2013/05/07 by Kaïs Ammari, Kais Ammari, Ammari, Kais
Mathematics · #17B20 #17B45 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B20 #msc:17B45
paper · pdf · doi:10.48550/arxiv.1305.1518
in French. arXiv admin note: text overlap with arXiv:1106.1730 by other authors
arxiv created 2013/05/07 · openalex publication_date 2013/05/07 · arxiv updated 2013/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbbK be an algebraically closed field of characteristic 0. A finite dimensional Lie algebra \mathfrakg over \mathbbK is said to be stable if there exists a linear form g∈\mathfrakg* and a Zariski open subset in \mathfrakg* 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.