algebroid structures and extended symmetries AlgebroidStructuresAndExtendedSymmetries 2009-01-10 20:30:30 2009-01-10 20:30:30 Definition double quantum algebroid quantum algebroid algebroid structures and extended symmetries % this is the default PlanetPhysics preamble. as your % almost certainly you want these \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} \def\baselinestretch{1.1} \hyphenation{prod-ucts} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \section{Algebroid structures and Quantum Algebroid Extended Symmetries.} \begin{definition} An \emph{algebroid structure} $A$ will be specifically defined to mean either a ring, or more generally, any of the specifically defined algebras, but \emph{with several objects} instead of a single object, in the sense specified by Mitchell (1965). Thus, an {\em algebroid} has been defined (Mosa, 1986a; Brown and Mosa 1986b, 2008) as follows. An \textit{$R$-algebroid } $A$ on a set of ``objects" $A_0$ is a directed graph over $A_0$ such that for each $x,y \in A_0,\; A(x,y)$ has an $R$-module structure and there is an $R$-bilinear function $$ \circ : A(x,y) \times A(y,z) \to A(x,z)$$ $(a , b) \mapsto a\circ b$ called ``composition" and satisfying the associativity condition, and the existence of identities. \end{definition} \begin{definition} A {\em pre-algebroid} has the same structure as an algebroid and the same axioms except for the fact that the existence of identities $1_x \in A(x,x)$ is not assumed. For example, if $A_0$ has exactly one object, then an $R$-algebroid $A$ over $A_0$ is just an $R$-algebra. An ideal in $A$ is then an example of a pre-algebroid. \end{definition} Let $R$ be a commutative ring. %%\cite{M1,M2,A} An \textit{$R$-category }$\A$ is a category equipped with an $R$-module structure on each \textit{hom} set such that the composition is $R$-bilinear. More precisely, let us assume for instance that we are given a commutative ring $R$ with identity. Then a small $R$-category--or equivalently an \emph{$R$-algebroid}-- will be defined as a category enriched in the monoidal category of $R$-modules, with respect to the monoidal structure of tensor product. This means simply that for all objects $b,c$ of $\A$, the set $\A(b,c)$ is given the structure of an $R$-module, and composition $\A(b,c) \times \A(c,d) \lra \A(b,d)$ is $R$--bilinear, or is a morphism of $R$-modules $\A(b,c) \otimes_R \A(c,d) \lra \A(b,d)$. If $\mathsf{G}$ is a \PMlinkname{groupoid}{Groupoids} (or, more generally, a category) then we can construct an \emph{$R$-algebroid} $R\mathsf{G}$ as follows. The object set of $R\mathsf{G}$ is the same as that of $\mathsf{G}$ and $R\mathsf{G}(b,c)$ is the free $R$-module on the set $\mathsf{G}(b,c)$, with composition given by the usual bilinear rule, extending the composition of $\mathsf{G}$. Alternatively, one can define $\bar{R}\mathsf{G}(b,c)$ to be the set of functions $\mathsf{G}(b,c)\lra R$ with finite support, and then we define the \emph{convolution product} as follows: \begin{equation} (f*g)(z)= \sum \{(fx)(gy)\mid z=x\circ y \} ~. \end{equation} As it is very well known, only the second construction is natural for the topological case, when one needs to replace `function' by \PMlinkname{`continuous function with compact support'}{SmoothFunctionsWithCompactSupport} (or \emph{locally compact support} for the \PMlinkname{QFT}{QFTOrQuantumFieldTheories} extended \PMlinkexternal{symmetry sectors}{http://planetmath.org/?op=getobj&from=books&id=153}), and in this case $R \cong \mathbb{C}$~. The point made here is that to carry out the usual construction and end up with only an algebra rather than an algebroid, is a procedure analogous to replacing a \PMlinkname{groupoid}{Groupoids} $\mathsf{G}$ by a semigroup $G'=G\cup \{0\}$ in which the compositions not defined in $G$ are defined to be $0$ in $G'$. We argue that this construction removes the main advantage of \PMlinkname{groupoids}{Groupoids}, namely the spatial component given by the set of objects. \textbf{Remarks:} One can also define categories of algebroids, $R$-algebroids, double algebroids , and so on. A `category' of $R$-categories is however a \PMlinkname{super-category}{Supercategory} $\S$, or it can also be viewed as a specific example of a \PMlinkname{metacategory}{AxiomsOfMetacategoriesAndSupercategories} (or $R$-supercategory, in the more general case of multiple operations--categorical `composition laws'-- being defined within the same structure, for the same class, $C$).