Quantization of parafermion vertex algebras

Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\ell$ be a positive integer. In this paper, we construct the quantization $K_{\hat{\mathfrak g},\hbar}^\ell$ of the parafermion vertex algebra $K_{\hat{\mathfrak g}}^\ell$ as an $\hbar$-adic quantum vertex subalgebra inside the simple quantum affine vertex algebra $L_{\hat{\mathfrak g},\hbar}^\ell$. We show that $L_{\hat{\mathfrak g},\hbar}^\ell$ contains an $\hbar$-adic quantum vertex subalgebra isomorphic to the quantum lattice vertex algebra $V_{\sqrt\ell Q_L}^{\eta_\ell}$, where $Q_L$ is the lattice generated by the long roots of ${\mathfrak g}$. Moreover, we prove the double commutant property of $K_{\hat{\mathfrak g},\hbar}^\ell$ and $V_{\sqrt\ell Q_L}^{\eta_\ell}$ in $L_{\hat{\mathfrak g},\hbar}^\ell$.