An analogue of the Kostant criterion for quadratic Lie superalgebras

TL;DR

This paper extends Kostant's criterion to quadratic Lie superalgebras, providing a necessary and sufficient condition for their structure and deriving an analogue of Parthasarathy's formula for the Dirac operator.

Contribution

It introduces a new criterion for quadratic Lie superalgebras, generalizing Kostant's condition from Lie algebras to superalgebras.

Findings

Established a necessary and sufficient condition for quadratic Lie superalgebras.

Proved an analogue of Parthasarathy's formula for the Dirac operator.

Extended classical results to the superalgebra setting.

Abstract

Let $\mathfrak{r}$ be a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form, let $\mathfrak{p}$ be a finite dimensional complex super vector space with a non-degenerate super-symmetric bilinear form, and let $\nu: \mathfrak{r}\rightarrow\mathfrak{osp}(\mathfrak{p})$ be a Lie superalgebra homomorphism. In this paper, we give a necessary and sufficient condition for $\mathfrak{r}\oplus\mathfrak{p}$ to be a quadratic Lie superalgebra. The criterion obtained is an analogue of a constancy condition given by Kostant in the Lie algebra setting. As an application, we prove an analogue of the Parthasarathy's formula for the square of the Dirac operator attached to a pair of quadratic Lie superalgebras.