Groupoid algebra

In mathematics, the concept of groupoid algebra generalizes the notion of group algebra.

Definition

Given a groupoid (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 , whenever this product is defined, and otherwise. The product is then extended by linearity.

Examples

Some examples of groupoid algebras are the following:

• Group rings
• Matrix algebras
• Algebras of functions

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.

See also

• Hopf algebra
• Partial group algebra

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.