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 the convolution product

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

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

General

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

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

This is indeed 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 $k$.

The following is proven in Gilmer 1992, p. 163:

Theorem

Let $A$ be an abelian group and $R$ a commutative ring. Then $R[A]$ is an integral domain iff $R$ is an integral domain and $A$ is torsion-free.

Of central extensions & Twisted group algebras

A quotient algebra of the group algebra of a central extension $G^\omega$ of a group $G$ corresponding to a group 2-cocycle $\omega \,\colon\, G \times G \to k$ is the $\omega$-twisted group algebra of $G$ (eg. Nachbin 1993, Ch 2, Thm. 4.1).

Example

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

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

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

$$\omega \,\colon\, G \times G \to \mathbb{R}/\mathbb{Z} \,,$$

which we may and do assume to be normalized:

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

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

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

is given by

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

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

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

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

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

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

(by normality of $\omega$) and hence

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

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

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

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

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

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

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

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

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

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

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

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

Example

(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)

Related concepts

• Cohn localization

• groupoid algebra, category algebra

• augmentation ideal

• pseudocompact ring

• Day convolution

• ∞-group ∞-ring

References

General

Original discussion:

• Hermann Weyl, §III.13 in: Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: The Theory of Groups and Quantum Mechanics, Dover (1950) [ISBN:0486602699, ark:/13960/t1kh1w36w]

Monographs

for the case of locally compact topological groups:

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

• Klaas Landsman, §C.18 in: Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open (2017) [doi:10.1007/978-3-319-51777-3, pdf]

for the case of 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)

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

• C. M. Edwards, $C^\ast$-algebras of central group extensions I, Annales de l’institut Henri Poincaré. Section A, Physique Théorique 10 3 (1969) 229-246 [numdam:AIHPA_1969__10_3_229_0]

• C. M. Edwards, John T. Lewis, Twisted group algebras I, Communications in Mathematical Physics 13 (1969) 119–130 [doi:10.1007/BF01649871, euclid:cmp/1103841535]

• C. M. Edwards, John T. Lewis, Twisted group algebras II, Communications in Mathematical Physics 13 (1969) 131–141 [doi:10.1007/BF01649872]

• Leopoldo Nachbin (ed.), Twisted Group Algebras, Chapter 2 in: Group Representations Volume 2, North-Holland Mathematics Studies 177 (1993) 65-103 [doi:10.1016/S0304-0208(09)70147-1]

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:

• Nicolaas P. Landsman, §3.4 in: Classical and quantum representation theory [arXiv:hep-th/9411172]

• Nicolaas P. Landsman, Ex. 2 in: Lie Groupoid $C^\ast$-Algebras and Weyl Quantization, Communications in Mathematical Physics 206 (1999) 367–381 [doi:10.1007/s002200050709]

• Nicolaas P. Landsman, B. Ramazan, Ex. 11.1 in: Quantization of Poisson algebras associated to Lie algebroids, in: Groupoids in Analysis, Geometry, and Physics, Contemporary Mathematics 282 (2001) [arXiv:math-ph/0001005, ams:conm/282]

See also:

• Klaas Landsman, Strict deformation quantization of a particle in external gravitational and Yang-Mills fields, Journal of Geometry and Physics 12 2 (1993) 93-132 [doi:10.1016/0393-0440(93)90010-C]

• Nicolaas P. Landsman, p. 27 in: Classical and quantum representation theory [arXiv:hep-th/9411172]

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

• Ernst Binz, Reinhard Honegger, Alfred Rieckers, Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization, International Journal of Pure and Applied Mathematics 38 1 (2007) [ijpam:2007-38-1/6, pdf]

• Reinhard Honegger, Alfred Rieckers, Heisenberg Group Algebra and Strict Weyl Quantization, Chapter 23 in: Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems, World Scientific (2015) [chapter:doi;10.1142/9789814696586_0023, book:doi:10.1142/9251-vol1]

On the supersymmetric WZW model using group algebra:

• Matthias Blau, George Thompson, Equivariant Kähler Geometry and Localization in the $G/G$ Model, Nucl. Phys. B 439 (1995) 367-394 [doi:10.1016/0550-3213(95)00058-Z, arXiv:hep-th/9407042]

Group algebras play a role in the relation between ordered abelian groups and divisibility in integral domains:

• Robert Gilmer: Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics (1992)