Vector fields on $\Pi$-symmetric flag supermanifolds

TL;DR

This paper computes the Lie superalgebras of holomorphic vector fields on complex -symmetric flag supermanifolds, showing most are generated by the natural Lie superalgebra action, with one notable exception.

Contribution

It provides the first explicit computation of holomorphic vector field Lie superalgebras on -symmetric flag supermanifolds, revealing their structure and fundamental nature.

Findings

Most vector fields are generated by the natural () action

Explicit Lie superalgebra structures are determined

One exceptional case where the vector fields are not fundamental

Abstract

The main result of this paper is the computation of the Lie superalgebras of holomorphic vector fields on the complex $\Pi$-symmetric flag supermanifolds, introduced by Yu.I.~Manin. We prove that with one exception any vector field is fundamental with respect to the natural action of the Lie superalgebra $\mathfrak q_n(\mathbb C)$.