Quantization of parafermion vertex algebras

Abstract

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 $K_{\hat{\mathfrak{g}},\hslash }^{\ell }$ of the parafermion vertex algebra $K_{\hat{\mathfrak{g}}}^{\ell }$ as an $\hslash$-adic quantum vertex subalgebra inside the simple quantum affine vertex algebra $L_{\hat{\mathfrak{g}},\hslash }^{\ell }$. We show that $L_{\hat{\mathfrak{g}},\hslash }^{\ell }$ contains an $\hslash$-adic quantum vertex subalgebra isomorphic to the quantum lattice vertex algebra $V_{\sqrt{\ell }{Q}_L}^{\hat{I}{\cdot }_{\ell }}$, where $Q_L$ is the lattice generated by the long roots of $\mathfrak{g}$. Moreover, we prove the double commutant property of $K_{\hat{\mathfrak{g}},\hslash }^{\ell }$ and $V_{\sqrt{\ell }{Q}_L}^{\hat{I}{\cdot }_{\ell }}$ in $L_{\hat{\mathfrak{g}},\hslash }^{\ell }$.