Quantization of parafermion vertex algebras

TL;DR

This paper constructs a quantum deformation of parafermion vertex algebras within quantum affine vertex algebras, revealing their structure and relationships with quantum lattice vertex algebras.

Contribution

It introduces the quantization of parafermion vertex algebras as $$-adic quantum vertex subalgebras and establishes their properties and connections with quantum lattice vertex algebras.

Findings

Construction of the quantized parafermion vertex algebra $K_{\u001b ext{g},}^$ as a subalgebra.

Identification of a quantum lattice vertex algebra inside the quantum affine vertex algebra.

Proof of the double commutant property between the quantized parafermion algebra and the quantum lattice vertex algebra.

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 $K_{\hat{\mathfrak g},\hbar}^\ell$ 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 $L_{\hat{\mathfrak g},\hbar}^\ell$.