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

Paul Terwilliger

paper

Using Catalan words and a $q$ -shuffle algebra to describe a PBW basis for the positive part of $U_{q} (sl_{2})$

arXiv:1806.11228

Abstract

The positive part $U_{q}^{+}$ of $U_{q} (sl_{2})$ has a presentation with two generators $A, B$ that satisfy the cubic $q$ -Serre relations. In 1993 I. Damiani obtained a PBW basis for $U_{q}^{+}$ , consisting of some elements ${E_{n δ + α_{0}}}_{n = 0}^{\infty}$ , ${E_{n δ + α_{1}}}_{n = 0}^{\infty}$ , ${E_{n δ}}_{n = 1}^{\infty}$ that are defined recursively. Our goal is to describe these elements in closed form. We achieve this goal using Catalan words and a $q$ -shuffle algebra.

arXiv comment13 pages

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Using Catalan words and a q-shuffle algebra to describe a PBW basis for the positive part of Uq​(sl2​)

Abstract

Cited by in corpus (4)

Using Catalan words and a $q$ -shuffle algebra to describe a PBW basis for the positive part of $U_{q} (sl_{2})$