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 Uq(g) with certain "quantum simple reflections" wi, adjoined. Let kW̃ be the group algebra of the standard covering W̃ of the Weyl group of g. Here k = C[[h(stroke)]]. We show that [formula] has the structure of a cocycle bicrossproduct, [formula] = kW̃ψ(bowtie)α,χUq(g) . It consists as an algebra of a cocycle semidirect product by a cocycle-action α of kW̃ on Uq(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 Uq(g) by the commutative algebra of functions on W̃ via a cocycle ψ*. © 1994 Academic Press, Inc.