Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras

TL;DR

This paper investigates the conditions under which biparabolic subalgebras in reductive Lie algebras are quasi-reductive, extending the classification to include exceptional cases and providing a comprehensive understanding of their structure.

Contribution

It completes the classification of quasi-reductive biparabolic subalgebras in reductive Lie algebras, including exceptional cases, building on prior work on parabolic subalgebras.

Findings

Classification of quasi-reductive biparabolic subalgebras completed

Exceptional cases of reductive Lie algebras addressed

Extended understanding of the structure of seaweed subalgebras

Abstract

We say that a finite dimensional Lie algebra is quasi-reductive if it has a linear form whose stabilizer for the coadjoint representation, modulo the center, is a reductive Lie algebra with a center consisting of semisimple elements. Parabolic subalgebras of a semisimple Lie algebra are not always quasi-reductive (except in types A or C by work of Panyushev). The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso. In this paper, we investigate the quasi-reductivity of biparabolic subalgebras of reductive Lie algebras. Biparabolic (or seaweed) subalgebras are the intersection of two parabolic subalgebras whose sum is the total Lie algebra. As a main result, we complete the classification of quasi-reductive parabolic subalgebras of reductive Lie algebras by considering the exceptional cases.