Orthogonal bases for two-parameter quantum groups

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.