Groupoid algebra

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

Definition[edit]

Given a groupoid and a field (in the sense of a category with all arrows invertible), it is possible to define the groupoid algebra as the algebra over formed by the vector space having the elements of (the arrows 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. 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.