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} $$A$$ 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 $$R$$-algebroid  $$A$$ on a set of ``objects" $$A_0$$ is a directed graph over $$A_0$$ such that for each $$x,y \in A_0,\; A(x,y)has anRR$$-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.

A 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. Let $$R$$ be a commutative ring.

An $$R$$-category  $$\A$$ is a category equipped with an $$R$$-module structure on each 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 $$R$$-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 $$\A(b,c) \times \A(c,d) \lra \A(b,d)isRR-modules\A(b,c) \otimes_R \A(c,d) \lra \A(b,d)$$.

If $$\mathsf{G}$$ is a groupoid (or, more generally, a category) then we can construct an $$R$$-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 \htmladdnormallink{convolution {http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} product} as follows:

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

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 $$R \cong \mathbb{C}$$~. 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 $$\mathsf{G}$$ 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 groupoids, namely the spatial component given by the set of objects.

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