On the geometry of algebras related to the Weyl groupoid

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.