Basic quasi-reductive root data and supergroups
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| Format: | Preprint |
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2023
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| author | Fioresi, Rita Shu, Bin |
| author_facet | Fioresi, Rita Shu, Bin |
| contents | We investigate pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd $G$-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup $\mathbb{G}$, that is, $\mathbb{G}_{\text{ev}}$ is isomorphic to $G$, and the quotient $\mathbb{G}/\mathbb{G}_{\text{ev}}$ is $G$-equivariantly isomorphic to $Y$. We prove that, if $Y$ satisfies certain conditions (basic quasi-reductive root data), then the question has a positive answer given by an existence and uniqueness theorem. The corresponding supergroups are said to be basic quasi-reductive, which can be classified, up to isogeny. We then decide the structure of connected quasi-reductive algebraic supergroups provided that: (i) the root system does not contain $0$; (ii) $\mathfrak{g}:=\text{Lie}(\mathbb{G})$ admits a non-degenerate even symmetric bilinear form. (iii) all odd reflections are invertible. Remarkably, those supergroups are exactly basic quasi-reductive supergroups of monodromy type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_18065 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Basic quasi-reductive root data and supergroups Fioresi, Rita Shu, Bin Representation Theory Algebraic Geometry Group Theory 17B05, 14M30, 58A50, 14A22 We investigate pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd $G$-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup $\mathbb{G}$, that is, $\mathbb{G}_{\text{ev}}$ is isomorphic to $G$, and the quotient $\mathbb{G}/\mathbb{G}_{\text{ev}}$ is $G$-equivariantly isomorphic to $Y$. We prove that, if $Y$ satisfies certain conditions (basic quasi-reductive root data), then the question has a positive answer given by an existence and uniqueness theorem. The corresponding supergroups are said to be basic quasi-reductive, which can be classified, up to isogeny. We then decide the structure of connected quasi-reductive algebraic supergroups provided that: (i) the root system does not contain $0$; (ii) $\mathfrak{g}:=\text{Lie}(\mathbb{G})$ admits a non-degenerate even symmetric bilinear form. (iii) all odd reflections are invertible. Remarkably, those supergroups are exactly basic quasi-reductive supergroups of monodromy type. |
| title | Basic quasi-reductive root data and supergroups |
| topic | Representation Theory Algebraic Geometry Group Theory 17B05, 14M30, 58A50, 14A22 |
| url | https://arxiv.org/abs/2303.18065 |