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 $U_q^{+}$.

22 pages. arXiv admin note: text overlap with arXiv:1902.00721

References in corpus (2)

• The alternating presentation of $U_q(\widehat{g{l}_2})$ from Freidel-Maillet algebras
• On the second realization for the positive part of $U_q(\widehat{s{l}_2})$ of equitable type

Cited by in corpus (3)

• Tridiagonal pairs, alternating elements, and distance-regular graphs
• A generating function associated with the alternating elements in the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Using a $q$-shuffle algebra to describe the basic module $V({\Lambda }_0)$ for the quantized enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$