Quasi-Coxeter Algebras, Dynkin Diagram Cohomology, and Quantum Weyl Groups
Quasi-Coxeter Algebras, Dynkin Diagram Cohomology, and Quantum Weyl Groups
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拟 Coxeter 代数、Dynkin 图上同调和量子 Weyl 群
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发表时间：
2010
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通讯作者：
V. T. Laredo
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文献类型：
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作者：
V. T. Laredo
作者[TL1，TL2]和独立的De Concini(未发表)猜想，[MTL]中引入的Casimir联络∇C的单调性是由Lusztig的量子Weyl群算符描述的。这一猜想在[TL1]中对李代数g=SLN的所有表示被证明，在[TL2]中对一些对(g，V)被证明，包括经典李代数的矢量和自旋表示以及所有复单李代数的伴随表示。本文及其续篇[TL4]的目的是对所有g证明这一猜想。我们的策略是受到Drinfeld证明g的Knizhnik-Zamolodchikov方程和来自量子群U~g的R-矩阵表示的单调性的等价性的启发。它依赖于拟Coxeter代数的使用，拟Coxeter代数对于g型广义辫子群的作用就像Drinfeld的拟三角拟双代数对于Artin的辫子群Bn的作用一样。利用这一概念和相应的形变上同调，我们称之为动态图上同调，我们将本文中的猜想归结为关于经典包络代数Ug的(非上同调)陈述，即它上存在一个拟Coxeter拟三角拟双代数结构，该结构介于联结∇C的单调背后的拟Coxeter代数结构和KZ方程的拟三角拟双代数结构之间.这种结构的存在将在[TL4]中得到证明。
The author [TL1, TL2], and independently De Concini (unpublished), conjectured that the monodromy of the Casimir connection∇C introduced in [MTL] is described by Lusztig’s quantum Weyl group operators. This conjecture was proved in [TL1] for all representations of the Lie algebra g = sln and in [TL2] for a number of pairs (g, V ) including vector and spin representations of classical Lie algebras and the adjoint representation of all complex, simple Lie algebras. The aim of this paper, and of its sequel [TL4] is to prove this conjecture for all g. Our strategy is inspired by Drinfeld’s proof of the equivalence of the monodromy of the Knizhnik–Zamolodchikov equations for g and the R–matrix representations coming from the quantum group U~g. It relies on the use of quasi–Coxeter algebras, which are to the generalised braid group of type g what Drinfeld’s quasitriangular quasibialgebras are to Artin’s braid groups Bn. Using this notion, and the associated deformation cohomology, which we call Dynkin diagram cohomology, we reduce the conjecture in this paper to a (non– cohomological) statement about the classical enveloping algebra Ug, namely the existence of a quasi–Coxeter, quasitriangular quasibialgebra structure on it interpolating between the quasi–Coxeter algebra structure underlying the monodromy of the connection ∇C and the quasitriangular quasibialgebra structure underlying that of the KZ equations. The existence of such a structure will be proved in [TL4].