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Some applications of the quantum Weyl groups. (English) Zbl 0729.17009
Let L be a simple complex Lie algebra, G a corresponding Lie group. The Weyl group \(W_ 1\) is the group generated by simple reflections in \(H^*\), H a Cartan subalgebra of L. \(W_ 1\) is isomorphic to \(W_ 2=N(T)/T\), T a maximal torus in G. The author gives a quantum analogue of \(W_ 2\) (not \(W_ 1)\), called the quantum Weyl group. It is a Hopf algebra. He applies this to obtain a formula for the universal quantum R- matrix of a simple Lie algebra. An application is given to Hecke algebras, namely that if L(\(\Lambda\)) is a q-analogue of the adjoint representation of L, and \(L(\Lambda)_ 0\subset L(\Lambda)\) is the space of zero-weight vectors, then the quantum Weyl group acts on \(L(\Lambda)_ 0\) as (generalized) Hecke algebra.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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