Some $q$-exponential formulas for finite-dimensional ${\square }_q$-modules

Abstract

We consider the algebra ${\square }_q$ which is a mild generalization of the quantum algebra $U_q({\mathfrak{sl}}_2)$. The algebra ${\square }_q$ is defined by generators and relations. The generators are $\{x_i{\}}_{i\in {\mathbb{Z}}_4}$, where ${\mathbb{Z}}_4$ is the cyclic group of order $4$. For $i\in {\mathbb{Z}}_4$ the generators $x_i$,$x_{i+1}$ satisfy a $q$-Weyl relation, and $x_i$,$x_{i+2}$ satisfy a cubic $q$-Serre relation. For $i\in {\mathbb{Z}}_4$ we show that the action of $x_i$ is invertible on each nonzero finite-dimensional ${\square }_q$-module. We view $x_i^{-1}$ as an operator that acts on nonzero finite-dimensional ${\square }_q$-modules. For $i\in {\mathbb{Z}}_4$, define ${\mathfrak{n}}_{i,i+1}=q(1-x_i{x}_{i+1})/(q-q^{-1})$. We show that the action of ${\mathfrak{n}}_{i,i+1}$ is nilpotent on each nonzero finite-dimensional ${\square }_q$-module. We view the $q$-exponential ${\exp }_q({\mathfrak{n}}_{i,i+1})$ as an operator that acts on nonzero finite-dimensional ${\square }_q$-modules. In our main results, for $i,j\in {\mathbb{Z}}_4$ we express each of of ${\exp }_q({\mathfrak{n}}_{i,i+1})x_j{\exp }_q({\mathfrak{n}}_{i,i+1}{)}^{-1}$ and ${\exp }_q({\mathfrak{n}}_{i,i+1}{)}^{-1}{x}_j{\exp }_q({\mathfrak{n}}_{i,i+1})$ as a polynomial in $\{x_k^{\pm 1}{\}}_{k\in {\mathbb{Z}}_4}$.

Cited by in corpus (2)

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• An Infinite-Dimensional ${\square }_q$-Module Obtained from the $q$-Shuffle Algebra for Affine ${\mathfrak{sl}}_2$