## Abstract

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)→Rep(OSp(1|2)) associated to any given element x∈Lie(G)_{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∈Lie(G)_{1¯} which define an embedding of supergroups OSp(1|2)→G so that x lies in the image of the corresponding Lie algebra homomorphism.

Original language | English |
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Article number | 108240 |

Journal | Advances in Mathematics |

Volume | 398 |

DOIs | |

State | Published - 26 Mar 2022 |

## Keywords

- Duflo Serganova functors
- Jacobson-Morozov Lemma
- Lie superalgebra
- Lie supergroup
- Semisimplification
- Tensor categories

## ASJC Scopus subject areas

- General Mathematics