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group algebra

Context

Algebra

Group Theory

Contents

Idea

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.

Definition

For discrete groups

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

Definition

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.

Remark

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

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} \,.
Remark

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, oftn 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.

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 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

(…)

Properties

General

Proposition

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.

Remark

There is an adjunction

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

between the category of 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.

Remark

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. And any homomorphism of groups p:GAut(V)p: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.

Theorem

(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 k.

Localizations

References

Lecture notes include

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

  • M. Farber, P. Vogel, The Cohn localization if 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

Revised on December 8, 2013 18:14:44 by David Roberts (129.127.252.10)