Spahn group algebra (Rev #1)

Definition

For a $k$-algebra $A$ and an (abelian) group $G$ the group

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 $\sum_{g\in G}n_ge_g$ 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,

Thoughts

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