Weyl groupoids and superalgebraic sets

TL;DR

This paper explores the geometry of algebras associated with Weyl groupoids, establishing a Nullstellensatz correspondence with superalgebraic sets, and characterizing Laurent supersymmetric polynomials to unify various algebraic definitions.

Contribution

It introduces a Nullstellensatz for algebras related to Weyl groupoids, characterizes superalgebraic sets, and unifies different algebraic frameworks for $J(G)$.

Findings

Bijective correspondence between radical ideals and superalgebraic sets

Explicit description of the smallest superalgebraic set containing a given set

Characterizations of Laurent supersymmetric polynomials

Abstract

This paper is a contribution to the study of the geometry of algebras related the Weyl groupoid initiated in \cite{M22}. The Nullstellensatz gives a bijection between radical ideals of such an algebra and their zero loci, the superalgebraic sets. Such sets are exactly the (Zariski) closed sets that are invariant under the action of a suitable groupoid, and the smallest superalgebraic set containing a given closed set can be described explicitly. Here we give several examples of superalgebraic sets. We also give several characterizations of Laurent supersymmetric polynomials. These adapt to unite several definitions of one of the algebras of interest, $J(G)$ that may be found in the literature.