Groupoid algebra

Given a groupoid $(G,\; \backslash cdot)$ (in the sense of a category with all morphisms invertible) 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 (the morphisms of) $G$ as generators and having the multiplication of these elements defined by $g\; *\; h\; =\; g\; \backslash cdot\; h$, whenever this product is defined, and $g\; *\; h\; =\; 0$ otherwise. The product is then extended by linearity.Dokuchaev, Exel & Piccione (2000), p. 7

Some examples of groupoid algebras are the following:da Silva & Weinstein (1999), [{{Google books|plainurl=y|id=2fcC1EGKz08C|page=97|text=groupoid algebras}} p. 97]

• Group rings
• Matrix algebras
• Algebras of functions

• 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.Khalkhali & Marcolli (2008), [{{Google books|plainurl=y|id=HsTkPOj0iusC|page=210|text=Groupoid algebra of a finite groupoid}} p. 210]

• Hopf algebra
• Partial group algebra

• {{cite book |last1=Khalkhali |first1=Masoud |title=Basic Noncommutative Geometry |series=EMS Series of Lectures in Mathematics |year=2009 |publisher=European Mathematical Society |isbn=978-3-03719-061-6 | url=https://books.google.com/books?id=UInc5AyTAikC&q=%22groupoid+algebra%22}}
• {{cite book |last1=da Silva |first1=Ana Cannas |last2=Weinstein |first2=Alan |title=Geometric models for noncommutative algebras |edition=2 |series=Berkeley mathematics lecture notes |volume=10 |year=1999 |publisher=AMS Bookstore |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 |pages=505–532 |publisher=Elsevier |issn=0021-8693 |doi= 10.1006/jabr.1999.8204|arxiv= math/9903129

|s2cid=14622598 }}

• {{cite book |last1=Khalkhali |first1=Masoud |last2=Marcolli |first2=Matilde | author-link2=Matilde Marcolli |title=An invitation to noncommutative geometry |year=2008 |publisher=World Scientific |isbn=978-981-270-616-4 }}

Category:Abstract algebra