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group algebra

Idea

The group algebra of a group $G$ over a ring $R$ is the associative algebra whose elements are formal linear combinations over $R$ of the elements of $G$ and whose mutliplication is given on these basis elements by the group operation in $G$.

Definition

For discrete groups

Let $G$ be a discrete group. Let $R$ be a commutative ring.

Definition

The group $R$-algebra $R[G]$ is the associative algebra over $R$

1. whose underlying $R$-module is the the free module over $R$ on the underlying set of $G$;

2. whose multiplication is given on basis elements by the group operation.

Remark

By the discussion at free module an element $r$ in $R[G]$ is a formal linear combination of basis elements in $G$ with coefficients in $R$, hence a formal sum

$$r=\sum _{g\in G}{r}_g\cdot g$$r = \sum_{g \in G} r_g \cdot g

with ${\forall }_{g\in G}(r_g\in R)$ and only finitely many of the coefficients different from $0\in R$.

The addition of algebra elements is given by the componentwise addition of coefficients

$$r+\tilde{r}=\sum _{g\in G}(r_g+{\tilde{r}}_g)g$$r + \tilde r = \sum_{g \in G} (r_g + \tilde r_g) g

and the multiplication is given by

$$\begin{array}{rl}r\tilde{r}& =\sum _{g\in G}\sum _{\tilde{g}\in G}(r_g{\tilde{r}}_{\tilde{g}})g\cdot \tilde{g}\\ & =\sum _{q\in G}(\sum _{k\in G}(r_{q\cdot k^{-1}}{r}_k))q\end{array}.$$\begin{aligned} r \tilde r & = \sum_{g \in G} \sum_{\tilde g \in G} (r_g \tilde r_{\tilde g}) g \cdot \tilde g \\ & = \sum_{q \in G} \left( \sum_{k \in G} (r_{q\cdot k^{-1}} r_k) \right) q \end{aligned} \,.

Remark

The formal linear combinations over $R$ of element in $G$ may equivalently be thought of as functions

$$r_{(-)}:U(G)\to U(R)$$r_{(-)} \colon U(G) \to U(R)

from the underlying set of $G$ to the underlying set of $R$ which have finite support. Accordingly, oftn the underlying set of the group $R$-algebra is written as

$$U(R[G])={\mathrm{Hom}}_{\mathrm{Set}}^{\mathrm{fin}\mathrm{supp}}(U(G),U(R))$$U(R[G]) = Hom_{Set}^{fin\;supp}(U(G), U(R))

and for the basis elements one writes

$${\chi }_g:U(G)\to U(R),$$\chi_g \colon U(G) \to U(R) \,,

the characteristic function of an element $g\in G$, defined by

$${\chi }_g:\tilde{g}\mapsto \{\begin{array}{cc}1& \mid g=\tilde{g}\\ 0& \mid \mathrm{otherwise}\end{array}.$$\chi_g \colon \tilde g \mapsto \left\{ \array{ 1 & | g = \tilde g \\ 0 & | otherwise } \right. \,.

In terms of this the product in the group algebra is called the convolution product on functions.

Remark

The notion of group algebra is a special case of that of a groupoid algebra, hence of category algebra.

For topological groups

(…)

Properties

Proposition

A group algebra is in particular a Hopf algebra and a $G$-graded algebra.

The following states a universal property of the construction of the group algebra.

Remark

There is an adjunction

$$(R[-]⊣(-{)}^{\times }):{\mathrm{Alg}}_R\stackrel{\stackrel{R[-]}{\leftarrow }}{\underset{(-{)}^{\times }}{\to }}\mathrm{Grp}$$(R[-]\dashv (-)^\times) \colon Alg_R \stackrel{\overset{R[-]}{\leftarrow}}{\underset{(-)^\times}{\to}} Grp

between the category of associative algebras over $R$ and that of groups, where $R[-]$ forms group rings and $(-{)}^{\times }$ assigns to an $R$-algebra its group of units.

Remark

Let $V$ be an abelian group. A homomorphism of rings $R[G]\to \mathrm{End}(V)$ of the group ring to the endomorphism ring of $V$ is equivalently a $R[G]$-module structure on $V$. And any homomorphism of groups $p:G\to \mathrm{Aut}(V)$ to the automorphism group of $V$ extends to to a morphism of rings. This observation is used extensively in the theory of group representations. See also at module – Abelian groups with G-action as modules over a ring.

Theorem

(Maschke's theorem)

Let $G$ be a finite group, let $R=k$ be a field.

Then $k[G]$ is a semi-simple algebra precisely if the order of $G$ is not divisible by the characteristic of k.

Related concepts

• groupoid algebra, category algebra

• augmentation ideal

References

Lecture notes include

• Kiyoshi Igusa, algebra II, part D: representations of groups, (pdf)