Let $G$ be a quasi-reductive supergroup (so its underlying algebraic group is reductive). To understand better odd elements in quasi-reductive Lie superalgebras, we consider two trivially-intersecting classes of odd elements, which turn out to be particularly convenient to work with: neat odd elements and balanced odd elements. Neat elements are always ad-nilpotent, and are the best analogues one can consider for nilpotent elements in a semisimple Lie algebra, since they may be embedded into subalgebras which are isomorphic to \(\mathfrak{osp}(1\vert{}2)\) , a simple Lie superalgebra whose underlying Lie algebra is\(\mathfrak{sl}(2)\) . Balanced odd elements, on the other hand, are natural generalization of the notion of a self-commuting element (an odd element such that \([x, x] = 0\) , and are used to define homology-type functors on the category of representations of $G$.We study the properties of balanced and neat odd elements in quasi-reductive Lie superalgebras, and show that any odd element $x$ may be written as a sum of a neat and a balanced odd element which commute with each other.

This theorem has a very nice categorical application. Let \(\mathfrak{g}(1\vert{}1)\) be the \(\mathfrak{1|1}\) -dimensional Lie superalgebra generated by an odd element $x$ . The semisimplification of the category of finite-dimensional super-representations of \(\mathfrak{g}(1\vert{}1)\) is a symmetric monoidal functor \(S \colon \operatorname{Rep}(\mathfrak{g}(1\vert{}1)) \to \operatorname{Rep}(\mathrm{OSp}(1\vert{}2))\). Any odd $x$ in \(\operatorname{Lie}(G)\) induces a homomorphism \(i \colon \mathfrak{g}(1\vert{}1) \to \operatorname{Lie}(G)\). Let $\Phi(x) \colon \operatorname{Rep}(G) \to \operatorname{Rep}(\mathrm{OSp}(1|2))$ be the composition of the restriction functor and the semisimplification functor $S$.We show that the functor $\Phi(x)$ may be described explicitly using the homology-type functor corresponding to the balanced part of $x$ n the above decomposition. These homology-type functors are known as Duflo–Serganova functors.

The talk is based on joint work with Inna Entova-Aizenbud.