nLab group algebra




Group Theory



The group algebra of a group GG over a ring RR is the associative algebra whose elements are formal linear combinations over RR of the elements of GG and whose multiplication is given on these basis elements by the group operation in GG.


For discrete groups

Let GG be a discrete group. Let RR be a commutative ring.


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

  1. whose underlying RR-module is the the free module over RR on the underlying set of GG;

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


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

r= gGr gg r = \sum_{g \in G} r_g \cdot g

with gG(r gR)\forall_{g \in G} (r_g \in R) and only finitely many of the coefficients different from 0R0 \in R.

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

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

and the multiplication is given by the convolution product

rr˜ = gG g˜G(r gr˜ g˜)gg˜ = qG( kG(r qk 1r k))q. \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}

The formal linear combinations over RR of element in GG may equivalently be thought of as functions

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

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

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

and for the basis elements one writes

χ g:U(G)U(R), \chi_g \colon U(G) \to U(R) \,,

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

χ g:g˜{1 |g=g˜ 0 |otherwise. \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.


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 GG be a profinite group.

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

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

For RR a pseudocompact ring, it is then easy to construct the corresponding pseudo-compact group algebra of GG over RR; 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




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

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


There is an adjunction

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

between the category of Algebras (associative algebras over RR) and that of Groups, where R[]R[-] forms group rings and () ×(-)^\times assigns to an RR-algebra its group of units.


Let VV be an abelian group. A homomorphism of rings R[G]End(V)R[G] \to End(V) of the group ring to the endomorphism ring of VV is equivalently a R[G]R[G]-module structure on VV.

Any homomorphism of groups p:GAut(V)p \colon G \to Aut(V) to the automorphism group of VV 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.


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

[G][V]Irreps (G) /End (V) \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)

For every representation VV, the defining group action

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

extends uniquely to an algebra homomorphism

[G]ϕ VEnd (V). \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 VV is irreducible.

We claim that the resulting homomorphism to the direct sum

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

This is indeed the case, by this property of the regular representation.


(Maschke's theorem)

Let GG be a finite group, let R=kR = k be a field.

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

Of central extensions & Twisted group algebras

A quotient algebra of the group algebra of a central extension G ωG^\omega of a group GG corresponding to a group 2-cocyle ω:G×Gk\omega \,\colon\, G \times G \to k is the ω\omega-twisted group algebra of GG (eg. Nachbin 1993, Ch 2, Thm. 4.1).


(complex group algebra of U(1)\mathrm{U}(1)-central extension)
For (G,()(),e)\big(G, (-)\cdot(-), \mathrm{e}\big) a discrete group, consider a central extension

1/G ωG1 1 \to \mathbb{R}/\mathbb{Z} \hookrightarrow G^\omega \twoheadrightarrow G \to 1

classified by a circle group-valued group 2-cocyle with underlying function

ω:G×G/, \omega \,\colon\, G \times G \to \mathbb{R}/\mathbb{Z} \,,

which we may and do assume to be normalized:

ω(g,e)=0,ω(e,g)=0. \omega(g,\mathrm{e}) \,=\, 0,\, \;\;\; \omega(\mathrm{e},g) \,=\, 0 \,.

In terms of this cocycle, the group operation on the underlying set

G ωG×/ G^\omega \;\simeq\; G \,\times\, \mathbb{R}/\mathbb{Z}

is given by

(g,r)(g,r)(gg,r+r+ω(r,r)). (g, r) \cdot (g',r') \;\equiv\; \big( g \cdot g' ,\, r + r' + \omega(r,r') \big) \,.

We regard /\mathbb{R}/\mathbb{Z} as a discrete group. Since the cocycle will typically (certainly if GG is finite) take values only in a finite cyclic subgroup /n/\mathbb{Z}/n \hookrightarrow \mathbb{R}/\mathbb{Z} one may want to take G ωG^\omega to be just the finite group given by the resulting /n\mathbb{Z}/n-central extension. The discussion here applies verbatim in either case.

If we denote the generators of the \mathbb{C} -valued group algebra of G ωG^\omega by

G ω (G ω) (g,r) U(g,r) \array{ G^\omega &\longrightarrow& \mathbb{C}\big(G^\omega\big) \\ (g,r) &\mapsto& U(g,r) }

then we have in (G ω)\mathbb{C}\big(G^\omega \big) the relations

U(g,r)=U(g,0)U(e,r)=U(e,r)U(g,0) U(g,r) \;=\; U(g,0) \cdot U(\mathrm{e},r) \;=\; U(\mathrm{e},r) \cdot U(g,0)

(by normality of ω\omega) and hence

U(g,0)U(g,0) = U(gg,ω(g,g)) = U(e,ω(g,g))U(gg,0). \begin{array}{rcl} U(g, 0) \cdot U(g', 0) &=& U\big(g g', \omega(g,g')\big) \\ &=& U\big(\mathrm{e}, \omega(g,g')\big) \cdot U(g g', 0) \,. \end{array}

This looks, up to the central correction factor U(e,ω(g,g))U\big(\mathrm{e}, \omega(g,g')\big), like the group algebra of GG.

To bring out this relation, consider now the quotient algebra of (G ω)\mathbb{C}\big(G^\omega\big) by the canonical augmentation ideal of the group algebra of the extension group, ie. by the two-sided ideal in (G ω)\mathbb{C}\big(G^\omega\big) generated by the kernel of the \mathbb{C}- algebra homomorphism

(/) ϵ U(e,r) e 2πir. \begin{array}{ccc} \mathbb{C}\big(\mathbb{R}/\mathbb{Z}\big) &\xrightarrow{\;\; \epsilon \;\;}& \mathbb{C} \\ U\big( \mathrm{e}, r \big) &\mapsto& e^{2 \pi \mathrm{i} r } \mathrlap{\,.} \end{array}

In other words, in this quotient of (G ω)\mathbb{C}\big(G^\omega\big) we enforce the relations

U(e,r) exp(2πir)U(e,0) = exp(2πir) \begin{array}{rcl} U(\mathrm{e}, r) &\sim& exp(2 \pi \mathrm{i} r ) \, U(\mathrm{e}, 0) \\ &=& exp(2 \pi \mathrm{i} r ) \end{array}

(using in the last step that (e,0)(\mathrm{e}, 0) is the neutral element of G ωG^\omega, so that U(e,0)U(\mathrm{e}, 0) is the unit element in its group algebra).

Therefore the quotient algebra (G ω)/ker(ϵ)\mathbb{C}\big(G^\omega\big)\big/ ker(\epsilon) is that generated by U(G)=U(G×{0})U(G) \,=\, U\big(G \times \{0\}\big) subject to the relations

U(g)U(g)=e 2πiω(g,g)U(gg). U(g) \cdot U(g') \;=\; e^{ 2 \pi \mathrm{i} \omega(g,g') } \, U(g\cdot g') \,.

This is the ω\omega-twisted group algebra ω(G)\mathbb{C}^\omega(G) of GG:

(G ω)/ker(ϵ) ω(G). \mathbb{C}\big( G^\omega \big) \big/ \mathrm{ker}(\epsilon) \;\; \simeq \;\; \mathbb{C}^\omega(G) \,.

With due care, this situation generalizes from discrete groups to suitable (eg. locally compact) topological groups (Edwards & Lewis 1969a, 1969b).


(Binz, Honegger & Rieckers 2007)
It is in this way that the group algebra of a (underlying discrete) Heisenberg group (which is a central extension of an abelian group) is related to the corresponding Weyl group (whose Weyl relations are those of a twisted additive group algebra).

Relation to universal enveloping algebras

Concerning group algebras of algebraic groups:

The universal enveloping algebra of a Lie algebra is the analogue of the usual group algebra of a group. It has the analogous function of exhibiting the category of Lie algebra modules as a category of modules for an associative algebra. This becomes more than an analogy when the universal enveloping algebra is viewed with its full Hopf algebra structure. By dualization, one obtains a commutative Hopf algebra which, in the case where the Lie algebra is that of an irreducible algebraic group over a field of characteristic 0, contains the algebra of polynomial functions of that group as a sub Hopf algebra in a natural fashion.

(quoted from Hochschild 1981, p. 221, see Thm. 3.1 on p. 230 there)



Original discussion:


for the case of locally compact topological groups:

for the case of finite groups:

Lecture notes:

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

  • Andrei Yafaev, Group algebras (pdf)

On twisted group algebras and their relation to plain group algebras of group extensions:

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)

On the case of profinite groups:

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

In physics

On group algebras as strict deformation quantizations of Lie-Poisson manifolds:

  • Marc A. Rieffel, Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures, American Journal of Mathematics 112 4 (1990) 657-685 [doi:10.2307/2374874, jstor:2374874]

  • Marc Rieffel, Ex. 7, Ex. 8 in: Deformation quantization and operator algebras, in: Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), 411-423, Proc. Sympos. Pure Math. 51, Part 1, Amer. Math. Soc. (1990) [pdf, pdf MR91h:46120]

Strengthening of the original result, including generalization to groupoid algebras of Lie groupoids integrating given Lie algebroids:

See also:

On group algebras of (underlying discrete) Heisenberg groups as strict deformation quantizations of pre-symplectic topological vector spaces by continuous fields of Weyl algebras:

On the supersymmetric WZW model using group algebra:

Last revised on December 8, 2023 at 11:15:16. See the history of this page for a list of all contributions to it.