Barbasch-Sahi algebras and Dirac cohomology
Barbasch-Sahi algebras and Dirac cohomology
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Barbasch-Sahi 代数和狄拉克上同调
DOI：
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发表时间：
2016
影响因子：
1.3
通讯作者：
J. Flake
中科院分区：
文献类型：
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作者：
J. Flake
我们定义了一类以 PBW 性质和正交性条件为特征的代数，我们将其称为 Hopf-Hecke 代数，因为它们概括了 Drinfeld 定义的 Drinfeld Hecke 代数。在研究正交条件的过程中，并与正交群类比，我们证明了 $\mathbb{C}$ 上具有正交模的共交换 Hopf 代数的 pin 覆盖的存在，或者更一般地，具有正交模的特征 $0$ 域上的指向共交换 Hopf 代数的存在。
根据 Dan Barbasch 和 Siddhartha Sahi 的建议，我们为 Hopf-Hecke 代数模定义了 Dirac 算子和 Dirac 上同调，将这些概念推广到连通半单李群、分级仿射 Hecke 代数和辛反射代数。使用 pin cover，我们证明了一类 Hopf-Hecke 代数的一般定理，我们称之为 Barbasch-Sahi 代数，该定理将具有非零狄拉克上同调的不可约模的中心特征与其狄拉克上同调中出现的中心特征联系起来，推广了连通半单李群的“Vogan 猜想”结果，类似于分级仿射 Hecke 的结果 代数以及辛反射代数和 Drinfeld Hecke 代数。
We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld. In the course of studying the orthogonality condition and in analogy to the orthogonal group we show the existence of a pin cover for cocommutative Hopf algebras over $\mathbb{C}$ with an orthogonal module or, more generally, pointed cocommutative Hopf algebras over a field of characteristic $0$ with an orthogonal module.
Following the suggestion of Dan Barbasch and Siddhartha Sahi, we define a Dirac operator and Dirac cohomology for modules of Hopf-Hecke algebras, generalizing those concepts for connected semisimple Lie groups, graded affine Hecke algebras and symplectic reflection algebras. Using the pin cover, we prove a general theorem for a class of Hopf-Hecke algebras which we call Barbasch-Sahi algebras, which relates the central character of an irreducible module with non-vanishing Dirac cohomology to the central characters occurring in its Dirac cohomology, generalizing a result called "Vogan's conjecture" for connected semisimple Lie groups, analogous results for graded affine Hecke algebras and for symplectic reflection algebras and Drinfeld Hecke algebras.
HopfâHecke 代数、无穷小 Cherednik 代数和狄拉克上同调
影响因子：
0.7
作者：
Flake, Johannes;Sahi, Siddhartha
通讯作者：
Sahi, Siddhartha