Lie algebras generated by reflections in types BCD

TL;DR

This paper investigates the Lie algebras generated by reflection elements within the complex group algebras of Weyl groups of types B and D, revealing their structural properties.

Contribution

It determines the structure of Lie subalgebras generated by reflections in the complex group algebras of types B and D Weyl groups.

Findings

Identifies the structure of Lie subalgebras generated by reflections.

Provides explicit descriptions for types B and D.

Enhances understanding of Lie algebra structures related to Weyl groups.

Abstract

We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of even-signed permutations), viewed as Lie algebras via the commutator bracket, and determine the structure of the Lie subalgebras generated by the sets of reflections.