Jacobson-Morozov Lemma for Algebraic Supergroups

Inna Entova-Aizenbud; Vera Serganova

doi:10.1016/j.aim.2022.108240

Jacobson-Morozov Lemma for Algebraic Supergroups

Inna Entova-Aizenbud, Vera Serganova

Department of Mathematics

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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
Article number	108240
Journal	Advances in Mathematics
Volume	398
DOIs	https://doi.org/10.1016/j.aim.2022.108240
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

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10.1016/j.aim.2022.108240

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title = "Jacobson-Morozov Lemma for Algebraic Supergroups",

abstract = "Given a quasi-reductive algebraic supergroup G, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor Φx:Rep(G)→Rep(OSp(1|2)) associated to any given element x∈Lie(G)1¯. For nilpotent elements x, we show that the functor Φx 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.",

keywords = "Duflo Serganova functors, Jacobson-Morozov Lemma, Lie superalgebra, Lie supergroup, Semisimplification, Tensor categories",

author = "Inna Entova-Aizenbud and Vera Serganova",

note = "Funding Information: We would like to thank Joseph Bernstein, Pavel Etingof and Victor Ostrik for helpful discussions, and Kevin Coulembier for helpful remarks. The authors were supported by the NSF-BSF grant NSFMath 2019694 . Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

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T1 - Jacobson-Morozov Lemma for Algebraic Supergroups

AU - Entova-Aizenbud, Inna

AU - Serganova, Vera

N1 - Funding Information: We would like to thank Joseph Bernstein, Pavel Etingof and Victor Ostrik for helpful discussions, and Kevin Coulembier for helpful remarks. The authors were supported by the NSF-BSF grant NSFMath 2019694 . Publisher Copyright: © 2022 Elsevier Inc.

PY - 2022/3/26

Y1 - 2022/3/26

N2 - Given a quasi-reductive algebraic supergroup G, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor Φx:Rep(G)→Rep(OSp(1|2)) associated to any given element x∈Lie(G)1¯. For nilpotent elements x, we show that the functor Φx 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.

AB - Given a quasi-reductive algebraic supergroup G, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor Φx:Rep(G)→Rep(OSp(1|2)) associated to any given element x∈Lie(G)1¯. For nilpotent elements x, we show that the functor Φx 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.

KW - Duflo Serganova functors

KW - Jacobson-Morozov Lemma

KW - Lie superalgebra

KW - Lie supergroup

KW - Semisimplification

KW - Tensor categories

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U2 - 10.1016/j.aim.2022.108240

DO - 10.1016/j.aim.2022.108240

M3 - Article

SN - 0001-8708

VL - 398

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108240

ER -

Jacobson-Morozov Lemma for Algebraic Supergroups

Abstract

Keywords

ASJC Scopus subject areas

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