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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 convolution product
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.
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 $G$-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 surjective, hence it is sufficient to observe that the dimension of the group algebra equals that of the right hand side, hence that
This is indeed 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 $k$.
The following is proven in Gilmer 1992, p. 163:
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.
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).
(complex group algebra of $\mathrm{U}(1)$-central extension)
For $\big(G, (-)\cdot(-), \mathrm{e}\big)$ a discrete group, consider a central extension
classified by a circle group-valued group 2-cocycle with underlying function
which we may and do assume to be normalized:
In terms of this cocycle, the group operation on the underlying set
is given by
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
then we have in $\mathbb{C}\big(G^\omega \big)$ the relations
(by normality of $\omega$) and hence
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
In other words, in this quotient of $\mathbb{C}\big(G^\omega\big)$ we enforce the relations
(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
This is the $\omega$-twisted group algebra $\mathbb{C}^\omega(G)$ of $G$:
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).
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:
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:
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:
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:
Group algebras play a role in the relation between ordered abelian groups and divisibility in integral domains:
Last revised on October 17, 2024 at 16:48:25. See the history of this page for a list of all contributions to it.