The alternating PBW basis for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

Abstract

The positive part $U^+_q$ of $U_q(\widehat{\mathfrak{sl}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. We introduce a PBW basis for $U^+_q$, said to be alternating. Each element of this PBW basis commutes with exactly one of $A$, $B$, $qAB-q^{-1}BA$. This gives three types of PBW basis elements; the elements of each type mutually commute. We interpret the alternating PBW basis in terms of a $q$-shuffle algebra associated with affine $\mathfrak{sl}_2$. We show how the alternating PBW basis is related to the PBW basis for $U^+_q$ found by Damiani in 1993.