Groupoid algebra

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

Definition

Given a groupoid [math]\displaystyle{ (G, \cdot) }[/math] (in the sense of a category with all arrows invertible) and a field [math]\displaystyle{ K }[/math], it is possible to define the groupoid algebra [math]\displaystyle{ KG }[/math] as the algebra over [math]\displaystyle{ K }[/math] formed by the vector space having the elements of (the arrows of) [math]\displaystyle{ G }[/math] as generators and having the multiplication of these elements defined by [math]\displaystyle{ g * h = g \cdot h }[/math], whenever this product is defined, and [math]\displaystyle{ g * h = 0 }[/math] 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. https://books.google.com/books?id=UInc5AyTAikC&q=%22groupoid+algebra%22. 
• da Silva, Ana Cannas; Weinstein, Alan (1999). Geometric models for noncommutative algebras. Berkeley mathematics lecture notes. 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 (Elsevier) 226: 505–532. doi:10.1006/jabr.1999.8204. ISSN 0021-8693. 
• Khalkhali, Masoud; Marcolli, Matilde (2008). An invitation to noncommutative geometry. World Scientific. ISBN 978-981-270-616-4.