Using Catalan words and a q-shuffle algebra to describe the Beck PBW basis for the positive part of [formula omitted

Abstract

We consider the positive part Uq+ of the quantized enveloping algebra Uq(slˆ2). The algebra Uq+ has a presentation involving two generators and two relations, called the q-Serre relations. There is a PBW basis for Uq+ due to Damiani, and a PBW basis for Uq+ due to Beck. In 2019 we used Catalan words and a q-shuffle algebra to express the Damiani PBW basis in closed form. In this paper we use a similar approach to express the Beck PBW basis in closed form. We also consider how the Damiani PBW basis and the Beck PBW basis are related to the alternating PBW basis for Uq+.