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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 \(\{x_{k}^{\pm 1}\}_{k\in \mathbb {Z}_{4}}\) .
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