Quantization of parafermion vertex algebras
Fei Kong
TL;DR
The paper develops an $\hbar$-adic quantum vertex algebra framework for parafermion structures in quantum affine settings. It constructs the quantized parafermion VA $K_{\hat{\mathfrak g},\hbar}^{\ell}$ inside the simple quantum affine VA $L_{\hat{\mathfrak g},\hbar}^{\ell}$ and shows an embedding of the quantum lattice VA $V_{\sqrt{\ell} Q_L}^{\eta_\ell}$ into this ambient algebra, establishing a quantum double-commutant relation akin to the classical case. The work proceeds from a thorough background on $\hbar$-adic VAs and quantum affine VAs to concrete constructions: a lattice embedding via the $\mathfrak{sl}_2$ case, and an inductive step using coproducts to handle general $\mathfrak g$. It further proves that $K_{\hat{\mathfrak g},\hbar}^{\ell}$ is a genuine $\hbar$-adic VA whose classical limit recovers the classical parafermion algebra, and demonstrates the invariance of the twisting action on this subalgebra. Overall, the results quantify and extend parafermion–lattice duality to the quantum, $\hbar$-adic setting, yielding a robust double-commutant structure with potential applications in quantum integrable systems and representation theory of quantum affine 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 $K_{\hat{\mathfrak g},\hbar}^\ell$ 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 $V_{\sqrt\ell Q_L}^{η_\ell}$, 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 $V_{\sqrt\ell Q_L}^{η_\ell}$ in $L_{\hat{\mathfrak g},\hbar}^\ell$.