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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>


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== 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

Template:Orphan

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

Definition

Given a groupoid (G,⋅) 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⋅h, whenever this product is defined, and g*h=0 otherwise. The product is then extended by linearity.[2]

Examples

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

Properties

See also

Notes

  1. ↑ Khalkhali (2009), Template:Google books quote
  2. ↑ Dokuchaev, Exel & Piccione (2000), p. 7
  3. ↑ da Silva & Weinstein (1999), Template:Google books quote
  4. ↑ Khalkhali & Marcolli (2008), Template:Google books quote

References

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