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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.

##### MSC:
 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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