Groupoid algebra

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

Definition

Given a groupoid $(G,\cdot )$ (in the sense of a category with all morphisms invertible) and a field $K$ , it is possible to define the groupoid algebra $KG$ as the algebra over $K$ formed by the vector space having the elements of (the morphisms of) $G$ as generators and having the multiplication of these elements defined by $g*h=g\cdot h$ , whenever this product is defined, and $g*h=0$ otherwise. The product is then extended by linearity.^[2]

Examples

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

Properties

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]

Notes

^ Khalkhali (2009), p. 48
^ Dokuchaev, Exel & Piccione (2000), p. 7
^ da Silva & Weinstein (1999), p. 97
^ Khalkhali & Marcolli (2008), p. 210

References

Khalkhali, Masoud (2009). Basic Noncommutative Geometry. EMS Series of Lectures in Mathematics. European Mathematical Society. ISBN 978-3-03719-061-6.
da Silva, Ana Cannas; Weinstein, Alan (1999). Geometric models for noncommutative algebras. Berkeley mathematics lecture notes. Vol. 10 (2 ed.). AMS Bookstore. ISBN 978-0-8218-0952-5.
Dokuchaev, M.; Exel, R.; Piccione, P. (2000). "Partial Representations and Partial Group Algebras". Journal of Algebra. 226. Elsevier: 505–532. arXiv:math/9903129. doi:10.1006/jabr.1999.8204. ISSN 0021-8693. S2CID 14622598.
Khalkhali, Masoud; Marcolli, Matilde (2008). An invitation to noncommutative geometry. World Scientific. ISBN 978-981-270-616-4.

[1] Khalkhali (2009), p. 48

[2] Dokuchaev, Exel & Piccione (2000), p. 7

[3] Silva & Weinstein (1999), p. 97

[4] Khalkhali & Marcolli (2008), p. 210

[1]

[2]

[3]

[4]