Cohomology of Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$

TL;DR

This paper explicitly computes the first and second cohomology groups of classical Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$ with various modules, showing the vanishing of certain cohomology groups and implications for algebra deformations.

Contribution

It provides explicit calculations of low-degree cohomology groups for these Lie superalgebras and demonstrates the absence of non-trivial deformations of their universal enveloping algebras.

Findings

First and second cohomology groups computed explicitly.

Second cohomology groups with coefficients in universal enveloping algebras vanish.

Universal enveloping algebras do not admit non-trivial Gerstenhaber deformations.

Abstract

We explicitly compute the first and second cohomology groups of the classical Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$ with coefficients in the finite dimensional irreducible modules and the Kac modules. We also show that the second cohomology groups of these Lie superalgebras with coefficients in the respective universal enveloping algebras (under the adjoint action) vanish. The latter result in particular implies that the universal enveloping algebras $U(sl_{m|n})$ and $U(osp_{2|2n})$ do not admit any non-trivial formal deformations of Gerstenhaber type.