Convex PBW-Type Lyndon Bases and Restricted Two-parameter Quantum Groups of Type B

TL;DR

This paper constructs Lyndon bases for two-parameter quantum groups of type B, analyzes their structure at roots of unity, and characterizes when these groups are ribbon Hopf algebras.

Contribution

It introduces convex PBW-type Lyndon bases for two-parameter quantum groups of type B and characterizes their Hopf algebra properties at roots of unity.

Findings

Restricted quantum groups are of Drinfeld double under certain conditions.

All Hopf isomorphisms of these quantum groups are classified.

Conditions for these groups to be ribbon Hopf algebras are established.

Abstract

We construct convex PBW-type Lyndon bases for two-parameter quantum groups U_{r,s}({so}_{2n+1}) with detailed commutation relations. It turns out that under a certain condition, the restricted two-parameter quantum group u_{r,s}({so}_{2n+1}) (r, s are roots of unity) is of Drinfeld double. All of Hopf isomorphisms of u_{r,s}({so}_{2n+1}), as well as u_{r,s}({sl}_n) are determined. Finally, necessary and sufficient conditions for u_{r,s}({so}_{2n+1}) to be a ribbon Hopf algebra are singled out by describing the left and right integrals.