symmetric monoidal (∞,1)-category of spectra
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 $G$.
Let $G$ be a discrete group. Let $R$ be a commutative ring.
The group $R$-algebra $R[G]$ is the associative algebra over $R$
whose underlying $R$-module is the the free module over $R$ on the underlying set of $G$;
whose multiplication is given on basis elements by the group operation.
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
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
and the multiplication is given by
The formal linear combinations over $R$ of element in $G$ may equivalently be thought of as functions
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
and for the basis elements one writes
the characteristic function of an element $g \in G$, defined by
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.
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 $\hat{\mathbb{Z}}$;
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.
(…)
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.
There is an adjunction
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.
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.
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:
For every representation $V$, the defining group action
extends uniquely to an algebra homomorphism
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
is an isomorphism: By the previous comment it is sujective, hence it is sufficient to observe that the dimension of the group algebra equals that of the right hand side, hence that
This is the case by this property of the regular representation.
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.
Textbook accounts:
Lecture notes include
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
Last revised on May 14, 2023 at 09:09:20. See the history of this page for a list of all contributions to it.