The alternating PBW basis for the positive part of Uq(sl^2)

Abstract

The positive part Uq+ of Uq(sl^2) has a presentation with two generators A, B that satisfy the cubic q-Serre relations. We introduce a PBW basis for Uq+, said to be alternating. Each element of this PBW basis commutes with exactly one of A, B, qAB − q−1BA. 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 sl2. We show how the alternating PBW basis is related to the PBW basis for Uq+ found by Damiani in 1993.