otimes_A\Omega^2(A)\;\to
A$ is half the value of the restriction of the first pairing to
$\Vect^2(A)\subset \TT^{0,2}$ and $\Omega^2(A)\subset \TT^{2,0}$, so that the Cartan structure equations read $\le R\stdd(u, v,s), \sts\re=\le u\wedge
v\otimes_A     s,
{\Dstst}^{\!\!\:\!\!\!\!\!\!2}\,\:\: \sts\re^{}_{\!\wedge}$ and
$\le T\stdd(u,v), \theta\re=-\le u \wedge v,
(\dd+\wedge\circ {}^{\!*}\std\,)\theta\re^{}_{\!\wedge}$.
\end{remark}

Using a dual basis $\{e_i,\omega^i : i=1\ldots, n\}$ of $\VV$

we define the curvature and torsion coefficients of a connection
$\std\in {}_A\Con(\VV)$,
$$
R^{}_{ijk}{}_{}^l:=\le R\stdd(e_i,e_j,
e_k),\omega^l\re~,~~T^{}_{ij}{}^l_{}:=\le T\stdd(e_i,e_j),\omega^l\re~
$$
and the curvature and torsion two forms (the signs are chosen to match the
commutative differential geometry case)
$$
\Rsf^{}_k{}_{}^l:= -\frac{1}{2}\omega^j\wedge\omega^i
R^{}_{ijk}{}_{}^l~,~~ \Tsf^l: =-\frac{1}{2}\omega^j\wedge\omega^i\:\! T^{}_{ij}{}^l~.
$$
Since $\omega^j\oA\omega^i\le e_i\oA e_j,
~~\re$ and $\omega^k\otimes_A\omega^j\oA\omega^i\le e_i\oA e_j\oA
e_k,~~~\re$ are the identity on   $\Omega(A)\oA\Omega(A)$ and
$\Omega(A)\oA\Omega(A)\oA\Omega(A)$ we  have the coefficient expression
of the Cartan structure equations
$$
{\Dstst}^{\!\!\:\!\!\!\!\!\!2}\,\:\:
\omega^l=\omega^k\otimes_A \frac{1}{2}\omega^j\wedge\omega^i R^{}_{ijk}{}_{}^l
=\omega^k\oA(-\:\!\Rsf^{}_k{}_{}^l)$$
$$
(\dd+\wedge\circ
{\!\!\:}^*\std\;\!)\omega^l=-\frac{1}{2}\omega^j\wedge\omega^i\:\! T^{}_{ij}{}^l= \Tsf^l
$$
We define a matrix $(\omega{}_k{}^l)$, with $k,l=1,\ldots n$, to be a
matrix of coefficients one forms of the
connection ${}^*\std\in \Con_A(\Omega(A))$, dual to $\std\in {}_A\Con(\VV)$,   if
${}^*\std\omega^l=\omega^k\otimes_A \omega{}_k{}^l$. In terms of these
coefficients we have
$$
{\Dstst}^{\!\!\:\!\!\!\!\!\!2}\,\:\: \omega^l=
\omega^k\oA(\dd\omega^{}_k{}{}_{}^l+\omega^{}_k{}^j_{}\wedge\omega^{}_j{}{}_{}^l)
$$
$$(\dd+\wedge\circ
{\!\!\:}^*\std\;\!)\omega^l=\dd\omega^l+\omega^j\wedge \omega^{}_j{}{}_{}^l$$
giving the full coefficient expression of the Cartan structure equations
$$
\omega^k\oA(\dd\omega^{}_k{}{}_{}^l+\omega^{}_k{}^j_{}\wedge\omega^{}_j{}{}_{}^l)=\omega^k\otimes_A (-\!\:\Rsf^{}_k{}_{}^l)$$
$$
\dd\omega^l+\omega^j\wedge \omega^{}_j{}{}_{}^l=
\Tsf^l~.
$$
As in commutative differential geometry, applying
$\id_{\Omega(A)}\otimes_A \dd$ to the first equation and differentiating
the second  we readily obtain the  Bianchi identities,
\begin{equation}\label{Bianchiid}
\begin{split}
&\omega^k\oA (\dd\Rsf_k{}^l+\omega_k{}^j\wedge\Rsf_j{}^l-\Rsf_k{}^j\wedge\omega_j{}^l)=0\\[.4em]
&\dd\Tsf^l-\Tsf^j\wedge\omega_j{}^l=\omega^j\wedge\Rsf_j{}^l~.
\end{split}
\end{equation}
The
only braiding present in these equations is within the wedge product.

Similarly, if we consider  a connection $\std\in{}_A\Con(\Gamma)$ on a
module $\Gamma$ in  $\catfp$, by writing
${}^{*\!}\std\:\sts^l=\sts^k\otimes_A \omega^{}_k{}_{}^l$ (with $\{s_i,\sts^i:
i=1\ldots, m\}$ a dual basis) we have the
Bianchi identity   $\sts^k\oA
(\dd\Rsf_k{}^l+\omega_k{}^j\wedge\Rsf_j{}^l-\Rsf_k{}^j\wedge\omega_j{}^l)=0$.

\section{Riemmanian geometry}\label{riemgeom}
We use the sum of connections (based on the tensor product of
internal morphisms in $\MMM^H$), the notion of dual connection and the
Cartan calculus results for the torsion in order to determine the Levi-Civita connection
associated with a  pseudo-Riemannian metric tensor on the algebra $A$.
The Ricci tensor is canonically introduced leading to the notion of
noncommutative Einstein space.
\subsection{Metric tensor}
Let  $A$ be a braided symmetric $H$-module algebra with $H$ triangular.
Let $\VV$ be the $A$-bimodule of braided derivations and
$\Omega(A)={}^*\VV={}_A\hom(\VV,A)$ the dual $A$-bimodule  of
forms. Consider  the morphism in $\cat$
\begin{equation}
\begin{split}\label{shVV}
\sharp: {}_A\hom(\VV\oA\VV,A) &\longrightarrow {}_A\hom(\VV,\Omega(A))\\
{}\tilde L & \longmapsto {\tilde L}^\sharp;~~{\tilde L}^\sharp(v)=\tilde
L(\,{\mbox{-}}\otimes_Av)~.
\end{split}
\end{equation}
A pseudo-Riemannian structure on $A$ or pseudo-Riemannian metric on
$\VV$  is a left $A$-linear map $\tilde G\in {}_A\hom(\VV\oA\VV,A)$
that is braided symmetric, i.e. $\tilde G=\tilde G\circ \tau$, and with
associated internal morphism ${\tilde G}^\sharp\in {}_A\hom(\VV,\Omega)$ that
is invertible.

If $\VV$ is finitely generated and projective,  Proposition \ref{propo:projhomiso} and Theorem \ref{Thmfgp123} give two isomorphisms
in $\catfp$, that with slight abuse of notation are both denoted by
$\flat$ (the second one being the inverse of \eqref{shVV}),
\begin{equation*}\begin{aligned}
\Omega(A)\otimes_A \Omega(A)&\stackrel{\flat_{}\,}{\longrightarrow}
                                     {}_A\hom(\Vect(A),\Omega(A))\stackrel{\flat}{\longrightarrow}
                                     {}_A\hom(\VV\oA\VV,A) \\
\g=\g^a\oA\g_a& \longmapsto ~\g^\flat:=\le\:\,,\g^a\re\oA \g_a
                                                                 \longmapsto ~\le\,{\mbox{-}}\oA{\mbox{-}}\,,\g\re=\le\,~,\le\,~,\g^a\re\g_a\re~.
\end{aligned}
\end{equation*}
Recalling that the dual of the braiding $\tau$ is the
braiding on the dual module
(cf. \ref{taudual}), we have the following equivalent definition of a metric on
the module $\VV$ in $\catfp$.
\begin{definition}
A pseudo-Riemannian metric on $\VV$ in $\catfp$ is a braided symmetric element
$\g\in\Omega(A)\otimes_A\Omega(A)$, i.e.,
$\g=\tau (\g)$, with associated internal morphism $\g^\flat\in
{}_A\hom(\Vect(A),\Omega(A))$ that is an isomorphism. We also simply  say that
$\g$ is a metric on $A$.
\end{definition}

 \subsection{Levi-Civita connection}
We prove existence and uniqueness of a metric compatible and torsion
free connection establishing a noncommutative Koszul formula.

In this section we simplify the notation of the braiding via the
$\RR$-matrix action
and set, for any  $w\in W$ with $W$ module in ${}^H\MMM$,  $\cat$ or $\Cat$,
${}^\al w=\oR^\al\ra w$ and ${}^{}_\al w=\oR_\al\ra w$ (for any $\alpha$). Hence, for
example, for all $u,v\in \Vect(A)$, $\theta\in \Omega(A)$,
\begin{equation*}
\begin{split}
&{}^\al v\oA{}_\al u=\oR^\al\ra v\:\!\oA\oR_\al\ra u=\tau(u\otimes_A v)~,\\[.4em]
&{}^\beta\theta\otimes (\!\:{}^{}_\beta{}^*\std)=\oR^\beta\ra
\theta\otimes(\oR_\beta\ra{}^*\std)~,~~({}^\al\std) \otimes {}_\al u
=(\oR^\al\ra^\cop \std) \otimes {}_\al u~.
\end{split}
\end{equation*}

Recall that considering sums of connections
(connections on tensor products) and dual connections a left connection $\std\in
{}_A\Con(\Vect(A))$ uniquely lifts to a left connection on
$\TT^{0,\bullet}(A)$ and to a dual right connection on $\TT^{\bullet,0}(A)$,
cf. Corollary \ref{corsumconn} and Proposition \ref{staroplus}.
For example
on the metric tensor we have
${}^{*}\std(\g)={}^{*}\std (\g^i\otimes_A \g_i)={}^{*}\std(\g^i)\otimes_A\g_i+
(\tau\oA\id_{\Omega(A)})({}^{\beta\;\!}\g^i\otimes_A({}_\beta{}^*\std)\g_i)$.
 Similarly for a right connection $\dst\in
\Con_A(\Omega(A))$. Moreover this latter is torsion free if its dual
 $\dst^*\in
{}_A\Con(\Vect(A))$ is torsion free, cf. Theorem \ref{CartanI} and \eqref{tordualCart}.
\begin{definition}Let $\g\in \Omega(A)\oA\Omega(A)$ be a
  pseudo-Riemannian metric. A connection
 $\dst\in \Con_A(\Omega(A))$ is metric compatible if it satisfies $\dst(\g)=0$.
A connection
$\std\in {}_A\Con(\Vect(A))$ is  metric compatible if
its dual ${}^{*}\std\in \Con_A(\Omega(A))$
is metric compatible.

A Levi-Civita connection is a  metric compatible and torsion free connection.
\end{definition}

For ease of the reader in the stamements of the next two theorems we  spell out the
condition that $\VV$ is a  module in $\catfp$.
\begin{theorem}[Uniqueness of Levi-Civita connection]\label{LCU}  Let
  $H$ be a triangular Hopf algebra,  $A$ a braided commutative
  $H$-module algebra, let the associated module in $\cat$ of braided derivations $\VV$ be finitely
  generated and projective and let $\g$ be a metric on $A$.
If a torsion free metric compatible left connection $\std\in
{}_A\Con(\Vect(A))$ exists, it is unique.
\end{theorem}
\begin{proof} Assume $\std\in
{}_A\Con(\Vect(A))$ is a torsion free metric compatible
connection.  Applying the contraction operator to the identity $$\dd\le v\oA z,\g\re=\le\std (v\oA z),\g\re+\le v\oA
z,{}^{*}\std \g\re= \le\std (v\oA z),\g\re$$ we have, for all
$u,v,z\in \Vect(A)$,
\begin{equation}\label{Luvzg}
\begin{split}\LLL_u\le v\oA z,\g\re&=
\le \std{}_u(v\oA z),\g\re\\[.4em]
&=\le\!\: {}^{\al\:\!\!}(\std{}_{{}^{\!}_\be u\,}{}_\gamma v)\oA{}_\al{}^{\beta\gamma\!\!\:}z
\,+\,
{}^\al v\oA \std{}_{{\!\!\:}^{\!}_\al u\,}z\,,\g\re\\[.4em]
 &= \le \:\! {}^{\al\:\!\!} (\std{}_{{}^\eta{}^{}_{\gamma\!\!\:} v}\,
{}_{\eta\beta}u + [{}^{}_\be u,{}^{}_\gamma v])\oA
{}_\al{}^{\beta\gamma\!\!\:}z
\,,\g\re
\,+\,
\le \!\:{}^\al v\oA\std{}_{{\!}^{\!}_\al u\,}z\,,\g\re \\[.4em]
&=\le {}^{\beta\gamma}z\oA \std{}_{{}^\eta{}^{}_{\gamma\!\!\:} v}\,
{}_{\eta\beta}u\,,\g\re\,+\, \le  {}^\al[{}^{}_\be u,{}^{}_\gamma v]\oA {}_\al{}^{\beta\gamma\!\!\:}z \,,\g\re
\,+\,
\le \!\:{}^\al v\oA\std{}_{{\!}^{\!}_\al u\,}z\,,\g\re \\[.4em]
&=\le {}^{\beta\gamma}z\oA \std{}_{{}_\beta{}^{\eta\!\!\:} v}\,
{}_{\gamma\eta}u\,,\g\re\,+\, \le  [ u, v]\oA z \,,\g\re
\,+\,
\le \!\:{}^\al v\oA\std{}_{{\!}^{\!}_\al u\,}z\,,\g\re
\end{split}
\end{equation}
where in the second line we used Corollary \ref{Leibcovderu} for the
braided derivation rule of the covariant derivative; in the third line we used the torsion free condition $T(u,v)=\std{}_uv-\std{}^{}_{{{\!}^\eta
  v}}{{}_{\,\eta\;\!\!} u}\,-[u,v]=0$; in the fourth
the braided symmetry of the metric
and that the adjoint of the braiding on forms is the braiding on
vector fields, cf. equation \eqref{taudual}; in the last line we used the
Yang-Baxter equation (in the form
$\RR^{-1}_{23}\RR^{-1}_{13}\RR^{-1}_{12}=\RR^{-1}_{12}\RR^{-1}_{13}\RR^{-1}_{23}$)  and that
the braided bracket $[~,~]$ is
$H$-equivariant.

We rewrite this identity for the cyclically permuted elements
$u\otimes v\otimes z\mapsto {}^{\al\beta} z\otimes {}_\al u\otimes
{}_\beta v$ and $u\otimes v\otimes z\mapsto {}^\eta v\otimes{}^\gamma z\otimes
{}_{\gamma\eta} u$. We then subtract the second from the first and
add the third thus obtaining (after using  the Yang-Baxter equation,
the  braided symmetry of the metric and the braided antisymmetry of the
braided Lie bracket of vector fields)
\begin{equation}\label{Koszul}
\begin{split}
2 \le \!\:{}^\al v\oA\std{}_{{\!}^{\!}_\al u\,}z\,,\g\re =&\;
\LLL_u\le v\oA z,\g\re-
\LLL_{{}_{}^\al v}\le {{}_\al u}\oA z,\g\re+
\LLL_{{}^{\al\beta\!\!\:} z}\le {{}_\al u}\oA {}^{}_\beta v,\g\re\\[.4em]
&\;-
 \le  [ u, v]\oA z \,,\g\re+
 \le  u\oA [v, z] \,,\g\re+
\le  [ u, {}^\beta z]\oA {}^{}_\beta v \,,\g\re~.
\end{split}
\end{equation}
The right hand side of this identity uniquely determines
the left hand side, that in turn, because of the exactness of the pairing
$\le~,~\re:\VV\otimes_A\Omega(A)\to A$ and the invertibility of $\g^\flat$,
uniquely determines the covariant derivative
$\std{}_u:\Vect(A)\to \Vect(A)$ for all $u\in\Vect(A)$. Recalling
Remark \ref{remcd} this proves uniqueness of the metric compatible and
torsion free connection $\std$.
\end{proof}

Let  $\kg:\Vect(A)
\otimes\Vect(A)\otimes \Vect(A)\to A$, $u\otimes v\otimes z\mapsto
\kg(u\otimes v\otimes z)$ be the $\Bbbk$-linear map defined by the right hand side
of equation
\eqref{Koszul}.
Existence of the Levi-Civita
connection is proven by studying the properties of this map.
Recall that ${}_A\hom({\Vect(A) \otimes_A\Vect(A)\otimes  \Vect(A),
  A})\subset {}_\Bbbk\hom({\Vect(A) \otimes_A\Vect(A)\otimes  \Vect(A),
  A})$ is the submodule in $\catm$ of left $A$-linear maps; it is
not a module in $\cat$ because $\Vect(A) \otimes_A\Vect(A)\otimes \Vect(A)$ is not in $\cat$.
\begin{lemma}
The $\Bbbk$-linear map $\kg$ is a left $A$-linear map
in ${}_A\hom({\Vect(A) \otimes_A\Vect(A)\otimes  \Vect(A), A})$
 and satisfies the derivation property, for all $u, v, z\in \Vect(A), a\in
A$,
\begin{equation}\label{kgder}
\kg(u\otimes_A v\otimes az)=\kg(u\otimes_A va\otimes z)+2\le{}^\al
v_{\!\;}\LLL_{{\!}^{}_\al\!\!\: u}(a)\otimes  z,\g\re~.
\end{equation}
\begin{proof}
We first show $\kg\in {}_\Bbbk\hom({\Vect(A) \otimes\Vect(A)\otimes
  \Vect(A), A})$, i.e. for all $h\in H$,
$h\ra(\kg(u\otimes v\otimes z))= (h_{(1)}\ra^\cop\kg)\big(h_{(2)}\ra
(u\otimes v\otimes z))$.
The map $\le~,\g\re:\Vect(A)\oA\Vect(A)\to A$ is easily seen to be in
${}_\Bbbk\hom(\Vect(A)\oA\Vect(A), A)$, the statement then follows
recalling  $H$-equivariance of $\LLL$, $[~,~]$, and of the braiding $\tau$
i.e.~quasi-cocommutativity of the
Hopf algebra $H$: $\Delta(h)\RR^{-1}=\RR^{-1}\Delta^\cop(
h)$, for all $h\in H$.

Next we show that the map $\kg$ is well defined on the
balanced tensor product $\Vect(A) \otimes_A\Vect(A)\otimes  \Vect(A)$.
The third addend is well defined because
$\LLL_{{}^{\al\beta\!} z}\le {{}_\al u}\oA {}^{}_\beta v,\g\re=
\LLL_{{}^{\al\!\!\:} z}\le {}^{}_{\,\al\!\!\;} (u\oA  v),\g\re$.
From the identity
$\LLL_u(v)=[u,v]=-[{}^\al v,{}_\al u]=-\LLL_{{}^\al v}({}_\al u)$
and the braided Leibniz rule of the Lie derivative we have
$$u\oA [v,z]+[u,{}^\beta z]\oA {}_\beta v=-\LLL_{{}^{\al\beta\!} z}({{}_\al u}\oA {}^{}_\beta v)=-\LLL_{{}^{\al\!}
  z}{\,}^{}_\al\!\;\!(u\oA v)$$
that implies that also the last two addends
of $\kg$ are well defined on $\Vect(A) \otimes_A\Vect(A)\otimes
\Vect(A)$.
We are left to prove the equality

$\kg(ua\otimes v\otimes
z)=\kg(u\otimes av\otimes z)$
for the sum of the first, second and fourth addend in $\kg$.
This is directly checked recalling that $\LLL_{av}=av=a\LLL_v$ on $A$,
and using the identities (the second one rewritten for $[\:\!^{\delta\!} a\!\; {}^{}_\delta u,v]$)
$$[u,av]=\LLL_u(av)=\LLL_u(a) v+{}^\al a_{\,}\LLL_{{\!}_\al u}(v)
=\LLL_u(a) v+{}^\al a[{}_\al u,v]~,$$
$$
[au, v]=[{}^{\al\beta}v,{}_\al a{\:\!}^{}_\beta
u]=\LLL_{{}^{\al\beta}v}({}_\al a){}_\beta u +a\!\:\LLL_{{}^\beta v\:\!}{}^{}_\beta
  u=a[u,v]+ \LLL_{{}^{\al\beta}v} ({}_\al a){}_\beta u~.$$
In order to show that $\kg\in {}_A\hom({\Vect(A)
  \otimes_A\Vect(A)\otimes \Vect(A), A})$  we are left to prove left
$A$-linearity of $\kg$.
This follows from left $A$-linearity of the
second plus fourth addend and of the third plus sixth addend in the
right hand side of \eqref{Koszul}.
\\

The derivation property is equivalent to
\begin{equation}\label{kgder2}
\kg(u\otimes_A v\otimes az)\,=\,{}^{\delta\eta}
a_{\,}\kg({}^{}_\delta u\otimes_A {}^{}_\eta v\otimes
z) +
2\le{}^\al
v_{\!\;}\LLL_{{\!}^{}_\al\!\!\: u}(a)\otimes  z,\g\re~.
\end{equation}
We use the braided Leibniz rule of the Lie derivative
on covariant and contravariant tensors (cf. \eqref{LonOm}) in the
first two addends of $\kg$ in \eqref{Koszul}
and, since the metric is braided symmetric,  we
also write the last addend of $\kg$ as
$\le  [ u, {}^\beta z]\oA {}^{}_\beta v \,,\g\re=\le{}^\al v\otimes_A
[{}_\al u,z],\g\re=\le{}^\al v\otimes_A
\LLL_{{}_\al u}z ,\g\re$, thus obtaining the following expression,
\begin{equation}
\begin{split}\label{kg2}
\kg(u\otimes_A v\otimes z)=\:&\le {}^\al v\otimes_A {}^\beta z,\LLL_{{}_{\!\beta\al}
  u}{}\g\re+\le[u,v]\otimes_A z,\g\re
-\le u\otimes_A {}^\al z,\LLL_{{}_\al
  v}\g\re\\[.4em]
&+\LLL_{{}^{\al} z}\le{}_\al (u\otimes_A
v),\g\re+2\le{}^\alpha v\otimes_A \LLL_{{}_\al u}z,\g\re~.
\end{split}
\end{equation}
We use this expression and the quantum Yang-Baxter equation  to compute the left hand side and the right
hand side of  \eqref{kgder2};
they are equal since each
of the first four addends in \eqref{kg2}
satisfies the homogeneous version of equation \eqref{kgder2}, while
the last one gives also the inhomogeneous term.
\end{proof}
\end{lemma}

\begin{theorem}[Levi-Civita connection] \label{LCE}
  Let
  $H$ be a triangular Hopf algebra,  $A$ a braided commutative
  $H$-module algebra, let the associated module in $\cat$  of braided derivations $\VV$ be finitely generated and projective and let $\g$ be a metric on $A$.
There exists a unique torsion free metric compatible left connection $\std\in
{}_A\Con(\Vect(A))$.
\begin{proof}
Uniqueness has been proven in Theorem \ref{LCU}. We are left to prove existence.
In the previous lemma we have seen that $\kg\in {}_A\hom(\VV\oA\VV
\otimes \VV,A)$; recalling
 the isomorphism $\sharp$ of
Theorem \ref{Thmfgp123} (equation \eqref{iso1e2}) we have the
isomorphism of modules in $\catm$,
$${}_A\hom(\VV\oA\VV \otimes \VV,A)\stackrel{\sharp}{\longrightarrow} {}_A\hom(\VV\otimes
\VV,\Omega(A))\stackrel{{\g_{}^\flat}^{-1}{\!\!\!\!}{}_{{}^\circ}{}\,}{\longrightarrow} {}_A\hom(\VV\otimes\VV,\VV)~.
$$
This shows that the map
$\std^{\cd}\in {}_A\hom(\VV\otimes\VV,\VV)$ defined by, for all $u,v,z\in
\VV$,
\begin{equation}\label{LCexists}2\le {}^\al v\otimes_A
\std^{\cd}({}_\al u\otimes z),\g\re=\kg(u\otimes_A v\otimes z)
\end{equation} is well given.
Indeed,  $\std^{\cd}$ is the image of $\kg\circ (\tau\otimes
\id_{\VV})$ under the above isomorphism, i.e.,
$\std^\cd:={\g_{}^\flat}^{-1}\!\circ \big(\kg\circ (\tau\otimes \id_{\VV})\big)^\sharp$. Explicitly, using a
dual basis of the finitely generated projective module $\VV$,
$\std^\cd(u\otimes z)={\g_{}^\flat}^{-1}(\omega^i
\kg({}^\alpha u\otimes_A {}_\alpha e_i\otimes z))$.

The map $\std^\cd$ is a covariant derivative (as defined in Remark \ref{remcd})
because the derivation property \eqref{kgder} of $\kg$ immediately
implies the Leibniz rule $\std^\cd(u\otimes az)=\std^\cd(ua\otimes
z)+\LLL_u(a)z$.
From  Remark \ref{remcd} it then follows that there exists a unique connection $\std\in {}_A\Con(\VV)$ such that
$\std{}_u(z)=\std^\cd(u\otimes z)$, for all $u,z\in \VV$.

We now show that the connection $\std$ is torsion free.  For all $u, v, z\in \VV$ consider the
permutation $u\otimes v\otimes z\mapsto {}^{\gamma\delta}z
\otimes {}^\eta{}_{\delta} v\otimes {}_{\eta\gamma} u =
 {}^{\gamma\delta}z
\otimes {}_\gamma{}^\eta{} v\otimes {}_{\delta\eta} u$.
On one hand, from \eqref{LCexists}, $$\kg(u\otimes_A v\otimes z)-\kg({}^{\gamma\delta}z\otimes {}^\eta{}_{\delta} v\otimes {}_{\eta\gamma} u)=
2\le {}^\al v\otimes_A (\std{}_{{}_{\!\al\!\!\:} u }z-\std{}_{{}^\beta z\,}{}^\beta{\!}^{}_\al u) ,\g\re~.$$
On the other hand, recalling the definition of $\kg$, and using the braided symmetry of the metric, the left hand side of the above expression equals $2\le [u,{}^\beta z]\otimes_A {}^{}_\beta v,\g\re=2\le {}^\al v\otimes_A
[{}_\al u, z],\g\re$, thus showing that $\std$ is torsionless.

We similarly show that  the connection $\std$ is metric compatible.
From  \eqref{LCexists} and Corollary \ref{Leibcovderu} we have
$$\kg(u\otimes_A v\otimes z)+\kg(u\otimes_A {}^\gamma z\otimes {}_\gamma v)=2\le \std{}_u(v\otimes z),\g\re=2\LLL_u\le v\oA z,\g\re-2\le
u\otimes_A v\otimes z , {}^*\std\g\re~.$$ From the definition of
$\kg$, cf.  \eqref{Koszul}, it easily follows that the left hand side of this expression simplifies to  $2\LLL_u\le v\otimes_A z,\g\re$, thus proving metric compatibility of $\std$.
\end{proof}
\end{theorem}

\subsection{Ricci tensor and Einstein spaces}
There is a canonical notion of trace in a ribbon
category. For an internal morphism $\tilde L\in {}_A\hom(\VV,\VV)$ in
$\catfp$ we have  {tr}$(\tilde L)= \le~,~\re'\circ
(\id_{\Omega(A)}\otimes_A \tilde L)\circ \coev$ that belongs to ${}_A\hom(A,A)$ in $\catfp$.
Using a dual basis it reads, {tr}$(\tilde L)=\le \omega^i, \tilde
L( e_i)\re'$. The Ricci tensor is the trace of the
Riemann tensor given by (recall  $(A\oA\VV\oA\VV\simeq\VV\oA\VV$),
\begin{equation}\label{defofric1}
\begin{split}&Ric:=\le~\,,\,~\re'\circ (\id_{\Omega(A)}\otimes_A
R)\circ(\coev\otimes_A\id_{\VV\otimes_A\VV}):\VV\otimes_A\VV\to A~,~~\\[.4em]
 &{Ric}(u,v)=\le \omega^i, R{\stdd} (e_i,u,v)\re'~.
\end{split}
\end{equation}
Since the coevaluation map  is a morphism in $\catfp$,
the trace is a morphism in $\catfp$ and we have that
$Ric\in{}_A\hom(\VV\otimes_A\VV, A)$, as it is immediate to see from
the definition.

We also define an Einstein metric on $A$ to be a metric $\g$
proportional to its Ricci tensor,
$$
Ric=\lambda \le\,{\mbox{-}}\oA{\mbox{-}}\,,\g\re~,~~(\lambda\in \Bbbk)~.
$$
This equation in ${}_A\hom(\VV\oA\VV,A)$ allows to study noncommutative Einstein spaces.

\appendix

\addtocontents{toc}{\protect\setcounter{tocdepth}{0}}
\section{Drinfeld twist of ribbon categories}\label{catequivF}

\begin{proposition}\label{prop:twist}
The ribbon category $\catfp$ is equivalent to the ribbon category
$\catfpFF$ via the braided monoidal functor $(F,\varphi)$, where the
functor $F$
assigns to  a module $\Gamma$ the module $F(\Gamma)=\Gamma^\FF$, and is the
identity on morphisms, while the natural transformation $\varphi$ is
given by the family of isomorphisms in $\catfpFF$
$$\varphi_{\Gamma,\Upsilon}:\Gamma^\FF\otimes^\FF\Upsilon^\FF\to
(\Gamma\otimes\Upsilon)^\FF~,~~ \varphi_{\Gamma,\Upsilon}(s\otimes^\FF
u)=\FF^{-1}\ra(s\otimes^\FF u)=\of^\al\ra s\otimes \of_\al\ra u~.$$
\begin{proof}
We first show that $(F,\varphi)$ is a braided monoidal equivalence
between $\cat$ and $\catFF$.
The functor $F$ is an equivalence because, since $\FF^{-1}$ is a twist
of $H^\FF$, we can twist back any module $\Theta$ in $\catFF$ to a
module $\Theta^{\FF^{-1}}$ in $\cat$. Similarly, $\FF^{-1}$ gives the
inverses of the
maps $\varphi_{\Gamma,\Upsilon}$. A proof that these maps
are morphisms in $\catFF$ is for example in \cite[Lemma 5.19]{AS} or in
\cite[Theorem 3.13]{BSS1}. Compatibility with the braiding in $\cat$
and in $\catFF$, $F(\tau(\RR))\circ
\varphi=\varphi\circ \tau({\RR^\FF})$, is straighforward since
$\RR^\FF=\FF_{21}\RR\FF^{-1}$.

Any monoidal functor preserves rigidity of objects, hence the equivalence $\cat\simeq\catFF$ restricts to the monoidal equivalence
$\catfp\simeq\catfpFF$.

For completeness an explicit proof follows. If $\Gamma$ has right
dual $\stG$ with evaluation and coevaluation maps  $\le~,~\re$ and
$\coev$, then $\Gamma^\FF$ has right dual $({\stG})^\FF$ with
evaluation and coevaluation maps
\begin{equation*}
\le~,~\re^\FF:=\le~,~\re\circ\varphi^{}_{\Gamma,\stG}~,~~\coev^\FF:=\varphi_{\stG,\Gamma}^{-1}\circ\coev
\end{equation*}
(we are using that $F$ on morphisms
is the identity hence we have written $\le~,~\re$ and $\coev$ rather than
$F(\le~,~\re)$ and $F(\coev)$).

Recalling that twisted modules equal the original ones as $\Bbbk$-modules,
if we set $\coev(1_A)=\sts^i\otimes_A s_i

$ we have
$\coev(1_{A^\FF})=\nof^\al\ra \sts^i\otimes_{A^\FF}\nof_\al \ra
s_i

$.
We show for example the first rigidity condition in \eqref{coerecoev}, that is, for all
$s\in \Gamma^\FF$, $s=\le s,\nof^\al\ra\sts^i\re^\FF \!\dotF\,\nof_{\!\al}
\ra s_i$.
Indeed,
\begin{equation*}
\begin{split}
\le s, \nof^\al\ra\sts^i\re^\FF\!\dotF\,\nof_{\!\al}\ra s_i&
=\of^\gamma\ra\le\of^\be\ra s,\of_{\!\be}\nof^\al\ra\sts^i\re\of_{\!\gamma}\nof_{\!\al}\ra s_
i\\
&=\le\of^\gamma_{(1)}\of^\be\ra
s,\of^\gamma_{(2)}\of_{\!\be}\nof^\al\ra\sts^i\re\of_{\!\gamma}\nof_{\!\al}\ra
s_i\\
&=\le\of^\delta\ra
s,\of_{\!\delta\,{(1)}}\ra\sts^i\re\of_{\!\delta\,(2)}\ra s_i  \\
&=\le s,\sts^i\re s_i=s
\end{split}
\end{equation*}
where in the third line we used the cocycle condition in the form
$(\Delta\otimes\id)\FF^{-1}(\FF^{-1}\otimes\id)=(\id\otimes\Delta)\FF^{-1}(\id\otimes\FF^{-1})$
and then simplified $\FF^{-1}\FF=1$. In the fourth line we used
$H$-equivariance of the coevaluation map in the form, for all $h\in H$,
$h\ra(\sts^i\otimes s_i)=\varepsilon(h)\sts^i\otimes s_i$ and then the
normalization condition of the twist.
\end{proof}
\end{proposition}

$\phantom{.}$\\
\noindent{\large{\bf{Acknowledgments}}}\\
The author would like to thank the organizers of the conference
``Noncommutative manifolds and their symmetries'' dedicated to Giovanni
Landi on the occasion of his 60th birthday (Scalea 16-20.9.2019) and
the guest of honour for the opportunity to present the main results of this research.

The author acknowledges partial support from INFN, CSN4, Iniziativa
Specifica GSS.  This research has a financial
support from Universit\`a del Piemonte Orientale. The author is
affiliated to INdAM-GNFM.
\medskip

config: arxiv