Groupoid algebra explained

In mathematics, the concept of groupoid algebra generalizes the notion of group algebra.^[1]

Definition

(G, ⋅ )

(in the sense of a category with all morphisms invertible) and a field

, it is possible to define the groupoid algebra

as the algebra over

formed by the vector space having the elements of (the morphisms of)

as generators and having the multiplication of these elements defined by

g*h=g ⋅ h

, whenever this product is defined, and

g*h=0

otherwise. The product is then extended by linearity.^[2]

Some examples of groupoid algebras are the following:^[3]

When a groupoid has a finite number of objects and a finite number of morphisms, the groupoid algebra is a direct sum of tensor products of group algebras and matrix algebras.^[4]

Khalkhali (2009), [{{Google books|plainurl=y|id=UInc5AyTAikC|page=48|text=groupoid algebra}} p. 48]
Dokuchaev, Exel & Piccione (2000), p. 7
da Silva & Weinstein (1999), [{{Google books|plainurl=y|id=2fcC1EGKz08C|page=97|text=groupoid algebras}} p. 97]
Khalkhali & Marcolli (2008), [{{Google books|plainurl=y|id=HsTkPOj0iusC|page=210|text=Groupoid algebra of a finite groupoid}} p. 210]

Book: Khalkhali . Masoud . Basic Noncommutative Geometry . EMS Series of Lectures in Mathematics . 2009 . European Mathematical Society . 978-3-03719-061-6 .
Book: da Silva . Ana Cannas . Weinstein . Alan . Geometric models for noncommutative algebras . 2 . Berkeley mathematics lecture notes . 10 . 1999 . AMS Bookstore . 978-0-8218-0952-5 .
Dokuchaev . M. . Exel . R. . Piccione . P. . 2000 . Partial Representations and Partial Group Algebras . Journal of Algebra . 226 . 505–532 . Elsevier . 0021-8693 . 10.1006/jabr.1999.8204. math/9903129. 14622598 .
Book: Khalkhali . Masoud . Marcolli . Matilde . Matilde Marcolli . An invitation to noncommutative geometry . 2008 . World Scientific . 978-981-270-616-4 .