Jacobson-Morozov Lemma for Algebraic Supergroups
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912835927277568 |
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| author | Entova-Aizenbud, Inna Serganova, Vera |
| author_facet | Entova-Aizenbud, Inna Serganova, Vera |
| contents | Given a quasi-reductive algebraic supergroup $G$, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor $Φ_x: Rep(G) \to Rep(OSp(1|2))$ associated to any given element $x \in \mathrm{Lie}(G)_{\bar 1}$. For nilpotent elements $x$, we show that the functor $Φ_x$ can be defined using the Deligne filtration associated to $x$.
We use this approach to prove an analogue of the Jacobson-Morozov Lemma for algebraic supergroups. Namely, we give a necessary and sufficient condition on odd nilpotent elements $x\in \mathrm{Lie}(G)_{\bar 1}$ which define an embedding of supergroups $OSp(1|2)\to G$ so that $x$ lies in the image of the corresponding Lie algebra homomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_08731 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Jacobson-Morozov Lemma for Algebraic Supergroups Entova-Aizenbud, Inna Serganova, Vera Representation Theory Given a quasi-reductive algebraic supergroup $G$, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor $Φ_x: Rep(G) \to Rep(OSp(1|2))$ associated to any given element $x \in \mathrm{Lie}(G)_{\bar 1}$. For nilpotent elements $x$, we show that the functor $Φ_x$ can be defined using the Deligne filtration associated to $x$. We use this approach to prove an analogue of the Jacobson-Morozov Lemma for algebraic supergroups. Namely, we give a necessary and sufficient condition on odd nilpotent elements $x\in \mathrm{Lie}(G)_{\bar 1}$ which define an embedding of supergroups $OSp(1|2)\to G$ so that $x$ lies in the image of the corresponding Lie algebra homomorphism. |
| title | Jacobson-Morozov Lemma for Algebraic Supergroups |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2007.08731 |