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 [[../TrivialGroupoid/|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 [[../Cod/|graph]] over $A_0$ such that for each $x,y\in A_0,A(x,y)<math>hasan$R$-modulestructureandthereisan$R</math>-bilinear [[../Bijective/|function]] $\circ :A(x,y)\times A(y,z)\to A(x,z)$ $(a,b)\mapsto a\circ b$ called ``[[../Cod/|composition]]" and satisfying the associativity condition, and the existence of [[../Cod/|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 [[../PAdicMeasure/|commutative ring]].

An $R$-category Failed to parse (unknown function "\A"): {\displaystyle \A} is a [[../Cod/|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/|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 $𝖦$ is a [[../Groupoids/|groupoid]] (or, more generally, a category) then we can construct an $R$-algebroid $R𝖦$ as follows. The object set of $R𝖦$ is the same as that of $𝖦$ and $R𝖦(b,c)$ is the free $R$-module on the set $𝖦(b,c)$, with composition given by the usual bilinear rule, extending the composition of $𝖦$.

Alternatively, one can define $\overline{R}𝖦(b,c)$ 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 [[../CoIntersections/|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 $R\cong ℂ$~. 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 [[../Groupoids/|groupoid]] $𝖦$ by a [[../TrivialGroupoid/|semigroup]] $G^{\prime }=G\cup \{0\}$ in which the compositions not defined in $G$ are defined to be $0$ in $G^{\prime }$. We argue that this construction removes the main advantage of [[../Groupoids/|groupoids]], namely the spatial component given by the set of objects.

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