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 $\mathsf{G}$ .

Alternatively, one can define $\bar{R}\mathsf{G}(b,c)$ to be the set of functions $\mathsf{G}(b,c){\longrightarrow}R$ with finite support, and then we define the convolution product as follows:

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

(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

1 I. C. Baianu , James F. Glazebrook, and Ronald Brown. 2009. Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review. SIGMA 5 (2009), 051, 70 pages. $arXiv:0904.3644$ , $doi:10.3842/SIGMA.2009.051$ , http://www.emis.de/journals/SIGMA/2009/051/Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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