Groupoid algebra

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

Definition

Given a groupoid ${\displaystyle (G,\cdot )}$ and a field ${\displaystyle K}$, it is possible to define the groupoid algebra ${\displaystyle KG}$ as the algebra over ${\displaystyle K}$ formed by the vector space having the elements of ${\displaystyle G}$ as generators and having the multiplication of these elements defined by ${\displaystyle g*h=g\cdot h}$, whenever this product is defined, and ${\displaystyle g*h=0}$ otherwise. The product is then extended by linearity.[2]

Examples

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

Notes

1. Khalkhali (2009), [[[:Template:Google books]] p. 48]
2. Dokuchaev, Exel & Piccione (2000), p. 7
3. da Silva & Weinstein (1999), [[[:Template:Google books]] p. 97]
4. Khalkhali & Marcolli (2008), [[[:Template:Google books]] p. 210]

References

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