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group algebra

Idea

Take the elements of a group, $G$, as labelling a basis for a vector space over a field $k$, then that multiplication of the group will extend to give that vector space the structure of an algebra over $k$. This is usually denoted $k[G]$.

If instead of a field we used the ring of integers, $\mathbb{Z}$, it is usual to call the result the group ring. For this we take the free abelian group on the set of elements of the group and extend the multiplication to give a ring structure on the result. This is usually denoted $\mathbb{Z}[G]$.

As a further generalisation, the field can be replaced by any commutative ring.

The multiplication:

If we denote by $e_g$, the generator corresponding to $g\in G$, then an arbitrary element of $k[G]$ can be written as ${\sum }_{g\in G}{n}_{\mathrm{ge}}{}_g$ where the $n_g$ are elements of $k$, and only finitely many of them are non-zero.

The multiplication is then by what is sometimes called a ‘convolution’ product, that is,

$$\left(\sum _{g\in G}{n}_{\mathrm{ge}}{}_g\right)\left(\sum _{g\in G}{m}_{\mathrm{ge}}{}_g\right)=\sum _{g\in G}\left(\sum _{g_1\in G}{n}_{g_1}{m}_{g_1^{-1}g}{e}_g\right).$$\Big(\sum_{g\in G}n_ge_g\Big)\Big(\sum_{g\in G}m_ge_g\Big) = \sum_{g\in G}\Big(\sum_{g_1\in G}n_{g_1}m_{g_1^{-1}g}e_g\Big).

Thoughts

Thinking of a group, $G$, as a special sort of category, the group algebra (denoted $k[G]$ or $kG$) of a group is just the category algebra of that category.

Extra structure

The group algebra is always a Hopf algebra.

The group algebra is always a graded 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.