Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups
Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups
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原投影代数、量子海森堡代数和二维复反射群的中心扩展
DOI：
10.1016/j.jalgebra.2006.01.005
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发表时间：
2005
影响因子：
0.9
通讯作者：
E. Rains
中科院分区：
文献类型：
--
作者：
P. Etingof;E. Rains
我们引入了有限 Dynkin 箭袋的前投影代数的中心扩展（取决于相应根系统的常规权重），其自然变形版本是平坦的（与前投影代数不同）。我们计算了中心扩张的希尔伯特多项式，并证明它是一个弗罗贝尼乌斯代数。作为推论，我们得到了通常变形预投影代数的希尔伯特级数，其中变形参数是变量，并证明该代数是 Gorenstein（尽管它不是参数环上的平坦模）。证明基于以下事实：我们对权重 \rho 的中心扩展是量子海森堡代数在张量函子下的量子 SL(2) 表示融合类别中的图像到半单代数上的双模。最后，我们解释了我们的代数如何与 2 阶复反射群的分圆 Hecke 代数联系起来，特别表明后者的泛型参数的维数等于群的阶数，正如 Broue、Malle 和 Rouquier 的猜想。
We introduce a central extension of the preprojective algebra of a finite Dynkin quiver (depending on a regular weight for the corresponding root system), whose natural deformed version is flat (unlike that for the preprojective algebra). We calculate the Hilbert polynomial of the central extension, and show that it is a Frobenius algebra. As a corollary, we obtain the Hilbert series of the usual deformed preprojective algebra in which the deformation parameters are variables, and show that this algebra is Gorenstein (although it is not a flat module over the ring of parameters). The proofs are based on the fact that our central extension for the weight \rho is the image of the quantum Heisenberg algebra in the fusion category of representations of quantum SL(2) under a tensor functor into bimodules over a semisimple algebra. Finally, we explain how our algebras are connected to cyclotomic Hecke algebras of complex reflection groups of rank 2, and in particular show that the dimension of the latter for generic parameters is equal to the order of the group, as conjectured by Broue, Malle, and Rouquier.