On Kostant Root Systems for Lie Superalgebras

TL;DR

This paper extends Kostant's classical results to Lie superalgebras by analyzing eigenspace decompositions under toral subalgebras, demonstrating that Kostant root systems share key properties with classical root systems using combinatorial methods.

Contribution

It introduces a combinatorial approach to study Kostant root systems in Lie superalgebras, avoiding reliance on the Killing form, and generalizes classical root system properties.

Findings

Kostant root systems inherit key properties of classical root systems.

The approach is purely combinatorial, utilizing associated graphs.

Classical results are reproved without using the Killing form.

Abstract

We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenspaces appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of classical root systems. Our approach is combinatorial in nature and utilizes certain graphs naturally associated with Kostant root systems. In particular, we reprove Kostant's results without making use of the Killing form.