Group Algebra
The group algebra K[G], where K is a field and G a group with the operation *, is the set of all linear combinations of finitely many elements of G with coefficients in K, hence of all elements of the form
| a_1g_1+a_2g_2+...+a_ng_n, |
(1)
|
where a_i in K and g_i in G for all i=1,...,n. This element can be denoted in general by
| [画像: sum_(g in G)a_gg, ] |
(2)
|
where it is assumed that a_g=0 for all but finitely many elements of g.
K[G] is an algebra over K with respect to the addition defined by the rule
the product by a scalar given by
and the multiplication
From this definition, it follows that the identity element of G is the unit of K[G], and that K[G] is commutative iff G is an Abelian group.
If the field K is replaced by a unit ring R, the addition and the multiplication defined above yield the group ring R[G].
If G=Z, and * is the usual addition of integers, the group ring R[G] is isomorphic to the ring R[x^(-1),x] formed by all sums
| [画像: sum_(i=n)^ma_ix^n, ] |
(6)
|
where n,m are integers, and a_i in R for all indices i=n,...,m.
Let G be a locally compact group and mu be a left invariant Haar measure on G. Then the Banach space L^1(G) under the product given by the convolution (f*g)(s)=int_Gf(t)g(t^(-1)s)dmu(t) for s in G is a commutative Banach algebra that is called the group algebra of G.
See also
Algebra, Group, Semigroup AlgebraPortions of this entry contributed by Margherita Barile
Portions of this entry contributed by Mohammad Sal Moslehian
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References
Bonsall, F. F. and Duncan, J. Complete Normed Algebras. New York: Springer-Verlag, 1973.Referenced on Wolfram|Alpha
Group AlgebraCite this as:
Barile, Margherita; Moslehian, Mohammad Sal; and Weisstein, Eric W. "Group Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GroupAlgebra.html