Bicrossproduct structure of the quantum Weyl group (Q1314247)

The authors apply some of the theory of the bicrossproducts [see \textit{S. Majid}, Isr. J. Math. 72, 133-148 (1990; Zbl 0725.17015)] and extensions [see \textit{W. M. Singer}, J. Algebra 21, 1-16 (1972; Zbl 0269.16011)] of Hopf algebras to the quantum Weyl group \(\widetilde {U_ q ({\mathfrak g})}\) associated to a complex simple Lie algebra \({\mathfrak g}\) in order to characterize the structure of \(\widetilde {U_ q ({\mathfrak g})}\). The authors show that \(\widetilde {U_ q ({\mathfrak g})}\) has the structure of a cocycle bicrossproduct, i.e. \[ \widetilde {U_ q ({\mathfrak g})} = k \widetilde W^ \psi \bowtie_{\alpha, \chi} U_ q ({\mathfrak g}) \] where \(k = C[[\hbar]]\) and \(\widetilde W\) is the standard covering of the Weyl group of \({\mathfrak g}\). Moreover, it consists as an algebra of a cocycle semi-direct product by a cocycle-action \(\alpha\) of \(k \widetilde W\) on \(U_ q ({\mathfrak g})\), defined with respect to a certain non-Abelian cocycle \(\chi\); it consists of a coalgebra of an extension by a non- Abelian dual cocycle \(\psi\). The dual of \(\widetilde {U_ q ({\mathfrak g})}\) is also a bicrossproduct and consists of an algebra of an extension of the dual of \(U_ q ({\mathfrak g})\) by the commutative algebra of functions on \(\widetilde W\) via a cocycle \(\psi^*\).