Using Catalan words and a $q$-shuffle algebra to describe the Beck PBW basis for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

DOI10.1016/J.JALGEBRA.2022.04.013arXiv2108.12708MaRDI QIDQ6376244

Publication date: 28 August 2021

Abstract: We consider the positive part

U_{q}^{+}

of the quantized enveloping algebra

U_{q} (w i d e h a t {m a t h f r a k s l}_{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}^{+}

.

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Quantum groups and related algebraic methods applied to problems in quantum theory (81R50) Associative rings and algebras arising under various constructions (16S99) Combinatorial aspects of groups and algebras (05E16)

This page was built for publication: Using Catalan words and a $q$-shuffle algebra to describe the Beck PBW basis for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6376244)