Stabilit\'e des sous-alg\`ebres paraboliques de so(n)

TL;DR

This paper investigates the stability of certain subalgebras of orthogonal Lie algebras, proving a conjecture that stability and quasi-reductivity are equivalent for these classes, with implications for Lie algebra classification.

Contribution

It proves Panyushev's conjecture for parabolic subalgebras of orthogonal Lie algebras and extends the equivalence of stability and quasi-reductivity to specific stabilizer Lie algebras.

Findings

Proved the conjecture for parabolic subalgebras of orthogonal Lie algebras.

Established equivalence between stability and quasi-reductivity in new classes of Lie algebras.

Enhanced understanding of the structure and classification of stable Lie algebras.

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic 0. A finite dimensional Lie algebra $\mathfrak{g}$ over $\mathbb{K}$ is said to be stable if there exists a linear form $g\in\mathfrak{g}^{*}$ and a Zariski open subset in $\mathfrak{g}^{*}$ containing $g$ in which all elements have their stabilizers conjugated under the connected adjoint group. It is well known that any quasi-reductive Lie algebra is stable. However, there are stable Lie algebras which are not quasi-reductive. This raises the question, if for some particular class of non-reductive Lie algebras, there is equivalence between stability and quasi-reductivity. In particular, it was conjectured by Panyushev that these two notions are equivalent for biparabolic subalgebras of a reductive Lie algebra. In this paper, we prove this conjecture for parabolic subalgebras of orthogonal Lie algebras and we answer positively to this question for certain Lie algebras which stabilize an alternating bilinear form of maximal rank and a flag in generic position.