arXiv subjects

Chenwei Ruan

Publications and source records attributed to Chenwei Ruan.

4 recordsLinked to original sources

arXiv2025

Freidel-Maillet type equations on fused K-matrices over the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

The positive part $U_q^+$ of the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$ has a reflection equation presentation of Freidel-Maillet type (Baseilhac 2022). Its defining K-matrix has size $2 \times 2$ and can be expressed, using Rosso's embedding of $U_q^+$ into a $q$-shuffle algebra, as generating functions whose coefficients are Terwilliger's alternating PBW basis elements. This and older PBW bases of $U_q^+$ due to Damiani and Beck are unified by linear combinations of Catalan words (Ruan 2025). In this paper, we use this unification to define K-matrices of any dimension $\geq 2$ whose entries are explicit generating functions over $U_q^+$. Our main result is that any pair of such K-matrices, possibly of different dimensions, satisfy a Freidel-Maillet type equation. This yields a family of algebraic relations over $U_q^+$ that generalize Baseilhac's equation and can be used to study integrable systems or higher-spin representations.

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arXiv2024

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

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.

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arXiv2023

A uniform approach to the Damiani, Beck, and alternating PBW bases for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

This paper is about the positive part $U_q^+$ of the $q$-deformed enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. The literature contains at least three PBW bases for $U_q^+$, called the Damiani, the Beck, and the alternating PBW bases. These PBW bases are related via exponential formulas. In this paper, we introduce an exponential generating function whose argument is a power series involving the Beck PBW basis and an integer parameter $m$. The cases $m=2$ and $m=-1$ yield the known exponential formulas for the Damiani and alternating PBW bases, respectively. The case $m=1$ appears in the author's previous paper. In the present paper, we give a comprehensive study of the generating function for an arbitrary integer $m$. We have two main results. The first main result gives a factorization of the generating function. In the second main result, we express the coefficients of the generating function in closed form.

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arXiv2023

A generating function associated with the alternating elements in the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

The positive part $U_q^+$ of $U_q(\widehat{\mathfrak{sl}}_2)$ admits an embedding into a $q$-shuffle algebra. This embedding was introduced by M. Rosso in 1995. In 2019, Terwilliger introduced the alternating elements $\{W_{-n}\}_{n \in \mathbb{N}}$, $\{W_{n+1}\}_{n \in \mathbb{N}}$, $\{G_{n+1}\}_{n \in \mathbb{N}}$, $\{\tilde{G}_{n+1}\}_{n \in \mathbb{N}}$ in $U_q^+$ using the Rosso embedding. He showed that the alternating elements $\{W_{-n}\}_{n \in \mathbb{N}}$, $\{W_{n+1}\}_{n \in \mathbb{N}}$, $\{\tilde{G}_{n+1}\}_{n \in \mathbb{N}}$ form a PBW basis for $U_q^+$, and he expressed $\{G_{n+1}\}_{n \in \mathbb{N}}$ in this alternating PBW basis. In his calculation, Terwilliger used some elements $\{D_n\}_{n \in \mathbb{N}}$ with the following property: the generating function $D(t)=\sum_{n \in \mathbb{N}}D_nt^n$ is the multiplicative inverse of the generating function $\tilde{G}(t)=\sum_{n \in \mathbb{N}}\tilde{G}_nt^n$ where $\tilde{G}_0=1$. Terwilliger defined $\{D_n\}_{n \in \mathbb{N}}$ recursively; in this paper, we will express $\{D_n\}_{n \in \mathbb{N}}$ in closed form.

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