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 $x,y\in A_{0},\;A(x,y)<math>hasan$R$-modulestructureandthereisan$R</math>-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 Failed to parse (unknown function "\A"): {\displaystyle \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 Failed to parse (unknown function "\A"): {\displaystyle \A} , the set Failed to parse (unknown function "\A"): {\displaystyle \A(b,c)} is given the structure of an $R$-module, and composition Failed to parse (unknown function "\A"): {\displaystyle \A(b,c) \times \A(c,d) \lra \A(b,d)<math> is } R$--bilinear,orisa[[../TrivialGroupoid/|morphism]]of$R$-modules$\A(b,c) \otimes_R \A(c,d) \lra \A(b,d)</math>.

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 Failed to parse (unknown function "\lra"): {\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:

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

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, $C$).