text:
\begin{document}

\title[Rapid decay]{Compact quantum metric spaces from quantum groups of rapid decay}

\author{Jyotishman Bhowmick}
\address{Indian Statistical Institute \\
203 Barrackpore Trunk Road \\
Kolkata 700 108 \\
India
}
\email{jyotishmanb@gmail.com}

\author{Christian Voigt}
\address{School of Mathematics and Statistics \\
         University of Glasgow \\
         15 University Gardens \\
         Glasgow G12 8QW \\
         United Kingdom
}
\email{christian.voigt@glasgow.ac.uk}

\author{Joachim Zacharias}
\address{School of Mathematics and Statistics \\
         University of Glasgow \\
         15 University Gardens \\
         Glasgow G12 8QW \\
         United Kingdom
}
\email{joachim.zacharias@glasgow.ac.uk}

\subjclass[2000]{16W30, 81R50}

\thanks{This work was supported by the Engineering and Physical Sciences Research Council Grants EP/G01419/2, EP/L013916/1 and
the UKIERI project Quantum Probability, Noncommutative Geometry and Quantum Information. J.B. wishes to thank Sergey Neshveyev and the Department of
Mathematics, University of Oslo, where he was a post doctoral fellow when this work started}

\maketitle

\begin{abstract}
We present a modified version of the definition of property RD for discrete quantum groups given by Vergnioux in order to accommodate examples of
non-unimodular quantum groups. Moreover we extend the construction of spectral triples associated to discrete groups with length functions, originally
due to Connes, to the setting of quantum groups. For quantum groups of rapid decay we study the resulting spectral triples from the point of view
of compact quantum metric spaces in the sense of Rieffel.
\end{abstract}

\section{Introduction}

In the theory of noncommutative geometry in the sense of Connes \cite{Connesbook}, spectral triples can be thought of as noncommutative analogues of
smooth Riemannian manifolds. A spectral triple $ (\A, \H, D) $ consists of a $ * $-algebra $ \A $, represented on a Hilbert space $ \H $, together with
an unbounded self-adjoint operator $ D $ on $ \H $. The basic requirements on this data are that $ D $ has compact resolvent and that
the commutators $ [D,a] $ are bounded for all $ a \in \A $. \\
The prototypical example of a spectral triple is given by the algebra $ \A = C^\infty(M) $ of smooth functions on a compact Riemannian spin
manifold $ M $, acting on the Hilbert space $ \H = L^2(M, S) $ of $ L^2 $-section of the spinor bundle $ S $ of $ M $, together with the associated
Dirac operator. Another class of examples, studied already by Connes \cite{Connescms}, arises from discrete groups equipped with length
functions. In this case $ \A = \CH[G] $ is the complex group algebra of the group $ G $, acting on the Hilbert space $ l^2(G) $,
and the operator $ D $ acts by multiplication with the length function. \\
Quantum groups can be viewed as noncommutative manifolds, and various examples of spectral triples
have been constructed in this context, see for instance \cite{CPspectraltriple}, \cite{DLSSVdirac}, \cite{NTDirac}.
In this paper we consider more elementary examples of spectral triples for quantum groups, motivated by the construction for
discrete groups with length functions mentioned above. Actually, the passage from discrete groups to discrete quantum groups
is essentially straightforward in this context. Although the resulting spectral triples are trivial from the point of view of $ K $-homology,
we show that they provide examples of quantum metric spaces in the sense of Rieffel \cite{Rieffelcqms}. In fact, in order to make a link to
the theory of Rieffel, we have to restrict to quantum groups of rapid decay, and follow the work of Antonescu-Christensen in the group
case \cite{ACmetricsgroup}. \\
The property of rapid decay for discrete quantum groups was introduced and studied by Vergnioux \cite{Vergniouxrd}, following the definition
for classical groups in \cite{Jolissaintrd}. Quantum groups of rapid decay in the sense of \cite{Vergniouxrd} are necessarily unimodular, which unfortunately
excludes some of the most studied examples, in particular those arising from $ q $-deformations of semisimple compact Lie groups.
The incompatibility of the theory in \cite{Vergniouxrd} with non-unimodularity is of course invisible in the classical setting of discrete
groups. For examples coming from $ q $-deformations it may appear somewhat surprising, because duals of classical compact Lie groups actually do have
property RD in the sense of \cite{Vergniouxrd}. \\
In the first part of this paper we explain how a slight modification of the definitions given in \cite{Vergniouxrd} allow to remedy this situation.
Our definitions agree with Vergnioux's for unimodular discrete quantum groups. On the other hand, we obtain a more interesting theory
in the non-unimodular case. \\
Let us explain how the paper is organised. In section \ref{secpreliminaries} we collect some definitions from the theory
of quantum groups and fix our notation. Section \ref{secrd} contains our modified definitions of rapid decay. In section
\ref{secamenable} we consider amenable quantum groups and compare our notion of rapid decay with a suitable notion of polynomial growth.
Actually, for polynomial growth the difference to the definition in \cite{Vergniouxrd} consists simply
in replacing quantum dimensions with ordinary dimensions. In section \ref{secspectraltriples} we
explain how the construction of a spectral triple from a group with a length function extends to the setting of quantum groups.
The aim of section \ref{seccompactquantummetricspace} is to review the definition of compact quantum metric spaces
in the sense of Rieffel, and to show that we obtain natural Lipschitz seminorms from the spectral triples defined in
section \ref{secspectraltriples}. In the final section \ref{seclipnorm} we prove the Lip-norm property for suitable
Lipschitz seminorms provided the underlying quantum group has property RD in our sense. This yields
a family of examples of compact quantum metric spaces associated to quantum groups. \\
Let us make some remarks on notation. We write $ \LH(\H) $ for the space of bounded operators on a Hilbert space $ \H $.
The closed linear span of a subset $ X $ of a Banach space is denoted by $ [X] $. Depending on the context, the symbol
$ \otimes $ denotes either the tensor product of Hilbert spaces, or the minimal tensor product of $ C^* $-algebras.
For operators on multiple tensor products we use the leg numbering notation.
We write $ \| \; \| = \|\; \|_{op} $ for the operator norm.

\section{Preliminaries} \label{secpreliminaries}

In this section we review some basic definitions concerning quantum groups. For more detailed information we refer to \cite{BSUM},
\cite{KVLCQG}, \cite{Woronowiczleshouches}. Our notation and conventions will follow \cite{NVpoincare}. \\
The main objects of study in this paper are discrete quantum groups. It is technically convenient to describe them using Hopf $ C^* $-algebras.
Recall first that a Hopf $ C^* $-algebra is a $ C^* $-algebra $ S $ together with an injective nondegenerate $ * $-homomorphism
$ \Delta: S \rightarrow M(S \otimes S) $, called comultiplication, such that $ (\Delta \otimes \id)\Delta = (\id \otimes \Delta) \Delta $
and $ [\Delta(S)(1 \otimes S)] = S \otimes S = [(S \otimes 1)\Delta(S)] $. \\
With this terminology, a discrete quantum group can be described by a pair of Hopf $ C^* $-algebras $ C_0(G) $ and
$ C^*_\red(G) $ together with a multiplicative unitary $ W \in M(C_0(G) \otimes C^*_\red(G)) $, satisfying certain axioms.
In particular, the algebra $ C^*_\red(G) $ is unital, and $ C_0(G) $ is a $ C^* $-algebraic direct sum of matrix algebras.
We write $ \Delta $ for the comultiplication of $ C_0(G) $ and $ \hat{\Delta} $ for the comultiplication of $ C^*_\red(G) $.
Some properties of the multiplicative unitary linking these two Hopf $ C^* $-algebras will be stated below.
We refer to $ C_0(G) $ as the algebra of functions on $ G $, and to $ C^*_\red(G) $ as the reduced group $ C^* $-algebra of $ G $.
The theory also provides a full group $ C^* $-algebra $ C^*_\max(G) $, which however will not show up explicitly in this paper.
At some points we will restrict attention to the case that $ G $ is amenable, which means that the canonical quotient
homomorphism $ C^*_\max(G) \rightarrow C^*_\red(G) $ is an isomorphism. \\
Inside the Hopf $ C^* $-algebra $ C_0(G) $ we have a canonical dense multiplier Hopf-$ * $-algebra $ C_c(G) $, compare \cite{vDadvances}.
More precisely, $ C_c(G) $ is the algebraic direct sum of matrix blocks defining $ C_0(G) $.
Moreover $ C_0(G) $ admits a left Haar weight $ \phi $, given by a positive linear functional $ \phi: C_c(G) \rightarrow \mathbb{C} $
satisfying $ (\id \otimes \phi)\Delta(f) = \phi(f) 1 $, and we let $ l^2(G) $ denote the GNS-construction of $ \phi $.
We write $ \Lambda(f) \in l^2(G) $ for the image of $ f \in C_c(G) $ under the GNS-map. \\
The multiplicative unitary $ W $ can be considered as an element of $ \LH(l^2(G) \otimes l^2(G)) $, and we have the explicit
formula
$$
W^*(\Lambda(f) \otimes \Lambda(g)) = (\Lambda \otimes \Lambda)(\Delta(g)(f \otimes 1))
$$
for its adjoint. Moreover, the $ C^* $-algebra of functions on $ G $ can be recovered from $ W $ as
$$
C_0(G) = [(\id \otimes \LH(l^2(G))_*)(W)],
$$
and the reduced group $ C^* $-algebra of $ G $ can be identified with
$$
C^*_\red(G) = [(\LH(l^2(G))_* \otimes \id)(W)].
$$
In fact, the latter formula can be taken as the definition of $ C^*_\red(G) $ if one constructs the multiplicative unitary first. \\
A finite dimensional unitary corepresentation of $ G $ is a unitary $ X \in C^*_\red(G) \otimes \LH(\H) $ satisfying
$ (\hat{\Delta} \otimes \id)(X) = X_{13} X_{23} $, here $ \H $ is a finite dimensional Hilbert space, and we are using leg numbering notation. Such
corepresentations form a semisimple $ C^* $-tensor category. We denote by $ \Irr(G) $ the set of equivalence classes of irreducible
corepresentations of $ G $, and we write $ \epsilon $ for the trivial corepresentation on $ \mathbb{C} $. \\
Using corepresentation theory we can identify
$$
C_c(G) \cong \bigoplus_{\alpha \in \Irr(G)} \LH(\H_\alpha) \cong \bigoplus_{\alpha \in \Irr(G)} M_{\dim(\alpha)}(\mathbb{C})
$$
as the algebraic direct sum of the endomorphism algebras of all irreducible corepresentations. The algebra $ C_0(G) $
is obtained by taking the $ C^* $-algebraic direct sum instead. Finally, we will also need the algebraic multiplier algebra $ C(G) $
of $ C_c(G) $, which can be written as
$$
C(G) \cong \prod_{\alpha \in \Irr(G)} \LH(\H_\alpha) \cong \prod_{\alpha \in \Irr(G)} M_{\dim(\alpha)}(\mathbb{C}),
$$
the algebraic direct product of all endomorphism algebras $ \LH(\H_\alpha) $ for $ \alpha \in \Irr(G) $. \\
The matrix coefficients of all irreducible corepresentations define a canonical dense
Hopf-$ * $-algebra $ \CH[G] \subset C^*_\red(G) $.
The algebras $ \mathbb{C}[G] \subset C^*_\red(G) $ and $ C_c(G) \subset C_0(G) $ are linearly spanned by
elements of the form $ (\omega \otimes \id)(W) $ and $ (\id \otimes \omega)(W) $, respectively, where
$ \omega = \omega_{\xi, \eta} \in \LH(l^2(G))_* $ is associated to vectors $ \xi, \eta \in \Lambda(C_c(G)) $. \\
If $ f \in C_c(G) $ and $ x \in \mathbb{C}[G] $ are represented by $ L_f, L_x \in \LH(l^2(G))_* $ in the sense
that $ (\id \otimes L_f)(W) = f $ and $ (L_x \otimes \id)(W) = x $, then we obtain a well-defined bilinear pairing
\begin{equation*}
\bra f, x \ket = \bra x, f \ket = (L_x \otimes L_f)(W) = L_f(x) = L_x(f)
\end{equation*}
between $ C_c(G) $ and $ \CH[G] $, see \cite{BSUM}. \\
We point out that the product of $ \CH[G] $ is dual to the coproduct of $ C_c(G) $, whereas the product of $ C_c(G) $ is dual to the opposite
coproduct of $ \CH[G] $. In other terms, we have for all $ f, g \in C_c(G)$ and $ x, y \in \CH[G] $ the relations
\begin{equation*}
\bra f, xy \ket = \bra f_{(1)}, x \ket \bra f_{(2)}, y \ket \quad \text{and} \quad
\bra fg, x \ket = \bra f, x_{(2)} \ket \bra g, x_{(1)} \ket
\end{equation*}
where we use the Sweedler notation $ \Delta(f) = f_{(1)} \otimes f_{(2)} $ and $ \hat{\Delta}(x) = x_{(1)} \otimes x_{(2)}$
for the comultiplications on $ C_c(G)$ and $ \CH[G] $.
Of course, this notation has to be interpreted with care, in particular, the coproduct $ \Delta(f) $ of an element $ f $ of the multiplier
Hopf $ * $-algebra $ C_c(G) $ can be represented only as an infinite sum of simple tensors in general. \\
We shall use the notations
\begin{align*}
(x \hit f)(y) = f(yx), &\qquad (f \hitby x)(y) = f(xy) \\
(f \hit x)(g) = x(fg), &\qquad (x \hitby f)(g) = x(gf)
\end{align*}
for the left and right regular actions of $ \CH[G] $ on $ C_c(G) $, and of $ C(G) $ on $ \CH[G] $, respectively. Remark
that these definitions are in accordance with our conventions for the comultiplications of $ C_c(G) $ and $ \CH[G] $. \\
From the duality theory of algebraic quantum groups \cite{vDadvances} it follows that there is a linear isomorphism $ \F: C_c(G) \rightarrow \CH[G] $
given by
$$
\F(f)(h) = \phi(hf).
$$
The inverse $ \F^{-1}: \CH[G] \rightarrow C_c(G) $ of this map is given by
$$
\F^{-1}(x)(y) = \hat{\phi}(S(x)y)
$$
where $ \hat{\phi} $ is the left and right invariant normalized Haar functional on $ \CH[G] $, and $ S $ denotes the antipode of $ \CH[G] $.
We fix the left invariant Haar functional $ \phi $ on $ C_c(G) $ such that
$$
\phi(\F^{-1}(x)) = \hat{\epsilon}(x)
$$
for all $ x \in \CH[G] $, where $ \epsilon: \CH[G] \rightarrow \mathbb{C} $ denotes the counit.
Then the map $ \F $ is isometric with respect to the standard scalar products
$$
\bra x, y \ket = \hat{\phi}(x^* y), \qquad \bra f, g \ket = \phi(f^*g)
$$
on $ \CH[G] $ and $ C_c(G) $, respectively. In fact, using $ \F $ we can identify $ l^2(G) $ with the GNS-representation of $ \hat{\phi} $,
and we will write $ \hat{\Lambda}(x) \in l^2(G) $ for the image of $ x \in \CH[G] $. \\
The modular function for $ G $ is a multiplier $ F \in C(G) $ which relates the left and right Haar integrals of $ C_c(G) $.
A discrete quantum group $ G $ is unimodular iff the modular function satisfies $ F = 1 $. This happens iff
the Haar state $ \hat{\phi} $ on $ \CH[G] $ is a trace. \\
In the general case, each component $ F^\alpha $ of $ F $ for $ \alpha \in \Irr(G) $ is a positive invertible matrix such
that $ \tr(F^\alpha) = \tr((F^\alpha)^{-1}) $, the latter being the quantum dimension of $ \alpha $, denoted by $ \dim_q(\alpha) $. One may fix bases
such that $ F^\alpha $ is a diagonal operator for all $ \alpha \in \Irr(G) $, and we will do this in the sequel. \\
Let us write $ u^\alpha_{ij} $ for the matrix coefficients of $ \alpha \in \Irr(G) $ with respect to such an orthonormal basis
of the representation space of $ \alpha $. Then the Schur orthogonality relations become
$$
\hat{\phi}(u^\alpha_{ij} (u^\beta_{kl})^*) = \delta_{\alpha\beta} \delta_{ik} \delta_{jl} \, \frac{(F^\alpha)_{jj}}{\dim_q(\alpha)},
\qquad \hat{\phi}((u^\alpha_{ij})^* u^\beta_{kl}) = \delta_{\alpha\beta} \delta_{ik} \delta_{jl} \, \frac{(F^{-1}_\alpha)_{ii}}{\dim_q(\alpha)},
$$
where $ \alpha, \beta \in \Irr(G) $. \\
The left and right Haar functionals for $ C_c(G) $ are given by
$$
\phi(f) = \sum_{\alpha \in \Irr(G)} \dim_q(\alpha) \tr(F p_\alpha f), \qquad \psi(f) = \sum_{\alpha \in \Irr(G)} \dim_q(\alpha) \tr(F^{-1} p_\alpha f),
$$
where we denote by $ p_\alpha \in C_c(G) $ the central projection corresponding to $ \alpha \in \Irr(G) $.

\section{Rapid decay for discrete quantum groups} \label{secrd}

In this section we review some definitions from \cite{Vergniouxrd} and introduce our notion of rapid decay. We also state some equivalent
characterisations of rapid decay, following Vergnioux. \\
Let us first recall the notion of a length for a discrete quantum group introduced in \cite{Vergniouxrd}.
\begin{definition}
Let $ G $ be a discrete quantum group. A length for $ G $ is a positive element $ L \in C(G) $ such that
\begin{bnum}
\item[a)] $ \epsilon(L) = 0 $.
\item[b)] $ S(L) = L $.
\item[c)] $ \Delta(L) \leq L \otimes 1 + 1 \otimes L $.
\end{bnum}
\end{definition}
These conditions reflect the classical definition of length functions. Note that $ L $ can be viewed as a sequence of positive matrices indexed
by the elements of $ \Irr(G) $. \\
We will be mainly interested in central lengths, that is, lengths $ L $ which are central elements
of the algebra $ C(G) $. Such central lengths are obtained from length functions $ l $ on $ \Irr(G) $ in the following sense.
\begin{definition}
Let $ G $ be a discrete quantum group. A length function for $ G $ is a function $ l: \Irr(G) \rightarrow [0, \infty) $ such that
\begin{bnum}
\item[a)] $ l(\epsilon) = 0 $ where $ \epsilon \in \Irr(G) $ is the trivial corepresentation.
\item[b)] $ l(\overline{\alpha}) = l(\alpha) $ for all $ \alpha \in \Irr(G) $.
\item[c)] $ \alpha \subset \beta \otimes \gamma $ implies $ l(\alpha) \leq l(\beta) + l(\gamma) $ for all $ \alpha, \beta, \gamma \in \Irr(G) $.
\end{bnum}
The length function $ l $ is called proper if for any $ n \in \mathbb{N} $ there are
only finitely many irreducible corepresentations $ \alpha \in \Irr(G) $ with $ l(\alpha) \leq n $ and $ l(\alpha) = 0 $ iff $ \alpha = \epsilon $.
\end{definition}
A length function $ l $ induces a central length $ L \in C(G) $ by the formula
$$
L = \sum_{\alpha \in \Irr(G)} l(\alpha) p_\alpha,
$$
recall that $ p_\alpha \in C_c(G) $ is the unit element in the matrix block corresponding to $ \alpha $. We can also view $ L $ as an unbounded
self-adjoint operator on $ l^2(G) $ in the obvious way. Any central length arises in this way, and we will freely pass from $ l $
to $ L $ in the sequel. \\
Basic examples of length functions are given by word length functions on finitely generated quantum groups. We recall that a subset $ D \subset \Irr(G) $
is said to generate the discrete quantum group $ G $ iff every corepresentation $ \alpha \in \Irr(G) $ is contained in some iterated tensor product
of corepresentations from $ D $. The quantum group $ G $ is called finitely generated provided there exists a finite subset $ D \subset \Irr(G) $ which
generates $ G $. \\
Given a finitely generated quantum group $ G $ with finite generating set $ D $, we obtain a proper length function $ l_D $ on $ \Irr(G) $ by
letting $ l_D(\alpha) $ be the smallest number $ k $ such that $ \alpha \subset \alpha_1 \otimes \cdots \otimes \alpha_k $ and $ \alpha_j \in D $ for all $ j $.
Although $ l_D $ clearly depends on $ D $, it can be shown that the definitions and results in the sequel
do not depend on the choice of the generating set in an essential way, compare lemma 3.3 and remark 3.6 in \cite{Vergniouxrd}. \\
Given a length function $ l $ on $ G $, we let $ p_n \in \LH(l^2(G)) $ be the sum of all projections $ p_\alpha $ for
$ \alpha \in \Irr(G) $ such that $ \|L p_\alpha \|_{op} \in (n - 1, n] $. Notice that $ p_n $ is a finite rank projection for all $ n \in \mathbb{N}_0 $
iff $ l $ is proper. \\
Let $ G $ be a discrete quantum group. In the sequel we shall work with the self-adjoint element $ C $ of $ C(G) $ given by
$$
C = \sum_{\alpha \in \Irr(G)} \frac{\dim_q(\alpha)}{\dim(\alpha)} F^\alpha p_\alpha.
$$
We remark that $ C = 1 $ iff $ G $ is unimodular.
If $ L $ is a length on $ G $, we define the associated Sobolev $ s $-norm for $ s \geq 0 $ and $ f \in C_c(G) $ by
\begin{align*}
\| f \|_{2,s}^2 &= \sum_{\alpha \in \Irr(G)} \frac{\dim_q(\alpha)^2}{\dim(\alpha)} \tr(p_\alpha ((1 + L)^s f F)^* (1 + L)^s f F) \\
&= \phi(((1 + L)^s f C^{1/2})^* (1 + L)^s f C^{1/2}) \\
&= \bra (1 + L)^s f C^{1/2}, (1 + L)^s f C^{1/2} \ket.
\end{align*}
Observe that $ \|\; \|_{2,0} = \|\;\|_2 $ iff $ G $ is unimodular.
Similarly, for $ s \geq 0 $ and $ x \in \CH[G] $ we define
\begin{align*}
\| x \|_{2,s}^2 &= \bra (1 + L)^s \F^{-1}(C^{1/2} \hit x), (1 + L)^s \F^{-1}(C^{1/2} \hit x) \ket \\
&= \sum_{\alpha \in \Irr(G)} \frac{\dim_q(\alpha)^2}{\dim(\alpha)} \tr(p_\alpha ((1 + L)^s \F^{-1}(F \hit x))^* (1 + L)^s \F^{-1}(F \hit x)),
\end{align*}
where we recall that $ \F^{-1}: \CH[G] \rightarrow C_c(G) $ is the Fourier transform given by $ \F^{-1}(x)(y) = \hat{\phi}(S(x)y) $
for $ x, y \in \mathbb{C}[G] $. Notice that
$$
\F(f C^{1/2})(h) = \phi(h f C^{1/2}) = \phi(C^{1/2} h f) = (C^{1/2} \hit \F(f))(h),
$$
which implies
$$
\F^{-1}(C^{1/2} \hit x) = \F^{-1}(x) C^{1/2}.
$$
Hence $ \F $ is an isometry with respect to the $ \| \; \|_{2,s} $-norms.
\begin{definition} \label{defrd}
Let $ G $ be a discrete quantum group. We say that $ G $ has property RD with respect to a central length $ L $ on $ G $ if there exist
constants $ c, s > 0 $ such that
$$
\| \F(f) \|_{op} \leq c \| f \|_{2,s}
$$
for all $ f \in C_c(G) $. \\
We say that $ G $ has property RD if it has property RD with respect to some central length.
\end{definition}
Since $ C = 1 $ for a unimodular discrete quantum group, our definition
of the Sobolev norms above reduces to the definitions in \cite{Vergniouxrd} in this case. In other words, for
unimodular quantum groups definition \ref{defrd} is equivalent to the definition of property RD given by Vergnioux. \\
As we will see below, this is not true for non-unimodular quantum groups. One might therefore call the property defined above
modular property RD, or something alike, in order to make a distinction with the original notion. However, since there
is no real conflict in terminology neither in the unimodular nor in the non-unimodular case - a non-unimodular quantum group
can only possibly have property RD according to the above version of the definition - we have refrained from introducing new
terminology. \\
Our above definition of Sobolev norms may appear somewhat arbitrary, but it can be motivated as follows. Essentially, the
operator $ C^{1/2} $ controls the deviation of the norm
$$
\|\hat{\Lambda}(u^\alpha_{ij})\| = \frac{(F^\alpha)^{-1/2}_{ii}}{\dim_q(\alpha)^{1/2}}
$$
from its value $ \dim(\alpha)^{-1/2} $ in the unimodular case. More precisely, we have
$$
\|\hat{\Lambda}(C^{1/2} \hit u^\alpha_{ij})\| = \frac{1}{\dim(\alpha)^{1/2}},
$$
so that the action of $ C^{1/2} $ compensates for the rescaling of Hilbert space norms. We remark that one could also
incorporate $ C^{1/2} $ in the definition of the Fourier transform, which would simplify some formulas in the sequel.
The remaining ingredients in the definition of the Sobolev norms are as in \cite{Vergniouxrd}, with the only difference that
we are working with left Haar weights. \\
If $ L $ is a length on $ G $ we denote by $ \H^s_L(G) $ the completion of $ \mathbb{C}[G] $ with respect to the
Sobolev $ s $-norm $ \|\; \|_{2,s} $. Moreover we define the associated Schwartz space $ \H^\infty_L(G) $ by
$$
\H^\infty_L(G) = \bigcap_{s \geq 0} \H^s_L(G).
$$
The space $ \H^\infty_L(G) $ is naturally a Fr\'echet space with respect to the topology given by the Sobolev seminorms.
Let us prove the following variant of proposition 3.5 in \cite{Vergniouxrd}.
\begin{prop} \label{RDchar}
Let $ G $ be a discrete quantum group and let $ L $ be a central length on $ G $. Then the following conditions are
equivalent.
\begin{bnum}
\item[a)] $ G $ has property RD with respect to $ L $.
\item[b)] There exists $ c, s > 0 $ such that $ \| x \|_{op} \leq c \| x \|_{2,s} $ for all $ x \in \CH[G] $.
\item[c)] The identity map $ \CH[G] \rightarrow \CH[G] $ induces a continuous linear embedding
$$
\H^\infty_L(G) \subset C^*_\red(G).
$$
\item[d)] There exists a polynomial $ p(x) \in \mathbb{R}[x] $ such that
$$
\| \F(f)\|_{op} \leq p(n) \| f \|_{2,0}
$$
for all $ n \in \mathbb{N} $ and $ f \in p_n C_c(G) $.
\item[e)] There exists a polynomial $ p(x) \in \mathbb{R}[x] $ such that
$$
\|p_l \F(f)p_k\|_{op} \leq p(n) \| f \|_{2,0}
$$
for all $ k,l,n \in \mathbb{N} $ and $ f \in p_n C_c(G) $.
\end{bnum}
\end{prop}
\proof The arguments follow precisely the pattern of \cite{Vergniouxrd}, for the convenience of the reader we shall include the details. \\
$ a) \Leftrightarrow b) $ This follows immediately from the fact that the Fourier transform $ \F: C_c(G) \rightarrow \CH[G] $
is a linear isomorphism preserving Sobolev norms. \\
$ b) \Rightarrow c) $ From the norm estimate in $ b) $ we obtain the existence of a continuous linear map $ \iota: \H^\infty_L(G) \rightarrow C^*_\red(G) $,
and we may compose with the embedding $ C^*_\red(G) \rightarrow l^2(G) \subset C(G) $. Since the completion of $ C_c(G) $
with respect to $ \|\; \|_s $ can be realised as a subspace of $ C(G) $ in a compatible way it follows that $ \iota $ is injective. \\
$ c) \Rightarrow b) $ is obvious. \\
$ a) \Rightarrow d) $ Let $ s $ be the natural number  and $ c > 0 $ such that $ \| \F(f) \|_{op} \leq c \| f \|_{2,s} $. Then for $ f \in p_n C_c(G) $
we have
\begin{align*}
\|\F(f)\|_{op} \leq c\|f\|_{2,s} = c\|(1 + L)^s f \|_{2,0} \leq c\|(1 + n)^s f \|_{2,0} \leq p(n) \|f\|_{2,0}
\end{align*}
where $ p(x) = c(1 + x)^s $. \\
$ d) \Rightarrow a) $ Let us choose constants $ c_1, s > 0 $ such that $ p(n) \leq c_1 (1 + n)^s $ for all $ n \in \mathbb{N} $. Then we obtain
\begin{align*}
\|\F(f) \|_{op} &\leq \sum_{n = 0}^\infty \|\F(p_n f)\|_{op} \\
&\leq c_1 \sum_{n = 0}^\infty (1 + n)^s \| p_n f \|_{2,0} \\
&= c_1 \sum_{n = 0}^\infty \frac{1}{1 + n} (1 + n)^{s + 1} \| p_n f \|_{2,0} \\
&\leq c_1 \biggl(\sum_{n = 0}^\infty \frac{1}{(1 + n)^2}\biggr)^{1/2} \biggl(\sum_{n = 0}^\infty (1 + n)^{2s + 2} \| p_n f \|_{2,0}^2 \biggr)^{1/2} \\
&\leq c_2 \|(1 + L)^{s + 1} f\|_{2,0} = c_2 \|f\|_{2,s + 1}
\end{align*}
for a suitable constant $ c_2 $, using the Cauchy-Schwarz inequality as in the classical case \cite{Haagerupnonnuclear}. \\
$ d) \Rightarrow e) $ is obvious. \\
$ e) \Rightarrow d) $ Let $ f \in p_n C_c(G) $. Then if $ p_l \F(f) p_k \neq 0 $ we have $ (k,l,n) \in \T $ where
$ \T \subset \mathbb{N}^3 $ is the set of all $ (k,l,n) $ such that $ \Delta(p_n)(p_k \otimes p_l) \neq 0 $. Indeed,
due to lemma 3.4 in \cite{Vergniouxrd} the set $ \T $ is stable under permutations and we have
$$
p_l \F(f) p_k = (\phi \otimes \id)((1 \otimes p_l) W (f \otimes p_k)) = (\phi \otimes \id)(W \Delta(p_l)(p_n f \otimes p_k)),
$$
using $ \F(f) \Lambda(h) = \bra S^{-1}(h_{(1)}), \F(f) \ket \Lambda(h_{(2)}) = \phi(S^{-1}(h_{(1)}) f) \Lambda(h_{(2)}) $ for
$ h \in C_c(G) $, and $ (\phi \otimes \id)(W (f \otimes 1)) \Lambda(h) = \phi(S^{-1}(h_{(1)})f) \Lambda(h_{(2)}) $. \\
For a unit vector $ \xi \in l^2(G) $ we obtain
\begin{align*}
\|\F(f) \xi \|^2 &= \sum_{l = 0}^\infty \|p_l \F(f) \xi\|^2 \\
&\leq \sum_{l = 0}^\infty \biggl(\sum_{k = 0}^\infty \|p_l \F(f) p_k \xi\| \biggr)^2 \\
&\leq p(n)^2 \|f\|_{2,0}^2 \sum_{l = 0}^\infty \biggl(\sum_{k \mid (k,l,n) \in \T} \|p_k \xi \| \biggr)^2
\end{align*}
using condition $ e) $.
According to lemma 3.4 in \cite{Vergniouxrd}, for fixed $ l \in \mathbb{N} $ the number of elements $ k $ such
that $ (k,l,n) \in \T $ is bounded above by $ 2n + 5 $. Hence by the Cauchy-Schwarz inequality we have
$$
\biggl(\sum_{k \mid (k,l,n) \in \T} \|p_k \xi \|\biggr)^2 \leq (2n + 5) \biggl(\sum_{k \mid (k,l,n) \in \T} \|p_k \xi \|^2 \biggr)
$$
for any $ l $. Therefore,
\begin{align*}
\sum_{l = 0}^\infty \biggl(\sum_{k \mid (k,l,n) \in \T} \|p_k \xi \|\biggr)^2
&\leq \sum_{l = 0}^\infty (2n + 5) \biggl(\sum_{k \mid (k,l,n) \in \T} \|p_k \xi \|^2 \biggr) \\
&\leq (2n + 5)^2 \sum_{k = 0}^\infty \|p_k \xi \|^2 = (2n + 5)^2 \|\xi\|^2,
\end{align*}
and we get $ \|\F(f)\|_{op} \leq (2n + 5) p(n) \|f\|_{2,0} $ as desired. \qed

\section{Property RD and polynomial growth} \label{secamenable}

In this section we study property RD for amenable quantum groups, and we show that it is equivalent to a suitable notion of
polynomial growth, again following the work of Vergnioux. However, we have to modify the definition of polynomial growth
introduced in \cite{Vergniouxrd}. We will comment on the relation between the various concepts below. \\
Let us start with the following definition.
\begin{definition} \label{defpolynomialgrowth}
Let $ G $ be a discrete quantum group. We say that $ G $ has polynomial growth with respect to a central length function $ l $ on $ G $ if there
exists a polynomial $ p(x) \in \mathbb{R}[x] $ such that
$$
\sum_{l(\alpha) \in (n - 1, n]} \dim(\alpha)^2 \leq p(n)
$$
for all $ n \in \mathbb{N} $. \\
We say that $ G $ has polynomial growth if it has polynomial growth with respect to some central length function on $ G $.
\end{definition}
A finitely generated discrete quantum group $ G $ has polynomial growth iff it has polynomial growth with respect to a word length
function $ l $ on $ G $. \\
If $ l $ is a central length function on $ G $ we shall write $ S^n \subset \Irr(G) $ for the set of all corepresentations $ \alpha $
satisfying $ l(\alpha) \in (n - 1, n] $. We note that $ G $ has polynomial growth with respect to $ l $ iff the sequences
$$
s_n = |S^n|, \qquad d_n = \sup_{\alpha \in S^n} \dim(\alpha)^2
$$
both have polynomial growth. In particular, a unimodular discrete quantum group has polynomial growth in the sense of \cite{Vergniouxrd} iff
it has polynomial growth in the sense of definition \ref{defpolynomialgrowth}. \\
For non-unimodular quantum groups the concept introduced above differs from the notion in \cite{Vergniouxrd}. Indeed, for polynomial growth
in the sense of Vergnioux one has to replace classical dimensions by quantum dimensions in definition \ref{defpolynomialgrowth}.
In order to distinguish the two notions one could refer to them as classical and quantum polynomial growth, respectively. As in the
case of property RD we shall however refrain from using new terminology.
We observe that a non-unimodular quantum group cannot have polynomial growth in the sense of \cite{Vergniouxrd}, so this should not lead
to confusion. \\
We have the following version of a result in \cite{Vergniouxrd}.
\begin{prop} \label{rdpgequiv}
Let $ G $ be a discrete quantum group and let $ l $ be a length function on $ G $.
\begin{bnum}
\item[a)] If $ G $ has polynomial growth with respect to $ l $ then $ G $ has property RD with respect to $ l $.
\item[b)] If $ G $ is amenable and has property RD with respect to $ l $, then $ G $ has polynomial growth with respect to $ l $.
\end{bnum}
In particular, rapid decay and polynomial growth are equivalent in the amenable case.
\end{prop}
\proof Again, we follow the arguments in \cite{Vergniouxrd}, and include the details for the convenience of the reader. \\
$ a) $ Recall that every element of $ \CH[G] \subset C^*_\red(G) $ can be written in the
form $ (\omega \otimes \id)(W) $ where $ W \in M(C_0(G) \otimes C^*_\red(G)) $ is the multiplicative unitary of $ G $ and
$ \omega $ a linear functional on $ C_0(G) $ of the form $ \omega = \F(f) $ for $ f \in C_c(G) $. Here $ \F $ denotes the Fourier transform as
above. Notice that $ (\omega \otimes \id)(W) = \F(f) \in C^*_\red(G) $ for $ \omega = \F(f) $, and
$ \|\F(f)\|_{op} = \|(\omega \otimes \id)(W) \| \leq \| \omega \| $. \\
Let us assume first that $ f \in p_n C_c(G) $ is such that $ \omega = \F(f) $ is a positive linear functional. The latter is equivalent to saying
that $ f F \in C_c(G) $ is positive. Taking an approximate identity $ (u_j)_{j \in J} $ for $ C_0(G) $ of central projections we
compute $ \|\omega\| = \lim \F(f)(u_j) = \lim \phi(u_j f) = \phi(f) $. Hence we obtain
\begin{align*}
\| \F(f) \|_{op} &\leq \|\omega\| \\
&= \phi(f) \\
&= \phi(p_n f) \\
&= \sum_{\alpha \in S^n} \dim_q(\alpha) \tr(p_\alpha f F) \\
&= \sum_{\alpha \in S^n} \dim_q(\alpha)^{1/2} \dim(\alpha)^{1/2} \tr(p_\alpha f F^{1/2} C^{1/2}) \\
&\leq \biggl(\sum_{\alpha \in S^n} \tr(\dim(\alpha) p_\alpha)\biggr)^{\frac{1}{2}}
\biggl(\sum_{\alpha \in S^n} \dim_q(\alpha) \tr(p_\alpha F^{\frac{1}{2}} (f C^{\frac{1}{2}})^* f C^{\frac{1}{2}} F^{\frac{1}{2}})\biggr)^{\frac{1}{2}} \\
&\leq \biggl(\sum_{\alpha \in S^n} \dim(\alpha)^2 \biggr)^{1/2}\|f\|_{2,0} \\
&\leq \sqrt{p(n)} \, \|f\|_{2,0}
\end{align*}
by the Cauchy-Schwarz inequality, where $ p(x) $ is the polynomial appearing in the polynomial growth estimate for $ G $. \\
Next assume that the functional $ \omega = \F(f) $ is hermitian, which is equivalent to saying that $ f F $ is self-adjoint. In this case
we may write $ f = f_+ - f_- $ where $ f_\pm F^{-1} $ are positive. In fact, we have $ f_\pm = \pm e_\pm f $ for suitable projections $ e_\pm $. The previous computation yields
\begin{align*}
\| \F(f) \|_{op} &\leq \| \F(f_+) \|_{op} + \| \F(f_-) \|_{op} \\
&\leq \sqrt{p(n)}(\|e_+ f\|_{2,0} + \|e_- f\|_{2,0}) \leq \sqrt{2 p(n)} \|f\|_{2,0},
\end{align*}
where the last step follows by inspecting the definition of the norm $ \|\; \|_{2,0} $. \\
Finally, consider an arbitrary element $ f \in p_n C_c(G) $ and write $ f $ as sum $ f = g + ih $ such that both $ gF $ and $ hF $
are self-adjoint. In this case, taking into account our previous computations, we obtain
\begin{align*}
\| \F(f) \|_{op} &\leq \| \F(g) \|_{op} + \| \F(h) \|_{op} \\
&\leq \sqrt{2p(n)} (\|g\|_{2,0} + \|h\|_{2,0}) \\
&\leq \sqrt{4 p(n)} \|f\|_{2,0}
\end{align*}
using that
$$
\|g\|_{2,0}^2 + \|h\|^2_{2,0} = \|g + ih\|_{2,0}^2 = \|f\|_{2,0}^2
$$
since
$$
\tr(p_\alpha (gF)^* hF) = \tr(p_\alpha gF hF) = \tr(p_\alpha(hF)^* gF)
$$
for any $ \alpha \in \Irr(G) $. It follows that $ G $ has property RD. \\
$ b) $ Due to amenability, the counit $ \hat{\epsilon}: \CH[G] \rightarrow \CH $ extends to a $ * $-homomorphism $ C^*_\red(G) \rightarrow \CH $. Moreover,
we have $ \hat{\epsilon}(\F(f)) = \phi(f) $ for all $ f \in C_c(G) $. Therefore
$$
|\phi(f)| = |\hat{\epsilon}(\F(f))| \leq \| \F(f)\|_{op} \leq c \|f \|_{2,s}
$$
for some constants $ c, s > 0 $. Let us define
$$
q_n = \sum_{\alpha \in \Irr(G)} \frac{\dim(\alpha)}{\dim_q(\alpha)} p_\alpha p_n F^{-1}
= \sum_{\alpha \in S^n} \frac{\dim(\alpha)}{\dim_q(\alpha)} p_\alpha F^{-1}
= \sum_{\alpha \in S^n} p_\alpha C^{-1},
$$
and observe that
$$
\phi(q_n) = \sum_{\alpha \in S^n} \dim(\alpha)^2.
$$
Hence $ q_n $ can be used to test polynomial growth. More precisely, we compute
\begin{align*}
\phi(q_n)^2 &\leq c^2 \|(1 + L)^s q_n\|_{2,0}^2 \\
&\leq c^2 (1 + n)^{2s} \sum_{\alpha \in S^n} \frac{\dim_q(\alpha)^2}{\dim(\alpha)} \tr(p_\alpha C^{-2} F^2) \\
&= c^2 (1 + n)^{2s} \sum_{\alpha \in S^n} \dim_q(\alpha) \tr(p_\alpha C^{-1} F) \\
&= c^2 (1 + n)^{2s} \phi(q_n)
\end{align*}
which implies $ \phi(q_n) \leq c^2 (1 + n)^{2s} $. That is, $ G $ has polynomial growth. \qed \\
Using proposition \ref{rdpgequiv} we conclude that all duals of $ q $-deformations of compact semisimple Lie groups have property RD.
Indeed, these discrete quantum groups are amenable \cite{Banicacompactsubfactor}, and the Weyl dimension formula
implies that they have polynomial growth in the sense of definition \ref{defpolynomialgrowth},
see example 4.5 in \cite{Vergniouxrd}. We record the following precise statement of this fact.
\begin{prop}
Let $ q \in (0,1] $ and let $ G_q $ be the standard deformation of a simply connected compact semisimple Lie group $ G $. Then the discrete dual
quantum group $ \hat{G}_q $ has property RD.
\end{prop}
This shows in particular that, in the non-unimodular case, our definition of property RD differs from the original definition given by Vergnioux.

\section{Spectral triples from length functions} \label{secspectraltriples}

In this section we explain how to associate spectral triples to discrete quantum groups equipped with proper length functions. Moreover we
study basic properties of the spectral triples obtained this way. \\
Let us first recall the definition of a spectral triple due to Connes, see \cite{Connesbook}, \cite{GFVbook}.
\begin{definition}
A spectral triple $ (\A, \H, D) $ consists of
\begin{bnum}
\item[a)] a $ * $-algebra $ \A $, faithfully represented on $ \H. $
\item[b)] a (graded) Hilbert space $ \H $, together with
\item[c)] an unbounded self-adjoint (odd) operator $ D $ in $ \H $
\end{bnum}
such that
\begin{bnum}
\item[a)] the commutators $ [D, a] $ are bounded for all $ a \in \A $
\item[b)] $ D $ has compact resolvent, that is,
$$
(1 + D^2)^{-1}
$$
is compact.
\end{bnum}
\end{definition}
In the above definition we have indicated how to include a grading in the general setup, but this will not be relevant in our examples. \\
The prototypical example of a spectral triple is given by $ \A = C^\infty(M) $, the algebra of smooth functions on a compact
Riemannian spin manifold, the Hilbert space $ \H = L^2(M, S) $ of $ L^2 $-sections of the spinor bundle of $ M $, and the
Dirac operator $ D $. In this case the formula
$$
d(x,y) = \sup_{f \in C^\infty(M) \mid \| [D, f] \| \leq 1}  |f(x) - f(y)|
$$
allows to express the Riemannian metric, or rather the corresponding geodesic distance function, in terms of operator theoretic data. \\
Connes studied spectral triples associated to discrete groups with length functions \cite{Connescms}. Let $ G $ be a discrete group and let $ l $ be a
proper length function on $ G $. Consider the Hilbert space $ l^2(G) $ and the unbounded operator $ D $ in $ l^2(G) $ defined
on $ \CH[G] \subset l^2(G) $ by
$$
D = \sum_{n \in \mathbb{N}_0} n p_n
$$
where $ p_n = q_n - q_{n - 1} $, and $ q_k $ denotes the projection onto the finite dimensional subspace of $ l^2(G) $ spanned by group elements of
length at most $ k $. If $ l $ comes from a word metric we may view $ D $ as multiplication by the length function $ l $. \\
It is not hard to check that one obtains a spectral triple in this way. In fact, we will consider a more general class of spectral
triples below, and for this it is convenient to work within the framework of filtered algebras. More precisely,
let $ \A $ be a complex unital dense $ * $-subalgebra of a unital $ C^* $-algebra $ A $.
Let us call $ \A $ a filtered algebra if we are given a family of finite dimensional subspaces $ (\A_m)_{m \in \mathbb{N}_0} $ of $ \A $ such that
\begin{bnum}
\item[a)] $ \A = \bigcup^{\infty}_{n = 0} \A_n $,
\item[b)] $ \A_0 = \mathbb{C} 1 $,
\item[c)] $ \A_m \subseteq \A_n $ for $ m < n $,
\item[d)] $ \A^*_n = \A_n $ for all $ n $,
\item[e)] $ \A_m \A_n \subseteq \A_{m + n} $ for all $ m,n $.
\end{bnum}
Assume further that $ \sigma $ is a state on $ A $ which is faithful on $ \A $, and let $ \H = L^2(A, \sigma) $ be the GNS-construction
of $ \sigma $. Then each $ \A_n $ is a finite dimensional, hence closed, subspace of $ \H $.
Let $ q_n $ denote the orthogonal projection of $ \H $ onto $ \A_n $.
Let $ p_n = q_n - q_{n - 1}  $ and $ p_0 = q_0 $. Then the formula
$$
D = \sum_{n \in \mathbb{N}_0} n p_n
$$
defines an unbounded operator $ D $ on $ \H $. \\
Filtrations and associated Dirac operators were first introduced by Voiculescu in \cite{Voiculescuquasicentral} and further studied in \cite{OzawaRieffel}.
The following general fact is a restatement of lemma 1.1. in \cite{OzawaRieffel}.
\begin{lemma} \label{ozawarieffelsptriple}
The triple $ (\A, \H, D) $ associated to a filtered algebra $ \A $ together with a state as above is a spectral triple.
\end{lemma}
We note that the $ K $-homology class of the resulting (odd) triple is trivial because the operator $ D $ is positive. That is, the
corresponding phase $ F $ is identically $ 1 $. In general, one should rather think of $ D $ as
the absolute value $ |D| $ of a true Dirac operator on the underlying noncommutative space, see \cite{Skandalisgnc}. \\
We recall that a spectral triple $ (\A, \mathcal{H}, D) $ is called regular if for each $ a \in \A $ both $ a $ and $ [D, a] $ are contained in the
domain of all powers of the derivation $ \delta $ given by
$$
\delta(T) = [|D|, T],
$$
see \cite{GFVbook}.
\begin{lemma} \label{regularityofozawarieffelsptriple}
The above spectral triple $ (\A, \H, D) $ associated to a filtered algebra $ \A $ with a state is regular.
\end{lemma}
\proof Notice again that we have $ |D| = D $ in this case. In order to verify regularity we will closely follow the proof of
lemma 1.1 of \cite{OzawaRieffel}. Let $ a $ be an element of $ \A_p $. Then we have
$$
[D, a] = \sum_{j \mid |j| \leq p} j T_j
$$
where $ T_j = \sum_m p_m a p_{m - j} $ is a bounded operator for each $ j $.
We consider $ [D, [D, a]] = \sum_{j \mid |j| \leq p} [D, T_j] $. Now
$$
[D, T_j] = [D, \sum_m p_m a p_{m - j}] = \sum_m j p_m a p_{m - j} = j T_j.
$$
Thus, $ [D, [D, a]] = \sum_{j \mid |j| \leq p} j^2 T_j. $ The statement of the corollary follows by repeated application of this technique. \qed \\
Now we turn to the case of quantum groups. Let $ G $ be a discrete quantum group and $ l $ be a proper length function on it. Imitating
Connes' construction in the group case, we construct a spectral triple with the Hilbert space $ l^2(G) $ where $ \mathbb{C}[G] $ acts by the regular representation. Our candidate for the Dirac operator is $ D $ where
$$
\dom(D) = \biggl\{\sum_{\alpha \in \Irr(G)} \sum_{i,j = 1}^{\dim(\alpha)} a^\alpha_{ij} \hat{\Lambda}(u^\alpha_{ij}) \mid
\sum_{\alpha,i,j} l(\alpha)^2 |a^\alpha_{ij} |^2 \|\hat{\Lambda}(u^\alpha_{ij})\|^2 < \infty \biggr\}
$$
and
$$
D\biggl(\sum_{\alpha, i, j} a^{\alpha}_{ij} \hat{\Lambda}(u^{\alpha}_{ij})\biggr) = \sum_{\alpha,i,j} l(\alpha) a^{\alpha}_{ij} \hat{\Lambda}(u^\alpha_{ij}).
$$
The algebra $ \A = \mathbb{C}[G] $ is naturally a filtered algebra with the filtration defined by letting $ \A_n $ be the linear span of
all matrix coefficients $ u^{\alpha}_{ij} $ such that $ l(\alpha) \leq n $. Since we assume that the length function $ l $ is proper there are only
finitely many irreducible corepresentations of length $ n $. Hence the subspaces $ \A_n $ are all finite dimensional. Moreover
$ \A_0 = \mathbb{C}1 $, and it is easy
to see from the properties of length functions that  $ \A^*_n = \A_n $ and $ \A_n \A_m \subset \A_{n + m} $.
\begin{lemma} \label{regularspectraltriple}
Let $ G $ be a discrete quantum group with a proper length function. Then the triple $ (\mathbb{C}[G], l^2(G), D) $ constructed above is a
regular spectral triple.
\end{lemma}
\proof Let $ A = C^*_\red(G) $ and $ \A = \mathbb{C}[G] $. The Hilbert space $ l^2(G) $ is the GNS-Hilbert space with respect to the Haar
state $ \hat{\phi} $, and $ \hat{\phi} $ is  faithful on $ \mathbb{C}[G] $. By the observations made above, the
filtration $ (\A_m)_{m \in \mathbb{N}_0} $ satisfies the conditions of lemma \ref{ozawarieffelsptriple}, and therefore $ (\mathbb{C}[G], l^2(G), D) $
is a spectral triple. Regularity follows from lemma \ref{regularityofozawarieffelsptriple}. \qed \\
Recall that a spectral triple $ (\A, \H, D) $ is called $ p $-summable if $ \Tr(\hat{D}^{-p}) < \infty $, where $ \hat{D} $ denotes the restriction
of $ |D| $ to the orthogonal complement of its kernel. \\
The following result is a variant of proposition 6 in \cite{Connescms}.
\begin{prop} \label{summability}
Let $ G $ be an amenable discrete quantum group of rapid decay, and fix a proper length function and constants $ c, s $ such that
$$
\|a\|_{op} \leq c \|a\|_{2,s}
$$
for all $ a \in \mathbb{C}[G] $. Then the associated spectral triple $ (\CH[G], l^2(G), D) $ constructed above is $ p $-summable for all $ p > 2s + 1 $.
\end{prop}
\proof Since $ G $ is amenable, we see from the proof of proposition \ref{rdpgequiv} that
$$
\sum_{l (\alpha) \in (n - 1, n]} \dim(\alpha)^2 = \phi(q_n) \leq c^2 (1 + n)^{2s}.
$$
Using this, we find
\begin{align*}
\Tr(\hat{D}^{-p}) &= \sum_{\alpha \neq \epsilon} \dim(\alpha)^2 l(\alpha)^{-p} \\
&\leq r \sum_{l(\alpha) \in (0,1]} \dim(\alpha)^2 + \sum_{n > 1} \sum_{l(\alpha) \in (n - 1, n]} \dim(\alpha)^2 (n - 1)^{-p} \\
&\leq r \sum_{l(\alpha) \in (0,1]} \dim(\alpha)^2 + \sum_{n > 1} \frac{c^2 (n + 1)^{2s}}{(n - 1)^p}
\end{align*}
where $ r $ is the inverse of the smallest nonzero value of $ l $ in the interval $ (0,1] $. This yields the claim. \qed \\
Notice that we cannot drop the amenability assumption in proposition \ref{summability} in general. For instance, according to
\cite{Vergniouxrd}, the free orthogonal quantum group $ \mathbb{F}O(n) $ has property RD, but the corresponding spectral triple
will fail to be finitely summable as soon as $ n > 2 $.

\section{Compact quantum metric spaces} \label{seccompactquantummetricspace}

We recall that the geodesic distance on a compact Riemannian spin manifold can be recovered from the associated spectral triple.
For a general spectral triple $ (\A, \H, D) $, one may take
$$
d(\mu, \nu) = \sup_{a \in \A \mid \| [D, a] \| \leq 1}  |\mu(a) - \nu(a)|
$$
as an Ansatz to define a metric on the state space $ S(A) $ of the $ C^* $-algebra closure $ A $ of $ \A $ in $ \LH(\H) $,
thus generalising the Monge-Kantorovich metric on probability measures.
We note that without further assumptions the above formula may yield $ d(\mu, \nu) = \infty $ for some states,
see \cite{Connesgravmat}. \\
The above considerations, along with further examples, motivated Rieffel to introduce the concept of a quantum metric
space. In \cite{Rieffelcqms} the theory is developed starting from order unit spaces instead of $ C^* $-algebras, but for
our purposes the following definition is sufficient.
\begin{definition} \label{defcqms}
Let $ A $ be a unital $ C^* $-algebra and let $ \A \subset A $ be a dense unital $ * $-subalgebra. A Lipschitz seminorm on $ \A $ is a seminorm
$ L: \A \rightarrow [0, \infty) $ such that $ L(a^*) = L(a) $ for all $ a \in \A $ and $ L(a) = 0 $ iff $ a \in \mathbb{C} 1 $.
A Lipschitz seminorm is called a Lip-norm if the topology on $ S(A) $ induced by
$$
d_L(\mu, \nu) = \sup_{a \in \A \mid L(a) \leq 1}  |\mu(a) - \nu(a)|
$$
coincides with the $ w^* $-topology. \\
A unital $ C^* $-algebra $ A $ together with a Lip-norm on a dense $ * $-subalgebra $ \A \subset A $ is called a compact quantum metric space.
\end{definition}
Starting from definition \ref{defcqms} it is natural to ask for conditions ensuring that a Lipschitz seminorm is indeed a Lip-norm.
Ozawa-Rieffel gave the following criterion, see proposition 1.3 in \cite{OzawaRieffel}.
\begin{prop} \label{ozawarieffellipnorm}
Let $ A $ be a unital $ C^* $-algebra, and let $ L $ be a Lipschitz seminorm on a dense unital $ * $-subalgebra $ \A \subset A $.
If $ \sigma $ is a state on $ A $ such that
$$
E = \{a \in A \mid L(a) \leq 1 \; \text{and} \; \sigma(a) = 0 \}
$$
is totally bounded with respect to the norm of $ A $, then $ L $ is a Lip-norm.
\end{prop}
Recall that a subset $ B $ of a metric space $ X $ is totally bounded if for any $ \epsilon > 0 $ there exists
finitely many balls of radius $ \epsilon $ whose union covers $ B $. \\
Let us now assume that $ (\A, \H, D) $ is the spectral triple associated to a filtered algebra as in section \ref{secspectraltriples}. One
may ask whether the resulting Lipschitz seminorm $ L(a) = \|\delta(a)\| = \|[D,a]\| $ is a Lip-norm on $ \A $, viewed as a dense subalgebra
of its norm closure $ A $. This seems to be unclear in general. \\
In \cite{OzawaRieffel}, Ozawa and Rieffel showed that $ L $ is indeed a Lip-norm provided a condition of the form
$$
\| p_m a p_n   \| \leq c \| a \|_2
$$
holds for $ a \in \A_k $ and all $ k,m,n $. They call such an inequality a Haagerup type condition. As remarked in section \ref{secspectraltriples},
Connes' spectral triples on group algebras coming from length functions are special cases of the Ozawa-Rieffel construction in
lemma \ref{ozawarieffelsptriple}. Examples of groups for which the Haagerup type condition hold include word hyperbolic
groups \cite{Connesbook}, \cite{OzawaRieffel} as well as free products of the form $ G_1 \ast G_2 $ where $ G_1 $ and $ G_2 $ satisfy the
Haagerup type condition and one works with tracial states on the group algebras of $ G_1 $ and $ G_2 $. \\
It is not clear, however, whether $ L $ is a Lip-norm for groups of rapid decay in general. In this case one has a weaker inequality of the form
$$
\| p_m a p_n \| \leq c P(k) \| a \|_2
$$
for all $ a \in \A_k $, where $ P $ is a polynomial. Antonescu and Christensen observed that one can easily prove
the Lip-norm property in this case if one works with a slightly different Lipschitz seminorm instead \cite{ACmetricsgroup}. More precisely,
for $ k \in \mathbb{N} $ and $ a \in \A = \mathbb{C}[G] $, consider
$$
L^k(a) = \| \delta^k(a) \| = \|[D, [D, \dots [D,a] \dots]\|,
$$
where the commutator is taken $ k $ times. \\
We will consider a similar construction in the case of discrete quantum groups, and prove some lemmas which are needed later on. However, in order
to accommodate non-unimodular discrete quantum groups, we need to use a twisted version of the seminorm $ L^k $.
Throughout, we assume that $ G $ is a discrete quantum group equipped with a proper length function $ l $.
\begin{lemma} \label{lemma1}
Using the same notation as before, we have
$ \delta^k(u^{\alpha}_{ij}) \hat{\Lambda}(1) = D^k \hat{\Lambda}(u^{\alpha}_{ij}) $ for all $ k \in \mathbb{N} $.
In particular, for a finite sum of the form
$ a = \sum_{\alpha,i,j} a^{\alpha}_{ij} u^{\alpha}_{ij} $, we have
$ \delta^k(a) \hat{\Lambda}(1) = \sum_{\alpha,i,j} l(\alpha)^k a^{\alpha}_{ij} \hat{\Lambda}(u^{\alpha}_{ij}) $.
\end{lemma}
\proof We proceed by induction. For $ k = 1 $, we have
\begin{align*}
\delta(u^{\alpha}_{ij}) \hat{\Lambda}(1) &= [D, u^{\alpha}_{ij}] \hat{\Lambda}(1) \\
&= D \hat{\Lambda}(u^{\alpha}_{ij}) - u^{\alpha}_{ij} D\hat{\Lambda}(1) = D\hat{\Lambda}(u^{\alpha}_{ij}).
\end{align*}
Let us now assume that $ \delta^k(u^{\alpha}_{ij}) \hat{\Lambda}(1) = D^k \hat{\Lambda}(u^{\alpha}_{ij}) $. Then
\begin{align*}
\delta^{k + 1}(u^{\alpha}_{ij}) \hat{\Lambda}(1) &= [D, \delta^k(u^{\alpha}_{ij})] \hat{\Lambda}(1) \\
&= D \delta^k (u^{\alpha}_{ij}) \hat{\Lambda}(1) - \delta^k(u^{\alpha}_{ij}) D \hat{\Lambda}(1) \\
&= D D^k \hat{\Lambda}(u^{\alpha}_{ij}) \\
&= D^{k + 1} \hat{\Lambda}(u^{\alpha}_{ij}),
\end{align*}
which yields the claim. \qed
\begin{lemma} \label{lemma2}
If $ a \in \mathbb{C}[G] $ is such that $ L^k(a) = 0 $, then $ a $ is a scalar multiple of the identity. Hence $ L^k $ is a
Lipschitz seminorm.
\end{lemma}
\proof Let $ a = \sum_{\alpha, i,j} a^{\alpha}_{ij} u^{\alpha}_{ij} \in \CH[G] $. By definition, the relation $ L^k(a) = 0 $
implies that $ \delta^k(a) = 0 $. In particular we have $ \delta^k(a) \hat{\Lambda}(1) = 0 $. Therefore, according to lemma \ref{lemma1} we obtain
$$
0 = \sum_{\alpha,i,j} a^{\alpha}_{ij} D^k \hat{\Lambda}(u^{\alpha}_{ij})
= \sum_{\alpha,i,j} l(\alpha)^k a^{\alpha}_{ij}  \hat{\Lambda}(u^{\alpha}_{ij}) = 0.
$$
Since the vectors $ \hat{\Lambda}(u^{\alpha}_{ij}) $ form a linearly independent set in $ l^2(G) $, we conclude $ l(\alpha)^k a^{\alpha}_{ij} = 0 $
for all $ \alpha,i,j $. If $ l(\alpha) \neq 0 $ then $ a^{\alpha}_{ij} = 0 $. Since $ l $ is a proper length function we conclude that $ a $ has
to be a scalar multiple of the identity. Moreover, we clearly have $ L^k(1) = 0 $ and $ L^k(a) = L^k(a^*) $.
Hence $ L^k $ is a Lipschitz seminorm. \qed \\
Fix a natural number $ k $ and consider the operator $ T = C^{\frac{1}{2k}} $. As before, we let $ \delta(a) = [D,a] $ and define
$$
\delta_{T} (a) = [D, T a T], \qquad L^k_{T} (a) = \| \delta^k_T(a) \|
$$
for $ a \in \A = \CH[G] $.
\begin{lemma}
Let $ k \in \mathbb{N} $. For any $ a \in \A $ we have $ \delta^k_T(a) = T^k \delta^k(a) T^k $, and $ L^k_T $ is a Lipschitz seminorm on $ \A $.
\end{lemma}
\proof Since $ T $ commutes with $ D $ we have $ \delta_T(a) = [D, T a T] = T [D,a] T $, and so $ \delta^k_T(a) = T^k \delta^k(a) T^k $.
Moreover
$$
L^k_T(a^*) = \|\delta_T^k(a^*)\| = \| T^k \delta^k(a^*) T^k \| = \| T^k \delta^k(a) T^k \| = L^k_T(a)
$$
since $ T $ is self-adjoint. For the Lipschitz seminorm property, we need to check that if $ L^k_T(a) = 0 $, then $ a $ is a scalar multiple of
the identity. But $ L^k_T(a) = 0 $ implies that $ T^k \delta^k(a) T^k = 0 $. Since $ T $ is invertible we conclude $ \delta^k(a) = 0 $
which means $ L^k(a) = 0 $. Hence the desired conclusion follows from lemma \ref{lemma2}. \qed
\begin{lemma} \label{labelledon21stapril}
Let $ k \in \mathbb{N} $. Then we have
\begin{align*}
\sum_{\alpha, i, j} \dim(\alpha)^{-1} l(\alpha)^{2k} | a^{\alpha}_{ij}|^2
&= \|\sum_{\alpha, i, j} \delta^{k}_T (a^{\alpha}_{ij} u^\alpha_{ij}) \hat{\Lambda}(1) \|^2_2 \\
&\leq L^k_T\biggl(\sum_{\alpha,i,j} a^{\alpha}_{ij} u^\alpha_{ij}\biggr)^2.
\end{align*}
\end{lemma}
\proof Using lemma \ref{lemma1} we compute
\begin{align*}
&\|\sum_{\alpha, i, j} \delta^{k}_T (a^{\alpha}_{ij} u^\alpha_{ij}) \hat{\Lambda}(1) \|^2_2
= \| \sum_{\alpha,i,j} a^{\alpha}_{ij} l(\alpha)^k T^k \hat{\Lambda}(u^{\alpha}_{ij}) \|^2_2 \\
&= \sum_{\alpha} \sum_{i,j} \sum_{i^{\prime},j^{\prime}}
\overline{a^{\alpha}_{ij}} a^{\alpha}_{i^{\prime}j^{\prime}} \frac{\dim_q(\alpha)}{\dim(\alpha)}
l(\alpha)^{2k} (F^{\alpha})^{\frac{1}{2}}_{ii} (F^{\alpha})^{\frac{1}{2}}_{i^{\prime}i^{\prime}}
\bra \hat{\Lambda}(u^{\alpha}_{ij}), \hat{\Lambda}(u^{\alpha}_{i^{\prime} j^{\prime}}) \ket \\
&= \sum_{\alpha,i,j} \dim(\alpha)^{-1} l(\alpha)^{2k} | a^\alpha_{ij}|^2.
\end{align*}
This yields the claim. \qed

\section{Rapid decay and the Lip norm property} \label{seclipnorm}

In this section we show that the Lipschitz norms considered above have the Lip norm property for all sufficiently large exponents
provided the quantum group under consideration has property RD. \\
For simplicity we concentrate on the case of a finitely generated discrete quantum group $ G $ equipped with a word length function $ l $.
Moreover we assume that $ G $ has property RD, and we fix constants $ c, s > 0 $ such that
$$
\| a \|_{op} \leq c \| a \|_{2,s}
$$
holds for all $ a $ in $ \CH[G] $ with respect to the Sobolev norms associated with $ l $. \\
Using the notation introduced in section \ref{seccompactquantummetricspace}, we will apply proposition \ref{ozawarieffellipnorm} by Ozawa-Rieffel to show
that there exists a positive integer $ k $ such that $ L^k_T $ has the Lip norm property. We start with the following lemma.
\begin{lemma} \label{lipnorm}
Let $ a = \sum_{\alpha,i,j} a^{\alpha}_{ij} u^\alpha_{ij} \in \CH[G] $. Then we have
$$
\|a\|^2_{2,s} = \sum_{\alpha, i,j} \frac{1}{\dim(\alpha)} (1 + l(\alpha))^{2s} |a^\alpha_{ij}|^2.
$$
\end{lemma}
\proof We compute
\begin{align*}
\| a \|^2_{2,s} &= \sum_{\alpha, i,j}
\bra (1 + L)^s a^\alpha_{ij} \F^{-1}(u^\alpha_{ij}) C^{1/2}, (1 + L)^s a^\alpha_{ij} \F^{-1}(u^\alpha_{ij}) C^{1/2} \ket \\
&= \sum_{\alpha, i,j} \frac{\dim_q(\alpha)}{\dim(\alpha)} (1 + l(\alpha))^{2s} |a^\alpha_{ij}|^2 (F^\alpha)_{ii}
\bra u^\alpha_{ij}, u^\alpha_{ij} \ket \\
&= \sum_{\alpha, i,j} \frac{1}{\dim(\alpha)} (1 + l(\alpha))^{2s} |a^\alpha_{ij}|^2.
\end{align*}
This yields the claim. \qed \\
Now, for a positive integer $ k > s $, let
$$
E = \{a \in \mathbb{C}[G] \mid L^{k}_T(a) \leq 1 \; \text{and} \; \hat{\phi}(a) = 0 \},
$$
which we view as a subset of $ \CH[G] \subset C^*_\red(G) $.
\begin{lemma} \label{termE1}
For all $ n \in \mathbb{N} $ there exists a constant $ c_n $ such that
$$
\| \sum_{\alpha, i, j \mid l(\alpha) \leq n} a^{\alpha}_{ij} u^{\alpha}_{ij}  \|_{op} \leq c_n
$$
for any $ a = \sum_{\alpha, i, j} a^{\alpha}_{ij} u^{\alpha}_{ij} \in E $.
\end{lemma}
\proof We recall that $ \epsilon $ denotes the trivial corepresentation. Using $ a^\epsilon_{11} = \hat{\phi}(a) = 0 $ and the Cauchy-Schwarz inequality we compute
\begin{align*}
&\sum_{\alpha, i, j \mid l(\alpha) \leq n} |a^{\alpha}_{ij}|
= \sum_{\alpha, i, j \mid 1 \leq l(\alpha) \leq n} l(\alpha)^{-k}
\sqrt{\dim(\alpha)} l(\alpha)^k |a^{\alpha}_{ij}|
\sqrt{\dim(\alpha)}^{\,-1} \\
&\leq \biggl(\sum_{\alpha, i, j \mid l(\alpha) \leq n}  l(\alpha)^{-2k}  \dim(\alpha) \biggr)^{\frac{1}{2}}
\biggl(\sum_{\alpha, i, j \mid l(\alpha) \leq n} l(\alpha)^{2k} |a^{\alpha}_{ij}|^2 \dim(\alpha)^{-1} \biggr)^{\frac{1}{2}} \\
&\leq c_n L_T^k(a) \leq c_n,
\end{align*}
where we choose $ c_n $ such that
$$
\biggl(\sum_{\alpha, i, j \mid l(\alpha) \leq n}  l(\alpha)^{-2k}  \dim(\alpha) \biggr)^{\frac{1}{2}} \leq c_n,
$$
and we use lemma \ref{labelledon21stapril}. \\
Since the matrices $ (u^\alpha_{ij}) $ are unitary we have $ \|u^{\alpha}_{ij}\|_{op} \leq 1 $ for all $ \alpha $ and $ i,j = 1, \dots, \dim(\alpha) $.
Hence we obtain
\begin{align*}
\| \sum_{\alpha, i, j \mid l(\alpha) \leq n} a^{\alpha}_{ij} u^{\alpha}_{ij} \|_{op}
&\leq \sum_{\alpha, i, j \mid l(\alpha) \leq n} |a^{\alpha}_{ij}| \|u^{\alpha}_{ij}\|_{op}
\leq \sum_{\alpha, i, j \mid l(\alpha) \leq n} |a^{\alpha}_{ij}| \leq c_n.
\end{align*}
This yields the claim. \qed
\begin{lemma} \label{termE2}
Let $ a = \sum_{\alpha,i,j} a^{\alpha}_{ij} u^\alpha_{ij} \in \CH[G] $ and fix a positive integer $ k > s $. Then we have
$$
\|\sum_{\alpha,i,j, l(\alpha) > n} a^{\alpha}_{ij} u^\alpha_{ij} \|^2_{op}
\leq c^2 2^{2s} n^{2(s - k)} \sum_{\alpha, i, j, l(\alpha) > n} \dim(\alpha)^{-1} l(\alpha)^{2 k} |a^{\alpha}_{ij}|^2
$$
for any $ n \in \mathbb{N} $.
\end{lemma}
\proof We follow Antonescu-Christensen \cite{ACmetricsgroup}. For $ \alpha $ satisfying $ l(\alpha) > n $ write
$$
(1 + l(\alpha))^{2s} \leq 2^{2s} l(\alpha)^{2s} \leq 2^{2s} n^{2s - 2 k} l(\alpha)^{2 k}.
$$
Therefore, by property RD and lemma \ref{lipnorm}, we have
\begin{align*}
&\|\sum_{\alpha,i,j, l(\alpha) > n} a^{\alpha}_{ij} u^\alpha_{ij} \|^2_{op}
\leq c^2 \| \sum_{\alpha,i,j, l(\alpha) > n} a^{\alpha}_{ij} u^\alpha_{ij} \|^2_{2,s} \\
&= c^2 \sum_{\alpha,i,j, l(\alpha) > n} \dim(\alpha)^{-1} (1 + l(\alpha))^{2s} |a^{\alpha}_{ij}|^2 \\
&\leq c^2 2^{2s} n^{2(s - k)} \sum_{\alpha, i, j, l(\alpha) > n} \dim(\alpha)^{-1} l(\alpha)^{2 k} |a^{\alpha}_{ij}|^2.
\end{align*}
This finishes the proof. \qed \\
We are now ready to prove the following result.
\begin{theorem}
For any positive integer $ k > s $, the algebra $ \CH[G] \subset C^*_\red(G) $ together with the seminorm $ L^{k}_T $ is a compact quantum metric
space.
\end{theorem}
\proof We have to show that $ L^{k}_T $ is a Lip norm. Fix $ \epsilon > 0 $ and choose $ n $ such that $ c 2^{s} n^{s - k} < \epsilon $
and write $ E = E_1 + E_2, $ where
$$
E_1 = \{x \in E \mid x = \sum_{\alpha, i, j \mid l (\alpha) \leq n} a^{\alpha}_{ij} u^{\alpha}_{ij} \}
$$
and
$$
E_2 = \{x \in E \mid x = \sum_{\alpha,i,j \mid l(\alpha) > n} a^{\alpha}_{ij} u^{\alpha}_{ij} \}.
$$
Using lemma \ref{termE1} we see that $ E_1 $ is a bounded subset of a finite dimensional normed space, and thus totally bounded.
Moreover, by our choice of $ n $ and lemma \ref{termE2}, the set $ E_2 $ is contained in the $ \epsilon $-ball around $ 0 $ in $ C^*_\red(G) $.
This completes the proof. \qed

\bibliographystyle{plain}

\bibliography{cvoigt}

\end{document}

config: arxiv


TITLE: Calculate values based on tilt of container, for volume and rate of change of angle of tilt.
QUESTION [1 upvotes]: Found this problem in a text based on exercises on calculus.
The problem statement is stated below:

In certain industrial processes it is necessary to control the rate at which a liquid is poured out of a container. For this problem, suppose the container is a rectangular box with no top, with square cross-section -- say 4 meters long, 2 meters wide and 2 meters top. We are going to pour some liquid out of it by leaving one long edge of the container on the floor and raising the other long edge, so that after awhile the side view of the container and the liquid in it looks as follows:
(a) Derive a formula for the liquid in the box (assuming that the liquid is initially full when theta= Pi/2 ($\theta=\frac{\pi}{2}$) as a function of theta ($\theta$). There will be two different formulas, depending on whether the left edge is against the left edge of the box, or against the bottom of the box.
(b) Calculate the derivative of the volume of the box wrt to theta ($\theta$).
(c) How fast should the theta  ($\theta$) change if the liquid is to be poured out of the box at a constant rate of 0.5 cubic meters per second (it is ok to leave the answer in terms of theta)?

My solution attempt is stated below:
(a) $\theta$ would be continuously changing from $\frac{\pi}{2}$ to $0$. The volume initially is $16\,m^3$, with $\theta=\frac{\pi}{2}$. When the $\theta$ changes to $0$, the volume becomes nil.
The dimension of the liquid affected by change in $\theta$. There are two changes induced by change in $\theta$ :
(i) height against the left edge of the box ($h_l$), that changes from $2$ meters to $0$ meters.
The below image states the angle $\theta_b = 45^0$.

(ii) once $h_l=0$, the height of liquid against the base ($h_b=2\sin(90^0-\theta_b)$ meters) starts decreasing from $4\cos(\theta_b)$ meters to $0$ meters. The start value of $\theta_b = 45^0$, i.e. $h_b$ decreases from $2\sqrt{2}$m to $0$m.
The below image states the angle $\theta_b = 45^0$.

Unable to pursue further.

Edit -
Based on the answer by @Andrei, have submitted diagram as below for the first part of the tilt (i.e., til the bottom edge point B is reached):

Also, as per the answer, my second diagram is wrong as shown below by necessary equality between base length ($L_1$) and height (h):

REPLY [2 votes]: Let's name several of the points in the figure:

$A$ is the corner of the container above the liquid
$B$ is the corner on the left
$C$ is the corner touching the ground
$D$ is on the right
$E$ is the point where the liquid touches $AB$

We define some lengths

$h=AB=CD$
$L_1=BC=AD$
$L_2$ the length of the container perpendicular to the figure plane

You will have two cases, where $L_1=4$ and $L_2=2$, or $L_1=2$ and $L_2=4$.
Now let's discuss what happens: you start tilting the container. Then point $E$ moves from $A$ to $B$. Note that $DE$ is always horizontal. That means that $\angle CDE=\theta$ and therefore $\angle DEA=\theta$. The area of $\triangle AED$ is then $\frac 12 AE\cdot AD=\frac 12 AD^2\cot\theta=\frac 12 L_1^2\cot\theta$. Then the volume of liquid that left the container is $\frac 12 L_1^2L_2\cot\theta$, so the volume left in the container is $$V=L_1L_2h-\frac 12 L_1^2L_2\cot\theta$$
This formula is only valid until the water level reaches $B$. Then, in $\triangle BCD$ we have $\angle CBD=\theta_0$. We can write $$\tan\theta_0=\frac{L_1}{h}$$
That means that your second figure is wrong. You get $\theta_0=45^\circ$ only if you have the case $L_1=h=2$.
If you increase the tilt, $E$ will move from $B$ to $C$, and, in a similar fashion as before, you can write $$V=\frac 12 L_2h^2\tan\theta$$
You can see from bot formulas that at $\theta=\theta_0$ you have $V=\frac 12 L_1L_2h$.
Hope this is enough to get you unstuck.


text:
\begin{document}

\title[A generalization of Watts's Theorem]{A generalization of Watts's Theorem: \\
Right exact functors on module categories}
\author{A. Nyman and S. P. Smith}
\address{Department of Mathematics, University of Montana, Missoula, MT 59812-0864}
\email{NymanA@mso.umt.edu} \keywords{Watts theorem}
\address{Department of Mathematics, University of
Washington, Box 354350, Seattle, Washington 98195}
\email{smith@math.washington.edu}
\date{\today}
\thanks{2000 {\it Mathematics Subject Classification. } Primary  18F99; Secondary 14A22, 16D90, 18A25}
\thanks{The first author was partially supported by the National Security Agency under grant NSA H98230-05-1-0021. The second author was supported by NSF grant DMS-0245724.}

\maketitle

\begin{abstract}
Watts's Theorem says that a right exact functor $F:\Mod R \to \Mod
S$ that commutes with direct sums is isomorphic to $-\otimes_R B$
where $B$ is the $R$-$S$-bimodule $FR$. The main result in this
paper is the following: if $\sA$ is a cocomplete abelian category and
$F:\Mod R \to \sA$ is a right exact functor commuting with direct
sums, then $F$ is isomorphic to $- \otimes_R \cF$ where $\cF$ is a
suitable $R$-module in $\sA$, i.e., a pair $(\cF,\rho)$ consisting
of an object $\cF \in \sA$ and a ring homomorphism $\rho:R \to
\Hom_\sA(\cF,\cF)$.  Part of the point is to give meaning to the
notation $-\otimes_R \cF$. That is done in the paper by Artin and
Zhang \cite{AZ} on Abstract Hilbert Schemes. The present paper is
a natural extension of some of the ideas in the first part of
their paper.
\end{abstract}

\pagenumbering{arabic}

\section{Introduction}
Let $R$ and $S$ be rings and let $\Mod R$ and $\Mod S$ denote the
category of right $R$-modules and right $S$-modules, respectively.
Watts's Theorem, which was proved by Eilenberg \cite{E} and Watts \cite{W} at about the same
time,  is the following:

\begin{thm} \label{thm.classical}
Suppose $F:\Mod R \to \Mod S$ is a right exact functor commuting
with direct limits.  Then $F \cong -\otimes_R B$ where $B$ is an
$R$-$S$-bimodule.
\end{thm}

Let $\sB(\Mod R,\Mod S)$ denote the full subcategory of the
category of functors from $\Mod R$ to $\Mod S$ consisting of right
exact functors commuting with direct limits.   The next result is a slightly more precise
version of Theorem  \ref{thm.classical}.

\begin{thm} \label{thm.generalization}
The functor $\Psi:\Mod (R^{\op} \otimes_\mathbb{Z} S) \to \sB(\Mod R,\Mod
S)$ induced by the assignment $B \mapsto -\otimes_R B$ is an
equivalence of categories.
\end{thm}
Theorem \ref{thm.classical} is then just the fact that the functor
$\Psi$ is essentially surjective.

The main result of this paper (Theorem \ref{thm.watts}) is that if
$\Mod S$ is replaced by an arbitrary cocomplete\footnote{An additive category is {\sf cocomplete}
if it has arbitrary direct sums. This is Grothendieck's condition Ab3.}
 category $\sA$,
then a version of Theorem \ref{thm.generalization} still holds.
One of the obvious hurdles in proving such a theorem is to have a
sensible notion of tensor product in this context.  We use the
tensor product functor that was defined in \cite[Thm. 3.7.1]{Po}
and investigated in detail in  \cite{AZ} (see Section
\ref{subsection.tensor}).

In Proposition \ref{prop.scheme}, we specialize our main result to
the case that $\sA$ is the category of quasi-coherent sheaves on a
scheme $Y$. This version of the main result is used extensively in
\cite{NS} to prove a structure theorem for right exact functors
between categories of quasi-coherent sheaves on schemes.

\section{Preliminaries}
Throughout this paper, $k$ is a fixed commutative ring, $R$ is a
$k$-algebra, and $\gamma:k \to R$ is the homomorphism giving $R$
its $k$-algebra structure.

\subsection{$k$-linearity}
 Let  $\sA$ be an additive category. We say $\sA$ is
 {\it $k$-linear} if for all objects $X$ and $Y$ in $\sA$,
$\Hom_\sA(X,Y)$ is a $k$-module and composition of morphisms
is $k$-bilinear.  Equivalently,  $\sA$ is $k$-linear if there is a ring homomorphism
$$
c: k \to \End(\id_\sA)
$$
from $k$ to the ring of natural transformations from the identity functor to itself.

The first definition tells us that for each object $X \in \sA$ and each $a \in k$ there is a morphism
$a_X:X \to X$ such that
\begin{equation} \label{eqn.lincompat}
a_Y \circ f =f \circ a_X
\end{equation}
 for all $a \in k$ and $f \in \Hom_\sA(X,Y)$.
The second definition tells us there are natural transformations
$c(a):\id_\sA \to \id_\sA$ for each $a \in k$, and therefore
associated morphisms $c(a)_X:X \to X$ for each $a \in k$ and $X \in \sA$.
The connection between the two definitions is that $$c(a)_X = a_X$$ for all $a \in k$ and $X$ in
$\sA$.

The $k$-linear structure on $\Mod R$ is given by
\begin{equation}
\label{ModR.k-linear}
a_M(m)= m.\c(a).
\end{equation}
for all $M \in \Mod R$, $m \in M$, and $a \in k$.

\subsection{$k$-linear functors}

Let $\sC$ and $\sA$ be $k$-linear categories. A functor $F:\sC \to
\sA$  is  {\it $k$-linear} if the natural maps $\Hom_\sC(X,Y) \to
\Hom_\sA(FX, FY)$ are $k$-linear for all $X$ and $Y$ in $\sC$.
Equivalently, $F$ is $k$-linear if $F$ is additive and
$$
F(a_Y)=a_{FY}
$$
for all $a \in k$ and $Y \in \Mod R$.

 We write
$$
\sB_k(\sC,\sA)
$$
for the full subcategory of the category of functors $\sC \to \sA$ consisting of $k$-linear right exact
functors that commute with direct limits. We use the letter $\sB$ to remind us of
bimodules.

It is surely well known that an adjoint to a $k$-linear functor is
again $k$-linear but we could not find a proof in the literature so provide one for completeness.

\begin{prop}
\label{prop.adj.linear}
Let $\sC$ and $\sA$ be $k$-linear categories.
Let $G:\sA \to \sC$ be a functor having a left adjoint $F$. If $G$ is $k$-linear so is $F$.
\end{prop}
\begin{pf}
Let $X \in \sC$, and let
$$
\nu:\Hom_\sA(FX,FX) \to \Hom_\sC(X,GFX)
$$ be the adjoint isomorphism. By the functoriality of the adjoint isomorphisms the diagrams
$$
\UseComputerModernTips
\xymatrix{
\Hom_\sA(FX,FX) \ar[rr]^\nu  \ar[d]_{- \circ Ff} &&  \Hom_\sC(X,GFX) \ar[d]^{-\circ f}
\\
\Hom_\sA(FX,FX) \ar[rr]_\nu  &&  \Hom_\sC(X,GFX)
}
$$
and
$$
\UseComputerModernTips
\xymatrix{
\Hom_\sA(FX,FX) \ar[rr]^\nu  \ar[d]_{g \circ -} &&  \Hom_\sC(X,GFX) \ar[d]^{Gg \circ -}
\\
\Hom_\sA(FX,FX) \ar[rr]_\nu  &&  \Hom_\sC(X,GFX)
}
$$
commute for all $X$ in $\sC$, all $f \in \Hom_\sC(X,X)$, and all $g \in \Hom_{\sA}(FX,FX)$.

Let $\theta \in \Hom_\sA(FX,FX)$ be an element in the top left corner of the diagrams.
Let $f=a_X$ and $g=a_{FX}$. The commutativity therefore gives
\begin{align*}
\nu(\theta \circ F(a_X)) & = \nu(\theta) \circ a_X \qquad \hbox{and}
\\
\nu(a_{FX} \circ \theta) & =G(a_{FX}) \circ \nu(\theta).
\end{align*}
But $\nu(\theta):X \to GFX$ is a $k$-linear morphism so $ \nu(\theta) \circ a_X  = a_{GFX} \circ \nu(\theta)$.
Since $G$ is $k$-linear, $G(a_{FX}) = a_{GFX}$. Hence
$$
\nu(\theta \circ F(a_X))  =a_{GFX} \circ \nu(\theta) = G(a_{FX})  \circ \nu(\theta) =  \nu(a_{FX} \circ \theta).
$$
But $\nu$ is an isomorphism so
$$
\theta \circ F(a_X)  = a_{FX} \circ \theta.
$$
Now take $\theta=\id_{FX}$ to get $F(a_X)=a_{FX}$, so showing that $F$ is $k$-linear.
\end{pf}

\subsection{The category $\sA_R$}

For the remainder of this paper, we let $\sA$ denote a $k$-linear
cocomplete category.

A  {\it left $R$-module in $\sA$} is a pair $(\cF,\rho)$ where $\cF$ is an object in $\sA$
and $\rho:R \to \End_\sA \cF$ is a $k$-algebra homomorphism.
Popescu \cite[p. 108]{Po} calls $(\cF,\rho)$ a left $R$-object of $\sA$.
Let $(\cF,\rho)$ and $(\cG,\rho')$ be left $R$-modules in $\sA$. We define the set of {\it $R$-module maps} from  $(\cF,\rho)$ to $(\cG,\rho')$ to be
$$
\Hom_R(\cF,\cG): = \bigl\{ \a \in \Hom_\sA(\cF,\cG) \; | \; \rho'(r) \circ \a=\a \circ \rho(r) \;
\hbox{ for all } \; r \in R\bigr\}.
$$
Using these $R$-module maps as morphisms we then obtain a category $\sA_R$,
 the category of left $R$-modules in $\sA$.

Suppose $(\cF,\rho) \in \sA_R$. If $\cG \in \sA$, then
$\Hom_\sA(\cF,\cG)$ becomes a right $R$-module through the
composition map
$$
\Hom_\sA(\cF,\cG) \times \Hom_\sA(\cF,\cF) \to \Hom_\sA(\cF,\cG),
$$
i.e.,
$$
\a.r:= \a \circ \rho(r)
$$
for $\a \in \Hom_\sA(\cF,\cG)$ and $r \in R$. This allows
us to view $\Hom_\sA(\cF,-)$ as a functor $\sA \to \Mod R$.

\smallskip

For each $x \in R$,  let  $\mu_x:R \to R$ be the right $R$-module
homomorphism  $\mu_x(r):=xr$.

\begin{lemma} \label{lemma.ar}
Suppose $F \in \sB_k(\Mod R,\sA)$. Define the ring homomorphism
$$
\rho:R \to \End_{\sA}FR, \qquad \rho(x):=F(\mu_x).
$$
Then $(FR,\rho) \in \sA_R$.
\end{lemma}

\begin{pf}
To prove the lemma it suffices to show that $\rho$ is a
$k$-algebra homomorphism, i.e., that $(\rho \circ \c)(a) = a_{FR}$
for all $a \in k$. But
$$
\rho(\c(a)) = F(\mu_{\c(a)}) = F(a_R) = a_{FR},
$$
where the second equality is due to (\ref{ModR.k-linear}). Hence the result.
\end{pf}

\subsection{The functor $-\otimes_R \cF$}
\label{subsection.tensor}
Recall the standing hypothesis that $\sA$ is cocomplete.

Let $(\cF,\rho) \in \sA_R$. By \cite[p.
108]{Po}, the functor $\Hom_\sA(\cF,-): \sA \to \Mod R$ has a left
adjoint.\footnote{It is essential that $\sA$ be cocomplete
for $-\otimes_R \cF$ to exist. For example, if  $R=\ZZ$ and $\sA$ consists of finitely generated abelian
groups and $\cF=\ZZ$, there is no adjoint. But the hypothesis of cocompleteness is absent from
\cite[p.108]{Po} and parts of  \cite{AZ}.}
  We fix a left adjoint and denote it by $-\otimes_R \cF$.  By
\cite[Proposition B3.1]{AZ}, the functor $-\otimes_R \cF$ is
unique up to isomorphism (of functors) such that
\begin{itemize}
\item{} $R \otimes_R \cF \cong \cF$, and

\item{} $-\otimes_R \cF$ is right exact and commutes with direct
sums.
\end{itemize}

Since the functor $\Hom_\sA(\cF,-)$ is $k$-linear for all $\cF \in
\sA$, Proposition \ref{prop.adj.linear} implies the following:

\begin{cor}
 \label{cor.klinear}
If $(\cF, \rho) \in \sA_R$, then $-\otimes_R \cF$ is $k$-linear.
\end{cor}

\section{The generalization of Watts's Theorem}

\begin{thm} \label{thm.watts}
The functor
$$
\Psi:\sA_R \to \sB_{k}({\sf Mod}R,\sA)
$$
induced by the assignment
$$
\qquad \Psi(\cF) = - \otimes_R \cF,
$$
is an equivalence of categories.
\end{thm}

\subsection{The proof that $\Psi$ is essentially surjective}

\begin{prop}\footnote{After we finished writing this paper we learned that a special case of this result had already
been proved by Brzezinski and Wisbauer \cite[39.3, p.410]{BW} under the hypothesis that the objects
of $\sA$ are abelian groups.}
\label{prop.general.watts} Let $F \in \sB_{k}({\sf Mod}R,\sA)$.
Then $F \cong -\otimes_R \cF$ where $\cF=FR$.
\end{prop}

\begin{pf}
Let $\theta_M : M \to \Hom_\sA(\cF,FM)$ be the composition
$$
\UseComputerModernTips
\xymatrix{
M \ar[r]^>>>>>{\L_M } & \Hom_R(R,M) \ar[r]^F & \Hom_{\sA}(\cF,FM)
}
$$
where $\L_M$ is the canonical isomorphism $m \to \l_m$ where
$\l_m(r):=mr$ for all $r \in R$.

Let
$$
\Theta_M:M \otimes_R \cF \to FM
$$
be the map that corresponds to $\theta_M$ under the adjoint isomorphism
$$
\Hom_R(M,\Hom_\sA(\cF,FM)) \cong \Hom_{\sA}(M \otimes_R \cF, FM).
$$

We will show that the $\Theta_M$s define a natural transformation, i.e., if $f:M \to N$
is a homomorphism of right $R$-modules, then the diagram
\begin{equation}
\label{nat.trans?}
\UseComputerModernTips
\xymatrix{
M   \otimes_R \cF\ar[r]^{f \otimes \cF} \ar[d]_{\Theta_M}& N  \otimes_R \cF \ar[d]^{\Theta_N}
\\
FM \ar[r]_{Ff} & FN
}
\end{equation}
commutes.
Define $\eta: \Hom_{\sA}(\cF,FM) \to  \Hom_{\sA}(\cF,FN)$ by $\eta(g):=Ff \circ g$. The left and right  squares in the diagram
$$
\UseComputerModernTips
\xymatrix{
M \ar[r]^>>>>>{\L_M } \ar[d]_f  & \Hom_R(R,M) \ar[r]^F \ar[d] & \Hom_{\sA}(\cF,FM) \ar[d]^\eta
\\
N \ar[r]_>>>>>{\L_N } & \Hom_R(R,N) \ar[r]_F & \Hom_{\sA}(\cF,FN)
}
$$
commute, so $\eta \circ \theta_M=\theta_N \circ f$.

We now consider the diagram
$$
\UseComputerModernTips
\xymatrix{
\Hom(M,\Hom(\cF,FM)) \ar[d] \ar[rr]^\sim &  &  \Hom(M\otimes \cF,FM)  \ar[d]
\\
\Hom(M,\Hom(\cF,FN))  \ar[rr]^\sim &&  \Hom(M \otimes \cF,FN)&&
\\
 \Hom(N,\Hom(\cF,FN))  \ar[u] \ar[rr]^\sim  &&  \Hom(N \otimes \cF,FN),   \ar[u]
}
$$
whose verticals are induced by $f$ and whose horizontals are the
adjoint isomorphism.  The top and bottom rectangles of this
diagram commute by the functoriality of the adjoint isomorphisms.
The maps $\theta_M$ and $\theta_N$ belong to the top and bottom
Hom-sets of the left-hand column and their images in $
\Hom(M,\Hom(\cF,FN))$ are the same because $\eta \circ \theta_M
=\theta_N \circ f$. It follows that the images of $\Theta_M$ and
$\Theta_N$ in $\Hom(M \otimes \cF,FN)$ are the same. In other
words,
$$
Ff \circ \Theta_M = \Theta_N \circ (f \otimes \cF)
$$
which proves that (\ref{nat.trans?}) commutes and hence that the $\Theta_M$s define a natural transformation
$$
\Theta: - \otimes_R \cF \to F.
$$

Because $(F \circ \L_R)(x)=F(\mu_x) = \rho(x)$, $\theta_R:R \to
\Hom_\sA(\cF,\cF)$ is the map giving $\cF$ its $R$-module
structure, so the corresponding map  $\Theta_R : R \otimes_R \cF
\to \cF$ is an isomorphism. Since the functors $-\otimes_R \cF$
and $F$ commute with direct sums, $\Theta_M$ is an isomorphism for
all free $R$-modules $M$. Since  $-\otimes_R \cF$ and $F$ are
right exact it follows that $\Theta_M$ is an isomorphism whenever
$M$ is the cokernel of a map between free $R$-modules. But every
$R$-module is of that form so $\Theta_M$ is an isomorphism for all
$M$. Hence $\Theta$ is an isomorphism of functors.\footnote{The argument in the last part of the proof is a result of B. Mitchell. See \cite[39.1, p.409]{BW} for more details.}
\end{pf}

Proposition \ref{prop.general.watts} says that the
functor $\Psi$ in Theorem \ref{thm.watts} is essentially
surjective.

\subsection{$R \otimes_R -:\sA_R \to \sA_R$ is isomorphic to the identity functor}

Let $(\cF,\rho) \in \sA_R$ and let $\cN \in \sA$. The composition
\begin{equation}
\label{defn.theta}
\UseComputerModernTips
\xymatrix{
\Hom_{\sA}(R \otimes_R \cF,\cN) \ar[r]^{\sim} &  \Hom_R(R,\Hom_\sA(\cF,\cN)) \ar[r]^\sim
& \Hom_\sA(\cF,\cN),
}
\end{equation}
where the first map is the adjoint isomorphism and the second is
the canonical isomorphism $\psi \mapsto \psi(1)$, induces an
isomorphism of functors $$\Hom_{\sA}(R \otimes_R \cF,-) \to
\Hom_\sA(\cF,-)$$ which, by the Yoneda Lemma, corresponds to a
unique isomorphism
$$
\UseComputerModernTips
\xymatrix{
\xi_\cF: \;   \cF   \ar[r]^\sim  &   R \otimes_R \cF.
}
$$

The next result is a slightly sharper form of \cite[Prop. B3.1(a)]{AZ}.

\begin{prop}
\label{prop.AZ.B3.1} The diagram
\begin{equation}
\label{eq.theta}
\UseComputerModernTips \xymatrix{
 R \otimes_R \cF   \ar[r]^{R \otimes \phi} &  R \otimes_R \cG
\\
\cF \ar[r]_{\phi}  \ar[u]^{\xi_\cF}  & \cG \ar[u]_{\xi_\cG}
}
\end{equation}
commutes for all $\cF,\cG \in \sA_R$ and all $\phi \in
\Hom_R(\cF,\cG)$. Therefore, the maps $\xi_\cF$ provide an
isomorphism
$$
\xi:\id_{\sA_R} \longrightarrow (R  \otimes_R -)
$$
of functors.
\end{prop}
\begin{pf}
By the Yoneda lemma, the commutivity of (\ref{eq.theta}) is
equivalent to the condition that for all $\cN \in \sA$ the outer
rectangle in the diagram
 \begin{equation}
 \label{eq.theta.nat.trans}
\UseComputerModernTips
\xymatrix{
\Hom_{\sA}(R \otimes_R \cF,\cN) \ar[d]_{\cong} &&  \ar[ll]_{-\circ(R \otimes \phi)}  \Hom_{\sA}(R \otimes_R \cG,\cN) \ar[d]^{\cong}
\\
\Hom_R(R,\Hom_\sA(\cF,\cN)) \ar[d]_{\cong} &&  \ar[ll]_{\G} \Hom_R(R,\Hom_\sA(\cG,\cN)) \ar[d]^{\cong}
\\
\Hom_\sA(\cF,\cN) && \ar[ll]^{-\circ \phi} \Hom_\sA(\cG,\cN)
}
\end{equation}
commutes, where the vertical arrows are the factorizations  in (\ref{defn.theta}) that are used
to define $\xi_\cF$ and $\xi_\cG$, and
$$
\G(\psi)(x) :=\psi(x) \circ \phi
$$
for all $x \in R$ and $\psi \in \Hom_R(R,\Hom_\sA(\cG,\cN))$.

The uppermost square of (\ref{eq.theta.nat.trans}) commutes by functoriality of
the adjoint isomorphism.  Going clockwise around the lower square,
the image in $\Hom_\sA(\cF,\cN)$ of  $\psi \in \Hom_R(R,\Hom_\sA(\cG,\cN))$ is $\psi(1) \circ \phi$.
Going counter-clockwise around the lower square,
the image of  $\psi$ in $\Hom_\sA(\cF,\cN)$ is $\Gamma(\psi)(1) = \psi(1) \circ \phi$.
Hence the lower square commutes.

It follows that the outer rectangle commutes.
\end{pf}

\subsection{The proof that $\Psi$ is full}
 Let $\cF$ and $\cG$ be objects in $\sA_R$ and let $\phi \in \Hom_R(\cF,\cG)$.
 Because (\ref{eq.theta}) commutes  $R \otimes \phi$ is non-zero if $\phi$ is non-zero. Hence
$\Psi$ is faithful.

\subsection{The proof that $\Psi$ is faithful}
To complete the proof of Theorem \ref{thm.watts}, it remains to
show that $\Psi$ is full. To that end,  let $$\tau:-\otimes_R \cF
\to -\otimes_R \cG$$ be a natural transformation. We must show
there is a homomorphism $\phi \in \Hom_R(\cF,\cG)$  such that
$\tau_M=M \otimes \phi$ for all $M \in \Mod R$.

Define
$$
\phi:= \xi_\cG^{-1} \circ \tau_R \circ \xi_{\cF}.
$$
It follows from the commutativity of (\ref{eq.theta}) that $R \otimes
\phi=\tau_R$. By Lemma \ref{lem.nat.trans} below, it follows that $M
\otimes \phi=\tau_M$ for all $M \in \Mod R$. In other words,
$\Psi(\phi)=\tau$.

\begin{lem}
\label{lem.nat.trans}
Let  $\sC$ be an abelian category and let $F,G:\Mod R \to \sC$ be right exact functors that commute
with direct sums. Let $\tau,\tau':F \to G$ be natural transformations. If $\tau_R=\tau'_R$, then
$\tau=\tau'$.
\end{lem}
\begin{pf}
Let $M_i$, $i \in I$, be a collection of right $R$-modules. Then there is a natural map
$$
\bigoplus_{i \in I} FM_i \to F \Bigl(\bigoplus_{i \in I} M_i\Bigr)
$$
and the fact that $F$ commutes with direct sums says that this map
is an isomorphism. By the universal property of colimits, there is
a commutative diagram
$$
\UseComputerModernTips
\xymatrix{
\bigoplus_{i \in I} FM_i \ar[r]   \ar[d]_{\oplus \tau_{M_i}} & F \Bigl(\bigoplus_{i \in I} M_i\Bigr) \ar[d]^{\tau_{\oplus M_i}}
\\
\bigoplus_{i \in I} GM_i \ar[r]  & G \Bigl(\bigoplus_{i \in I}
M_i\Bigr). }
$$
Since the horizontal maps are isomorphisms,  if $\tau_{M_i}=\tau'_{M_i}$ for all $i$, then
$$
\tau_{\oplus M_i}=\tau'_{\oplus M_i}.
$$
In particular, it follows that $\tau_P=\tau'_P$ for all free $R$-modules $P$.

Let $M$ be a right $R$-module and let $P \to Q \to M \to 0$ be an exact sequence in which $P$ and
$Q$ are free $R$-modules.  Then there is a commutative diagram
$$
\UseComputerModernTips
\xymatrix{
FP \ar[d]_{\tau_P} \ar[r] & FQ  \ar[d]_{\tau_Q} \ar[r] & FM   \ar[r] & 0
\\
GP   \ar[r] & GQ   \ar[r] & GM    \ar[r] & 0,
}
$$
 and a unique map $FM \to GM$ making the diagram commute, namely $\tau_M$.
 Since $\tau_P=\tau'_P$ and $\tau_Q=\tau'_Q$, it follows that $\tau_M=\tau'_M$.
\end{pf}

\section{An application}
Throughout this section, let $X$ denote a $k$-scheme.  If
$X=\operatorname{Spec }R$, we let
$$
\widetilde{(-)}:{\sf Mod }R \to {\sf Qcoh }X
$$
be the quasi-inverse to the global sections functor defined in
\cite[II, Definition, p. 110]{Hart}.

\begin{example}
Let $f:Y \to X$ be a morphism from an arbitrary scheme to an
affine scheme $X = \Spec R$.  Then $f^* \circ \widetilde{(-)}:\Mod
R \to \Qcoh Y$ is a right exact functor commuting with direct
sums. Proposition \ref{prop.general.watts} says that $f^* \circ
\widetilde{(-)} \cong - \otimes_R \cO_Y$ where $\cO_Y$ is made
into an $R$-module via the ring homomorphism
$$
R \to \Hom_R(R,R) \to
\Hom_Y(f^*\cO_X,f^*\cO_X) \to \Hom_Y(\cO_Y,\cO_Y)
$$
where the first map sends $r \in R$ to multiplication by $r$, the
second map is induced by $f^{*}\circ \widetilde{(-)}$ and the
third isomorphism is induced by the natural isomorphism
$f^{*}\cO_X \cong O_Y$.
\end{example}

The motivation for this present paper lies in our paper \cite{NS}.
There we consider $k$-schemes $X$ and $Y$ and $k$-linear
functors  $F:\Qcoh X \to \Qcoh Y$ that are right exact and commute with direct sums.
One source of such functors is the following. Let $\cF$ be a quasi-coherent sheaf on $X \times_k Y$, and
define
\begin{equation}
-\otimes_{\mathcal{O}_{X}}\cF :=
\operatorname{pr}_{2*}(\operatorname{pr}_{1}^{*}(-)
\otimes_{\mathcal{O}_{X \times_{k} Y}} \cF)
\end{equation}
where $\operatorname{pr}_{i}:X \times_{k} Y \to X,Y$, $i=1,2$, are
the obvious projections.

Two warnings are necessary. First, a functor of the form
$-\otimes_{\mathcal{O}_{X}}\cF$ is not always in
$\sB_{k}({\sf Qcoh }X,{\sf Qcoh }Y)$.  This happens, for example,
if $Y=\Spec k$, $X=\PP^1_k$ and
$\cF=\mathcal{O}_{X \times Y} \cong
\Gamma(X,-)$.

Second, an object in $\sB_{k}({\sf Qcoh }X,{\sf Qcoh
}Y)$ is not always isomorphic to one of the form
$-\otimes_{\mathcal{O}_{X}}\cF$.  For example, when
$Y=\Spec k$ and $X=\PP^1_k$, the functor $H^1(X,-)$ is not of this form
\cite[Proposition  5.4]{NS}.

The question motivating \cite{NS} is whether $F$ is isomorphic to
a functor of the form $-\otimes_{\mathcal{O}_{X}}\cF$.  It follows
from Theorem \ref{thm.watts} that this is always the case if $X$
is affine, as we now show.

\begin{prop}
 \label{prop.scheme}
 Let $R$ be a commutative $k$-algebra and  $Y$ a $k$-scheme.
 Write $X := \operatorname{Spec }R$.
Then the inclusion functor
$$
{\sf Qcoh }(X \times_{k} Y ) \to \sB_{k}({\sf Qcoh }X,{\sf Qcoh }Y),
\qquad
\cF \mapsto
-\otimes_{\mathcal{O}_{X}}\cF,
$$
is an equivalence of  categories.
\end{prop}
\begin{pf}
By \cite[II, exercise 5.17e]{Hart}, the functor
$$
\operatorname{pr}_{2*}:{\sf Qcoh }(X \times_{k} Y )\to {\sf Qcoh
}(\operatorname{pr}_{2*}\mathcal{O}_{X \times_{k}Y})
$$
is an equivalence, where ${\sf Qcoh
}(\operatorname{pr}_{2*}\mathcal{O}_{X \times_{k}Y})$ denotes the
category of quasi-coherent $\mathcal{O}_{Y}$-modules with
$\operatorname{pr}_{2*}\mathcal{O}_{X \times_{k}Y}$-module
structure.  Furthermore, it is straightforward to check that the
functor
$$
{\sf Qcoh }(\operatorname{pr}_{2*}\mathcal{O}_{X \times_{k}Y}) \to
({\sf Qcoh }Y)_R
$$
induced by the assignment $\cE \mapsto (\cE,\rho)$, where $\rho:R
\to \Hom_Y(\cE,\cE)$ is defined through the
$\operatorname{pr}_{2*}\mathcal{O}_{X \times_{k}Y}$-structure of
$\cE$, is an equivalence.  By Theorem \ref{thm.watts}, the functor
$$
({\sf Qcoh }Y)_R \to {\sf B}_{k}(\Mod R,{\sf Qcoh }Y)
$$
induced by the assignment $(\cE,\rho) \mapsto -\otimes_R \cE$ is
an equivalence.  Therefore, the functor
$$
{\sf Qcoh }(X \times_{k} Y) \to {\sf B}_{k}(\Mod R,{\sf Qcoh }Y)
$$
induced by the assignment $\cF \mapsto -\otimes_R
\operatorname{pr}_{2*}\cF$ is an equivalence.  By the uniqueness
properties of the functor $-\otimes_R \cE$ described in Section
\ref{subsection.tensor}, we have an isomorphism of functors
$$
\UseComputerModernTips
\xymatrix{
-\otimes_R  \operatorname{pr}_{2*}\cF
\ar[rr]^{\sim}    &&
\widetilde{(-)} \otimes_{\mathcal{O}_{X}}\cF
}
$$
in ${\sf B}_{k}(R,{\sf Qcoh }Y)$.  It follows that the functor
$$
{\sf Qcoh }(X \times_{k} Y) \to {\sf B}_{k}(\Mod R,{\sf Qcoh }Y)
$$
induced by the assignment $\cF \mapsto \widetilde{(-)}
\otimes_{\mathcal{O}_{X}}\cF$ is an equivalence.  The claim
follows easily from this.
\end{pf}


TITLE: In Newtonian pressure, what type of function is force?
QUESTION [0 upvotes]: This is pressure in Newtonian mechanics:
$$P=\frac {dF}{dA}.$$
What does this mean?
(Doesn't it mean that force is a function of area?)
What type of function is force?

REPLY [6 votes]: The ratio is meant to denote
$$ p = \lim_{\Delta A\to 0} \frac{\Delta F}{\Delta A}$$
where $\Delta A$ is the area of a particular piece of the surface whose magnitude we send to zero; and $\Delta F$ is the correspondingly small force that acts on this small area. In practice, it's enough to choose $\Delta A$ small enough so that the pressure is constant (the force is linear in the area) over that small area within the error margin. In principle, we want $\Delta A$ to be really infinitely small.
While the formula above still looks pretty much identical to the definition of a derivative, the ratio is not a derivative in any useful sense because $F$ isn't a natural function of area $A$. We're dealing with one particular physical situation in which the total area $A$ is fixed and it is not a variable at all.
But in fact, we may describe the situation in such a way that it becomes possible to interpret $dF/dA$ as a derivative. Just divide the total area $A$ to (almost) infinitely many (nearly) infinitesimal areas $\Delta A$. Now, the key unnatural point is to order these pieces of area $\Delta A$; choose an order or schedule how these areas are added one by one, starting from $0$ and finally reaching the total area $A$. For example, you divide the rectangular area $A$ to a grid and add the small squares one by one, a row after a row.
If $A$ is the symbol for the total area of the squares in the subset that have already been added and $F$ represents the force that acts on this subset of the (original, whole) $A$ composed of the areas $\Delta A$ that were already added, then $F=F(A)$ is a function of $A$ and $dF/dA$ evaluated at the value of $A$ corresponding to the point at which we added a particular $\Delta A$ (or $dA$, if you wish) is precisely the pressure at the same point of space (or point of the surface) where this $\Delta A$ is located.