Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of type G₂

Naihong Hu and Xiuling Wang

Vol. 241 (2009), No. 2, 243–273

DOI: 10.2140/pjm.2009.241.243

Abstract

From the 2-parameter quantum group U_r,s(G₂) defined by Hu and Shi in 2007, we construct finite-dimensional pointed Hopf algebras u_r,s(G₂) (that is, restricted 2-parameter quantum groups); these turn out to be Drinfel’d doubles. Crucial is a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G₂ in the 2-parameter quantum version. We exhibit the possible commutation relations among quantum root vectors. Then we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions, by determining their left and right integrals. We also determine all the Hopf algebra isomorphisms of u_r,s(G₂) by describing its sets of left (right) skew-primitive elements.

Keywords

restricted 2-parameter quantum groups, Lyndon basis, Drinfel’d double, integrals, ribbon Hopf algebra

Mathematical Subject Classification 2000

Primary: 17B37, 81R50

Secondary: 17B35

Milestones

Received: 16 October 2008

Accepted: 18 February 2009

Published: 1 June 2009