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

Definition edit

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.[2]

Examples edit

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

Properties edit

See also edit

Notes edit

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

References edit

  • 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.