On the geometry of algebras related to the Weyl groupoid

TL;DR

This paper explores the algebraic geometry of certain algebras associated with the Weyl groupoid of Lie superalgebras, establishing Nullstellensatz properties and describing superalgebraic sets in relation to groupoid orbits.

Contribution

It investigates the structure of algebras related to the Weyl groupoid, proving Nullstellensatz-like results and characterizing superalgebraic sets for these algebras.

Findings

Algebras satisfy the Nullstellensatz in many cases.

Superalgebraic sets correspond to unions of groupoid orbits.

Provides descriptions of minimal superalgebraic sets containing given closed sets.

Abstract

Let $\mathtt{k}$ be an algebraically closed field of characteristic zero. Let $\mathfrak{g} $ be a finite dimensional classical simple Lie superalgebra over $\mathtt{k}$ or $\mathfrak{g} l(m,n)$. In the case that $\mathfrak{g} $ is a Kac-Moody algebra of finite type with set of roots $\Delta$, Sergeev and Veselov introduced the Weyl groupoid $\mathfrak{W}=\mathfrak{W}(\Delta)$, which has significant connections with the representation theory of $\mathfrak{g} $. Let $\mathfrak{h}$, $W$ and $Z(\mathfrak{g} )$ be a Cartan subalgebra of $\mathfrak{g} _0$, the Weyl group of $\mathfrak{g} _0$ and the center of $U(\mathfrak{g} )$ respectively. Also let $G$ be a Lie supergroup with Lie $G =\mathfrak{g} $. There are several important commutative algebras related to $\mathfrak{W}$. Namely \begin{itemize} \item The image $I(\mathfrak{h} )$ of the injective Harish-Chandra map $Z(\mathfrak{g} ){\longrightarrow} S(\mathfrak{h} )^W$. \item The supercharacter $\mathbb Z$-algebras $J(\mathfrak{g} )$ and $J(G)$ of finite dimensional representations of $\mathfrak{g} $ and $G$. \end{itemize} Let $\mathcal A = \mathcal A(\mathfrak{g})$ be denote either $I(\mathfrak{h} )$ or $J(G) \otimes_{\mathbb Z}{\mathtt{k}}$. The purpose of this paper is to investigate the algebraic geometry of $\mathcal A.$ In many cases, the algebra $\mathcal A$ satisfies the Nullstellensatz. This gives a bijection between radical ideals in $\mathcal A$ and superalgebraic sets (zero loci of such ideals). Any superalgebraic set is uniquely a finite union of irreducible superalgebraic components. In the non-exceptional Kac-Moody case, we describe the smallest superalgebraic set containing a given (Zariski) closed set, and show that the superalgebraic sets are exactly the closed sets that are unions of groupoid orbits.