Commutative quantum current operators, semi-infinite construction and functional models

We construct the commutative current operator $\bar x^+(z)$ inside $U_q(\hat{\mathfrak{sl}}(2))$. With this operator and the condition of quantum integrability on the quantum currents of $U_q(\hat{\mathfrak{sl}}(2))$, we derive the quantization of the semi-infinite construction of integrable modules of $\hat{\mathfrak{sl}}(2)$ which has been previously obtained by means of the current operator $e(z)$ of $\hat{\mathfrak{sl}}(2)$. The quantization of the functional models for $\hat{\mathfrak{sl}}(2)$ is also given.