The Plesken Lie algebra for associative algebras with anti-involution: semisimple cellular algebras

Abstract

Cohen and Taylor, following an idea of Plesken, introduced a Lie algebra to the complex group algebra of a finite group and determined its structure, based on the character theory of the group. We show how the definition of this Plesken Lie algebra can be extended to any associative algebra with an anti-involution. After some examples we consider semisimple cellular algebras and prove that their Plesken Lie algebras are direct sums of orthogonal Lie algebras, the sizes of which are determined by the dimensions of the cell modules of the cellular algebra.