Jacobson-Morozov Lemma for Algebraic Supergroups

TL;DR

This paper extends the Jacobson-Morozov Lemma to algebraic supergroups by constructing a symmetric monoidal functor from the representation category of a supergroup to that of OSp(1|2), characterizing odd nilpotent elements.

Contribution

It introduces a new approach using semisimplifications and Deligne filtration to establish an analogue of the Jacobson-Morozov Lemma for algebraic supergroups.

Findings

Established a criterion for embedding OSp(1|2) into supergroups based on odd nilpotent elements.

Defined a symmetric monoidal functor linking representations of supergroups and OSp(1|2).

Proved the necessary and sufficient conditions for the existence of such embeddings.

Abstract

Given a quasi-reductive algebraic supergroup $G$, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor $\Phi_x: Rep(G) \to Rep(OSp(1|2))$ associated to any given element $x \in \mathrm{Lie}(G)_{\bar 1}$. For nilpotent elements $x$, we show that the functor $\Phi_x$ 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.