The Dynkin diagram cohomology of finite Coxeter groups

Details The Dynkin diagram cohomology of finite Coxeter groups The Dynkin diagram cohomology of finite Coxeter groups

2008-04-07

arxiv.org/abs/0804.0947v2

Journal of Algebra 323 (2010), 59-82

Mathematics

Quantum Algebra

Final version, 25 pages, to appear in the Journal of Algebra. Minor word changes. A new section (6) has been added where the r

Scientific paper

10.1016/j.jalgebra.2009.09.023

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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