Groupoid algebra

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

Definition

Given a groupoid ${\displaystyle (G,\cdot )}$ (in the sense of a category with all morphisms invertible) 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 (the morphisms 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]

• Group rings
• 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

• Hopf algebra
• Partial group algebra