Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras
Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras
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Mirković–Vilonen 多面体和 Khovanov–Lauda–Rouquier 代数
作者：
P. Tingley;Ben Webster
我们描述了如何利用Khovanov-Lauda-Rouquier （KLR）代数从李代数的分类中自然产生Mirković-Vilonen （MV）多交体。给出了KLR代数的简单表示与MV多面体之间的唯一晶体同构的明确描述。由仿射格拉斯曼几何定义的MV多面体仅在有限型下有意义。另一方面，我们的构造给出了所有可对称的Kac-Moody代数从无限晶体到多面体的映射。然而，为了使映射注入并在图像上具有定义良好的晶体算子，我们通常必须用一些额外的信息来修饰多面体。我们认为所得到的“KLR多面体”是MV多面体的一般类型类似物。我们给出了在所有仿射情况下所得到的修饰多面体的组合描述，并表明这恢复了Baumann， Kamnitzer和第一作者最近在对称仿射类型中定义的仿射MV多面体。我们还简要讨论了仿射型以外的情况。
We describe how Mirković–Vilonen (MV) polytopes arise naturally from the categorification of Lie algebras using Khovanov–Lauda–Rouquier (KLR) algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of KLR algebras and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense in finite type. Our construction on the other hand gives a map from the infinity crystal to polytopes for all symmetrizable Kac–Moody algebras. However, to make the map injective and have well-defined crystal operators on the image, we must in general decorate the polytopes with some extra information. We suggest that the resulting ‘KLR polytopes’ are the general-type analogues of MV polytopes. We give a combinatorial description of the resulting decorated polytopes in all affine cases, and show that this recovers the affine MV polytopes recently defined by Baumann, Kamnitzer, and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.