Some applications of the quantum Weyl groups
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.
scientific article; zbMATH DE number 4204613