On extensions of the Jacobson-Morozov theorem to even characteristic
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911758261682176 |
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| author | Stewart, David I. Thomas, Adam R. |
| author_facet | Stewart, David I. Thomas, Adam R. |
| contents | Let G be a simple algebraic group over an algebraically closed field k of characteristic 2. We consider analogues of the Jacobson-Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3-dimensional Lie overalgebra in $\mathfrak{g} := \text{Lie}(G)$ and also those with overalgebras isomorphic to the algebras $\text{Lie}(\text{SL}_2)$ and $\text{Lie}(\text{PGL}_2)$. This leads us to calculate the dimension of Lie automiser $\mathfrak{n}_\mathfrak{g}(k\cdot e)/\mathfrak{c}_\mathfrak{g}(e)$ for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07303 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On extensions of the Jacobson-Morozov theorem to even characteristic Stewart, David I. Thomas, Adam R. Representation Theory Group Theory Rings and Algebras 17B45 Let G be a simple algebraic group over an algebraically closed field k of characteristic 2. We consider analogues of the Jacobson-Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3-dimensional Lie overalgebra in $\mathfrak{g} := \text{Lie}(G)$ and also those with overalgebras isomorphic to the algebras $\text{Lie}(\text{SL}_2)$ and $\text{Lie}(\text{PGL}_2)$. This leads us to calculate the dimension of Lie automiser $\mathfrak{n}_\mathfrak{g}(k\cdot e)/\mathfrak{c}_\mathfrak{g}(e)$ for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny. |
| title | On extensions of the Jacobson-Morozov theorem to even characteristic |
| topic | Representation Theory Group Theory Rings and Algebras 17B45 |
| url | https://arxiv.org/abs/2401.07303 |