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.