论文标题
在与Weylsoid相关的代数的几何形状上
On the geometry of algebras related to the Weyl groupoid
论文作者
论文摘要
令$ \ mathtt {k} $为特征零的代数封闭字段。令$ \ mathfrak {g} $为有限的维度经典简单谎言superalgebra，超过$ \ mathtt {k} $或$ \ mathfrak {g} l（m，n）$。在这种情况下，$ \ mathfrak {g} $是有限类型的kac-moody代数，带有一组根$δ$，Sergeev和Veselov介绍了Weyl groupoid $ \ Mathfrak {w} = \ Mathfrak {w} = \ Mathfrak {w}（w}（δ）$，它与$ Mathak的代表理论有很大的联系。令$ \ mathfrak {h} $，$ w $和$ z（\ mathfrak {g}）$为$ \ mathfrak {g} _0 $的cartan subalgebra，$ \ m athfrak {g mathfrak {g} _0 $的Weyl grout of $ u（\ mathfrak {另外，让$ g $为Lie SuperGroup，lie $ g = \ mathfrak {g} $。与$ \ mathfrak {w} $有关的几个重要的交换代数。即\ begin {initizize} \项目$ i（\ mathfrak {h}）$的harish-chandra map $ z（\ mathfrak {g}）{\ longrightArrow} s（\ mathfrak {h h}）^w $。 \ item shustracter $ \ mathbb z $ -Algebras $ j（\ Mathfrak {g}）$和$ j（g）$ $ \ mathfrak {g} $和$ g $的有限维表示。 \ end {initizize}令$ \ mathcal a = \ mathcal a（\ mathfrak {g}）$表示为$ i（\ mathfrak {h}）$或$ j（g）\ otimes _ {\ otimes _ {\ Mathbb z} {\ Mathtt {\ Mathtt {k} {k}} $。本文的目的 是为了调查$ \ Mathcal A的代数几何。这提供了$ \ Mathcal中的激进理想和超级式套件（此类理想的零基因座）之间的两者。任何超级实体集都是独特的不可约合超级分量的有限结合。在非评估的Kac-Moody情况下，我们描述了包含给定（Zariski）封闭套件的最小超级式套件，并表明Superalgebraic Sets正是封闭的集合集，即Goletoid Orbits的工会。
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 $Δ$, Sergeev and Veselov introduced the Weyl groupoid $\mathfrak{W}=\mathfrak{W}(Δ)$, 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.
