On a super-analog of the Schur-Weyl Duality

TL;DR

This paper explores two super-analogues of the Schur-Weyl duality involving Lie superalgebras and symmetric groups, constructing isomorphisms and describing dualities between their centers.

Contribution

It introduces a special symmetrization isomorphism between symmetric and universal enveloping algebras of Lie superalgebras, advancing understanding of their dualities.

Findings

Constructed an isomorphism called special symmetrization.

Explicitly described the duality between centers of universal enveloping algebras and group algebras.

Analyzed dualities involving Lie superalgebras and symmetric groups.

Abstract

Two super-analogs of the Schur-Weyl duality are considered: the duality of actions in $(\mathbb{C}^{m|n})^{\otimes N}$ of the Lie superalgebra $\mathfrak{gl}(m,n)$ and the symmetric group $S_N$, and the duality of actions of the Lie superalgebra $Q(n)$ and a certain finite group $Se(N)$ in $(\mathbb{C}^{n|n})^{\otimes N}$. We construct an isomorphism of symmetric and universal enveloping algebras of these Lie superalgebras called special symmetrization. Using this isomorphism of vector spaces we describe explicitly the duality between the centers of the corresponding universal enveloping algebras and the group algebras.