The alternating PBW basis for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$

Abstract

The positive part $U_q^{+}$ of $U_q({\widehat{\mathfrak{sl}}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. We introduce a PBW basis for $U_q^{+}$, said to be alternating. Each element of this PBW basis commutes with exactly one of $A$, $B$, $qAB-q^{-1}BA$. This gives three types of PBW basis elements; the elements of each type mutually commute. We interpret the alternating PBW basis in terms of a $q$-shuffle algebra associated with affine ${\mathfrak{sl}}_2$. We show how the alternating PBW basis is related to the PBW basis for $U_q^{+}$ found by Damiani in 1993.

37 pages

References in corpus (2)

• The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
• An action of the free product ${\mathbb{Z}}_2\star {\mathbb{Z}}_2\star {\mathbb{Z}}_2$ on the $q$-Onsager algebra and its current algebra

Cited by in corpus (7)

• The alternating central extension of the $q$-Onsager algebra
• The alternating presentation of $U_q(\widehat{g{l}_2})$ from Freidel-Maillet algebras
• The $q$-Onsager algebra and its alternating central extension
• On the second realization for the positive part of $U_q(\widehat{s{l}_2})$ of equitable type
• Using Catalan words and a $q$-shuffle algebra to describe the Beck PBW basis for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Tridiagonal pairs, alternating elements, and distance-regular graphs
• A generating function associated with the alternating elements in the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$