Bicrossproduct Structure of the Quantum Weyl Group Original Research Article

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" imagei, adjoined. Let kimage be the group algebra of the standard covering image of the Weyl group of g. Here k = image[[image]]. We show that [formula] has the structure of a cocycle bicrossproduct, [formula] = kimageψbowtieα,χUq(g). It consists as an algebra of a cocycle semidirect product by a cocycle-action α of kimage 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 image via a cocycle ψ*.