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

## TL;DR

This paper extends the Plesken Lie algebra concept to associative algebras with anti-involution, especially focusing on semisimple cellular algebras, revealing their Lie algebra structure as a direct sum of orthogonal Lie algebras.

## Contribution

It generalizes the Plesken Lie algebra construction to a broader class of algebras and characterizes the structure of these Lie algebras for semisimple cellular algebras.

## Key findings

- Plesken Lie algebra can be defined for any associative algebra with anti-involution.
- Semisimple cellular algebras have Plesken Lie algebras as direct sums of orthogonal Lie algebras.
- Sizes of orthogonal components relate to dimensions of cell modules.

## 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.