Groupoid algebra
In mathematics, the concept of groupoid algebra generalizes the notion of group algebra.[1]
Definition
Given a groupoid and a field , it is possible to define the groupoid algebra as the algebra over formed by the vector space having the elements 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
Some examples of groupoid algebras are the following:[3]
- Group algebras
- 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
Notes
- ↑ Khalkhali (2009), p. 48
- ↑ Dokuchaev, Exel & Piccione (2000), p. 7
- ↑ da Silva & Weinstein (1999), p. 97
- ↑ 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. 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.
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