algebroid structures and extended symmetries

0.1 Algebroid Structures and Algebroid Extended Symmetries

Definition 0.1.

An algebroid structure $A$ will be specifically defined to mean either a ring, or more generally, any of the specifically defined algebras  , but 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 an $R$-module structure  and there is an $R$-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  .

Definition 0.2.

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 $\mathcal{A}$ 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 $\mathcal{A}$, the set $\mathcal{A}(b,c)$ is given the structure of an $R$-module, and composition $\mathcal{A}(b,c)\times\mathcal{A}(c,d){\longrightarrow}\mathcal{A}(b,d)$ is $R$–bilinear, or is a morphism  of $R$-modules $\mathcal{A}(b,c)\otimes_{R}\mathcal{A}(c,d){\longrightarrow}\mathcal{A}(b,d)$.

If $\mathsf{G}$ is a groupoid    (http://planetmath.org/Groupoids) (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 $\mathsf{G}$.

 $(f*g)(z)=\sum\{(fx)(gy)\mid z=x\circ y\}~{}.$ (0.1)

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’ (http://planetmath.org/SmoothFunctionsWithCompactSupport) (or locally compact support   for the QFT (http://planetmath.org/QFTOrQuantumFieldTheories) extended http://planetmath.org/?op=getobj&from=books&id=153symmetry  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 (http://planetmath.org/Groupoids) $\mathsf{G}$ by a 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 (http://planetmath.org/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   (http://planetmath.org/Supercategory) $\mathbb{S}$, or it can also be viewed as a specific example of a metacategory (http://planetmath.org/AxiomsOfMetacategoriesAndSupercategories) (or $R$-supercategory, in the more general case of multiple operations  categorical ‘composition laws’– being defined within the same structure, for the same class, $C$).

References

 Title algebroid structures and extended symmetries Canonical name AlgebroidStructuresAndExtendedSymmetries Date of creation 2013-03-22 18:13:55 Last modified on 2013-03-22 18:13:55 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 50 Author bci1 (20947) Entry type Topic Classification msc 81T25 Classification msc 81T18 Classification msc 81T13 Classification msc 81T10 Classification msc 81T05 Classification msc 81R50 Classification msc 55U35 Synonym extensions of quantum operator algebras Related topic HamiltonianAlgebroids Related topic QFTOrQuantumFieldTheories Related topic LieAlgebroids Related topic RCategory Related topic RAlgebroid Related topic AxiomsOfMetacategoriesAndSupercategories Related topic MonoidalCategory Related topic Groupoids Related topic ETAS Defines algebroid structure Defines convolution product Defines pre-algebroid Defines algebroid extended symmetries Defines set of functions with finite support