Odd nilpotent element and \mathfrakosp(1|2)-subalgebra in \mathfrakgl(m|n)

2025/10/24 by Jeonghwan Ko, Ko, Junseo
Mathematics · #17B10 (Primary) #17B20 #17B30 (Secondary) #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Mathematics and Applications #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2510.21477

openalex publication_date 2025/10/24 · openalex created_date 2025/10/28 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the conditions under which an odd nilpotent element in \mathfrakgl(m|n) lies inside an \mathfrakosp(1|2)-subalgebra. In the case of the classical Lie algebra \mathfrakgl_m, every nilpotent element can be embedded into an \mathfraksl₂-subalgebra, which is the result of the Jacobson-Morozov Theorem. In the case of the Lie superalgebra \mathfrakgl(m|n), we define super Jordan matrices and prove that an odd nilpotent element e is contained in an \mathfrakosp(1|2)-subalgebra if and only if e lies in the orbit of a super Jordan matrix consisting only of super Jordan blocks of odd size.

Citations

• Jacobson-Morozov Lemma for Algebraic Supergroups
• The Nilpotent Cone for Classical Lie Superalgebras

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