Orthogonal bases for two-parameter 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 “Martin–Tsymbaliuk [SIGMA 21 (2025), paper no. 064]”. Following the ideas of “Leclerc [Math. Z. 246 (2004), no. 4, 691–732]” and “Clark–Hill–Wang [Quantum Topol. 7 (2016), no. 3, 553–638]”, 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.