Transitive irreducible Lie superalgebras of vector fields

TL;DR

This paper characterizes the structure of certain Lie superalgebras of vector fields over flag varieties, showing their simplicity and isomorphism to supervector fields on superpoints under specific conditions.

Contribution

It establishes conditions under which the Lie superalgebra of superderivations is transitive, irreducible, and simple, and identifies it with the supervector fields on a superpoint.

Findings

The Lie superalgebra is transitive and irreducible when associated with an irreducible P-module.

It is proven to be simple under the given conditions.

The algebra is isomorphic to the Lie superalgebra of vector fields on a superpoint.

Abstract

Let $\mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf of sections of the exterior algebra of the homogeneous vector bundle $E$ over the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie group and $P$ its parabolic subgroup. Then, $\mathfrak{d}$ is transitive and irreducible whenever $E$ is defined by an irreducible $P$-module $V$ such that the highest weight of $V^*$ is dominant. Moreover, $\mathfrak{d}$ is simple; it is isomorphic to the Lie superalgebra of vector fields on the superpoint, i.e., on a $0|n$-dimensional supervariety.