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 $U_{q}(\frak {sl}_{2})$. 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 every 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 every nonzero finite-dimensional $\square _{q}$-module. We view the q-exponential $\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})$ 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 $\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})x_{j}\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})^{-1}$ and $\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})^{-1}x_{j}\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})$ as a polynomial in $\{x_{k}^{\pm 1}\}_{k\in \mathbb {Z}_{4}}$.