Groupoid algebra

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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

Definition[edit]

Given a groupoid (G, \cdot) and a field K, it is possible to define the groupoid algebra KG as the algebra over K formed by the vector space having the elements of G as generators and having the multiplication of these elements defined by g * h = g \cdot h, whenever this product is defined, and g * h = 0 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 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. 
  • Khalkhali, Masoud; Marcolli, Matilde (2008). An invitation to noncommutative geometry. World Scientific. ISBN 978-981-270-616-4.