nLab 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 multiplication 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

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

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

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

and the multiplication is given by

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

Remark

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

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

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

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

and for the basis elements one writes

$$\chi_g \colon U(G) \to U(R) \,,$$

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

$$\chi_g \colon \tilde g \mapsto \left\{ \array{ 1 & | g = \tilde g \\ 0 & | otherwise } \right. \,.$$

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 profinite groups

The completed group ring of a profinite group is a pseudocompact ring. Let $\hat{\mathbb{Z}}$ be the profinite completion of the ring of integers, $\mathbb{Z}$, then $\hat{\mathbb{Z}}$ is itself a pseudocompact ring as it is the inverse limit of its finite quotients. Now let $G$ be a profinite group.

The completed group algebra, $\hat{\mathbb{Z}}[\![G]\!]$, of $G$ over $\hat{\mathbb{Z}}$ is the inverse limit of the ordinary group algebras, $\hat{\mathbb{Z}}[G/U]$, of the finite quotients, $G/U$ (for $U$ in the directed set, $\Omega(G)$, of open normal subgroups of $G$), over $\hat{\mathbb{Z}}$;

$$\hat{\mathbb{Z}}[\![G]\!] = lim_{U\in \Omega(G)} \hat{\mathbb{Z}}[G/U].$$

For $R$ a pseudocompact ring, it is then easy to construct the corresponding pseudo-compact group algebra of $G$ over $R$; see the paper by Brumer.

For topological groups

Discussion of group algebras in the generality of locally compact topological groups:

e.g. Dixmier (1977) §13.2

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

$$(R[-]\dashv (-)^\times) \;\colon\; Alg_R \underoverset { \underset{ \;\;\; (-)^\times \;\;\; }{ \longrightarrow } } { \overset{ \;\;\; R[-] \;\;\; }{ \leftarrow } } {} Grp$$

between the category of Algebras (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 $R[G] \to End(V)$ of the group ring to the endomorphism ring of $V$ is equivalently a $R[G]$-module structure on $V$.

Any homomorphism of groups $p \colon G \to Aut(V)$ 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.

Proposition

For $G$ a finite group with isomorphism classes of irreducible representations $[V] \in Irreps_{\mathbb{C}}(G)_{/\sim}$ over the complex numbers, the complex group algebra of $G$ is isomorphic to the direct sum of the linear endomorphism algebras of the complex vector spaces underlying the irreps:

$$\mathbb{C}[G] \;\simeq\; \underset{ [V] \in Irreps_{\mathbb{C}}(G)_{/\sim} }{ \oplus } End_{\mathbb{C}}(V)$$

(e.g. Fulton-Harris 91, Prop. 3.29)

Proof

For every representation $V$, the defining group action

$$G \overset{}{\longrightarrow} Aut_{\mathbb{C}}(V) \hookrightarrow End_{\mathbb{C}}(V)$$

extends uniquely to an algebra homomorphism

$$\mathbb{C}[G] \overset{ \phi_V }{\longrightarrow} End_{\mathbb{C}}(V) \,.$$

Observe that this is a surjection, since if it were not then we could split off a non-trivial cokernel, contradicting the assumption that $V$ is irreducible.

We claim that the resulting homomorphism to the direct sum

$$\mathbb{C}[G] \overset{ (\phi_V)_{[V]} }{\longrightarrow} \underset{ [V] \in Irreps_{\mathbb{C}}(G)_{/\sim} }{\oplus} End_{\mathbb{C}}(V)$$

is an isomorphism: By the previous comment it is surjective, hence it is sufficient to observe that the dimension of the group algebra equals that of the right hand side, hence that

$$dim\big( \mathbb{C}[Sym(G)] \big) \;=\; \underset{[V]}{\sum} \big( dim(V)\big)^2 \,.$$

That is the case by this property of the regular representation.

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

• Cohn localization

• groupoid algebra, category algebra

• augmentation ideal

• pseudocompact ring

• Day convolution

• ∞-group ∞-ring

References

Textbook accounts:

for locally compact topological groups:

• Jacques Dixmier, §13.2 in $C^\ast$-algebras, North Holland (1977)

for finite groups

• William Fulton, Joe Harris, Section 3.4 of: Representation Theory: a First Course, Springer (1991) [doi:10.1007/978-1-4612-0979-9]

Lecture notes:

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

• Andrei Yafaev, Group algebras (pdf)

The universal localization of group rings (see also at Snaith's theorem) is discussed in

• M. Farber, Pierre Vogel, The Cohn localization of the free group ring, Math. Proc. Camb. Phil. Soc. (1992) 111, 433 (pdf)

• Davidson, Nicholas, Modules Over Localized Group Rings for Groups Mapping Onto Free Groups (2011). Boise State University Theses and Dissertations. Paper 170. (web)

For the case of profinite groups, see

• A. Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra 4 (1966) 442-470, MR202790, doi pdf