Generalized Matrix Algebras

Abstract

The algebras considered here arose in the investigation of an algebra connected with the orthogonal group. We consider an algebra of dimension mn over a field K of characteristic zero, and possessing a basis {eij} (1 ≤ i ≤ m; 1 ≤ j ≤ n) with the multiplication property1,The field elements ϕij form a matrix Φ = (ϕij) of order n × m. It will be called the multiplication matrix of the algebra relative to the basis {ϕij}.

A characteristic free approach to Brauer algebras

“…We have introduced this concept for general cellular algebras in [KX2] and shown (theorem 4.1 in [KX2]) that actually an algebra is cellular if and only if it is such an iterated inflation. Special cases of inflations can be found in the papers of Graham and Lehrer [GL], Hanlon and Wales [HW1] and even in the early studies of Brown [Brow1,Brow2,Brow3], who considered 'generalised matrix algebras' which coincide in his case with our inflations.…”

“…In this section we use matrices to present and study the inflated algebras (see Brown's generalised matrix algebras [Brow1]). We assume that V is a non-zero vector space with a basis {v 1 , · · · , v m } and that B is an arbitrary k-algebra with identity.…”

“…Define E( [Brow1] we will prove that the radical of A has the kdimension l 2 s 2 − r 2 , where r is the rank of the matrix Φ. We fix a left ideal L µ0 in B and its basis …”

On the radical of Brauer algebras

“…For such an A, the following hold (cf. [Bw1]): (1) either A is simple, or A has non-zero radical Rad (A) , and A Rad (A) is simple; (2) A is simple if and only if it has an identity element; (3) dim k (A) = h 2 , for some h ∈ N , and dim k Rad (A) = h 2 − rk Φ(A) 2 ; (4) the nilpotency degree of Rad (A) is at most 3 .…”

“…We recall (from [Bw1]) the notion of generalized matrix algebra: this is any associative k-algebra A with a finite basis { e ij } i,j∈I for which the multiplication table looks like e ij · e pq = σ * jp e iq for some σ * jp ∈ k ∀ i, j, p, q ∈ I . Then we set Φ(A) := σ * ij i,j∈I .…”