PlanetPhysics/Algebroid Structures and Extended Symmetries

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Algebroid structures and Quantum Algebroid Extended Symmetries.

An \htmladdnormallink{algebroid {http://planetphysics.us/encyclopedia/Algebroids.html} structure} ${\displaystyle A}$  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 algebroid has been defined (Mosa, 1986a; Brown and Mosa 1986b, 2008) as follows. An ${\displaystyle R}$ -algebroid ${\displaystyle A}$  on a set of objects" ${\displaystyle A_{0}}$  is a directed graph over ${\displaystyle A_{0}}$  such that for each ${\displaystyle x,y\in A_{0},\;A(x,y)$hasan}$ R${\displaystyle -modulestructureandthereisan}$ R$-bilinear function ${\displaystyle \circ :A(x,y)\times A(y,z)\to A(x,z)}$  ${\displaystyle (a,b)\mapsto a\circ b}$  called composition" and satisfying the associativity condition, and the existence of identities.

A pre-algebroid has the same structure as an algebroid and the same axioms except for the fact that the existence of identities ${\displaystyle 1_{x}\in A(x,x)}$  is not assumed. For example, if ${\displaystyle A_{0}}$  has exactly one object, then an ${\displaystyle R}$ -algebroid ${\displaystyle A}$  over ${\displaystyle A_{0}}$  is just an ${\displaystyle R}$ -algebra. An ideal in ${\displaystyle A}$  is then an example of a pre-algebroid. Let ${\displaystyle R}$  be a commutative ring.

An ${\displaystyle R}$ -category $\displaystyle \A$ is a category equipped with an ${\displaystyle R}$ -module structure on each hom set such that the composition is ${\displaystyle R}$ -bilinear. More precisely, let us assume for instance that we are given a commutative ring ${\displaystyle R}$  with identity. Then a small ${\displaystyle R}$ -category--or equivalently an ${\displaystyle R}$ -algebroid -- will be defined as a category enriched in the monoidal category of ${\displaystyle R}$ -modules, with respect to the monoidal structure of tensor product. This means simply that for all objects ${\displaystyle b,c}$  of $\displaystyle \A$ , the set $\displaystyle \A(b,c)$ is given the structure of an ${\displaystyle R}$ -module, and composition $\displaystyle \A(b,c) \times \A(c,d) \lra \A(b,d)$is$ R${\displaystyle --bilinear,orisa[[../TrivialGroupoid/|morphism]]of}$ R${\displaystyle -modules}$ \A(b,c) \otimes_R \A(c,d) \lra \A(b,d)$.

If ${\displaystyle {\mathsf {G}}}$  is a groupoid (or, more generally, a category) then we can construct an ${\displaystyle R}$ -algebroid ${\displaystyle R{\mathsf {G}}}$  as follows. The object set of ${\displaystyle R{\mathsf {G}}}$  is the same as that of ${\displaystyle {\mathsf {G}}}$  and ${\displaystyle R{\mathsf {G}}(b,c)}$  is the free ${\displaystyle R}$ -module on the set ${\displaystyle {\mathsf {G}}(b,c)}$ , with composition given by the usual bilinear rule, extending the composition of ${\displaystyle {\mathsf {G}}}$ .

Alternatively, one can define ${\displaystyle {\bar {R}}{\mathsf {G}}(b,c)}$  to be the set of functions $\displaystyle \mathsf{G}(b,c)\lra R$ with finite support, and then we define the \htmladdnormallink{convolution {http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} product} as follows:

${\displaystyle (f*g)(z)=\sum \{(fx)(gy)\mid z=x\circ y\}~.}$

As it is very well known, only the second construction is natural for the topological case, when one needs to replace function' by continuous function with compact support' (or \emph{locally compact support} for the QFT extended symmetry sectors), and in this case ${\displaystyle R\cong \mathbb {C} }$ ~. 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 groupoid ${\displaystyle {\mathsf {G}}}$  by a semigroup ${\displaystyle G'=G\cup \{0\}}$  in which the compositions not defined in ${\displaystyle G}$  are defined to be ${\displaystyle 0}$  in ${\displaystyle G'}$ . We argue that this construction removes the main advantage of groupoids, namely the spatial component given by the set of objects.

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