Jacobson-Morozov Lemma for Algebraic Supergroups

Inna Entova-Aizenbud, Vera Serganova

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

Original languageEnglish
Article number108240
JournalAdvances in Mathematics
Volume398
DOIs
StatePublished - 26 Mar 2022

Keywords

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

ASJC Scopus subject areas

  • Mathematics (all)

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