Using Catalan words and a $q$shuffle algebra to describe the Beck PBW basis for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
Abstract
We consider the positive part $U^+_q$ of the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation involving two generators and two relations, called the $q$Serre relations. There is a PBW basis for $U^+_q$ due to Damiani, and a PBW basis for $U^+_q$ 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 $U^+_q$.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.12708
 Bibcode:
 2021arXiv210812708T
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Combinatorics;
 17B37
 EPrint:
 22 pages. arXiv admin note: text overlap with arXiv:1902.00721