Jacobson-Morozov Lemma for Algebraic Supergroups
Inna Entova-Aizenbud, Vera Serganova
TL;DR
The paper develops an odd Jacobson–Morozov theory for quasi-reductive algebraic supergroups by introducing neatness for odd nilpotent elements and constructing a universal semisimple quotient via a symmetric monoidal functor $\Phi_x = S\circ R_x$ from $\mathrm{Rep}(G)$ to $\mathrm{Rep}(OSp(1|2))$. It proves that a nonzero neat odd element $x$ in $\mathrm{Lie}(G)_{\bar{1}}$ yields an embedding of $OSp(1|2)$ into $G$ extending the corresponding $\mathbb{G}_a^{(1|1)}$-embedding, with uniqueness up to $G_{\bar{0}}$-conjugacy. The work develops a Deligne filtration realization of $\Phi_x$, relates the construction to the Duflo–Serganova functor, and introduces the minuscule supergroup $\mathbb{M}$ as a universal emitter of neat indecomposables whose semisimplification recovers $Rep(OSp(1|2))$. It further investigates the structure of neat elements, the finiteness of neat orbits, and reductive envelopes, providing a robust categorical framework for extending odd nilpotent data to full supergroup embeddings with applications to representation theory of 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 $Φ_x: Rep(G) \to Rep(OSp(1|2))$ associated to any given element $x \in \mathrm{Lie}(G)_{\bar 1}$. For nilpotent elements $x$, we show that the functor $Φ_x$ 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 $x\in \mathrm{Lie}(G)_{\bar 1}$ which define an embedding of supergroups $OSp(1|2)\to G$ so that $x$ lies in the image of the corresponding Lie algebra homomorphism.