论文标题
在重量模块之间移动
Moving between weights of weight modules
论文作者
论文摘要
在谎言理论中，部分总和属性（PSP）说，对于任何kac-moody代数中的根系，每个积极根源都是一个有序的简单根总和，其部分总和都是根。在本文中，我们提出了两个概括： 1）“抛物线概括”：如果$ i $是简单根的子集，则每个带有$ i $ $ $ height的根。实际上，我们通过证明每个根空间都由由$ i $ -Height 1的根矢量形成的谎言单词跨越的谎言代数级别。作为一个应用程序，我们为每一个（不可汇总）简单的最高权重模块在任何Kac-Moody Algebra上提供了一个“最小”描述。即使在有限类型中，这似乎也是新颖的。 2）对重量模块的重量的概括：PSP给出了一个（固定）和任何正根之间的根链。我们将其推广到重量模块的重量，以在任意两个可比的重量之间获得一系列权重。库马尔（S.在本文中，我们将此结果扩展到（i）所有kac-moody代数$ \ mathfrak {g} $上的大量最高权重模块，其中包含$ \ mathfrak {g} $的所有简单最高权重模块; （ii）更一般而言，对于非最大的重量模块，例如$ \ mathfrak {g} $本身（伴随表示）和寄生虫Verma模块的任意子模块，$ \ mathfrak {g} $; （iii）半胶合$ \ mathfrak {g} $上的任意集成模块。此外，我们还证明了第二次概括的“抛物线”概括。 我们还找到了所有最高权重模块，它们的权重集与抛物线Verma模块相同的权重，并为kac-moody $ \ mathfrak {g} $的任意最高权重模块的权重提供Minkowski差异公式。
In Lie theory the partial sum property (PSP) says that for a root system in any Kac-Moody algebra, every positive root is an ordered sum of simple roots whose partial sums are all roots. In this paper, we present two generalizations: 1) "Parabolic generalization": if $I$ is a subset of simple roots, every root with positive $I$-height is an ordered sum of roots of $I$-height 1, whose partial sums are all roots. In fact we show this on the Lie algebra level, by showing that every root space is spanned by the Lie words formed from root vectors of $I$-height 1. As an application, we provide a "minimal" description for the set of weights of every (non-integrable) simple highest weight module over any Kac-Moody algebra. This seems to be novel even in finite type. 2) Generalization to weights of weight modules: the PSP gives a chain of roots between 0 (fixed) and any positive root. We generalize this to the weights of weight modules to get a chain of weights between any two comparable weights. This was shown by S. Kumar for any finite-dimensional simple module over a semisimple Lie algebra. In this paper, we extend this result to (i) a large class of highest weight modules over any Kac-Moody algebra $\mathfrak{g}$, which includes all simple highest weight modules over $\mathfrak{g}$; (ii) more generally, for non-highest weight modules such as $\mathfrak{g}$ itself (adjoint representation) and arbitrary submodules of parabolic Verma modules over $\mathfrak{g}$; (iii) arbitrary integrable modules over semisimple $\mathfrak{g}$. Additionally, we also prove the "parabolic" generalizations of this second generalization to the best possible extent. We also find all the highest weight modules which have their sets of weights same as those of parabolic Verma modules and provide a Minkowski difference formula for weights of arbitrary highest weight modules over Kac-Moody $\mathfrak{g}$.
