Bicrossproduct Structure of the Quantum Weyl Group

✍ Scribed by S. Majid

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 826 KB
• Volume: 163
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1994.1004

✦ Synopsis

The quantum Weyl group (\widetilde{U_{q}(g)}) associated to a complex simple Lie algebra (g) consists of the quantum group (U_{4}(g)) with certain "quantum simple reflections" (\bar{u}), adjoined. Let (k \tilde{W}) be the group algebra of the standard covering (\tilde{W}) of the Weyl group of (g). Here (k=\mathbb{C}[[h]]). We show that (\widehat{U_{4}(g)}) has the structure of a cocycle bicrossproduct, (\widehat{U_{q}(g)}=k W^{\psi} \bowtie_{\alpha, \gamma} U_{4}(g)). It consists as an algebra of a cocycle semidirect product by a cocycle-action (x) of (k W) on (U_{4}(g)), defined with respect to a certain non-Abelian cocycle (\chi). It consists as a coalgebra of an extension by a non-Abelian dual cocycle (\psi). The dual of (\widehat{U_{4}(g)}) 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 (\mathscr{W}) via a cocycle (\psi^{*}). '. 1994 Academic Press, Inc.