Jacobson-Morozov Lemma for Algebraic Supergroups

Given a quasi-reductive algebraic supergroup $G$, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor ${\mathrm{\Phi }}_x:Rep(G)\to Rep(OSp(1\mathrm{\mid }2))$ associated to any given element $x\in \mathrm{L}\mathrm{i}\mathrm{e}(G{)}_{\bar{1}}$. For nilpotent elements $x$, we show that the functor ${\mathrm{\Phi }}_x$ 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 $x\in \mathrm{L}\mathrm{i}\mathrm{e}(G{)}_{\bar{1}}$ which define an embedding of supergroups $OSp(1\mathrm{\mid }2)\to G$ so that $x$ lies in the image of the corresponding Lie algebra homomorphism.