有限Coxeter群的Dynkin图上同调
The Dynkin diagram cohomology of finite Coxeter groups
摘要：
设D是一个连通图。第二作者在[TL 2]中引入了D-代数A的Dynkin复形CD（A）来控制A上拟Coxeter代数结构的形变。本文研究了当A是Coxeter群W的群代数，D是W的Dynkin图时，该复形的上同调.我们计算这个上同调时，W是有限的，并证明特别是准Coxeter代数结构的刚性千瓦。对于任意的W，我们计算了上上同调群，得到了W是仿射的一些附加的部分结果。我们的计算是通过用D的子图的顶点数过滤CD（A）来进行的。相应的分次复形与W的所有标准不可约抛物子群的Coxeter复形之和是对偶的。
英文摘要：
Let D be a connected graph. The Dynkin complex CD(A) of a D-algebra A was introduced by the second author in [TL2] to control the deformations of quasi-Coxeter algebra structures on A. In the present paper, we study the cohomology of this complex when A is the group algebra of a Coxeter group W and D is the Dynkin diagram of W. We compute this cohomology when W is finite and prove in particular the rigidity of quasi-Coxeter algebra structures on kW. For an arbitrary W, we compute the top cohomology group and obtain a number of additional partial results when W is affine. Our computations are carried out by filtering CD(A) by the number of vertices of subgraphs of D. The corresponding graded complex turns out to be dual to the sum of the Coxeter complexes of all standard, irreducible parabolic subgroups of W.
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