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In [[mathematics]], the concept of '''groupoid algebra''' generalizes the notion of [[group algebra]].<ref>Khalkhali (2009), {{Google books quote|id=UInc5AyTAikC|page=48|text=groupoid algebra|p. 48}}</ref> | |||
== Definition == | |||
Given a [[groupoid]] <math>(G, \cdot)</math> and a [[field (mathematics)|field]] <math>K</math>, it is possible to define the groupoid algebra <math>KG</math> as the [[Algebra over a field|algebra]] over <math>K</math> formed by the [[vector space]] having the elements of <math>G</math> as [[generator (mathematics)|generator]]s and having the [[multiplication]] of these elements defined by <math>g * h = g \cdot h</math>, whenever this product is defined, and <math>g * h = 0</math> otherwise. The product is then extended by [[linearity]].<ref>Dokuchaev, Exel & Piccione (2000), p. 7</ref> | |||
== Examples == | |||
Some examples of groupoid algebras are the following:<ref>da Silva & Weinstein (1999), {{Google books quote|id=2fcC1EGKz08C|page=97|text=groupoid algebras|p. 97}}</ref> | |||
* [[Group algebra]]s | |||
* [[Matrix ring|Matrix algebra]]s | |||
* [[Algebra of functions|Algebras of functions]] | |||
== Properties == | |||
* When a groupoid has a [[Wikt:finite|finite]] number of [[Object (category theory)|objects]] and a finite number of [[morphism (category theory)|morphisms]], the groupoid algebra is a [[direct sum]] of [[tensor product]]s of group algebras and matrix algebras.<ref>Khalkhali & Marcolli (2008), {{Google books quote|id=HsTkPOj0iusC|page=210|text=Groupoid algebra of a finite groupoid|p. 210}}</ref> | |||
== See also == | |||
* [[Hopf algebra]] | |||
* [[Partial group algebra]] | |||
== Notes == | |||
<references /> | |||
== References == | |||
* {{cite book |last1=Khalkhali |first1=Masoud |authorlink1= |last2= |first2= |authorlink2= |title=Basic Noncommutative Geometry |url= |edition= |series=EMS Series of Lectures in Mathematics |volume= |year=2009 |publisher=European Mathematical Society |location= |isbn=978-3-03719-061-6 }} | |||
* {{cite book |last1=da Silva |first1=Ana Cannas |authorlink1= |last2=Weinstein |first2=Alan |authorlink2= |title=Geometric models for noncommutative algebras |url= |edition=2 |series=Berkeley mathematics lecture notes |volume=10 |year=1999 |publisher=AMS Bookstore |location= |isbn=978-0-8218-0952-5 }} | |||
* {{cite journal |last1=Dokuchaev |first1=M. |last2=Exel |first2=R. |last3=Piccione |first3=P. |year=2000 |title=Partial Representations and Partial Group Algebras |journal=Journal of Algebra |volume=226 |issue= |pages=505–532 |publisher=Elsevier |issn=0021-8693 |doi= 10.1006/jabr.1999.8204|url=|arxiv= math/9903129 | |||
}} | |||
* {{cite book |last1=Khalkhali |first1=Masoud |authorlink1= |last2=Marcolli |first2=Matilde |authorlink2= |title=An invitation to noncommutative geometry |url= |edition= |series= |volume= |year=2008 |publisher=World Scientific |location= |isbn=978-981-270-616-4 }} | |||
[[Category:Algebra]] | |||
{{algebra-stub}} |
Revision as of 14:59, 15 July 2013
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]
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), Template:Google books quote
- ↑ Dokuchaev, Exel & Piccione (2000), p. 7
- ↑ da Silva & Weinstein (1999), Template:Google books quote
- ↑ Khalkhali & Marcolli (2008), Template:Google books quote
References
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