Orthogonal bases for two-parameter quantum groups

TL;DR

This work constructs dual PBW-type bases for the positive and negative parts of two-parameter quantum groups of classical types by embedding $U^{+}_{r,s}(g)$ into a two-parameter shuffle algebra and organizing a canonical basis via dominant Lyndon words. It proves these bases are orthogonal with respect to the Hopf pairing and computes explicit pairing constants, enabling a product decomposition of the $R$-matrix consistent with prior factorization results. The paper then treats root vectors and their pairings across types $A$–$D$, providing detailed formulas, and finally translates the shuffle-orthogonal framework back to $U_{r,s}(g)$ to establish the main dual PBW results. The approach blends combinatorial Lyndon-word methods with shuffle-algebra techniques to overcome the lack of Lusztig braid-group action in the multiparameter setting, yielding concrete orthogonal bases and explicit constants. These results underpin a structured, factorization-friendly understanding of finite and affine $R$-matrices in the two-parameter context and advance the construction of canonical bases in multiparameter quantum groups.

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.

Theorems & Definitions (136)

• Theorem
• Definition 2.1
• Proposition 2.2
• Remark 2.3
• Lemma 2.4
• Lemma 2.5
• Proposition 2.6
• proof
• Proposition 2.7
• proof