Groupoid algebra

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

Definition

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

Examples

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

  • 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.[4]

See also

  • Hopf algebra
  • Partial group algebra

Notes

  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

  • 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.