Bicrossproduct Structure of the Quantum Weyl Group

Abstract

The quantum Weyl group [formula] associated to a complex simple Lie algebra g consists of the quantum group U q ( g) with certain "quantum simple reflections" w i , adjoined. Let kW̃ be the group algebra of the standard covering W̃ of the Weyl group of g. Here k = C [[ħ]]. We show that [formula] has the structure of a cocycle bicrossproduct, [formula] = kW̃ ψ⋈ α,χ U q ( g). It consists as an algebra of a cocycle semidirect product by a cocycle-action α of kW̃ on U q ( g), defined with respect to a certain non-Abelian cocycle χ. It consists as a coalgebra of an extension by a non-Abelian dual cocycle ψ. The dual of [formula] is also a bicrossproduct and consists as an algebra of an extension of the dual of U q ( g) by the commutative algebra of functions on W̃ via a cocycle ψ*.