The Nullstellensatz for supersymmetric polynomials

TL;DR

This paper establishes a Nullstellensatz for supersymmetric polynomials, linking radical ideals to superalgebraic sets invariant under a specific Weyl groupoid, despite the algebra's non-Noetherian nature.

Contribution

It proves a Nullstellensatz for supersymmetric polynomials, including Laurent variants, and applies this to resolve a conjecture about maximal ideals in Lie superalgebra enveloping algebras.

Findings

Bijection between radical ideals and superalgebraic sets.

Decomposition of superalgebraic sets into irreducible components.

Proof of a conjecture on maximal ideals in Lie superalgebra enveloping algebras.

Abstract

In this paper we prove a Nullstellensatz for supersymmetric polynomials. This gives a bijection between radical ideals and superalgebraic sets. These are algebraic sets which are invariant under the Weyl groupoid of Sergeev and Veselov, \cite{SV2}. Note that the algebra of supersymmetric polynomials is not Noetherian, so the usual Nullstellensatz does not apply. However it deos satisfy the ascending chain condition on radical ideals and this allows for the decomposition of superalgebraic sets into irreducible components. Analogous results hold for the a ring of Laurent supersymmetric polynomials. As an application, we give a proof of conjecture 13.5.1 from \cite{M}. This concerns the maximal ideals in the enveloping algebra of the general linear and orthosymplectic Lie superalgebras. The center is closely related to the algebra of supersymmetric polynomials and the result can be thought of as an analog of the weak Nullstellensatz.