The Joseph ideal for $\mathfrak{sl}(m|n)$

TL;DR

This paper extends the characterization of the Joseph ideal to the Lie superalgebra sl(m|n), defining a quadratic ideal that is primitive for m-n>2 and relates to previous constructions.

Contribution

It generalizes the Joseph ideal characterization to Lie superalgebras, specifically sl(m|n), and proves primitivity under certain conditions.

Findings

The quadratic ideal is primitive when m-n>2.

The ideal can be characterized similarly to Garfinkle's construction.

Extension of deformation theory methods to Lie superalgebras.

Abstract

Using deformation theory, Braverman and Joseph obtained an alternative characterisation of the Joseph ideal for simple Lie algebras, which included even type A. In this note we extend that characterisation to define a remarkable quadratic ideal for sl(m|n). When m-n>2 we prove the ideal is primitive and can also be characterised similarly to the construction of the Joseph ideal by Garfinkle.