text:
\section{Conclusions}
In this chapter, we find that the inhomogeneous LWR model \refe{system} exhibits nonlinear resonance. For this system, we discuss the Riemann problem and find 10 types of wave solutions and the formula for the boundary flux related to each type of wave solutions. Then we obtain a general formula for the boundary flux with the definition of upstream demand and downstream supply. We also conclude that the results obtained here are consistent with those in existing literature. Since the inhomogeneous LWR model can be written in a conservation form \refe{system}, Godunov's method is efficient for its numerical solutions. Godunov's method for the inhomogeneous LWR model is briefly described in section \ref{go}.

In this chapter, we consider the inhomogeneity as variable number of lanes. Similarly we can solve the inhomogeneous LWR model with other inhomogeneities, by changing the functions $v(a,x)$ and $f(a,x)$ corresponding to the new inhomogeneities. For all kinds of inhomogeneity, equation \refe{ds} is always the general formula for computing the boundary flux.

In this chapter, we consider an inhomogeneous LWR model, which is a first-order traffic flow model. However, we want to point out that the method provided here  can be extended to solve  inhomogeneous higher-order traffic flow models such as Zhang's model discussed in Chapter 3 and the PW model discussed in Chapter 4.

config: arxiv


TITLE: Rotation in the x-t plane
QUESTION [4 upvotes]: I am currently studying special relativity using tensors. My lecture notes (which happen to be publicly accessible, see top of page 99) say that the standard configuration can be viewed as a rotation in the x-t plane. Can anyone explain this a bit? Is there a good way to visualize it?
The Standard configuration;

I would prefer an intuitive (not rigorous) explanation over a maths one, but if you have a maths one then for completeness it would be good to post it.
All other notes from these lectures are available here.

REPLY [0 votes]: ... Another way to look at it is to use geometry which includes the vertical time axis, the horizontal space axis, and the stacking of both motion vectors( c, v ) and length scalars. Rotation determines the direction of travel in space-time, thus determines the velocity across space and the rate of the ticking of time.
Here, twin spaceship A is a rest in space, but is in motion across time at c, and twin spaceship B is moving at a velocity of 260,000 km/s. From this simple geometry we determine the Length Contraction equation, the Time Dilation equation, and we can also quickly derive the Velocity Addition equation, along with the Lorentz Transformation equations, all in a matter of minutes.


text:
\begin{document}

\title[Algebraic exponentiation for Lie algebras]{Algebraic exponentiation for Lie algebras}

	\author{Xabier García-Martínez}
\address[Xabier García-Martínez]{Departamento de Matemáticas, Esc.\ Sup.\ de Enx.\ Informática, Campus de Ourense, Universidade de Vigo, E--32004, Ourense, Spain
	\newline and\newline
	Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B--1050 Brussel, Belgium}
\email{xabier.garcia.martinez@uvigo.gal}

\author{James R. A. Gray}
\address[James Gray]{Mathematics Division, Department of Mathematical Sciences Stellenbosch University, Private Bag X1,7602 Matieland, South Africa}
\email{jamesgray@sun.ac.za}

\thanks{The first author was supported by Ministerio de Economía y Competitividad (Spain), with grant number MTM2016-79661-P and is a Postdoctoral Fellow of the Research Foundation–Flanders (FWO).
The second author was supported in part by the National Research Foundation of South Africa (Grant Number 109203)}

\begin{abstract}
It is known that the category of Lie algebras over a ring admits \emph{algebraic exponents}.
The aim of this paper is to show that the same is true for the category of internal Lie algebras in an
additive, cocomplete, symmetric, closed, monoidal category.
In this way, we add some new examples to the brief list of known locally algebraically cartesian closed categories,
including the categories of Lie superalgebras and differentially graded Lie algebras amongst others.
\end{abstract}

\subjclass[2010]{18E99, 18A40, 18D15, 18D10, 17B99}
\keywords{locally algebraically cartesian closed, semi-abelian category; algebraic exponentiation}

\maketitle
\section{Introduction}
Let $\C$ be a finitely complete category. Given an object $B$ of~$\C$, we write $\Pt_{B}(\C)$ for the \emph{category of points over $B$} in which an object $(A, p, s)$ is a split epimorphism~$p\colon {A\to B}$ in $\C$, together with a chosen section $s\colon {B\to A}$, so that $p\circ s=1_{B}$.
A morphism of $\Pt_B(\C)$ from $(A,p, s)$ to $(A', p', s')$ is a morphism $f \colon A \to A'$ in $\C$ such that $p' \circ f = p$ and $f \circ s = s'$. For each morphism $q \colon E \to B$, there is a \emph{change-of-base} functor $q^{*} \colon \Pt_B(\C) \to \Pt_E(\C)$, that takes a point $(A, p, s)$ to the point $(E\times_{B} A, \pi_1,\langle \id,sq\rangle)$ obtained from the pullback
\[
\xymatrix{E\times_{B}A \ar[r]^-{\pi_2} \ar@<0ex>[d]^-{\pi_1}  \pullback & A \ar@<.5ex>[d]^-{p}  \\
	E \ar[r]_-{q} \ar@{..>}@<1ex>[u]^-{\langle \id, sp\rangle} & B \ar@{..>}@<.5ex>[u]^-{s}}
\]

If for each morphism $q$ in $\C$, the change-of-base functor $q^{*}$ reflects isomorphisms, the category is said to be \emph{Bourn-protomodular} \cite{Bourn1991}. If $\C$ is pointed, this is equivalent to the same condition restricted to morphisms whose domain is the zero object. Note that the composition of the change-of-base functor $\gnab_B^{*} \colon \Pt_B(\C) \to \Pt_0(\C)$, where $\gnab_B \colon 0\to B$ is the unique morphism from the zero object to $B$, with the isomorphism $\Pt_0(\C) \cong \C$ is the kernel functor $\Ker_B$ sending each point $(A,p,s)$ to the kernel of $p$. Therefore, a finitely complete pointed category is Bourn-protomodular if and only if for each $B$ in $\C$ the kernel functor reflects isomorphisms, or equivalently, the split short five lemma holds. Recall that a pointed protomodular category which is Barr-exact and has finite coproducts is called a \emph{semi-abelian} category \cite{Janelidze-Marki-Tholen}.

If the change-of-base functor $q^{*}$ has a right adjoint for each morphism $q$, then~$\C$ is said to be \emph{locally algebraically cartesian closed} (\LACC{} for short) \cite{Bourn-Gray, GrayPhD, Gray2012}.
If~$\C$ is also protomodular and pointed, then by \cite[Theorem 5.1]{Gray2012} it is sufficient to check if for each $B$ in $\C$ the kernel functor $\Ker_B \colon \Pt_B(\C) \to \C$ has a right adjoint.

If~$\C$ is a \LACC, semi-abelian category, it implies several categorical-algebraic properties are satisfied by $\C$, such as \emph{peri-abelianness}~\cite{Bourn-Peri}, \emph{strong protomodularity}~\cite{B1}, the \emph{Smith is Huq} condition~\cite{MFVdL}, \emph{normality of Higgins commutators}~\cite{CGrayVdL1}, and \emph{algebraic coherence}.

As we can see, \LACC{} is a strong condition where the short list of known semi-abelian examples includes groups, Lie algebras, crossed modules, cocommutative Hopf algebras over a field and all abelian categories.
In fact, it was shown in~\cite{GM-Vdl-2} that \LACC{} characterizes Lie algebras amongst all varieties of non-associative algebras over an infinite field of characteristic not equal to two. Furthermore, for an infinite field of characteristic two there are exactly two non-abelian \LACC{} sub-varieties: Lie algebras and quasi-Lie algebras; that is the variety of algebras obtained from Lie algebras by replacing the identity $xx=0$ by $xy=-yx$.

It is not known whether there is any relation between \LACC\ and \emph{action representability}~\cite{BJK2}. They coincide in non-associative algebras over an infinite field (they both characterize Lie algebras~\cite{GTVV}), but it is not longer true over finite fields, since Boolean algebras are action representable but not \LACC~\cite{BJK2, Gray2012}.

The aim of the present paper is to show that, under certain conditions, the category of Lie algebras in a monoidal category is \LACC. In doing so, we will add the categories of Lie superalgebras, Lie colour algebras and differentially graded Lie algebras, amongst others, to the brief list of known \LACC{} examples. Another interesting example, is the category of Lie algebras in the Loday-Pirashvili category~\cite{Loday-Pirashvili}. In this way, we will show that the category of Leibniz algebras (which is not \LACC~\cite{GM-Vdl}) is a full reflective subcategory of a \LACC{} category.

Let us briefly explain our approach, which is motivated by \cite{GrayLie} where it is shown that the category of Lie algebras over a commutative ring is \LACC. Let $\Cb$ be a symmetric monoidal closed category with underlying category additive and small complete.
We show that for an object $B$ in $\LieC$, the category of internal Lie algebras in $\Cb$, the category $\Pt_B(\LieC)$ is equivalent to the category $\Lie(\Db)$ of internal Lie algebras in the symmetric monoidal category $\Db$ with underlying category the category of \emph{internal monoid actions} of $U(B)$, the \emph{universal enveloping monoid} of~$B$, acting on the objects of $\Cb$, and with monoidal structure induced by $\Cb$ together with a natural comonoid structure on~$U(B)$. Moreover, we show that the kernel functor $\Ker_B \colon \Pt_B(\LieC)\to \LieC$ factors, via this isomorphism, through a forgetful functor $V\colon\Lie(\Db) \to \LieC$. This reduces the problem of finding a right adjoint of $\Ker_B$ to finding a right adjoint of $V$, which can be found by an adjoint lifting theorem.

The paper is organized as follows: In Section~\ref{algebras} after recalling the definition of various relevant types of internal algebras we construct adjoint functors between certain categories of such internal algebras, which we will use in Section~\ref{actions}. In Section~\ref{monoidal functors} we recall the necessary background and then prove the adjoint lifting theorem which we use to produce the right adjoint of the functor $V$ mentioned above. The adjoint functors previously defined are used in Section~\ref{actions} to produce the symmetric monoidal category $\Db$ as well as the equivalence of categories between~$\Pt_B(\LieC)$ and  $\Lie(\Db)$ mentioned before. Finally, in Section~\ref{examples} we study some applications of the obtained result.

\section{Internal algebras}\label{algebras}

Throughout the rest of paper we denote by $\Cb = (\C, \t, I, \az, \lz, \rz, \sz)$ a \emph{symmetric monoidal category}. We refer to \cite{MacLane} for basic knowledge about this topic.
Recall that~$\Cb$ is said to be \emph{closed} if for each object $X$ in $\C$ the endofunctor $X \t - \colon \C\to \C$ is a left adjoint. A chosen right adjoint will be denoted by $(-)^{X}$ and the counit by~$\ev$.
As the main non-cartesian example we have the symmetric monoidal category $\Vect_{\K}$ of vector spaces with the tensor product $\t_{\K}$.
The canonical isomorphism between $\Hom_{\Vect_{\K}}(X \t_{\K} Z, Y)$ and $\Hom_{\Vect_{\K}}(Z, \Hom_{\Vect_{\K}}(X, Y))$ gives us the right adjoint~$(-)^{X} = \Hom_{\Vect_{\K}}(X, -)$. In this particular case, the counit of the adjunction $\ev_Y \colon Y^{X} \t X \to Y$ maps $f \t x$ to the evaluation $f(x)$.

The setting we need in this manuscript is the following: $\Cb$ is a symmetric monoidal closed category with underlying category $\C$ additive and cocomplete.

Recall that a \emph{monoid} $(A, m, u)$ in $\Cb$ is a triple, where $A$ is an object of $\C$, $m \colon A \t A \to A$ and $u \colon I \to A$ are morphisms of $\C$, such that the diagrams
\[
\xymatrix{
A \t (A \t A) \ar[rr]^-{\az} \ar[d]_-{\id \t m} &&  (A \t A) \t A \ar[d]^-{m\t \id}   & A \t I \ar[r]^-{\id \t u} \ar[rd]_-{\rz} & A \t A \ar[d]^-{m} & I \t A \ar[l]_-{u \t \id} \ar[ld]^-{\lz}  \\
A \t A \ar[r]^-{m} & A & A\t A \ar[l]_-{m} & &A &
}
\]
commute \cite{MacLane}.
If in addition the diagram
\[
\xymatrix{
A \t A \ar[rr]^-{\sz} \ar[rd]_-{m} & & A \t A \ar[dl]^-{m} \\
& A &
}
\]
commutes, then it is said to be a \emph{commutative monoid}. The categories of such objects will be denoted by $\MonC$ and $\cMonC$, respectively. Recall that since~$\Cb$ is a symmetric monoidal category, given two monoids $(A,m,u)$ and $(A',m',u')$ in $\Cb$ the triple $(A\t A', (m\t m') i, (u\t u')\lz)$, where $i$ is the \emph{middle interchange isomorphism}, is a monoid which we denote by $(A,m,u)\t (A',m',u')$ (which is commutative whenever both $(A,m,u)$ and $(A',m',u')$ are). Moreover, with this definition of tensor product the symmetric monoidal structure of $\Cb$ lifts to both~$\MonC$ and~$\cMonC$. Let us write $\Monu{\Cb}$ and $\cMonu{\Cb}$ for the respective symmetric monoidal categories.
The duals of the notions of monoid and commutative monoid are \emph{comonoid} and \emph{cocommutative comonoid}, respectively, and the categories of such objects are denoted $\CoMonC$ and $\cCoMonC$, respectively. By duality the symmetric monoidal structure on $\Cb$ lifts to both $\CoMonC$ and $\cCoMonC$, producing the symmetric monoidal categories $\CoMonu{\Cb}$ and $\cCoMonu{\Cb}$, respectively.

A \emph{bimonoid} can be defined as a comonoid in the symmetric monoidal category~$\MonC$ (or equivalently as a monoid in~$\CoMonC$). We present a bimonoid as a quintuple $(A, m, u, d, e)$ where $(A, m, u)$ is an monoid, $(A, d, e)$ is a comonoid and $d\colon (A,m,e) \to (A,m,u)\t (A,m,u)$ and~$e\colon (A,m,u) \to (I,\lz,\id)$ are monoid morphisms, or equivalently $m$ and $u$ are comonoid morphisms. The category of bimonoids will be denoted by $\BiMonC$.
A bimonoid is called \emph{commutative} if the monoid part is commutative, and it is called \emph{cocommutative} if the comonoid part is cocommutative.

Under the conditions assumed on $\Cb$ it is well known that free monoids exist, in fact much milder assumptions are needed (see e.g.~\cite{Lack:free_monoids} and the references therein).
\begin{proposition}\label{prop:free_monoid}
The forgetful functor $G\colon\MonC \to \C$ has a left adjoint which we will denote by $F$.
\end{proposition}

A \emph{Lie algebra} in $\Cb$ is a pair $(X, b)$ where $X$ is an object and  $b \colon X \t X \to X$ is a morphism in $\C$  making the diagrams
\begin{equation}
\label{dia:lie}
\vcenter{
\xymatrix{
X\t X \ar[d]_-{0} \ar[r]^-{\id + \sz} & X\t X \ar[dl]^-{b} & X\t (X\t X) \ar[rr]^-{\id + \sz\az + (\sz\az)^2} \ar[d]_-{0} && X\t (X\t X)\ar[d]^-{\id \t b}\\
X  & & X && X\t X\ar[ll]^-{b}
}
}
\end{equation}
commute.
Note that we can sum morphisms since we are assuming that $\C$ is additive. The category of Lie algebras in $\Cb$ will be denoted by $\LieC$.
As expected, when $\Cb$ is the category of vector spaces (with the usual monoidal structure) Lie algebras in $\Cb$ are usual Lie algebras.

Throughout this paper we will denote the direct sum of $A$ and $B$ by $A\oplus B$ with projections $\pi_1\colon A\oplus B\to A$ and $\pi_2\colon A\oplus B\to B$ and inclusions $\iota_1\colon A\to A\oplus B$ and $\iota_2\colon B\to B\oplus A$. For morphisms $f\colon W\to A$, $g\colon W\to B$ and $h\colon A\to C$ and~$i\colon B\to C$ we will denote by $\langle f,g \rangle \colon W\to A\oplus B$ and $[h,i]\colon A\oplus B\to C$ the unique morphisms with $\pi_1 \langle f,g\rangle =f$, $\pi_2 \langle f,g\rangle =g$, $[h,i]\iota_1=h$ and $[h,i]\iota_2=i$.

Given a monoid $(A,m,u)$ in $\Cb$, the pair $(A,m(\id-\sigma))$ is an object in $\LieC$ and this assignment determines the object map of a functor $L \colon \MonC \to \LieC$ which is the identity on morphisms (see e.g.~\cite{GrayLie}).
The following proposition is probably known, but we couldn't find a reference so we included a proof.
\begin{proposition}
The functor $L \colon \MonC \to \LieC$ has a left adjoint, which we denote by $U$.
\end{proposition}
\begin{proof}
Let us write $\eta$ for the unit of the adjunction $F \dashv G$ from Proposition~\ref{prop:free_monoid}.
Let $(B,b)$ be an object in $\LieC$ and $(A,m,u)$ an object in $\MonC$. For a morphism $f\colon B\to A$ in $\C$  let $\bar f$ be the corresponding monoid morphism from $F(B)=(T(B),m_B,u_B)$ to $(A,m,u)$ obtained via the adjunction $F \dashv G$, that is the unique monoid morphism such that $G(\bar f)\eta =f$. By considering the diagram
\[
\xymatrix{
B^{\t 2}\ar[rr]^-{b} \ar[rd]^-{f ^{\t 2}} \ar[d]_-{\id - \sz} \ar@{}[ddrrr]|(0.55)*+[F]{1} && B\ar[dr]^-{\eta} \ar[dd]_-{f}&\\
B^{\t 2} \ar[rd]^-{f^{\t 2}}\ar[dd]_-{\eta \t \eta} &A^{\t 2}\ar[d]^-{\id -\sz} && T(B)\ar[dl]^-{\bar f}\\
 &A^{\t 2}\ar[r]^-{m} & A &\\
T(B)^{\t 2} \ar[ru]^-{\bar f ^{\t 2}} \ar[r]_-{m_B} &T(B)\ar[ru]_-{\bar f}&
}
\]
ones sees that the sub-diagram $\xybox{(0,0.2)*+[F]{1}}$ commutes if and only if the outer arrows commute.  This implies that $f$ is a morphism from $(B,b)$ to $L(A,m,u)$ in $\LieC$ if and only if the diagram
\[
\xymatrix@C=10ex{
B^{\t 2} \ar@<0.5ex>[r]^-{\eta b}\ar@<-0.5ex>[r]_-{m_B(\eta\t \eta)(\id-\sz)} & T(B) \ar[r]^-{\bar f} & A
}
\]
is a fork in $\C$. However, the previous diagram is a fork if and only if the diagram
\[
\xymatrix@C=10ex{
B\oplus B^{\t 2} \ar@<0.5ex>[r]^-{[\eta,\eta b]}\ar@<-0.5ex>[r]_-{[\eta,m_B(\eta\t \eta)(\id-\sz)]} & T(B) \ar[r]^-{\bar f} & A
}
\]
is a fork.
Therefore, writing $r, s\colon F(B\oplus B^{\t2})\to F(B)$ for the corresponding monoid morphisms induced by $[\eta,\eta b]$ and $[\eta,m_B(\eta\t \eta)(\id -\sz)]$, respectively, we see that previous diagram is a fork if and only if the diagram
\[
\xymatrix@C=10ex{
F(B\oplus B^{\t2})\ar@<0.5ex>[r]^-{r}\ar@<-0.5ex>[r]_-{s} & F(B) \ar[r]^-{\bar f} & (A,m,u)
}
\]
is a fork in $\MonC$. Since the tensor $\t$ preserves coequalizers in each argument, it is well-known that it preserves reflexive coequalizers, and the forgetful functor~${\MonC\to \C}$ creates reflexive coequalizers. Noting that the parallel pair of morphisms in the previous diagram is reflexive it easily follows that if~$q\colon F(B)\to U(B,b)$ is the coequalizer of $r$ and $s$ in $\MonC$, then $U(B,b)$ together with the morphism $q\eta$ from $(B,b)$ to $L(U(B,b))$ in $\LieC$ is the universal morphism from $(B,b)$ to $L$.
\end{proof}
Next we show that for any $B$ in $\LieC$ the monoid $U(B)$ is the underlying monoid of a cocommutative bimonoid (in fact of a cocommutative Hopf monoid)~$\widetilde U(B)$, and the assignment $B \mapsto \widetilde U(B)$ is the object map of a functor which is left adjoint of a functor $P$ which we construct below. This functor $P$ generalizes the construction of the primitive Lie algebra of a bialgebra.

Recall, as mentioned above, that since $\Cb$ is symmetric monoidal the monoidal structure lifts to $\MonC$.
\begin{proposition}
For each monoid $A=(A,m,u)$ the morphism $\delta_A \colon A\to A\t A$ defined by $\delta_A = (u\t \id)\lz^{-1} + (\id\t u)\rz^{-1}$ is a morphism from $L(A)$ to $L(A\t A)$ in~$\LieC$, natural in $A$.
\end{proposition}
\begin{proof}
Let $A=(A,m,u)$ be a monoid in $\Cb$ and let $\delta_A \colon A\to A\t A$ be the morphism~$\delta_A= l_A + r_A$ where $l_A=(u\t \id) \lz^{-1}$ and $r_A=(\id \t u)\rz^{-1}$. The naturality of~$\delta$ easily follows from the naturality of $l$ and $r$.
Therefore, it only remains to show that $\delta_A$ is a morphism from $L(A)$ to $L(A\t A)$. We have
\begin{align*}
(m\t m) i (l\t r) &= (m\t m) i ((u\t \id)\t (\id \t u) (\lz^{-1} \t \rz^{-1}) \\
&= (m\t m) ((u\t \id) \t (\id \t u)) i (\lz^{-1}\t \rz^{-1})\\
& = (\lz \t \rz) i (\lz^{-1}\t \rz^{-1})\\
&= \sz
\end{align*}
and
\begin{align*}
(m\t m) i\sz (l\t r) &= (m\t m) i\sz ((u\t \id)\t (\id \t u) (\lz^{-1} \t \rz^{-1}) \\
&= (m\t m) ((\id \t u) \t (u \t \id)) i\sz (\lz^{-1}\t \rz^{-1})\\
& = (\rz \t \lz) i\sz (\lz^{-1}\t \rz^{-1})\\
&= \sz
\end{align*}
and hence
\[
(m\t m) i (\id -\sz) (l \t r)  =0
\]
and
\[
(m \t m) i (\id -\sz)(r\t l) =(m \t m) i (\sz -\id)\sz(r\t l)=-(m \t m) i (\id -\sz)(l\t r)\sz=0.
\]
On the other hand it is easy to check that $l_A$ and $r_A$ are monoid morphisms from $A$ to $A\t A$ and hence they are morphisms from $L(A)$ to $L(A\t A)$ in $\LieC$. Therefore
\begin{align*}
(m\t m) i (\id-\sz) (\delta_A\t \delta_A) &= (m\t m) i (\id-\sz) (l_A\t l_A)+(m\t m) i (\id-\sz)(l_A\t r_A)\\
&\hspace{-0.03cm}+ (m\t m) i (\id-\sz)(r_A\t l_A) + (m\t m) i (\id-\sz)(r_A\t r_A)\\
&= l_Am(\id-\sz)+0+0+r_Am(\id-\sz)\\
&=\delta_A m(\id-\sz)
\end{align*}
as desired.
\end{proof}
Recall that a bimonoid $(A,m,u,d,e)$ is called a \emph{Hopf monoid} if there is a morphism $s\colon A\to A$ making the diagram
\[
\xymatrix@C=3ex@R=4ex{
&&A\t A \ar[rr]^-{\id \t s} && A\t A\ar[drr]^-{m}\\
A \ar[drr]_-{d}\ar[urr]^-{d}\ar[rrr]^-{e} &&& I \ar[rrr]^-{u} &&& A \\
&&A\t A \ar[rr]^-{s \t \id} && A\t A\ar[urr]_-{m}
}
\]
commute. Recall also that such a morphism $s$ is unique (whenever it exists) and is called the \emph{antipode} of the Hopf monoid $(A,m,u,d,e)$.
Note that Goyvaerts and Vercruysse have studied a generalization of the functor $P$, below,  in \cite{Goyvaerts_Vercruysse}.
\begin{proposition}
Let $H$ be the forgetful functor from $\BiMonC\to \MonC$.
For a  bimonoid $A$, with comultiplication $d_A\colon A\to A\t A$, the morphism denoted by~${j_A\colon P(A) \to L(H(A))}$ forming part of the equalizer diagram
\[
\xymatrix{
P(A) \ar[r]^-{j_A} & L(H(A)) \ar@<0.5ex>[r]^-{\delta_{H(A)}}\ar@<-0.5ex>[r]_-{L(d_A)} & L(H(A)\t H(A)),
}
\]
is the component of a natural transformation from a functor~$P\colon\BiMonC \to \LieC$ to the functor $LH$. Moreover, the functor $P$ is part of an adjunction with left adjoint $\widetilde{U}$ and unit $\widetilde{\nu}$ such that: $H\widetilde U=U$ and $(j\circ \widetilde U) \widetilde \nu=\nu$ where $\nu$ is the unit of the adjunction $U \dashv L$. Furthermore, for each object $X$ in $\LieC$ the bimonoid~$\widetilde U(X)$ is a cocommutative Hopf monoid.
\end{proposition}
\begin{proof}
It is easy to see that the definition of $P$, on objects above, is extended uniquely, making it into a functor in such a way that $j$ is a natural transformation~$P\to LH$.
Now let $\nu$ be the unit of the adjunction $U \dashv L$.
For an object $X$ in~$\LieC$ by adjunction the morphism $\delta_{U(X)}\nu_X \colon X \to L(U(X)\t U(X))$ determines a monoid morphism $d \colon U(X)\to U(X)\t U(X)$ such that $L(d) \nu_X = \delta_{U(X)} \nu_X$. Note that this means that $d\nu_X = \delta_X \nu_X = (u_{U(X)}\t \nu_X) \lz^{-1} + (\nu_X \t u_{U(X)})\rz^{-1}$ where~$u_{U(X)}$ is the unit of $U(X)$. In what follows we will drop subscripts when there is little risk of confusion. To show that $d$ is coassociative it is sufficient to show that $\az (\id\t d) d \nu = (d \t \id)d \nu$.
A straightforward calculation shows that:
\begin{align*}
\az(\id \t d) d \nu
&= ((u\t u) \t \nu)\az (\id \t \lz^{-1}) \lz^{-1} + ((u \t \nu) \t u)\az(\id \t \rz^{-1})\lz^{-1}\\
&\ \  + ((\nu \t u)\t u)\az(\id \t \lz^{-1})\rz^{-1}\\
\intertext{and}
(d \t \id) d \nu
&= ((u\t u) \t \nu) (\lz^{-1}\t \id) \lz^{-1} + ((u \t \nu) \t u)(\lz^{-1}\t \id)\lz^{-1}\\
&\ \  + ((\nu \t u)\t u)(\rz^{-1}\t \id)\lz^{-1},
\end{align*}
and hence by coherence $\az (\id d) d \nu = (d \t \id)d \nu$. Trivially $d$ is cocommutative. Let~$e\colon U(X)\to I$ be the unique monoid morphism such that $L(e) \nu =0$. Since
\begin{align*}
(e \t \id) d \nu & = (e \t \id)((u\t \nu)\lz^{-1} + (\nu \t u)\rz^{-1})\\
&= (\id \t  \nu) \lz^{-1} + (0\t u)\rz^{-1}\\
&= \lz^{-1} \nu
\end{align*}
it follows that $e$ is the counit of $U(X)$. Let us write $\widetilde U(X)$ for the bimonoid with underlying monoid $U(X)$ and comonoid structure as described above. Letting~$s\colon U(X)\to U(X)$ be the unique monoid morphism such that $L(s)\nu = -\nu$ we see that
\begin{align*}
m(s\t \id) d \nu &= m(s\t \id) \delta \nu \\
&= m(s\t \id)((u\t \nu)\lz^{-1} + (\nu\t u)\rz^{-1})\\
&= m((u\t \nu)\lz^{-1} - (\nu\t u) \rz^{-1})\\
&= m((u\t \id)\lz^{-1}\nu) - m((\id \t u) \rz^{-1}\nu)\\
&= \nu - \nu\\
&=0\\
&=e \nu
\end{align*}
and hence $\widetilde U(X)$ is a cocommutative Hopf monoid with antipode $s$. Supposing that~$A$ is a bimonoid we will show that the natural isomorphism $\hom(X,LH(A)) \cong \hom(U(X),H(A))$ induces a natural isomorphism $\hom(X,P(A))\cong \hom(\widetilde U(X),A)$. For a monoid morphism $f\colon U(X)\to H(A)$ since the outer arrows of diagram
\[
\xymatrix{
X \ar[r]^-{\nu}\ar[d]_-{\nu} & L(U(X) \ar[r]^-{L(f)}\ar[d]_-{L(d_{\widetilde U(X)})} \ar@{}[dr]|*+[F]{1}& L(H(A))\ar[d]^-{L(d_A)}\\
U(X) \ar[r]^-{\delta_{U(X)}}\ar[dr]_-{L(f)} & L(U(X)^{\t 2}) \ar[r]_-{L(f^{\t 2})} & L(H(A)^{\t 2})\\
& L(H(A)) \ar[ur]_-{\delta_{H(A)}} &
}
\]
commute if and only if the square $\xybox{(0,0.2)*+[F]{1}}$ commutes, it follows that $f$ preserves the comultiplication if and only if the morphism $L(f) \nu$ factors through $j_A \colon P(A)\to L(H(A))$. Now suppose that $f$ preserves comultiplication. Since in any bimonoid the identity $e = e m d$ holds it follows that
\begin{align*}
L(e f)\nu &= L(e) L(m) L(d) L(f)\nu\\
 &= L(e) L(f) L(m) L(d) \nu\\
 &= L(ef) L(m) \delta \nu\\
 &= L(ef) (\nu+\nu)\\
\end{align*}
and hence $L(e f)\nu =0$ which implies $e f =e$ making $f$ a bimonoid morphism. As mentioned above we have that $L(d)\nu_X=\delta_{U(X)}\nu_X$. This produces a unique morphism $\widetilde \nu_X \colon X \to P(\widetilde U(X))$ such that $j_{\widetilde U (X)} \widetilde \nu_X = \nu_X$ which one easily shows is the $X$ component of the unit of $P\dashv \widetilde U$.
\end{proof}

\section{Monoidal functors and lifting of adjoints}\label{monoidal functors}

Given two monoidal categories $\Cb=(\C,\t,I,\az,\lz,\rz)$ and $\Cb'=(\C',\t',\az',\lz',\rz')$ a \emph{lax monoidal functor} from  $\Cb$ to $\Cb'$ is a triple $(F,\theta,\phi)$ where $F\colon\C\to \C'$ is a functor, $\theta\colon I'\to F(I)$ is a morphism in $\C'$, and $(\phi_{A,B} \colon F(A)\t' F(B) \to F(A\t B))_{A,B \in \C}$ is a natural transformation, such that for all $A$, $B$ and $C$ in $\C$ the diagrams
\[
\xymatrix{
F(A)\t'(F(B)\t'F(C)) \ar[r]^-{\az'}\ar[d]_-{\id \t' \phi} & (F(A)\t' F(B))\t' F(C) \ar[d]^-{\phi \t' \id}\\
F(A)\t' F(B\t C) \ar[d]_-{\phi} & (F(A\t B) \t'  F(C)\ar[d]^-{\phi}\\
F(A\t(B\t C)) \ar[r]_-{F(\az)} & F((A\t B)\t C)
}
\]
\[
\xymatrix{
F(A) \ar[r]^-{\lz'}\ar[ddr]_-{F(\lz)} & I'\t' F(A)\ar[d]^-{\theta \t' \id} & F(A)\t' I'\ar[d]_-{\id \t' \theta}  & F(A)\ar[l]_-{\rz'}\ar[ddl]^-{F(\rz)}\\
& F(I)\t'F(A)\ar[d]^-{\phi} & F(A)\t F(I) \ar[d]_-{\phi} &\\
& F(I\t A) & F(A\t I)
}
\]
commute. If $\Cb$ and $\Cb'$ are symmetric monoidal categories with \emph{symmetry} isomorphisms $\sz$ and $\sz'$ respectively, then a lax monoidal functor $(F,\theta,\phi)$ between $\Cb$ and $\Cb'$ is called a lax symmetric monoidal functor as soon as the diagram
\[
\xymatrix{
F(A)\t' F(B) \ar[r]^-{\sz'}\ar[d]_-{\phi} & F(B)\t' F(A)\ar[d]^-{\phi}\\
F(A\t B) \ar[r]_-{F(\sz)} & F(B\t A)
}
\]
commutes,
for every $A$ and $B$ in $\C$. A lax symmetric monoidal functor $(F,\theta,\phi)$ is called strong when $\theta$ and $\phi$ are isomorphisms, and strict when they are identity morphisms.

Recall also that (symmetric) monoidal structures on categories lift to both functor categories and products of categories (componentwise). In particular, if $\Cb$ is a (symmetric) monoidal category, then by $\Cb \times \Cb$ we will mean the (symmetric) monoidal category with underlying category $\C\times \C$, with tensor product defined by~$(A,B) \t (A',B') = (A\t A',B\t B')$, and with remaining structure defined componentwise. For a symmetric monoidal category $\Cb$ there is via the coherence theorem a unique natural isomorphism $i_{A,B,C,D} \colon (A\t B)\t (C\t D) \to (A\t C)\t (B\t D)$ built from $\az$, $\lz$, $\rz$ and $\sz$ which we will call (middle) interchange isomorphism. Note that this produces a strong symmetric monoidal functor $(\t, \lz, i) \colon \Cb \t \Cb \to \Cb$.

It is well known that if $\Cb=(\C,\t,I,\az,\lz,\rz)$ and $\Cb'=(\C',\t',\az',\lz',\rz')$ are monoidal categories, and $(F,\theta,\phi)$ is a lax monoidal functor $\Cb \to \Cb'$, then the assignment $(A,m,u) \mapsto (F(A),F(m)\phi, F(u)\theta)$ is the object map of a functor ${\Mon(\Cb)\to \Mon(\Cb')}$ which sends a morphism $f$ to $F(f)$.
In a similar way, when~$\C$,~$\C'$ and~$F$ are also additive, the assignment $(B,b) \mapsto (F(B),F(b)\phi)$ is the object map of a functor $\Lie(\Cb) \to \Lie(\Cb')$ which sends a morphism $f$ to $F(f)$. Let us denote this induced functor by $\Lie(F)$. The following fact may be known, but we couldn't find a refence:
\begin{proposition}\label{prop:adjunction_lifts_to_lie}
Let $\Cb=(\C,\t,I,\az,\lz,\rz)$ and $\Cb'=(\C',\t',\az',\lz',\rz')$ be monoidal categories such that $\C$ and $\C'$ are additive, and let $F\colon \Cb \to \Cb'$ be a strict monoidal functor which is additive. If $F\colon\C\to \C'$ has a right adjoint, then $\Lie(F)$ has a right adjoint.
\end{proposition}
\begin{proof}
Suppose $F$ has a right adjoint $G$, and the unit and counit of the associated adjunction are denoted by $\eta$ and $\epsilon$, respectively. According to~\cite{Kelly:doctrinal} we know that defining $\theta=\eta_I$,
 and for each $A'$ and $B'$ in $\C'$, $\phi_{A',B'} \colon  G(A')\t G(B') \to G(A'\t'B')$ to be the composite $G(\epsilon_{A'}\t'\epsilon_{B'}) \eta_{G(A')\t G(B')}$
produces a lax monoidal functor $(G,\theta,\phi) \colon \Cb'\to \Cb$, and furthermore, these data make the triangle in \eqref{diag:right adjoint lifts}, below, commute. As explained above this lax functor determines a functor $\Lie(G) \colon \Lie(\Cb')\to \Lie(\Cb)$. The claim now follows by observing that for objects $(X,b)$ and $(X',b')$ in $\LieC$ and $\Lie(\Cb')$, respectively, and for $f\colon X\to G(X')$ a morphism in $\C$, either (and hence both) of the statements: $f \colon (X,b)\to (G(X),G(b')\phi)$ is a morphism in $\LieC$; and $\epsilon F(f)\colon (F(X),F(b)) \to (X',b')$ is a morphism in $\Lie(\Cb')$; are equivalent to the commutativity of the diagram
\begin{equation}
\label{diag:right adjoint lifts}
\vcenter{
\xymatrix@C=12ex{
 F(X)\t' F(X)\ar@{=}[d] \ar[r]^-{F(f)\t' F(f)} & FG(X')\t' FG(X') \ar[ddr]^-{\epsilon_{X'}\t'\epsilon_{X'}}\ar@{=}[d] & \\
F(X\t X) \ar[r]^-{F(f\t f)}\ar[dd]_-{F(b)} & F(G(X')\t G(X'))\ar[d]^-{F(\phi_{X',X'})} & \\
& FG(X'\t' X') \ar[r]^-{\epsilon_{X'\t' X'}}\ar[d]_-{FG(b')} & X'\t'X'\ar[d]^-{b'}\\
F(X)\ar[r]_-{F(f)} & FG(X') \ar[r]_-{\epsilon_{X'}} & X'.
}
}
\end{equation}
\end{proof}

\section{Actions}\label{actions}

Recall that if $A=(A,m,u)$ is a monoid in $\Cb$ and $X$ is an object in $\C$ an action of $A$ on $X$ is a morphism $\phi \colon A\t X \to X$ making the diagrams
\[
\xymatrix{
(A\t A) \t X \ar[rr]^-{\az^{-1}} \ar[d]_-{m \t \id} && A \t (A \t X) \ar[d]^-{\id \t \phi}\\
A \t X \ar[r]_-{\phi} &X &A\t X\ar[l]^-{\phi}
}
\xymatrix{
& I\t X \ar[dr]_-{\lz} \ar[r]^-{u \t \id} & A \t X \ar[d]^-{\phi}\\
& & X
}
\]
commute.
Given an object $B=(B,b)$ in $\LieC$ and an object $X$ in $\C$ we say that a morphism $\theta \colon B\t X \to X$ is an action of $B$ on $X$ if
the diagram
\begin{equation}\label{eq:lie_act}
\vcenter{
\xymatrix{
(B\t B)\t X \ar[rr]^-{\az^{-1}((\id - \sz)\t \id)} \ar[d]_-{b\t \id} && B\t (B\t X) \ar[d]^-{\id \t \theta}\\
B\t X \ar[r]_-{\theta} &X &B\t X\ar[l]^-{\theta}
}
}
\end{equation}
commutes.
Let us write $E(X)$ for the monoid with underlying object $X^X$ and with multiplication $m \colon X^X\t X^X\to X^X$ and $u \colon I \to X^X$ the unique morphisms making the diagrams
\[
\xymatrix{
(X^X\t X^X) \t X \ar[rr]^-{\az^{-1}} \ar[d]_-{m \t \id} && X^X \t (X^X \t X) \ar[d]^-{\id \t \ev}\\
X^X \t X \ar[r]_-{\ev} &X &X^X\t X\ar[l]^-{\ev}
}
\xymatrix{
& I\t X \ar[dr]_-{\lz} \ar[r]^-{u \t \id} & X^X \t X \ar[d]^-{\ev}\\
& & X
}
\]
commute. Let $f \colon A' \to A$ be a morphism in $\MonC$, $g\colon B'\to B$ be a morphism in~$\LieC$, $\phi\colon A\t X\to X$ be an action of $A$ on $X$, and $\theta \colon B\t X\to X$ be an action of $B$ on $X$, then $\phi(f\t \id)$ is an action of $A'$ on $X$ and $\theta(g\t \id)$ is an action of $B'$ on $X$. We have
\begin{proposition}
For an object $X$ in $\C$ there are natural isomorphisms between the following functors:
\begin{enumerate}
\item the functor assigning to each $B$ in $\LieC^{\op}$ the set of actions of $B$ on $X$;
\item  $\hom(-,L(E(X))\colon\LieC^{\op}\to \Set$;
\item $\hom(U(-),E(X))\colon\LieC^{\op} \to \Set$;
\item the functor assigning to each $B$ in $\LieC^{\op}$ the set of monoid actions of $U(B)$ on $X$.
\end{enumerate}
 For an object $B$ in $\LieC$, under these natural isomorphisms,
an action $\theta$ of $B$ on $X$, a morphism $\bar \theta\colon B\to L(E(X))$ in $\LieC$, a morphism $\bar \phi\colon U(B)\to E(X)$ in~$\MonC$, and a monoid action $\phi$ of $U(B)$ on $X$ are related when:
$\theta = \ev (\bar \theta \t \id)$, $\bar \theta = \bar \phi \nu$, $\phi = \ev(\bar \phi \t \id)$,
and (hence) $\theta = \phi(\nu \t \id)$, where $\nu$ is the unit of the adjunction $U \dashv L$.
\end{proposition}
\begin{proof}
The natural isomorphism between the functors in (3) and (4) is standard, while the natural isomorphism between the functors in (2) and (3) is obtained from the adjunction $U \dashv L$. Given a morphism $\theta \colon B\t X\to X$, let $\bar \theta \colon B\to X^X$ be the unique morphism such that $\ev (\bar \theta \t \id)=\theta$. One easily observes that if the map~$\bar \theta \colon B\to L(E(X))$ is a morphism in $\LieC$, then the diagram $\xybox{(0,0.2)*+[F]{1}}$ below is commutative, making the whole diagram
\[
\xymatrix@C=7ex@R=4ex{
(B\t B)\t X \ar[r]^-{(\id-\sz)\t\id}\ar[rd]_-{(\bar \theta\t\bar \theta)\t \id}\ar[ddd]_-{b\t \id}\ar@{}[dddr]|*+[F]{1}&(B\t B)\t X\ar[r]^-{\az^{-1}}&B\t(B\t X)\ar[dd]_-{\bar \theta \t (\bar \theta \t \id)}\ar[rdd]^-{\id \t \theta} &\\
&(X^X\t X^X)\t X \ar[d]^-{(\id-\sz)\t \id}&&\\
&(X^X\t X^X)\t X\ar[d]^-{m \t \id}\ar[r]^-{\az^{-1}}&X^X\t(X^X\t X)\ar[d]^-{\id \t \ev}&B\t X\ar[ldd]^-{\theta}\\
B\t X \ar[r]^-{\bar \theta \t \id}\ar@/_2ex/[rrd]_-{\theta}&X^X\t X\ar[rd]^-{\ev}&X^X\t X\ar[d]^-{\ev}&\\
&&X&
}
\]
commute and hence $\theta$ is an action of $B$ on $X$. The converse follows immediately from the universal property of $\ev$.
\end{proof}
Let $\two$ denote the category with two objects $0$ and $1$ and one non-identity morphism $0\to 1$. Recall that the functor category $\C^{\two}$ can be made into a symmetric monoidal category $\Cb^\two$ with monoidal structure defined componentwise. Recall also, that when $\C$ has pullbacks the symmetric monoidal category $\Cb^\two$ is monoidal closed. Since the functor $L \colon \Lie(\Cb^\two) \to \Mon(\Cb^\two)$ is, up to isomorphism, the same as the functor $L^\two \colon \LieC^{\two} \to \MonC^\two$, it follows from the previous proposition applied to $\Cb^\two$ that for a morphism $g\colon B\to B'$ in $\LieC$, a morphism $f\colon X\to X'$ in $\C$, and monoid actions $\phi$ and $\phi'$ of $U(B)$ on $X$ and $U(B')$ on $X'$ respectively, the outer arrows of the diagram
\[
\xymatrix{
B\t X \ar[r]^-{\nu \t \id}\ar[d]_-{g\t f} & U(B)\t X \ar[d]^-{U(g)\t f} \ar[r]^-{\phi} & X\ar[d]^-{f}\\
B'\t X' \ar[r]_-{\nu \t \id} & U(B')\t X' \ar[r]_-{\phi'} & X'
}
\]
commute if and only if the right hand square commutes. In particular, writing~$\LieAct{B}{\Cb}$ and $\MonAct{A}{\Cb}$ for the categories of $B$ actions and $A$ actions respectively (for some objects $B$ in $\LieC$ and $A$ in $\MonC$), this implies:
\begin{proposition}\label{prop:mon_act_isomorphic_lie_act}
For an object $B$ in $\LieC$ the assignment $(X,\phi) \mapsto (X,\phi(\nu \t \id))$ is the object map of an isomorphism $\MonAct{U(B)}{\Cb} \to \LieAct{B}{\Cb}$ which is identity on morphisms.
\end{proposition}
Now suppose that $A$ is a bimonoid with comultiplication $d\colon A\to A\t A$. If $\phi$ and $\phi'$ are actions of the underlying monoid of $A$ on $X$ and $Y$, respectively, then $\phi \t^{d} \phi' \colon A\t (X\t Y) \to (X\t Y)$ defined by $\phi \t^{d} \phi' = (\phi \t \phi') i (d \t \id)$ is a monoid action.
To see why, note that the monoidal functor $(\t,\lz,i) \colon \Cb\times \Cb \to \Cb$ sends the action $(\phi,\phi')$ of the monoid~$(A,A)$ on $(X,Y)$ to the action $(\phi \t \phi') i$ of the monoid~$A\t A$ on $X\t Y$. However, since $d \colon A\to A\t A$ is a monoid morphism it follows that $\phi \t^{d} \phi' = (\phi \t \phi')i(d\t \id)$ is an action of $A$ on $X\t Y$.

Noting that the counit of the bimonoid $A$ determines a monoid action of $A$ on $I$ one easily establishes that the symmetric monoidal structure of $\Cb$ lifts to $\MonAct{A}{\Cb}$ and hence the forgetful functor $G_A\colon \MonAct{A}{\Cb} \to \Cb$ is strict monoidal. Let us write~$\MonActu{A}{\Cb}$ for this monoidal category. It is well known that when $\Cb$ is monoidal closed $G_A$ has a right adjoint. The object map of this right adjoint assigns to each object $X$ the object $X^A$ with action $\phi \colon A\t X^A \to X^A$, the unique morphism making the diagram
\[
\xymatrix{
(A\t X^A) \t A \ar[rr]^-{\phi \t \id}\ar[d]_-{(\id \t \sz)\az(\sz\t \id)} && X^A\t A \ar[d]^-{\ev}\\
X^A\t (A\t A) \ar[r]_-{\id \t m} & X^A \t A \ar[r]^-{\ev} & X
}
\]
commute.
Therefore, since the categories $\C$ and $\MonAct{A}{\Cb}$ are additive and so is the functor $G_A$, by Proposition~\ref{prop:adjunction_lifts_to_lie}, it follows that:
\begin{proposition}\label{prop:forget_lmonact_right_adj}
The induced forgetful functor $\Lie(G_A)\colon \LMonAct{A}{\Cb} \to \LieC$ has a right adjoint.
\end{proposition}
Suppose $B$ is an object in $\LieC$. Let us write $\LieActu{B}{\Cb}$ for the monoidal category obtained by translating the monoidal structure from $\MonActu{U(B)}{\Cb}$ via the isomorphism given in Proposition~\ref{prop:mon_act_isomorphic_lie_act}.
Trivially:
\begin{proposition}\label{prop:LLieAct_iso_LMonAct}
For an object $B$ in $\LieC$ the assignment given by $((X,\phi),b) \mapsto ((X,\phi(\nu\t \id)),b)$ is the object map of an isomorphism $\LMonAct{U(B)}{\Cb} \to \LLieAct{B}{\Cb}$ which is the identity on morphisms.
\end{proposition}

Let us calculate explicitly what the tensor product in $\LieActu{B}{\Cb}$ is. Suppose that $(X,\theta)$ and $(X',\theta')$ are objects in $\LieAct{B}{\Cb}$. Letting $(X,\phi)$ and $(X',\phi')$ be the corresponding objects in $\MonAct{U(B)}{\Cb}$ we see that the diagram
\[
\xymatrix@C=-4ex@R=3.5ex{
&&U(B)^{\t 2}\t (X\t X')\ar[dd]^-{i}&&\\
&(B\t I)\t (X\t X')\ar[ru]^-(0.4){(\nu\t u)\t \id}\ar[dd]^-{i}&&(I\t B)\t (X\t X')\ar[lu]_-(0.4){((u\t \nu)\t \id}\ar[dd]^-{i}&\\
B\t (X\t X')\ar[ru]^-{\rz^{-1}\t \id}\ar@/_17.22pt/[rddd]_-{\az}&&(U(B)\t X)\t(U(B)\t X')\ar[dd]^-{\phi\t \phi'}&&B\t (X \t X') \ar[lu]_-{\lz^{-1}\t \id}\ar@/^17.22pt/[lddd]^-{\sz \az (\id \t \sz)}\\
&(B\t X)\t (I\t X')\ar[dd]^-{\id \t \lz}\ar[ru]|{(\nu\t \id)\t (u \t \id)}&&(I\t X)\t(B\t X')\ar[lu]|{(u\t \id)\t(\nu \t \id)}\ar[dd]^-{\lz\t \id}&\\
&&X\t X'&&\\
&(B\t X)\t X'\ar[ru]_-{\theta \t \id}&&X\t(B\t X')\ar[lu]^-{\id \t \theta'}&
}
\]
commutes. This implies that
\begin{align*}
(\phi \t^{d} \phi') (\nu \t \id) &= (\phi \t \phi') i (d \nu \t \id)\\
&=(\phi \t \phi') i (\delta \nu \t \id) \\
&= (\phi \t \phi') i (((u\t \nu)\lz^{-1} + (\nu\t u)\rz^{-1}) \t \id)\\
&= (\theta \t \id) \az + (\id \t \theta') \sz \az (\id \t \sz)
\end{align*}
and hence the translated tensor product of $\LieActu{B}{\Cb}$ is defined by
\[
(X,\theta)\t (X',\theta') = (X\t X', \theta * \theta')
\]
where $\theta * \theta'= (\theta \t \id) \az + (\id \t \theta') \sz \az (\id \t \sz)$.
One easily observes that this means that an object in $\LLieAct{B}{\Cb}$ can be identified with a pair $((X,b), \theta)$ where~$(X,b)$ is an object in $\LieC$, and $\theta$ is an action of $B$ on $X$ making the diagram
\begin{equation}\label{eq:lie_lie_act}
\vcenter{
\xymatrix{
B\t (X\t X) \ar[r]^-{\id \t b} \ar[d]_-{\theta * \theta} & B\t X\ar[d]^-{\theta}\\
X\t X \ar[r]_-{b} & X
}
}
\end{equation}
commute.
Writing $\overline{G_B} \colon \LieActu{B}{\Cb} \to \C$ for the forgetful functor we have:
\begin{proposition}
For an object $B$ in $\LieC$ the map assigning to each object $(A,p,s)$ in $\Pt_B(\LieC)$ the pair $(X,\theta)$ where $X$ is the domain of $k\colon X\to A$ the kernel of $p$ and $\theta \colon B\t X \to X$ is the unique morphism in $\C$ with $k \theta = b_A (s\t k)$, is the object map of a functor $W\colon \Pt_B(\LieC) \to \LLieAct{B}{\Cb}$ which is part of an equivalence of categories
making the diagram
\begin{equation}
\label{diag:equiv_pts_llieact}
\vcenter{
\xymatrix@C=1ex{
\Pt_B(\LieC) \ar[rr]^-{W} \ar[dr]_-{\Ker_B} && \LLieAct{B}{\Cb} \ar[dl]^-{\Lie(\overline{G_B})} \\
& \LieC
}
}
\end{equation}
commute.
\end{proposition}
\begin{proof}
Let $B$ be an object in $\LieC$, for an object $(A,p,s)$ in $\Pt_B(\LieC)$, defining $W(A,p,s)=(X,\theta)$ where $X$ and $\theta$ are defined as above. One can check that the commutativity of diagrams \eqref{eq:lie_act} and \eqref{eq:lie_lie_act} follow from  commutativity of diagrams \eqref{dia:lie} and coherence.

If $f\colon (A,p,s)\to (A',p',s')$ is a morphism in $\Pt_B(\LieC)$, then  an easy calculation shows that the induced map between the kernels of $p$ and $p'$ lifts to a morphism $W(A,p,s)\to W(A',p',s')$ producing a functor $W\colon\Pt_B(\LieC)\to \LLieAct{B}{\Cb}$ making diagram~\eqref{diag:equiv_pts_llieact} commute. On the other hand, given $(X,\theta)$ in $\LLieAct{B}{\Cb}$ letting  $b \colon (B\oplus X)\t (B\oplus X) \to B\oplus X$ be the unique morphism with $b(\iota_1\t \iota_1) = \iota_1 b_B$, $b(\iota_1\t\iota_2)=\theta$, $b(\iota_2\t \iota_1)=-\theta \sz$ and $b(\iota_2\t \iota_2)= \iota_2 b_X$ makes $((B\oplus X,b),\pi_1,\iota_1)$ into an object of $\Pt_B(\LieC)$ such that $W((B\oplus X,b),\pi_2,\iota_2) = (X,\theta)$. Therefore, noting that $W$ is faithful, to show that it is part of an equivalence of categories we need only show it is full.

To that end suppose that $(A,p,s)$ and $(A',p',s')$
are objects in $\Pt_B(\LieC)$, $k\colon X\to A$ and $k'\colon X'\to A'$ are the kernels of
$p$ and $p'$, respectively, $\theta\colon B\t X\to X$ and $\theta'\colon B\t X'\to X'$
are morphisms such that $W(A,p,s)=(X,\theta)$ and $W(A',p',s')=(X',\theta')$,
and $f$ is a morphism from $W(A,p,s)$ to $W(A',p',s')$. Letting $l\colon A\to X$ be the unique morphism
in $\C$ such that $kl=\id-sp$, using the fact that $k$ and $s$ are jointly epimorphic,
one can easily check that if $g = k'fl+s'p$ then $g$ is a morphism from $(A,p,s)$ to $(A',p',s')$ such that $W(g)=f$.
\end{proof}
Combining the previous proposition with Proposition~\ref{prop:LLieAct_iso_LMonAct} we obtain:
\begin{proposition}
For each $B$ in $\LieC$ the kernel functor $\Ker_B \colon \Pt_B(\LieC) \to \C$ factors through the forgetful functor $\Lie(G_{U(B)}) \colon \Lie(\MonActu{U(B)}{\Cb}) \to \LieC$ via an equivalence of categories.
\end{proposition}
We are now ready to prove our main theorem:
\begin{theorem}\label{theorem:main}
For a symmetric monoidal closed category $\Cb = (\C,\t,I,\az,\lz,\rz,\sz)$ such that $\C$ is cocomplete and additive, the category $\LieC$ is \LACC.
\end{theorem}
\begin{proof}
Combining the previous proposition and Proposition~\ref{prop:forget_lmonact_right_adj} we see that for each~$B$ in $\LieC$ the functor $\Ker_B \colon \Pt_B(\LieC) \to \LieC$ has a right adjoint. The claim now follows by \cite[Theorem 5.1]{Gray2012}.
\end{proof}

\section{Examples}\label{examples}
\begin{example}
Let $R$ be a commutative ring and consider the category of chain complexes of $R$-modules. If $(V, d)$ and $(V', d')$ are chain complexes, the tensor product of $(V,d)$ and $(V',d')$ is the chain complex $(V\t V', \delta)$ where
\[
(V \t V')_i = \bigoplus_{j + k = i}V_j \t V'_k,
\]
and $\delta$ is defined for all $v \in V_j$ and $v' \in V'_k$ by
\[
\delta(v \t v') = d(v) \t v' + (-1)^{j} v \t d'(v').
\]
The symmetry isomorphism $\sz \colon V \t V' \to V' \t V$ is defined for all $v \in V_j$ and $v' \in V'_k$ by $\sz(v \t v') = (-1)^{ij} v' \t v$.
With this tensor product, symmetry isomorphism and the remaining structure defined canonically, it forms a symmetric monoidal closed category whose underlying category is cocomplete and abelian.

Its category of internal Lie algebras is equivalent to the category whose objects are \emph{differential graded Lie algebras}, i.e., a $\mathbb{Z}$-graded $R$-module $V$, with a linear map $d \colon V \to V$ of degree $-1$ such that $d \circ d = 0$, and a bilinear map $[-, -] \colon V \t V \to V$ of degree zero satisfying: for all homogeneus $x, y, z$ in $V$
\begin{enumerate}
	\item $[x, y] = -(-1)^{|x||y|} [y, x]$,
	\item $(-1)^{|x||z|} [x, [y, z]] + (-1)^{|y||x|} [y, [z, x]] + (-1)^{|z||y|} [z, [x, y]]$,
	\item $d([x, y]) = [d(x), y] + (-1)^{|x|}[x, d(y)]$
\end{enumerate}
where $|v|$ denotes the degree of $v$.
By Theorem~\ref{theorem:main} it follows that the category of differential graded Lie algebras is \LACC.
\end{example}

\begin{example}
If in the previous example we consider the full subcategory formed by chain complexes where $d = 0$, internal Lie algebras are essentially \emph{Lie colour algebras}.
Applying Theorem~\ref{theorem:main} again we know that the category of Lie colour algebras is \LACC.

In a similar way, the category of Lie superalgebras over a commutative ring can be seen as the category of internal Lie algebras in the symmetric monoidal closed category of \emph{super $R$-modules} and hence it is \LACC.
\end{example}

\begin{example}
Let us consider now the category whose objects are linear maps $f \colon V \to W$ and morphisms are homomorphisms of $R$-modules making the diagram commutative
\[
\xymatrix{
V \ar[r] \ar[d]_-{f} & V' \ar[d]^-{f'} \\
W \ar[r] & W'
}
\]
There is a tensor product, called the \emph{infinitesimal tensor product} defined as
\[
\left(\vcenter{\xymatrix{ V \ar[d]^-{f} \\ W }}\right) \t \left(\vcenter{\xymatrix{ V' \ar[d]^-{f'} \\ W' }}\right) =
\left(\vcenter{\xymatrix{ (V \t W') \oplus (W \t V') \ar[d]^-{[f \t \id, \id \t f']} \\ W \t W' }}\right)
\]

This monoidal category was introduced in \cite{Loday-Pirashvili} as \emph{the tensor category of linear maps}, and in more recent papers was renamed to the \emph{Loday-Pirashvili category}~\cite{FCGMLa, GMLa, Ro}, denoted by $\mathcal{LP}$. An internal Lie algebra in this category is a linear map $f \colon M \to \mathfrak{g}$, where~$\mathfrak{g}$ is a Lie algebra, $M$ is a right $\mathfrak{g}$-module and $f$ preserves the $\mathfrak{g}$-action. Once more, as a consequence of Theorem~\ref{theorem:main}, the category of Lie algebras in $\mathcal{LP}$ is \LACC. Furthermore, the category of Leibniz algebras can be found as a full reflective subcategory of internal Lie algebras in $\mathcal{LP}$, but it is known that the category of Leibniz algebras is not \LACC~\cite{GM-Vdl}.
\end{example}

\begin{example}
	Whereas the categories of \emph{Hom-Lie algebras} and \emph{multiplicative Hom-Lie algebras} are known to be semi-abelian categories satisfying very few categorical-algebraic properties (see~\cite{CaGM}), it was shown in~\cite{GoVe} that \emph{regular Hom-Lie algebras} can be found as internal categories in a certain symmetric monoidal category, and therefore, again by Theorem~\ref{theorem:main}, it is \LACC.
\end{example}

\section*{Acknowledgements}
The first author is grateful to Stellenbosch University for their kind hospitality during his stay there.


TITLE: How to find fixed field of this particular Automorphism
QUESTION [0 upvotes]: Consider the following problem part of my abstract algebra course assignment:

Let F be an algebraic closure of $\mathbb{Z}_{p}$ (p prime) then prove that F is algebraic and Galois over $\mathbb{Z}_p$.

F is algebraic closure and so all roots of irreducible polynomials in $\mathbb{Z}_p$[x] are in F and but to prove that F is algebraic i need to choose any element say p which is in F  and then prove that
p is a root of element in $\mathbb{Z}_p $[x] but as F is given as algebraically closed so  it is algebraic over $\mathbb{Z}_p$ .

But The problem is that I am not able to prove it to be Galois over $\mathbb{Z}_p$.  I need to prove that Fixed field of $Aut_{\mathbb{Z}_p} F $  is $\mathbb{Z_p}$ itself. SO, If I choose an automorphism $f\in  Aut_{\mathbb{Z}_p} F $ I need to prove that only for x $\in \mathbb{Z_p}$ we have $f(x) =x $ . So , For any superfield K" of $\mathbb{Z}_p$ I need to show that there exists atleast 1 x  $f(x)\neq x$ .

But I am unable to form ideas on how should I proceed towards it. So, do u mind giving any advice.
Thank you!

REPLY [2 votes]: I assume that $\mathbb{Z}_p$ refers to the field with $p$ elements; I'll use the more common notation $\mathbb{F}_p$.
An algebraic closure of a field $K$ is algebraic over $K$ by definition (it is unique up to isomorphisms fixing $K$, but it's irrelevant here).
Let $F_0$ be the fixed field of $\operatorname{Aut}(F)$.
The Frobenius homomorphism $\varphi\colon F\to F$, $\varphi(a)=a^p$ is an automorphism of $F$, because $x^p-b$ has a root for every $b\in F$. Since $F_0$ is the fixed field, $\varphi(a)=a$ for every $a\in F_0$. Thus $F_0$ consists of roots of $x^p-x$ and so it has at most $p$ elements; on the other hand, it contains $\mathbb{F}_p$.


TITLE: A projective-cofibrant replacement in $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ such that $\operatorname{ev}_0Q(S) \cong \operatorname{ev}_0S$
QUESTION [3 upvotes]: Consider the category of simplicial presheaves $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ endowed with the projective model structure, i.e. weak equivalences and fibrations are point-wise.

I need a functorial cofibrant replacement
$Q:\mathsf{sSet}^{\mathcal{C}^{\text{op}}} \to
 \mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ with the property that
$\operatorname{ev}_0Q(S) \cong \operatorname{ev}_0S$ are isomorphic in
$\mathsf{Set}^{\mathcal{C}^{\text{op}}}$ for every simplicial presheaf
$S$.
Actually, I would be satisfied if it worked just for all $S$ constant in the simplicial direction, so just for all $S:n \mapsto P$ for some $P \in
 \mathsf{Set}^{\mathcal{C}^{\text{op}}}$.

Question 1
Does such a cofibrant replacement exist? If it does, a reference too would be awesome.
Question 2
Before someone answers I add a related less general question. Does there exist a cofibrant replacement (not necessarily functorial) of the point $* \in \mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ with the property that $\operatorname{ev}_0Q(*) \cong * \in \mathsf{Set}^{\mathcal{C}^{\text{op}}}$ ?

REPLY [3 votes]: Before someone answers I add a related less general question. Does there exist a cofibrant replacement (not necessarily functorial) of the point ∗∈sSetCop with the property that ev0Q(∗)≅∗∈SetCop ?

No.  Cofibrations in the projective model structure on simplicial presheaves on C are retracts of transfinite compositions of cobase changes of generating cofibrations, which are given by tensoring a representable presheaf on C with a generating cofibration ∂Δ^n→Δ^n of simplicial sets.
In particular, if a cofibrant replacement of the terminal simplicial presheaf has a terminal presheaf of sets in degree 0, then the terminal presheaf is a retract of a coproduct of representables.
It is easy to construct examples of C for which there no maps from the terminal presheaf to a coproduct of representables.
Indeed, the latter amounts to saying the taking the limit of such a coproduct (consider as a functor from C^op) produces the empty presheaf.
This argument can be leveraged to give a complete description of projectively cofibrant simplicial presheaves, see Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves.


text:
\begin{document}

\title[Noninertial relativity consistent with Heisenberg algebra]{Noninertial Relativity group with invariant Minkowski metric consistent with Heisenberg quantum commutation relations}
\author{Stephen G. Low}
\address{Austin, Texas }
\email{Stephen.Low@alumni.utexas.net}
\date{\today}
\keywords{noninertial, Heisenberg group, Born reciprocity, quaplectic group, reciprocal relativity,special relativitic quantum mechanics, maximal acceleration}
\subjclass[2000]{81R05,81R60,83E99,83A05,51N25}
\maketitle
\begin{abstract}

The inhomogeneous Lorentz group defines the transformations between
inertial states and special relativistic quantum mechanics is defined
in terms of its projective representations. Special relativity does
not address how noninertial states are related. If the noninertial
system is due to gravity, general relativity resolves this through
a curved manifold where particles under the action of gravity follow
geodesics that are locally inertial trajectories. However, general
relativity also does not address the issue of how the states of
noninertial particles on a flat space due to a force other than
gravity are related.\ \ We study this by starting with a quantum
system with physical observables of position, time, energy and momentum
that are the Hermitian representation of the generators of the algebra
of the Weyl-Heisenberg group.\ \ We require that this is true for
any states related by the projective representation of the relativity
group. We show that this results in a consistency condition that
requires the relativity group to be a subgroup of the group of automorphisms
of the Weyl-Heisenberg algebra and consider the relativity groups
that also leaves invariant a Minkowski line element. This defines
the expected noninertial relativistic transformations and that have
the expected classical limit as $c\rightarrow \infty $ .\ \ In a
companion paper, a quantum mechanics for this noninertial relativity
group is formulated in terms of the projective representations of
the inhomogeneous group using the same approach as for special relativistic
quantum mechanics.
\end{abstract}

\section{Introduction}

The inhomogeneous Lorentz group defines the relation between inertial
states.\ \ Clocks locally at rest to a state are related to the
clocks of other inertial observers through the Minkowski proper
time line element.\ \ Quantum states are rays in a Hilbert space
and therefore inertial states are related through the projective
representation of the inhomogeneous Lorentz group. As projective
representations are equivalent to the unitary representations of
the central extension,\ \ these are the unitary representations
of the Poincar\'e group that is the cover of the inhomogeneous Lorentz
group \cite{wigner},\cite{bargmann},\cite{mackey2},\cite{Weinberg1}.

The equivalence principle of general relativity enables the noninertial
frames of a particle accelerating under gravity to be understood
as locally inertial frames on a curved manifold. Particles under
gravity follow geodesics and neighboring locally inertial frames
are related by the connection. The clock locally at inertial rest
is related to the local clocks of other neighboring observers in
the gravitating system through the Riemannian proper time\ \ line
element.

Neither general relativity nor special relativity addresses the
issue of noninertial states that are not due to gravity, but rather
one of the other forces, and therefore the underlying manifold is
flat. Consider for example an electron in a region that gravity
is negligible that encounters an electromagnetic field and therefore
perturbs to a noninertial trajectory and is observed also by an
observer in an apparent inertial frame.\ \ How is the clock of this
noninertial state related to the clocks of other observers?

We hypothesize that the noninertial relativity group relating these
states is the most general group consistent with the requirements
that

1) the Heisenberg uncertainty principle holds in the noninertial
as well as inertial states

2) the proper time given by the Minkowski line element that is invariant
in noninertial states

To make this more precise, we first consider a quantum system in
which the position, momentum, energy and time degrees of freedom
are represented by the Hermitian representation of the algebra of
the Weyl-Heisenberg group $\mathcal{H}( n+1) $ where the number
of spacial dimensions is $n=3$. The requirement that the algebra
transforms into itself under the action of the relativity group
means that the relativity group a subgroup of the automorphism group
of the Weyl-Heisenberg algebra.\ \ This automorphism group is \cite{folland,Low7}
\begin{equation}
{\mathcal{A}ut}_{\mathcal{H}}\simeq \mathbb{Z}_{2}\otimes _{s}\mathcal{D}\otimes
_{s}\overline{\mathcal{H}\mathcal{S}p}( 2n+2) ,
\label{nirg: autH definition}
\end{equation}

\noindent where\ \ $\mathcal{H}\mathcal{S}p( 2n+2) \simeq \mathcal{S}p(
2n+2) \otimes _{s}\mathcal{H}( n+1) $. $\mathbb{Z}_{2}$ is the 2
element discrete group, $\mathcal{D}$ is the abelian group isomorphic
to the reals under multiplication, $\mathcal{S}p( 2n+2) $ is the
symplectic group and $\mathcal{H}( n+1) $ is the Weyl-Heisenberg
group.

The Minkowski line element is\ \ $d \tau ^{2 }=d t^{2}-\frac{1}{c^{2}}d
q^{2}$. This is an invariant for states that are inertially related
and the second assertion is that this continues to be true for general
noninertial states.\ \ \ \

We will show that the homogeneous relativity group that is a subgroup
of the automorphism group of the Weyl-Heisenberg group\ \ that leaves
the Minkowski line element invariant is
\begin{equation}
\mathcal{U}b( 1,n) \simeq \mathcal{O}( 1,n) \otimes _{s}\mathcal{A}(
m) ,
\end{equation}

\noindent where $m=(n+1)(n+2)/2$ and $\mathcal{A}( m) $ is the abelian
group isomorphic to $\mathbb{R}^{m}$ under addition.\ \ The additional
generators of the abelian group behave as a power-force stress tensor
that is the proper time derivative of the energy-momentum stress
tensor. We show that this leads to expected relativistic results
in transforming to noninertial states \cite{Low5}.\ \

This relativistic theory must lead to expected classical results
in the limit $c\rightarrow \infty $ where the Minkowski line element
reduces to the invariant Newtonian time line element $d t^{2}$.
We have previously studied the most general group that leaves invariant
the Newtonian time line element $d t^{2}$ that is a subgroup of
the automorphisms of the Weyl-Heisenberg group. This results in
a group that leads directly to Hamilton's equations and, with the\ \ additional
requirement of orthonormal position frames,\ \ \ describes the Hamilton
relativity group for noninertial transformations in a classical
context \cite{Low7}, \cite{Low8}.

\section{Consistency between a relativity group and quantum mechanics}

States in quantum mechanics are represented by rays $\Psi $ in a
Hilbert space $\text{\boldmath $\mathrm{H}$}$ that are the equivalence
class of states in the Hilbert space related by a phase
\begin{equation}
\Psi \simeq \left\{  e ^{i \omega }\left| \psi \right\rangle  |\omega
\in \mathbb{R}\right\} ,
\end{equation}

\noindent where $|\psi \rangle \in \text{\boldmath $\mathrm{H}$}$.\ \ A
relativity group $g\in \mathcal{G}$ acts on the states through a
projective representation $\pi $, $\tilde{\Psi }=\pi ( g) \Psi $,\ \ with
the property that
\begin{equation}
\pi ( \tilde{g}\cdot g) = e ^{i \omega ( \tilde{g},g) }\pi ( \tilde{g})
\pi ( g) , \omega ( \tilde{g},g) \in \mathbb{R}\text{}.
\end{equation}

\noindent Projective representations are equivalent to the unitary
representations $\varrho $ of the central extension $\check{\mathcal{G}}$
of the group $\mathcal{G}$ \cite{bargmann}, \cite{mackey2} that
act on the states as
\begin{equation}
\left. \tilde{\left. \left| \psi \right. \right\rangle  }=\varrho
( g) \left| \psi \right. \right\rangle  ,\ \ g\in \check{\mathcal{G}},\ \ \ \ \ \ \left.
\left| \psi \right. \right\rangle  \in {\text{\boldmath $\mathrm{H}$}}^{\varrho
}.
\end{equation}

\noindent The Hilbert space is determined by the unitary representation
$\varrho $ and so we label it as ${\text{\boldmath $\mathrm{H}$}}^{\varrho
}$.\ \ Observables corresponding to a the relativity group $\mathcal{G}$
are represented by the Hermitian representations $\varrho ^{\prime
}$ of the algebra of a group $ \check{\mathcal{G}}$ , $\hat{Z}=\varrho
^{\prime }( Z) $. The action of the group element $g\in \check{\mathcal{G}}$
on these observables is
\begin{equation}
\hat{\tilde{Z}}\tilde{\left| \psi \right\rangle  }=\varrho ( g )
\hat{Z}\left| \psi \right\rangle  =\varrho ( g ) \hat{Z}{\varrho
( g ) }^{-1}\varrho ( g ) \left| \psi \right\rangle  =\varrho (
g ) \hat{Z}{\varrho ( g ) }^{-1}\tilde{\left| \psi \right\rangle
}
\label{nirg: projective tx of algebra}
\end{equation}

\noindent and so
\begin{equation}
\hat{\tilde{Z}}=\varrho ^{\prime }( \tilde{Z}) =\varrho ( g ) \hat{Z}{\varrho
( g ) }^{-1}=\varrho ( g ) \varrho ^{\prime }( Z) {\varrho ( g )
}^{-1}=\varrho ^{\prime }( g Z g^{-1}) .
\end{equation}

\noindent Therefore, if the representation $\varrho $ is faithful,
we have that
\begin{equation}
\tilde{Z}=g Z g^{-1}
\label{nirg: transformation of an observable}
\end{equation}

\noindent and otherwise this is an equivalence up to the kernel
of the homomorphism.

 Position, momentum, energy and time observables are the Hermitian
representation of the algebra of the Weyl-Heisenberg group $\mathcal{H}(
n+1) $\ \ with a general element given by $\begin{array}{rl}
 Z & =z^{\alpha }Z_{\alpha }
\end{array}$, $\alpha =1,.. 2n+2$ where $\{z^{\alpha }\}\in \mathbb{P}\simeq
\mathbb{R}^{2n+2}$ and $Z_{\alpha }$ are a dimensionless basis for
the Weyl-Heisenberg algebra that satisfy the commutation relations\ \
\begin{equation}
\left[ Z_{\alpha },Z_{\beta }\right] = \zeta _{\alpha ,\beta }I,
\end{equation}

\noindent and $\zeta _{\alpha ,\beta }$ are the components of a
symplectic metric.\ \ The Hermitian representation of the algebra
satisfies
\begin{equation}
\left[ {\hat{Z}}_{\alpha },{\hat{Z}}_{\beta }\right] =i \zeta _{\alpha
,\beta }\hat{I},
\end{equation}

\noindent where $\hat{Z_{\alpha }}=\varrho ^{\prime }( Z_{\alpha
}) $ and $\hat{I}=\varrho ^{\prime }( I) $ is the unit operator
on the Hilbert space. Set $\{\hat{Z_{\alpha }}\}=\{{\hat{P}}_{i},{\hat{Q}}_{i},\hat{E},\hat{T}\}$
with $i=1,..n$.\ \ \ These are the familiar Hermitian representations
that in a basis with position and time diagonal, are
\begin{equation}
\begin{array}{ll}
 \left. \left. \left\langle  q,t\right. \right| {\hat{Q}}_{i}\left|
\psi \right. \right\rangle  =q^{i}\psi ( q,t) , & \left. \left.
\left\langle  q,t\right. \right| \hat{T}\left| \psi \right. \right\rangle
=t \psi ( q,t) , \\
 \left. \left. \left\langle  q,t\right. \right| {\hat{P}}_{i}\left|
\psi \right. \right\rangle  =i \hbar \partial \psi ( q,t) /\partial
q^{i}, & \left. \left. \left\langle  q,t\right. \right| \hat{E}\left|
\psi \right. \right\rangle  =-i \hbar \partial \psi ( q,t) /\partial
t.
\end{array}
\end{equation}

Bases that diagonalize other commuting sets $\psi ( p,t) =\langle
q,t|\psi \rangle $, $\psi ( p,e) =\langle q,e|\psi \rangle $, $\psi
( q,e) =\langle q,e|\psi \rangle $ and the corresponding representations
of the operators\ \ $\{{\hat{P}}_{i},{\hat{Q}}_{i},\hat{E},\hat{T}\}$
in these bases are equally valid \cite{dirac}.\ \ Generally, our
bias is to diagonalize the position time basis $\psi ( q,t) =\langle
q,t|\psi \rangle $.

The basic physical assumption is that the Heisenberg commutation
relations are satisfied by any basis related by a relativity group
$ \mathcal{G}$.\ \ That is,\ \ position, momentum, energy and time
observables satisfying the Heisenberg quantum commutation relations
will also satisfy the Heisenberg quantum commutation relations for
any states related by the projective representations of the relativity
group (6). This implies using (8) that if $\{Z_{\alpha },I\}$ are
a basis of the Weyl-Heisenberg algebra, then $\{{\tilde{Z}}_{\alpha
},\tilde{I}\}$ are also a basis of the Weyl-Heisenberg algebra where
\begin{equation}
\tilde{Z_{\alpha }}=g Z_{\alpha } g^{-1}, \tilde{I}=g I g^{-1}=I.
\end{equation}

\noindent and $g\in \check{\mathcal{G}}$ and $\varrho$ is a faithful
representation.\ \ The maximal group for which this property is
true is the automorphism group of the Weyl-Heisenberg group.\ \ This
results in basic consistency condition that the central extension
$\check{\mathcal{G}}$ of the relativity group $\mathcal{G}$ must
be a subgroup of the automorphism group of the Weyl-Heisenberg algebra
${\mathcal{A}ut}_{\mathcal{H}( n+1) }$,
\begin{equation}
\check{\mathcal{G}}\subseteq {\mathcal{A}ut}_{\mathcal{H}( n+1)
}.
\end{equation}

 The automorphism group of the Weyl-Heisenberg group is \cite{folland}
\begin{equation}
{\mathcal{A}ut}_{\mathcal{H}( n+1) }={\overline{\mathcal{O}\mathcal{A}ut}}_{\mathcal{H}(
n+1) }\otimes _{s}\mathcal{H}( n+1)
\end{equation}

\noindent where the Heisenberg group itself are the inner automorphisms.\ \ The
outer automorphisms are
\begin{equation}
{\mathcal{O}\mathcal{A}ut}_{\mathcal{H}( n+1) }\simeq \mathbb{Z}_{2}\otimes
\mathcal{D}\otimes \mathcal{S}p( 2n+2) .
\end{equation}

\noindent The matrix realization of this group and the group properties
are given in Appendix A.\ \ The central extension is
\begin{equation}
{\check{\mathcal{O}\mathcal{A}ut}}_{\mathcal{H}( n+1) }\simeq {\overline{\mathcal{O}\mathcal{A}ut}}_{\mathcal{H}(
n+1) }\simeq \mathbb{Z}_{2}\otimes \mathcal{D}\otimes \overline{\mathcal{S}p}(
2n+2) .
\end{equation}

\noindent Therefore, the relativity group may always be written
as
\begin{equation}
\check{\mathcal{G}}\subseteq \check{\mathcal{K}}\otimes _{s}\mathcal{N}
\end{equation}

\noindent where $\mathcal{K}$ is the homogeneous relativity group
that is a subgroup of the outer automorphisms, $\mathcal{K}\subseteq
{\mathcal{O}\mathcal{A}ut}_{\mathcal{H}( n+1) }$ and $\mathcal{N}\subseteq
\mathcal{H}( n+1) $ and $\check{\mathcal{G}}\subseteq {\mathcal{O}\mathcal{A}ut}_{\mathcal{H}(
n+1) }$.

\section{Homogeneous relativity group }

We determine in this section the homogeneous relativity group for
noninertial frames that satisfies two conditions.

1) It leaves invariant the Minkowski proper time line element. The
line element that is the invariant also of the inertial frames of
special relativity is valid also for the noninertial case.

2) It is a subgroup of the automorphism group of the Weyl-Heisenberg
group. Therefore, the Heisenberg commutation relations hold in all
states related by this relativity group and therefore from the previous
section must be a subgroup of ${\mathcal{O}\mathcal{A}ut}_{\mathcal{H}(
n+1) }$.

We name the relativity group that satisfies these two conditions
$\mathcal{U}b( 1,n) $ and use it to study relativistic noninertial
transformations.

The group $\mathcal{U}b( 1,n) $ is dependent on the scale $c$.\ \ We
show a homomorphism parameterized by $c$ satisfies the conditions
to define a In\"on\"u-Wigner contraction \cite{Glimore2}.\ \ This
contraction results in the\ \ Hamilton group that we have previously
shown is the relativity group for noninertial frames in the classical
($c\rightarrow \infty )$ context.\ \

\subsection{The group $\mathcal{U}b( 1,n) $ and its algebra}

The postulates of special relativity requires the invariance of
the Minkowski proper time line element
\begin{equation}
d \tau ^{2} = \eta _{a,b}d x^{a}d x^{b}
\end{equation}

\noindent with $a,b..=0,..n$ and $\eta $ is the diagonal matrix\ \ $\eta
=[\eta _{a,b}]=\mathrm{diag}\{-1,1,...1\}$ and units where $c=1$.

Consider the $2n+2$ dimensional time, position, energy, momentum
space\ \ $\mathbb{P}\simeq \mathbb{R}^{2n+2}$ with coordinates $\{z^{\alpha
}\}=\{x^{a},p^{a}\}$ where $\alpha ,\beta =1,...2n+2$, $a,b=0,1..n$.\ \ The
Minkowski metric may be considered to be a degenerate line element
on the cotangent space $T_{z}^{*}\mathbb{P}$
\begin{equation}
d \tau ^{2} = {\tilde{\eta }}_{\alpha ,\beta }d z^{\alpha }d z^{\beta
}
\label{nirg: Minkowski line element}
\end{equation}

\noindent where ${\tilde{\eta }}_{\alpha ,\beta }$ are the components
of the $(2n+2)\times (2n+2)$ dimensional matrix $\tilde{\eta }$
\[
\tilde{\eta }= \left[ {\tilde{\eta }}_{\alpha ,\beta }\right] =\left(
\begin{array}{ll}
 \left[ \eta _{a,b}\right]  & 0 \\
 0 & 0
\end{array}\right) .
\]

The group $\mathcal{G}\mathcal{L}( 2n+2,\mathbb{R}) $ of nonsingular
$(2n+2)\times (2n+2)$ matrices acts naturally on the cotangent space
$T_{z}^{*}\mathbb{P}$ with basis $\{d z^{\alpha }|_{z}\}$.\ \ Elements
$\Gamma $ of the subgroup $\mathcal{O}b( 1,n)  \subset \mathcal{G}\mathcal{L}(
2n+2,\mathbb{R}) $ that leave invariant the degenerate line element\ \ (19)
satisfy\ \ \cite{Glimore2}
\begin{equation}
{}^{t}\Gamma  \tilde{\eta } \Gamma =\tilde{\eta }.
\label{nirg: orthogonal line element condition}
\end{equation}

\noindent $\Gamma $ may be written in terms of\ \ $(n+1)\times (n+1)$
submatrices as $\Gamma =(\begin{array}{ll}
 \Lambda  & B \\
 \Xi  & A
\end{array})$ and therefore using (20)
\begin{equation}
\begin{array}{rl}
 \left( \begin{array}{ll}
 \eta  & 0 \\
 0 & 0
\end{array}\right)  & =\left( \begin{array}{ll}
 {}^{t}\Lambda  & {}^{t}\Xi  \\
 {}^{t}B & {}^{t}A
\end{array}\right) \left( \begin{array}{ll}
 \eta  & 0 \\
 0 & 0
\end{array}\right) \left( \begin{array}{ll}
 \Lambda  & B \\
 \Xi  & A
\end{array}\right)  \\
  & =\left( \begin{array}{ll}
 {}^{t}\Lambda  \eta  \Lambda  & {}^{t}\Lambda  \eta  B \\
 {}^{t}B \eta  \Lambda  & {}^{t}B \eta  B
\end{array}\right) .
\end{array}
\label{nirg: group leaving minkowski invariant}
\end{equation}

\noindent It follows immediately that $B=0$ and $\Lambda \in \mathcal{O}(
1,n) $ and as the $\det  \Gamma =\mathrm{det\Lambda } \det  A $,
$\det  A\neq 0$.\ \ \ Therefore,
\begin{equation}
\mathcal{O}b( 1,n)  \simeq \left( \mathcal{O}( 1,n) \otimes \mathcal{G}\mathcal{L}(
n+1,\mathbb{R}) \right) \otimes _{s}\mathcal{A}( {\left( n+1\right)
}^{2})
\end{equation}

\noindent with elements $\Gamma $\ \ and $\Gamma ^{-1}$ of the form
\begin{equation}
\Gamma =\left( \begin{array}{ll}
 \Lambda  & 0 \\
 \Xi  & A
\end{array}\right) ,\ \ \ \ \Gamma ^{-1}=\left( \begin{array}{ll}
 \Lambda ^{-1} & 0 \\
 -A^{-1} \Xi  \Lambda ^{-1} & A^{-1}
\end{array}\right) .
\label{nirg: Ob element and inverse}
\end{equation}

The homogeneous relativity group $\mathcal{U}b( 1,n) $ must be a
subgroup of the group of outer automorphisms ${\mathcal{O}\mathcal{A}ut}_{\mathcal{H}(
n+1) }$ and also the group $\mathcal{O}b( 1,n) $ that leaves the
degenerate line element invariant,\ \
\begin{equation}
\mathcal{U}b( 1,n) ={\mathcal{O}\mathcal{A}ut}_{\mathcal{H}( n+1)
}\cap \mathcal{O}b( 1,n) .
\end{equation}

The elements of the outer automorphism group are of the form $\Delta
\Sigma $\ \ where $\Delta \in \mathbb{Z}_{2}\otimes \mathcal{D}$
and $\Sigma \in \mathcal{S}p( 2n+2) $ as given in Appendix A. The
symplectic matrices satisfy the condition ${}^{t}\Sigma  \zeta
\Sigma  =\zeta $ and so $\Sigma ^{-1}=-\zeta {}^{t}\Sigma  \zeta
$ with $\zeta =(\begin{array}{ll}
 0 & \eta  \\
 -\eta  & 0
\end{array})$. This may also be written in terms of $(n+1)\times
(n+1)$ submatrices $\Sigma _{\mu ,\nu }$\ \ $\mu ,\nu =1,2$\ \ with
the matrix and inverse having the form
\begin{equation}
\Sigma =\left( \begin{array}{ll}
  \Sigma _{1,1} & \Sigma _{1,2} \\
 \Sigma _{2,1} & \Sigma _{2,2}
\end{array}\right) ,\ \ \ \Sigma ^{-1}=\left( \begin{array}{ll}
 \eta {}^{t}\Sigma _{2,2}\eta  & -\eta  {}^{t}\Sigma _{1,2}\eta
\\
 {}-{\eta  }^{t}\Sigma _{2,1}\eta  & \eta  {}^{t}\Sigma _{1,1}\eta
\end{array}\right) .
\end{equation}

Therefore, if $\Gamma $ in (23) is a subgroup of the outer automorphism
group, we have $\Sigma =\Delta ^{-1}\Gamma $ and so
\begin{equation}
\left( \begin{array}{ll}
  \Sigma _{1,1} & \Sigma _{1,2} \\
 \Sigma _{2,1} & \Sigma _{2,2}
\end{array}\right) =\Delta ^{-1}\left( \begin{array}{ll}
 \eta  \Lambda  \eta  & 0 \\
 -\eta  \Xi  \eta  & \eta A\eta
\end{array}\right)
\end{equation}

\noindent and\ \ as $\Gamma ^{-1 }=\Sigma ^{-1}\Delta ^{-1}$, we
also have
\begin{equation}
\left( \begin{array}{ll}
 {}^{t}\Sigma _{2,2} & {}^{t}\Sigma _{1,2} \\
 {}^{t}\Sigma _{2,1} &  {}^{t}\Sigma _{1,1}
\end{array}\right) =\Delta ^{-1}\left( \begin{array}{ll}
 {}{\eta  }^{t}A \eta  & 0 \\
 {}{\eta  }^{t}\Xi  \eta  & \eta  {}^{t}\Lambda  \eta
\end{array}\right)  \Delta ^{-1}=\Delta ^{-2}\left( \begin{array}{ll}
 {}{\eta  }^{t}A \eta  & 0 \\
 {}{\eta  }^{t}\Xi  \eta  & \eta  {}^{t}\Lambda  \eta
\end{array}\right) .
\end{equation}

\noindent Finally, equating to the inverse $\Gamma ^{-1}$ previously
calculated in (23)
\begin{equation}
\ \ \ \Gamma ^{-1}=\left( \begin{array}{ll}
 \Lambda ^{-1} & 0 \\
 -\Lambda ^{-1} \Xi  A^{-1} & A^{-1}
\end{array}\right) =\Delta ^{-2}\left( \begin{array}{ll}
 {}{\eta  }^{t}A \eta  & 0 \\
 {}-{\eta  }^{t}\Xi  \eta  & \eta  {}^{t}\Lambda  \eta
\end{array}\right)
\label{nirg: Ob element and inverse}
\end{equation}

\noindent from which it follows that $\Lambda ^{-1}=\Delta ^{-2}{}\eta
\ \ {}^{t}A \eta $ and $A^{-1}=\Delta ^{-2}\eta  {}^{t}\Lambda
\eta $.\ \ This has a solution if and only if $\Delta =\pm I_{n}\in
\mathbb{Z}_{2}\subset \mathcal{D}$ and ${}^{t}A=\eta  \Lambda ^{-1}\eta
$.\ \ Noting that $\Lambda ^{-1}=\eta {}^{t}\Lambda  \eta $ this
gives ${}^{t}A={}^{t}\Lambda $ and therefore $A=\Lambda $. Finally,
\begin{equation}
{}^{t}\Xi =\eta  \Lambda ^{-1} \Xi  \Lambda ^{-1}\eta = {}^{t}\Lambda
\eta  \Xi  \eta \ \ {}^{t}\Lambda .
\end{equation}

 Thus elements of\ \ $\mathcal{U}b( 1,n) $ have the form
\begin{equation}
\Gamma ( \Lambda ,\Xi ) =\left( \begin{array}{ll}
 \Lambda  & 0 \\
 \Xi  & \Lambda
\end{array}\right) .
\end{equation}

\noindent In this expression, $\Lambda \in \mathcal{O}( 1,n) $.
The group multiplication and inverse of $\mathcal{U}b( 1,n) $ are
\begin{gather}
\begin{array}{rl}
 \Gamma ( \Lambda ,\Xi )  & =\Gamma ( \Lambda ^{\prime },\Xi ^{\prime
}) \Gamma ( \Lambda ^{{\prime\prime}},\Xi ^{{\prime\prime}})  \\
  & =\Gamma ( \Lambda ^{\prime } \Lambda ^{{\prime\prime}},\Xi ^{\prime
}\Lambda ^{{\prime\prime}}+{\Lambda  }^{\prime }\Xi ^{{\prime\prime}})
\end{array},
\label{nirg: Ub group transformation}
\end{gather}
\begin{equation}
{\Gamma ( \Lambda ,\Xi ) }^{-1}= \Gamma ( \Lambda ^{-1},-{\Lambda
}^{-1}\Xi  \Lambda ^{-1}) .
\end{equation}

The Lorentz group is the subgroup $\Gamma ( \Lambda ,0) $.\ \ The
matrix components of the Lorentz matrices may be given as the usual
expressions in regular and hyperbolic trigonometry terms of the
rotation angles and boost angles.

The elements $\Gamma ( I_{n}, \Xi  ) $ define an abelian normal
subgroup with group multiplication,\ \ inverse and automorphisms
given by
\begin{gather}
\begin{array}{l}
 \Gamma ( I_{n},\Xi ^{\prime }) \Gamma ( I_{n},\Xi ^{{\prime\prime}})
=\Gamma ( I_{n},\Xi ^{\prime }+\Xi ^{{\prime\prime}}) , \\
 {\Gamma ( I_{n},\Xi ) }^{-1}= \Gamma ( I_{n},- \Xi  ) ,
\end{array}
\end{gather}
\begin{equation}
\begin{array}{rl}
 \Gamma ( \Lambda ^{\prime },\Xi ^{\prime }) \Gamma ( I_{n},\Xi
) {\Gamma ( \Lambda ^{\prime },\Xi ^{\prime }) }^{-1} & =\Gamma
( I_{n},\Lambda ^{\prime } \Xi  {\Lambda ^{\prime }}^{-1})
\end{array}.
\label{nirg: inner automoprhisms of Ub}
\end{equation}

Also, for this subgroup ${}^{t}\Xi =\eta  \Xi  \eta $ and the matrix
components of $\Xi $ are the $(n+1)(n+2)/2$ real parameters $\xi
_{b}^{a}=\eta ^{a,d}\eta _{b,c}\xi _{d}^{c}$.\ \ Therefore $\Gamma
( I_{n},M) \in \mathcal{A}( (n+1)(n+2)/2) $. Consequently the full
group is
\begin{equation}
\mathcal{U}b( 1,n)  \simeq \mathcal{O}( 1,n) \otimes _{s}\mathcal{A}(
\left( n+1\right) \left( n+2\right) /2) .
\label{nirg: Ub definition}
\end{equation}

 It can be shown that it does not admit an algebraic central extension
and therefore the central extension of this group is simply its
cover
\begin{equation}
\overline{\mathcal{U}b}( 1,n) \simeq \overline{\mathcal{O}}( 1,n)
\otimes _{s}\mathcal{A}( \left( n+1\right) \left( n+2\right) /2)
.
\label{nirg: definition of Ub}
\end{equation}

A general element of the algebra of $\mathcal{U}b( 1,n) $ is $Z=\lambda
^{a,b}L_{a,b}+\xi ^{a,b}M_{a,b}$ . Note that as $\xi ^{a,b}=\xi
^{b,a}$, that $M_{a,b}=M_{b,a}$.\ \ The\ \ Lie algebra relations
may be directly computed to be
\begin{equation}
\begin{array}{l}
 \left[ L_{a,b},L_{c,d}\right] =-L_{b,d} \eta _{a,c}+L_{b,c} \eta
_{a,d}+L_{a,d} \eta _{b,c}-L_{a,c} \eta _{b,d}, \\
 \left[ L_{a,b},M_{c,d}\right] =-M_{b,d} \eta _{a,c}-M_{b,c} \eta
_{a,d}+M_{a,d} \eta _{b,c}+M_{a,c} \eta _{b,d}, \\
 \left[ M_{a,b},M_{c,d}\right] =0.
\end{array}
\label{nirg: Ub algebra}
\end{equation}

 The $M_{a,b}$ abelian generators transform as a symmetric $(0,2)$
tensor under the Lorentz generators $L_{a,b}$ .

Returning to the group, the transformation equations are\ \ $d \tilde{z}
=\Gamma  d z$.\ \ Using the definition of\ \ $\Gamma $ in (35)\ \ results
in
\begin{equation}
\begin{array}{l}
 d \tilde{x} = \Gamma  d x, \\
 d \tilde{p} = \Gamma  d p + \Xi  d x,
\end{array}
\end{equation}

\noindent that in component form are (with units where $c=1$)
\begin{equation}
\begin{array}{l}
 d {\tilde{x}}^{a} =\lambda _{b}^{a} d x^{b}, \\
 d {\tilde{p}}^{a} =\lambda _{b}^{a} d p^{b}+ \xi _{b}^{a} d x^{b}.
\end{array}
\end{equation}

\noindent Then, the proper time line element is invariant as required
by construction\ \
\begin{equation}
\begin{array}{rl}
 d \tau ^{2} & =\eta _{a,b}d {\tilde{x}}^{a}d {\tilde{x}}^{b}=\eta
_{a,b}\lambda _{c}^{a} d x^{c} \lambda _{d}^{b} d x^{d} \\
  & =\eta _{a,b}d x^{a}d x^{b}.
\end{array}
\end{equation}

The $\lambda _{c}^{a} $ are the components of the Lorentz transformation
that as usual depend on the relative rotation angle and hyperbolic
boost\ \ angle.\ \ The mass $\mu $ satisfies
\begin{equation}
\begin{array}{rl}
 c^{2 }d {\tilde{\mu }}^{2} & =\eta _{a,b}d {\tilde{p}}^{a}d {\tilde{p}}^{b}
\\
  & =\eta _{a,b}( \lambda _{c}^{a} d p^{c}+\xi _{c}^{a} d x^{c})
\left(  \lambda _{d}^{c} d p^{d} + \xi _{d}^{b} d x^{d}\right)
\\
  & =c^{2 }d \mu ^{2}+\eta _{a,b}\xi _{c}^{a}\xi _{d}^{b} d x^{c}d
x^{d}+2 \eta _{a,b}\xi _{c}^{a} \lambda _{d}^{b} d x^{c}d p^{d}.
\end{array}
\end{equation}

\noindent From basic dimensional analysis, the $\xi _{c}^{a}$ have
the dimensions of force or power (in units with $c=1$ these are
the same).\ \ It is a symmetric tensor satisfying $\xi _{b}^{a}=\eta
^{a,c}\eta _{b,d}\xi _{c}^{d}$ that transforms as an (1,1) tensor
under the Lorentz transformation
\begin{equation}
{\tilde{\xi }}_{b}^{a}=\lambda _{c}^{a}\lambda _{b}^{d}\xi _{d}^{c}.
\end{equation}

These are the properties of a power-force stress tensor that is
the proper time derivative of the energy-momentum stress tensor.

\noindent The rate of change of the mass squared with respect to
the proper time is given by
\begin{equation}
\frac{d {\tilde{\mu }}^{2}}{d \tau ^{2}}=\frac{d \mu ^{2}}{d \tau
^{2}}+\frac{1}{c^{2}}\eta _{a,b}\xi _{c}^{a}V^{c}( \xi _{d}^{b}
V^{d}+2\ \ \lambda _{d}^{b} F^{d})
\end{equation}

\noindent where $V^{a}=\frac{d x^{a}}{d \tau }$ is the {\itshape
four} velocity and $F^{a}= \frac{d p^{a}}{d \tau }$ is the {\itshape
four} force for the case $n=3$.\ \ \

\subsection{Three notation}

Further insight into the physical meaning of the group may be obtained
by converting to $n+1$ notation that for $n=3$ is the familiar {\itshape
three} notation $\{x^{a}\}=\{t,\frac{1}{c}q^{i}\}$, $\{p^{a}\}=\{\frac{1}{c}e,p^{i}\}$,
$i,j=1,..n$.\ \ The Lorentz matrix $\Lambda ( \alpha ,\beta ) $
parameterized by rotation angles $\alpha ^{i,j}=-\alpha ^{j,i}$
and hyperbolic boost rotations $\beta ^{i}$ that have the usual
form.\ \ For simplicity, we give here only the case $\alpha ^{i,j}=0$,
\begin{equation}
\Lambda ( 0,\beta ) = \left( \begin{array}{ll}
 \lambda _{0}^{0} & \lambda _{i}^{0} \\
 \lambda _{0}^{j} & \lambda _{i}^{j}
\end{array}\right) =\left( \begin{array}{ll}
 \cosh  \left( \beta \right)  & \sinh  \left( \beta \right)  \frac{
\beta _{i}}{c \beta }  \\
 c \sinh  \left( \beta  \right) \frac{\beta ^{j} }{\beta }\ \ \
& {\delta  }_{i}^{j} +\left(  \cosh ( \beta ) -1\right)  \frac{
\beta ^{j}\beta _{i}}{\beta ^{2}}
\end{array}\right) ,
\end{equation}

\noindent where $\beta ^{2}=\beta _{i}\beta ^{i}$. Indices are raised
and lowered with the kronecker delta $\delta _{i,j}$. As usual,
we identify velocity as $v^{i} =c \frac{\beta ^{i} }{\beta } \tanh
(\beta  ) $ and define\ \ $\gamma ( \beta ) = \cosh ( \beta ) =\lambda
_{0}^{0}$ or equivalently $\gamma ( v) ={(1-{(\frac{v}{c})}^{2})}^{-\frac{1}{2}}$.\ \

The velocity {\itshape four} vectors are given as usual by $\{V^{0},V^{i}\}=\{\gamma
,\gamma  v^{i}\}=\gamma \{1,\frac{d x^{i}}{d t}\}$ where\ \ $\gamma
=\frac{d t}{d \tau }$.\ \ The {\itshape four} force likewise is
$\{F^{0},F^{i}\}=\{\gamma  r,\gamma  f^{i}\}$ where $f^{i}=\frac{d
p^{i}}{d t}$ and $r=\frac{d e}{d t}$ and\ \ $f^{i}$ has the dimensions
of force and $r$ has the dimensions of power.\ \ The power-force-stress
components are
\begin{equation}
\Xi = \left( \begin{array}{ll}
 \xi _{0}^{0} & \xi _{i}^{0} \\
 \xi _{0}^{j} & \xi _{i}^{j}
\end{array}\right) =\gamma \left( \begin{array}{ll}
 \frac{1}{ c}r & -f_{i} \\
 f^{j} & \frac{1}{ c}m^{j,i}
\end{array}\right) .
\end{equation}

Therefore, the transformation equations for the position, time,
momentum, energy basis is
\begin{equation}
\begin{array}{l}
 d\tilde{t}= \gamma  d t+\frac{1}{c} \lambda _{i}^{0}\ \ d q^{i},
\\
 d{\tilde{q}}^{i}=\lambda _{j}^{i}d q^{j} +c \lambda _{0}^{i}\ \ \ d
t, \\
 d{\tilde{p}}^{i}=\lambda _{j}^{i}d p^{j} +\lambda _{i}^{0}\ \ \ d
t, \\
 d\tilde{e}= \gamma  d e+c \lambda _{i}^{0}\ \ d p^{i}-\gamma  f_{i}
d q^{i}+c \gamma  r d t\ \ .
\end{array}
\label{nirg: genral tx equations}
\end{equation}

\noindent For $n=1$ these are simply
\begin{equation}
\begin{array}{l}
 d\tilde{t}= \gamma ( v)  \left( \ \ d t+\frac{1}{c^{2}} v d q \right)
, \\
 d\tilde{q}=\gamma ( v) \left( d q +v d t \right) , \\
 d\tilde{p}=\gamma ( v) \left( d p^{j} +\frac{1}{c^{2}}v d e+f d
t+\frac{1}{c^{2}}\ \ m\ \ d q\right) , \\
 d\tilde{e}= \gamma ( v)  \left( d e+v d p-f d q+r d t \right)
.
\end{array}
\end{equation}

\noindent and the corresponding group parameter transformations
using (31) are
\begin{gather}
\gamma ( v) \left( \begin{array}{ll}
 1 & v \\
 v & 1
\end{array}\right) =\gamma ( v^{{\prime\prime}}) \gamma ( v^{\prime
}) \left( \begin{array}{ll}
 1 & v^{\prime } \\
 v^{\prime } & 1
\end{array}\right) \left( \begin{array}{ll}
 1 & v^{{\prime\prime}} \\
 v^{{\prime\prime}} & 1
\end{array}\right)
\end{gather}
\begin{equation}
\gamma ( v) \left( \begin{array}{ll}
 r & f \\
 f & m
\end{array}\right) =\gamma ( v^{{\prime\prime}}) \gamma ( v^{\prime
}) \left( \left( \begin{array}{ll}
 r^{\prime } & f^{\prime } \\
 f^{\prime } & m^{\prime }
\end{array}\right) \left( \begin{array}{ll}
 1 & v^{{\prime\prime}} \\
 v^{{\prime\prime}} & 1
\end{array}\right) +\left( \begin{array}{ll}
 1 & v^{\prime } \\
 v^{\prime } & 1
\end{array}\right) \left( \begin{array}{ll}
 r^{{\prime\prime}} & f^{{\prime\prime}} \\
 f^{{\prime\prime}} & m^{{\prime\prime}}
\end{array}\right) \right)
\end{equation}

\noindent and so
\begin{equation}
\begin{array}{l}
 v=\left( v^{{\prime\prime}}+ v^{\prime } \right) /\left( 1+ \frac{v^{\prime
}v^{{\prime\prime}}}{c^{2}}\right) , \\
 f=\left( f^{{\prime\prime}}+f^{\prime } \mathrm{+}\frac{1 }{c^{2}}\left(
r^{\prime } v^{{\prime\prime}}-v^{\prime } r^{{\prime\prime}}\right)
\right) /\left( 1+ \frac{v^{\prime }v^{{\prime\prime}}}{c^{2}}\right)
, \\
 r=\left( r^{{\prime\prime}}\mathrm{+}r^{\prime }-f^{\prime } v^{{\prime\prime}}+v^{\prime
}f^{{\prime\prime}} \right) /\left( 1+ \frac{v^{\prime }v^{{\prime\prime}}}{c^{2}}\right)
. \\
 m=\left( m^{{\prime\prime}}\mathrm{+}m^{\prime }+f^{\prime } v^{{\prime\prime}}-v^{\prime
}f^{{\prime\prime}} \right) /\left( 1+ \frac{v^{\prime }v^{{\prime\prime}}}{c^{2}}\right)
.
\end{array}
\end{equation}

Consider next the algebra,\ \ we have the infinitesimal parameter
correspondence
\begin{equation}
\lambda ^{0,i}=\frac{1}{c}\beta ^{i}, \lambda ^{j,i}=\alpha ^{i,j},
\xi ^{0,0}=\frac{1}{ c}r,\xi ^{j,0}=f^{j}, \xi ^{j,i}=\frac{1}{
c}m^{j,i}
\end{equation}

\noindent where $\alpha ^{i,j}=-\alpha ^{i,j}$ and $m^{i,j}=m^{j,i}$
with the corresponding generators
\begin{equation}
L_{0,j}=c K_{j}, L_{i,j}=J_{i,j}, M_{0,0}=c R, M_{i,0}=N_{i}, M_{i,j}=c
M \mbox{}^{\circ}_{i,j}.
\end{equation}

A general element of the algebra is\ \
\begin{equation}
Z= \alpha ^{i,j}J_{i,j}+\beta ^{i}K_{i}+f^{i}N_{i} +r R+ m^{i,j}M
\mbox{}^{\circ}_{i,j}.
\end{equation}

 The nonzero commutators of the Lie algebra (37) written in terms
of these generators is
\begin{equation}
\begin{array}{l}
 \left[ J_{i,j},J_{k,l}\right] =-J_{j,l} \delta _{i,k}+J_{j,k} \delta
_{i,l}+J_{i,l} \delta _{j,k}-J_{i,k} \delta _{j,l}, \\
 \left[ J_{i,j},J_{k,l}\right] =-J_{j,l} \delta _{i,k}+J_{j,k} \delta
_{i,l}+J_{i,l} \delta _{j,k}-J_{i,k} \delta _{j,l}, \\
 \left[ J_{i,j},K_{k}\right] =-K_{j} \delta _{i,k}+K_{i} \delta
_{j,k},\ \ \ \left[ K_{i},K_{k}\right] =\frac{1}{c^{2}}J_{i,k},
\\
 \left[ J_{i,j},N_{k}\right] =-N_{j} \delta _{i,k}+N_{i} \delta
_{j,k},\ \ \ \left[ K_{i},N_{k}\right] =-M_{i,k}-R \delta _{i,k},
\\
 \left[ K_{i},R\right] =-\frac{2 }{c^{2}}N_{i}, \\
 \left[ J_{i,j},M \mbox{}^{\circ}_{k,l}\right] =M \mbox{}^{\circ}_{j,l}
\delta _{i,k}-M \mbox{}^{\circ}_{j,k} \delta _{i,l}+M \mbox{}^{\circ}_{i,l}
\delta _{j,k}+M \mbox{}^{\circ}_{i,k} \delta _{j,l}, \\
 \left[ K_{i},M \mbox{}^{\circ}_{k,l}\right] =\frac{-1}{c^{2}}\left(
N_{l} \delta _{i,k}+N_{k} \delta _{i,l}\right) .
\end{array}
\label{nirg: Ub Algebra in three form}
\end{equation}

\subsection{Contraction in the limit $c\rightarrow \infty $}

 The scaling with $c$ given in (54) satisfies the conditions for
an In\"on\"u-Wigner contraction \cite{Glimore2} $c\rightarrow \infty
$ for which the nonzero contracted commutators are
\begin{equation}
\begin{array}{l}
 \left[ J_{i,j},J_{k,l}\right] =-J_{j,l} \delta _{i,k}+J_{j,k} \delta
_{i,l}+J_{i,l} \delta _{j,k}-J_{i,k} \delta _{j,l}, \\
 \left[ J_{i,j},K_{k}\right] =-K_{j} \delta _{i,k}+K_{i} \delta
_{j,k}, \\
 \left[ J_{i,j},N_{k}\right] =-N_{j} \delta _{i,k}+N_{i} \delta
_{j,k}, \\
 \left[ K_{i},N_{k}\right] =-M \mbox{}^{\circ}_{i,k}-R \delta _{i,k},
\\
 \left[ J_{i,j},M \mbox{}^{\circ}_{k,l}\right] =-M \mbox{}^{\circ}_{j,l}
\delta _{i,k}-M \mbox{}^{\circ}_{j,k} \delta _{i,l}+M \mbox{}^{\circ}_{i,l}
\delta _{j,k}+M \mbox{}^{\circ}_{i,k} \delta _{j,l}.
\end{array}
\end{equation}

 The subgroup spanned by $\{J_{i,j}, K_{i}, N_{i}, R\}$ is the algebra
of the Hamilton group $\mathcal{H}a( n) $.\ \ The full\ \ algebra
of the group that we call $\mathcal{U}bc( n) $ that is defined by
\begin{equation}
\mathcal{U}bc( n) = \mathcal{O}( n) \otimes _{s}\mathcal{A}( n(
n+1) /2) \otimes _{s}\mathcal{H}( n)
\end{equation}

\noindent where $\{J_{i,j}\}$ are the generators of $\mathcal{O}(
n) $, $\{M_{i,j}\}$ are the generators of $\mathcal{A}( n( n+1)
/2) $ and $\{K_{i},N_{i},R\}$ are the generators of the Weyl-Heisenberg
group $\mathcal{H}( n) $.\ \

Furthermore, we take the limit\ \ $c\rightarrow \infty $ so that
$\beta \rightarrow 0$ in such a way that $c \beta =\tilde{\beta
}$ is finite,
\begin{equation}
v=\operatorname*{\lim }\limits_{c\,\rightarrow \:\infty }c \tanh
\frac{\tilde{\beta }}{c}\ \ =\tilde{\beta },\ \ \ \ \ \ \operatorname*{\lim
}\limits_{c\,\rightarrow \:\infty } \gamma ( \frac{\tilde{\beta
}}{c}) =1
\end{equation}

\noindent The basis transformation equations (46) contract to\ \ the
expected transformation equations in the limit \cite{Low7}
\begin{equation}
\begin{array}{l}
 d\tilde{t}=\ \ d t, \\
 d{\tilde{q}}^{i}={\lambda ( \alpha ) }_{j}^{i}d q^{j} +v^{i}\ \ d
t, \\
 d{\tilde{p}}^{i}={\lambda ( \alpha ) }_{j}^{i}d p^{j} +f^{i}d t,
\\
 d\tilde{e}= d e+v_{i}d p^{i}-f_{i} d q^{i}+r d t.
\end{array}
\end{equation}

\noindent where the ${\lambda ( \alpha ) }_{j}^{i}$ are now the
components of a rotation matrix, $\mathcal{O}( n) $.

\subsection{Contraction from $\mathcal{U}( 1,n) $ of reciprocal
relativity in limit $b\rightarrow \infty $}

Similar considerations using the Born-Green metric
\begin{equation}
d s^{2}= d x^{a}d x^{b }+\frac{1}{b^{2}}d p^{a}d p^{b}
\end{equation}

\noindent instead of the degenerate Minkowski line element results
in a reciprocal relativity theory where the homogeneous group is
$\mathcal{U}( 1,n) $ \cite{Low5},\cite{Low6}.\ \ This theory requires
the introduction of a constant $b$ with the dimensions of force
that may be taken to be on of the three universal constants that
form the natural dimensional basis $\{c,b,\hbar \}$.\ \ This is
instead of the usual choice $\{c,G,\hbar \}$ where $G$ is the gravitational
coupling constant. $G$ may be defined in terms of $b$ (or vice versa)
as $G=\alpha _{b}\frac{c^{4}}{b}$ where $\alpha _{b}$ is the dimensionless
gravitational coupling constant.

The algebra for the $\mathcal{U}( 1,n) $ group is
\begin{equation}
\begin{array}{l}
 \left[ L_{a,b},L_{c,d}\right] =-L_{b,d} \eta _{a,c}+L_{b,c} \eta
_{a,d}+L_{a,d} \eta _{b,c}-L_{a,c} \eta _{b,d}, \\
 \left[ L_{a,b},M_{c,d}\right] =-M_{b,d} \eta _{a,c}-M_{b,c} \eta
_{a,d}+M_{a,d} \eta _{b,c}+M_{a,c} \eta _{b,d}, \\
 \left[ M_{a,b},M_{c,d}\right] =-\frac{1}{b^{2}}\left( L_{b,d} \eta
_{a,c}+L_{b,c} \eta _{a,d}+L_{a,d} \eta _{b,c}+L_{a,c} \eta _{b,d}\right)
.
\end{array}
\label{nirg: Ub algebra}
\end{equation}

This satisfies the condition for an In\"on\"u-Wigner contraction
to the algebra for the $\mathcal{U}b( 1,n) $ group given in (37)
in the limit $b\rightarrow \infty $. (This is where the notation
'$\mathcal{U}b$' originates.)

The $\mathcal{U}b( 1,n) $ group defines the limiting behavior of
reciprocal relativity in the limit of small interactions where the
systems are {\itshape almost inertial.\ \ }These are the {\itshape
expected} transformations from an analysis of noninertial frames
in a special relativistic context.

\section{ Summary}

We started by noting that neither special relativity nor general
relativity address the problem of how clocks of noninertial states
due to forces other than gravity where gravity is negligible and
therefore the manifold is flat.

The hypothesis that the Minkowski proper time line element is invariant
in these noninertial states (that includes the inertial states of
special relativity) and also requiring that the Heisenberg commutation
relations hold in all noninertial states results in the noninertial
relativity group $\mathcal{U}b( 1,n) $.\ \ This group give expected
transformations to noninertial states in terms of a power-force
stress tensor that is the proper time derivative of the energy-momentum
stress tensor.\ \ A general formula for the {\itshape non-quantum
classical} decay rate of mass for noninertial frames is derived.

The $\mathcal{U}b( 1,n) $ group is also the $b\rightarrow \infty
$ of the $\mathcal{U}( 1,n) $ group of reciprocal relativity described
in \cite{Low5},\cite{Low6}.\ \ This gives an understanding of the
behavior of reciprocal relativity in the small interaction limit
(that is, small forces relative to $b$)\ \ that is analogous to
the manner in which the Euclidean group that is the homogeneous
group of the Galilei group gives the small velocity limit, relative
to $c$, of the Lorentz group.

Spacetime is an invariant subspace under the actions of the $\mathcal{U}b(
1,n) $ group and therefore is observer independent or absolute.
In this limit,\ \ there is an apparent global inertial frame that
all observers agree on.\ \ Forces appear to be relative to this
frame rather than being strictly relative to particle states. Forces
and the power-force-stress energy tensor are simply additive and
unbounded. Velocities are bounded by $c$ and strictly relative to
particle states.

In the $c\rightarrow \infty $ limit yields the classical {\itshape
nonrelativistic} Hamilton theory that describes particles undergoing
general noninertial motion.\ \ \ In this case, there is an apparent
global inertial rest frame that all observers agree on.\ \ Forces
and velocities appear to be relative to this frame rather than being
strictly relative to particle states.\ \ Forces and velocities are
simply additive and unbounded.

In a companion paper, the quantum mechanics that results from the
projective representations of the inhomogeneous $\mathcal{U}b( 1,n)
$ group for these noninertial states is studied using the same method
of projective representations of the inhomogeneous Lorentz group
for the inertial states of special relativistic quantum mechanics
\cite{wigner},\cite{Weinberg1}.

\section{Appendix A: Automorphisms of the Weyl-Heisenberg group
$\mathcal{H}( m) $}\label{nirg: Appendix A automorphisms of Heisenberg
group}

The Weyl-Heisenberg group $\mathcal{H}( m) $ has the matrix realization
as a subgroup of\ \ $\mathcal{G}\mathcal{L}( 2m+2) $ given by
\[
 \Upsilon ( z,\iota ) =\left( \begin{array}{lll}
 I_{2m} & 0 & z \\
  ^{t}\left( \zeta  z\right)   & 1 & \iota  \\
 0 & 0 & 1
\end{array}\right)
\]

\noindent where $z\in \mathbb{R}^{2m}$, $\iota \in \mathbb{R}$.\ \ The
group multiplication and inverse are
\begin{equation}
\Upsilon ( z^{\prime },\iota ^{\prime }) \cdot \Upsilon ( z,\iota
) =\Upsilon ( z+z^{\prime },\iota +\iota ^{\prime }+z^{\prime }\cdot
\zeta  \cdot z) ,\ \ {\Upsilon ( z,\iota ) }^{-1}=\Upsilon ( -z,-\iota
) .
\end{equation}

$\mathcal{H}( m) $ has a group manifold diffeomorphic to $\mathbb{R}^{2m+1}$
and is therefore simply connected and is its own cover, $\overline{\mathcal{H}}(
m) \simeq \mathcal{H}( m) $,

Elements $\Omega $ of the linear automorphism group ${\mathrm{aut}}_{\mathcal{H}(
m) }\subset \mathcal{G}\mathcal{L}( 2m+2) $\ \ that is a matrix
group that may be represented by $(2m+2)\times (2m+2)$ nonsingular
matrices $\Omega $ that satisfy
\begin{equation}
\Omega  \Upsilon ( w^{\prime },\iota ^{\prime })  {\Omega }^{-1}=\Upsilon
( w^{{\prime\prime}},\iota ^{{\prime\prime}}) .
\label{nirg: automorphism condition}
\end{equation}

The proof given by Folland \cite{folland} shows that the most general
matrix group with this property is $\Omega \in \mathcal{A}_{\mathcal{H}}$
where
\begin{equation}
\mathcal{A}_{\mathcal{H}}\simeq \mathbb{Z}_{2}\otimes _{s}\mathcal{D}\otimes
_{s}\mathcal{H}\mathcal{S}p( 2n+2) .
\label{nirg: autH definition}
\end{equation}

 This can be shown by direct matrix computation that the most general
elements of\ \ $\mathcal{G}\mathcal{L}( 2m+2) $ satisfying (62)
are
\begin{equation}
\Omega ( \epsilon ,\delta ,\Sigma ,z,\iota ) =\left( \begin{array}{lll}
 \delta  \Sigma  & 0 & z \\
 -^{t}z \zeta  \Sigma  & \epsilon  \delta ^{2} & \iota  \\
 0 & 0 & \epsilon
\end{array}\right)  ,
\label{matrix reps of AutH}
\end{equation}

\noindent where $A\in \mathcal{S}p( 2m) $, $z\in \mathbb{R}^{2m}$,
$\delta ,\iota  \in \mathbb{R}$, $\epsilon =\pm 1$\ \ and $\zeta
$ is the $2m\times 2m$ symplectic matrix. The group multiplication
and inverse are
\begin{equation}
\begin{array}{l}
 \begin{array}{rl}
 \Omega ( \epsilon ^{{\prime\prime}},\delta ^{{\prime\prime}},\Sigma
^{{\prime\prime}},z^{{\prime\prime}},\iota ^{{\prime\prime}})  &
=\Omega ( \epsilon ,\delta ,\Sigma ,z,\iota ) \Omega ( \epsilon
^{\prime },\delta ^{\prime },\Sigma ^{\prime },z^{\prime },\iota
^{\prime })
\end{array} \\
 =\Omega ( \epsilon  \epsilon ^{\prime },\delta  \delta ^{\prime
},\Sigma  \Sigma ^{\prime },\epsilon ^{\prime }z+\delta  \Sigma
z^{\prime },\epsilon ^{\prime }r+\epsilon  \delta ^{2}r^{\prime
}{- }^{t}z \zeta  \Sigma  z^{\prime } )  \\
 {\Omega ( \epsilon ,\delta ,\Sigma ,z,r) }^{-1}=\Omega ( \epsilon
,\delta ^{-1},\delta ^{-1}\Sigma ^{-1},-\epsilon  \delta ^{-1}\Sigma
^{-1}z,-\delta ^{-2}r)
\end{array}
\end{equation}

\noindent Note that
\begin{equation}
\begin{array}{l}
 \Omega ( 1,1,I_{2n},z,\iota ) \simeq \Upsilon ( z,\iota ) \in \mathcal{H}(
m)  \\
 \Omega ( 1,1,\Sigma ,0,0) \simeq \Sigma \in \mathcal{S}p( 2n)
\\
 \Omega ( \epsilon ,\delta ,1,0,0) \simeq \Delta ( \epsilon ,\delta
) \in \mathcal{D}\otimes \mathbb{Z}_{2}
\end{array}
\end{equation}

\noindent with $\mathrm{Det} \Omega =\delta ^{2m+2}$ where
\begin{equation}
\Delta ( \epsilon ,\delta ) =\left( \begin{array}{lll}
 \delta   & 0 & 0 \\
 0 & \epsilon  \delta ^{2} & 0 \\
 0 & 0 & \epsilon
\end{array}\right)  ,\ \ \Sigma \simeq \left( \begin{array}{lll}
 \Sigma   & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & 1
\end{array}\right)  .
\label{matrix reps of AutH}
\end{equation}

The automorphism group may be written as
\begin{equation}
\Omega ( \epsilon ,\delta ,\Sigma ,\tilde{z},\tilde{\iota }) =\Delta
( \epsilon ,\delta ) \Sigma  \Upsilon ( z,\iota )
\end{equation}

\noindent where
\begin{equation}
\tilde{z}= \delta  \Sigma  z,\ \ \tilde{\iota }= \epsilon  \delta
^{2}\iota .
\end{equation}

The above discussion gives the automorphism group $\mathcal{A}_{\mathcal{H}}$
that is a matrix group through direct computation of (62) with the
matrix group $\mathcal{H}( m) $.\ \ The central extension of this
group also defines automorphisms of $\mathcal{H}( m) $. This is
true because these elements are in the center of the group and as
they commute with all elements, they always satisfy (8).\ \ The
central extension of a matrix group is not necessarily a matrix
group \cite{Hall}.\ \ This is true in particular for $\mathcal{A}_{\mathcal{H}}$
which is why the matrix calculation does not give the central elements.
Therefore, to obtain the full automorphism group, we must calculate
the central extension\ \ ${\mathcal{A}ut}_{\mathcal{H}}={\check{\mathcal{A}}}_{\mathcal{H}}$.

The method of determining the central extension is given in \cite{Azcarraga},\cite{Weinberg1},\cite{Low8}.
It first requires the determination of the algebraic central extension
of the Lie algebra. Using the methods given in these references,
it may be shown that the algebra of ${\mathrm{aut}}_{\mathcal{H}}$
does not have a central extension.

\noindent Therefore, the central extension is simply the universal
cover of the group. $\mathcal{D}$ has a group manifold diffeomorphic
to $\mathbb{R}$ and\ \ $\mathcal{H}( m) $ has a group manifold diffeomorphic
to $\mathbb{R}^{2m+1}$.\ \ The fundamental homotopy group for the
symplectic group is the integers under addition and so $\mathcal{S}p(
m) \simeq \overline{\mathcal{S}p}( m) /\mathbb{Z}$.\ \ \ Therefore,
\begin{equation}
\begin{array}{rl}
 {\mathcal{A}ut}_{\mathcal{H}( m) } & \simeq {\check{\mathcal{A}}}_{\mathcal{H}(
m) }\simeq {\overline{\mathcal{A}}}_{\mathcal{H}( m) } \\
  & =\left( \mathbb{Z}_{2}\otimes \mathcal{D}\otimes \overline{\mathcal{S}p}(
2m) \right) \otimes _{s}\mathcal{H}( m)
\end{array}
\end{equation}

\noindent This issue of the central extensions is important for
the quantum mechanical treatment where the projective representations
are required.


TITLE: About $\mathcal{L}(V,W)$
QUESTION [3 upvotes]: Let $V,W$ are two vector space and let $S\subseteq V$. Define: $$S^{0}=\{T\in\mathcal{L}(V,W)\mid~T(x)=0, \forall x\in S\}$$ The problem aks me to verify $S^{0}$ is a subspace of $V$ and if $V_1,V_2$ are two subspaces, then $$(V_1+V_2)^{0}=V^0_1\cap V^0_2$$ Please help me to overcome the second part. I did the first one. Thank you!

REPLY [3 votes]: You can do this by a double inclusion, suppose that $W\in(V_1+V_2)^0$, then for every $x\in V_1$ we have that $W(x)=0$ and similarly for $x\in V_2$, therefore $W\in V_1^0$ and $W\in V_2^0$, so $W$ is in the intersection.
The opposite direction is similar, and I leave it to you. Note that you will have to use the definition of $V_1+V_2$ and the linearity of $W$ here.