In mathematics, the concept of groupoid algebra generalizes the notion of group algebra.
Given a groupoid
(
G
,
⋅
)
and a field
K
, it is possible to define the groupoid algebra
K
G
as the algebra over
K
formed by the vector space having the elements of
G
as generators and having the multiplication of these elements defined by
g
∗
h
=
g
⋅
h
, whenever this product is defined, and
g
∗
h
=
0
otherwise. The product is then extended by linearity.
Some examples of groupoid algebras are the following:
Group algebras
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.
