Orthogonal bases for two-parameter quantum groups

TL;DR

This paper constructs dual PBW bases for positive and negative parts of two-parameter quantum groups of classical types, using shuffle algebra and Lyndon word combinatorics, advancing algebraic understanding.

Contribution

It introduces a two-parameter shuffle algebra framework and establishes dual PBW bases for two-parameter quantum groups, extending prior algebraic structures.

Findings

Dual PBW bases constructed for positive and negative subalgebras.

Relation established between shuffle algebra and quantum group subalgebras.

Utilization of Lyndon words for combinatorial basis construction.

Abstract

In this note, we construct dual PBW bases of the positive and negative subalgebras of the two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ in classical types, as used in our earlier work arXiv:2407.01450. Following the ideas of Leclerc and Clark-Hill-Wang, we introduce the two-parameter shuffle algebra and relate it to the subalgebras above. We then use the combinatorics of dominant Lyndon words to establish the main results.