Spahn group algebra

Definition

For a $k$-algebra $A$ and an (abelian) group $G$, the space of group morphisms with finite support

$$A[G]:=hom_{fin\,supp}(G,A)$$

(i.e. space containing just those morphisms which send only finitely many elements of $G$ not to $0$) equipped with the multiplication defined by convolution of functions is called group algebra of $A$ over $G$. With pointwise addition $K[G]$ is an associative $A$-algebra.

Equivalently one can think of the multiplication inthe following way

Take the elements of a group, $G$, as labelling a module basis and define the multiplication of two elements by $x_g \cdot x_h:=x_{gh}$.

If $A$ is a ring one calls this construction also a group ring.

The notion of group algebra is a special case of that of a category algebra?.

Any group algebra is in particular a Hopf alebra? and a graded algebra?.

The convolution product

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,

$$\Big(\sum_{g\in G}n_ge_g\Big)\Big(\sum_{g\in G}m_ge_g\Big) = \sum_{g\in G}\Big(\sum_{g_1\in G}n_{g_1}m_{g_1^{-1}g}e_g\Big).$$

Properties

Remark

There is an adjunction?

$$(R[-]\dashv (-)^\times):Alg_R \stackrel{(-)^\times}{\to}Grp$$

where $R[-]$ forms group rings and $(-)^\times$ assigns to an $R$-algebra its group of units?.

Remark

Let $V$ be an abelian group. A morphism of rings $K[G]\to End(V)$ of the group ring to the endomorphism ring of $V$ is a $K[G]$-module. And any morphism of groups $p:G\to End(V)$ can by extended to a morphism of rings $P:K[G]\to End (V)$ by $p(g)\mapsto P(e_g)$. This observation is used extensively in the theory of group representations?.

Remark

(Maschke’s theorem) Let $G$ be a finite group, let $K$ be a field.

Then $K[G]$ is a semi-simple algebra? iff the order of $G$ is not divisible by the characteristic? of K.