Groupoid algebra

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

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

• Group rings
• Matrix algebras
• Algebras of functions

• 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]

• Hopf algebra
• Partial group algebra

• 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. Elsevier. 226: 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.