Groupoid algebra

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

Definition

Given a groupoid $(G,\cdot )$ (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 $g*h=g\cdot h$, whenever this product is defined, and $g*h=0$ otherwise. The product is then extended by linearity.

Examples

Some examples of groupoid algebras are the following:

• 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.

See also

• Hopf algebra
• Partial group algebra