Talk:PlanetPhysics/Algebroid Structures and Extended Symmetries

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%%% Primary Title: algebroid structures and extended symmetries
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\begin{document}

 \section{Algebroid structures and Quantum Algebroid Extended Symmetries.}
\begin{definition}
An \emph{\htmladdnormallink{algebroid}{http://planetphysics.us/encyclopedia/Algebroids.html} structure} $A$ will be specifically defined to mean
either a ring, or more generally, any of the specifically defined algebras, but \emph{with several
\htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}} 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 \htmladdnormallink{graph}{http://planetphysics.us/encyclopedia/Cod.html} 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
\htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $$ \circ : A(x,y) \times A(y,z) \to A(x,z)$$ $(a , b)
\mapsto a\circ b$ called ``\htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html}" and satisfying the
associativity condition, and the existence of \htmladdnormallink{identities}{http://planetphysics.us/encyclopedia/Cod.html}.
\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 \htmladdnormallink{commutative ring}{http://planetphysics.us/encyclopedia/PAdicMeasure.html}.

An \textit{$R$-category }$\A$ is a \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} 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 \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} 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 \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of $R$-modules $\A(b,c) \otimes_R \A(c,d) \lra \A(b,d)$.

If $\mathsf{G}$ is a \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/Groupoids.html} (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{\htmladdnormallink{convolution}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} 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 \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} case, when one needs to replace `function' by
`continuous function with compact support' (or \emph{locally
compact support} for the QFT extended
\htmladdnormallink{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
\htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/Groupoids.html} $\mathsf{G}$ by a \htmladdnormallink{semigroup}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $G'=G\cup \{0\}$ 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
\htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/Groupoids.html}, namely the spatial component given by the set of
objects.

\textbf{Remarks:}
One can also define categories of algebroids, $R$-algebroids, \htmladdnormallink{double algebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html} , and so on.
A `category' of $R$-categories is however a \htmladdnormallink{super-category}{http://planetphysics.us/encyclopedia/Supercategory.html} $\S$, or it can also be viewed as a specific example of a \htmladdnormallink{metacategory}{http://planetphysics.us/encyclopedia/AxiomsOfMetacategoriesAndSupercategories.html} (or $R$-supercategory, in the more general case of multiple operations--categorical `\htmladdnormallink{composition laws}{http://planetphysics.us/encyclopedia/Identity2.html}' being defined within the same structure, for the same class, $C$).

\end{document}