Convex PBW-type Lyndon Basis and Restricted Two-parameter Quantum Group of Type G_2

TL;DR

This paper constructs finite-dimensional restricted two-parameter quantum groups of type G_2 as Drinfel'd doubles, providing a detailed combinatorial basis and analyzing their algebraic properties including ribbon structure and isomorphisms.

Contribution

It introduces a detailed convex PBW-type Lyndon basis for 2-parameter quantum groups of type G_2 and explores their structure as ribbon Hopf algebras.

Findings

Constructed finite-dimensional pointed Hopf algebras _{r,s}(G_2) as Drinfel'd doubles.

Established conditions under which these quantum groups are ribbon Hopf algebras.

Classified all Hopf algebra isomorphisms of _{r,s}(G_2).

Abstract

We construct finite-dimensional pointed Hopf algebras \mathfrak u_{r,s}(G_2) (i.e. restricted 2-parameter quantum groups) from the 2-parameter quantum group U_{r,s}(G_2) defined in \cite{HS}, which turn out to be of Drinfel'd doubles, where a crucial point is to give a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G_2 in 2-parameter quantum version. After furnishing possible commutation relations among quantum root vectors, we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions through determining their left and right integrals. Besides these, we determine all of the Hopf algebra isomorphisms of u_{r,s}(G_2) in terms of the description of the sets of its left (right) skew-primitive elements.