Related papers: On the Duflo-Serganova functor for the queer Lie s…

We prove that the Duflo-Serganova functor $DS_x$ attached to an odd nilpotent element $x$ of $\mathfrak{osp}(m|2n)$ is semisimple, i.e. sends a semisimple representation $M$ of $\mathfrak{osp}(m|2n)$ to a semisimple representation of…

We discuss several points regarding symmetries of the Duflo-Serganova functor. In particular we give new constructions of Lie superalgebras, Lie supergroups, and associative superalgebras which act on the Duflo-Serganova functor. We connect…

We review old and new results concerning the $DS$ functor and associated varieties for Lie superalgebras. These notions were introduced in the unpublished manuscript arXiv:math/0507198 by Michel Duflo and the third author. This paper…

We study Duflo-Serganova functor for non-twisted affine Lie superalgebras and affine vertex superalgebras.

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo-Serganova functor. Given a simple $\mathfrak{p}(n)$-module $L$ and a certain element $x\in \mathfrak{p}(n)$ of rank $1$, we give an explicit…

We develop a fragment of the theory of Duflo-Serganova functor over a field of odd characteristic. We elaborate a method of computing the symmetry supergroup $\widetilde{\mathbb{G}_x}$ of this functor, recently introduced by A.Sherman, for…

Duflo--Serganova functors play an important role in the representation theory of Lie superalgebras. While it is desirable to understand the images of modules under DS, little is known beyond finite-dimensional representations. For general…

We show that the Duflo-Serganova functor on the category of finite-dimensional modules over a finite-dimensional contragredient Lie superalgebra induces a ring homomorphism on a natural quotient of the Grothendieck ring, which is isomorphic…

In this paper we study extensions between finite-dimensional simple modules over classical Lie superalgebras $\mathfrak{gl}(m|n), \mathfrak{osp}(M|2n)$ and $\mathfrak{q}_m$. We consider a simplified version of the extension graph which is…

In this paper, we study Lie superalgebras of $2\times 2$ matrix-valued first-order differential operators on the complex line. We first completely classify all such superalgebras of finite dimension. Among the finite-dimensional…

For the exceptional finite-dimensional modular Lie superalgebras $\mathfrak{g}(A)$ with indecomposable Cartan matrix $A$, and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd…

We establish an explicit formula for the character of an irreducible finite-dimensional representation of $\mathfrak{gl}(m|n)$. The formula is a finite sum with integer coefficients in terms of a basis $\mathcal{E}_{\mu}$ (Euler characters)…

The Dulfo-Serganova functors DS are tensor functors relating representations of different Lie superalgebras. In this paper we study the behaviour of various invariants, such as the defect, the dual Coxeter number, the atypicality and the…

We define and study cohomological tensor functors from the category $T_n$ of finite-dimensional representations of the supergroup $Gl(n|n)$ into $T_{n-r}$ for $0 <r \leq n$. In the case $DS: T_n \to T_{n-1}$ we prove a formula $DS(L) =…

We systematically apply semisimplification functors in modular representation theory. Motivated by the Duflo--Serganova functor in Lie superalgebras, we construct various functors of interest. In the setting of finite groups, we refine the…

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…

We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the `Irreducibility theorem' and `Duflo theorem', to much wider classes of algebras. Our general version of Irreducibility theorem says…

We prove a direct analogue of the classical Duflo formula in the case of $L_\infty$-algebras. We conjecture an analogous formula in the case of an arbitrary Q-manifold. When $G$ is a compact connected Lie group, the Duflo theorem for the…

Let $\mathfrak g=\mathfrak g_{\bar 0}\oplus\mathfrak g_{\bar 1}$ be the queer Lie superalgebra and let $L$ be a finite-dimensional non-trivial irreducible $\mathfrak g$-module. Restricting the $\mathfrak g$-action on $L$ to $\mathfrak…

Let $L$ be a finite-dimensional Lie algebra over a field of non-zero characteristic and let $S$ be a subalgebra. Suppose that $X$ is a finite set of finite-dimensional $L$-modules. Let $D$ be the category of all finite-dimensional…