NetNews Usenet Archive 1992 #23/NN_1992_23.iso/spool/sci/math/13086

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From: [e-mail] (Frederick W. Chapman)
Newsgroups: sci.math
Subject: Re: Matrices as Group Sub-Algebras ?
Message-ID: <[e-mail]>
Date: 13 Oct 92 02:16:16 GMT
Organization: Lehigh University
Lines: 49

In article <[e-mail]>,
[e-mail] (BORIS Borcic) writes:

>There are probably other ways to look into. My question
>boils down to : are there any (standard, simple) ways
>to construct endomorphisms from matrix algebra into
>group algebra ?

As stated, your question seems to ask: does an arbitrary group algebra
contain a subalgebra isomorphic to a matrix algebra; or put another
way, can we embed a matrix algebra in a given group algebra.  I really
have no idea.  If you ask the question the other way, however, the
answer is easy: can we embed a group algebra in a matrix algebra?
Yes, always.

Let A be ANY n-dimensional algebra with identity 1 over a field F, and
let M(n,F) denote the algebra of n x n matrices over F.  The regular
matrix representations of A give embeddings of A in M(n,F) as
follows.  Let x and y belong to A, and define lambda_x : A --> A
by labmda_x (y) := xy; similarly, define rho_x : A --> A by
rho_x (y) := yx.  lambda_x is left-multiplication by x, and rho_x is
right-multiplication by x; both mappings are linear transformations
from A to itself (where A is viewed as an F-vectorspace).  Fix a basis
g_1, ..., g_n of A (the choice of a basis is obvious when A is a group
algebra -- the elements of the group, by definition, form a basis of
the group algebra).  Let L_x denote the matrix of the linear
transformation lambda_x with respect to the fixed basis; likewise, let
R_x denote the matrix of the linear transformation rho_x.  The mapping
A --> M(n,F) given by x |--> L_x is called the left regular matrix
representation of the algebra A; the mapping given by x |--> R_x is
called the right regular matrix representation.  Both left and right
regular representations are algebra homomorphisms.  Furthermore, the
presence of an identity element 1 in A makes both the left and right
regular representations "faithful", which means one-to-one/injective.
Thus, the left and right regular matrix representations give
embeddings of the algebra A into the matrix algebra M(n,F).

I know this is not exactly what you asked, but I hope the information
is of some use.  For more information, look for books on linear
representations of finite groups and linear representations of
finite-dimensional algebras over a field.
--

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