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 algebrasPlanetmathPlanetmath, 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” A0 is a directed graphMathworldPlanetmath over A0 such that for each x,yA0,A(x,y) has an R-module structureMathworldPlanetmath and there is an R-bilinear function


(a,b)ab called “compositionMathworldPlanetmathPlanetmath” and satisfying the associativity condition, and the existence of identitiesPlanetmathPlanetmath.

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 1xA(x,x) is not assumed. For example, if A0 has exactly one object, then an R-algebroid A over A0 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 productPlanetmathPlanetmath. 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)×𝒜(c,d)𝒜(b,d) is R–bilinear, or is a morphismMathworldPlanetmath of R-modules 𝒜(b,c)R𝒜(c,d)𝒜(b,d).

If 𝖦 is a groupoidPlanetmathPlanetmathPlanetmathPlanetmath ( (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 R¯𝖦(b,c) to be the set of functions 𝖦(b,c)R with finite supportPlanetmathPlanetmath, and then we define the convolution productPlanetmathPlanetmath as follows:

(f*g)(z)={(fx)(gy)z=xy}. (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 functionPlanetmathPlanetmath with compact support’ ( (or locally compact supportMathworldPlanetmathPlanetmath for the QFT ( extended sectors), and in this case R . 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 ( 𝖦 by a semigroup G=G{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-categoryPlanetmathPlanetmathPlanetmath ( 𝕊, or it can also be viewed as a specific example of a metacategory ( (or R-supercategory, in the more general case of multiple operationsMathworldPlanetmathcategorical ‘composition laws’– being defined within the same structure, for the same class, C).


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