nLab
group algebra

A group is a special sort of category, and the group algebra (denoted $k[G]$ or $kG$) of a group $G$ is just the category algebra of this category.

The group algebra is always a Hopf algebra.

Sometimes instead of working over a ground field $k$, one allows $k$ to be a commutative unital ring. Then we talk about group ring (though it is in fact a commutative unital $k$-algebra). The integer group ring $\mathbb{Z}G$ is the most important example, extensively used in the representation theory of finite groups.