81T25-AlgebroidStructuresAndExtendedSymmetries.tex

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%\pmkeywords{symmetry sectors}
%\pmkeywords{groupoids}
%\pmkeywords{continuous function with compact support}
%\pmkeywords{convolution product}
%\pmkeywords{extensions of  quantum operator algebras}
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\pmdefines{algebroid extended symmetries}
\pmdefines{set of functions with finite support}

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\begin{document}
\subsection{Algebroid Structures and 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}

%%\cite{M1,M2,A}
 Let $R$ be a commutative ring. 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} $\mathbb{S}$, 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, $C$).  


\begin{thebibliography}{9}
\bibitem{BBG2009}
I. C. Baianu , James F. Glazebrook, and Ronald Brown. 2009. Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review. \emph{SIGMA} 5 (2009), 051, 70 pages.      $arXiv:0904.3644$, $doi:10.3842/SIGMA.2009.051$, 
\PMlinkexternal{Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)}{http://www.emis.de/journals/SIGMA/2009/051/}
\end{thebibliography}

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