Doubly alternating words in the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

TL;DR

This paper investigates the structure of the positive part of the quantum affine algebra $U_q(\\mathfrak{sl}_2)$, identifying special classes of words called alternating and doubly alternating, and explores their algebraic relations.

Contribution

It characterizes all words in the $q$-shuffle algebra that belong to the positive part and introduces the concept of doubly alternating words, linking them to existing alternating words.

Findings

Identified all words in the $q$-shuffle algebra contained in $U_q^+$.

Defined and studied doubly alternating words within the algebra.

Established relations between doubly alternating and alternating words.

Abstract

This paper is about the positive part $U_q^+$ of the $q$-deformed enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U_q^+$ admits an embedding, due to Rosso, into a $q$-shuffle algebra $\mathbb{V}$. The underlying vector space of $\mathbb{V}$ is the free algebra on two generators $x,y$. Therefore, the algebra $\mathbb{V}$ has a basis consisting of the words in $x,y$. Let $U$ denote the image of $U_q^+$ under the Rosso embedding. In our first main result, we find all the words in $x,y$ that are contained in $U$. One type of solution is called alternating. The alternating words have been studied by Terwilliger. There is another type of solution, which we call doubly alternating. In our second main result, we display many commutator relations involving the doubly alternating words. In our third main result, we describe how the doubly alternating words are related to the alternating words.