Odd nilpotent element and $\mathfrak{osp}(1|2)$-subalgebra in $\mathfrak{gl}(m|n)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915578957004800 |
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| author | Ko, Junseo |
| author_facet | Ko, Junseo |
| contents | In this paper, we investigate the conditions under which an odd nilpotent element in $\mathfrak{gl}(m|n)$ lies inside an $\mathfrak{osp}(1|2)$-subalgebra. In the case of the classical Lie algebra $\mathfrak{gl}_m$, every nilpotent element can be embedded into an $\mathfrak{sl}_2$-subalgebra, which is the result of the Jacobson-Morozov Theorem. In the case of the Lie superalgebra $\mathfrak{gl}(m|n)$, we define super Jordan matrices and prove that an odd nilpotent element $e$ is contained in an $\mathfrak{osp}(1|2)$-subalgebra if and only if $e$ lies in the orbit of a super Jordan matrix consisting only of super Jordan blocks of odd size. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21477 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Odd nilpotent element and $\mathfrak{osp}(1|2)$-subalgebra in $\mathfrak{gl}(m|n)$ Ko, Junseo Representation Theory 17B10 (Primary), 17B20, 17B30 (Secondary) In this paper, we investigate the conditions under which an odd nilpotent element in $\mathfrak{gl}(m|n)$ lies inside an $\mathfrak{osp}(1|2)$-subalgebra. In the case of the classical Lie algebra $\mathfrak{gl}_m$, every nilpotent element can be embedded into an $\mathfrak{sl}_2$-subalgebra, which is the result of the Jacobson-Morozov Theorem. In the case of the Lie superalgebra $\mathfrak{gl}(m|n)$, we define super Jordan matrices and prove that an odd nilpotent element $e$ is contained in an $\mathfrak{osp}(1|2)$-subalgebra if and only if $e$ lies in the orbit of a super Jordan matrix consisting only of super Jordan blocks of odd size. |
| title | Odd nilpotent element and $\mathfrak{osp}(1|2)$-subalgebra in $\mathfrak{gl}(m|n)$ |
| topic | Representation Theory 17B10 (Primary), 17B20, 17B30 (Secondary) |
| url | https://arxiv.org/abs/2510.21477 |