Affine Mirković-Vilonen polytopes
Affine Mirković-Vilonen polytopes
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Affine Mirković-Vilonen 多胞体
DOI：
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发表时间：
2011
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通讯作者：
P. Tingley
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文献类型：
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作者：
Pierre Baumann;J. Kamnitzer;P. Tingley
可对称化的Kac-Moody李代数$mathfrak{g}$的每个可积最低权表示都有一个Kashiwara意义下的晶体，它描述了它的组合性质。对于给定的$mathfrak{g}$，存在一个极限晶体，通常用B（−∞）表示，它包含所有其他晶体。当$mathfrak{g}$是有限维时，一个凸多面体，称为Mirković-Vilonen多面体，可以与B（−∞）中的每个元素相关联。这个多面体位于$mathfrak{g}$的Cartan子代数的对偶空间中，它的边平行于$mathfrak{g}$的根。在本文中，我们推广这种结构的情况下，$mathfrak{g}$是一个对称仿射Kac-Moody代数。然而，多面体的基准必须由平行于虚根δ的边的分割来补充。我们证明了这些装饰多面体的特点是他们的正常球迷和他们的2面的条件。此外，我们还讨论了我们的多面体如何提供仿射Kac-Moody代数的Lusztig数据的概念的模拟。我们的主要工具是由Lusztig和Kashiwara和Saito构建的B（−∞）的代数几何模型，基于与$mathfrak{g}$相同类型的完备预投射代数Λ的表示。在我们构造的基础上，借助于Buan，Iyama，Reiten和Scott的Lambda范畴的倾斜理论描述了多面体 ext {upshape -}mathrm {mod}$。我们所需要的划分来自于对维数向量为δ的倍数的半稳定Λ-模范畴的研究。
Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra $mathfrak{g}$ has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given $mathfrak{g}$, there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When $mathfrak{g}$ is finite dimensional, a convex polytope, called the Mirković-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of $mathfrak{g}$, and its edges are parallel to the roots of $mathfrak{g}$. In this paper, we generalize this construction to the case where $mathfrak{g}$ is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as $mathfrak{g}$. The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category $Lambda ext {upshape -}mathrm {mod}$. The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.