The two-parameter quantum group of exceptional type G₂ and Lusztig’s symmetries

Naihong Hu and Qian Shi

Vol. 230 (2007), No. 2, 327–345

DOI: 10.2140/pjm.2007.230.327

Abstract

We give the defining structure of the two-parameter quantum group of type G₂ defined over a field ℚ(r,s) (with r≠s), and prove the Drinfel’d double structure as its upper and lower triangular parts, extending a result of Benkart and Witherspoon in type A and a recent result of Bergeron, Gao, and Hu in types B,C,D. We further discuss Lusztig’s ℚ-isomorphisms from U_r,s(G₂) to its associated object U_{s⁻¹,r⁻¹}(G₂), which give rise to the usual Lusztig symmetries defined not only on U_q(G₂) but also on the centralized quantum group U_q^c(G₂) only when r = s⁻¹ = q. (This also reflects the distinguishing difference between our newly defined two-parameter object and the standard Drinfel’d–Jimbo quantum groups.) Some interesting (r,s)-identities holding in U_r,s(G₂) are derived from this discussion.

Keywords

2-parameter quantum group, Hopf skew-dual pairing, Hopf dual pairing, Drinfel’d quantum double, Lusztig symmetries

Mathematical Subject Classification 2000

Primary: 17B37, 81R50

Secondary: 17B35

Milestones

Received: 16 August 2005

Accepted: 21 January 2006

Published: 1 April 2007