Automorphism group of parafermion vertex operator algebras
Fix an integer \(k\geq 2\). Let \(\mathcal{L}(k,0)\) be the level \(k\) integrable highest weight module for the affine Kac-Moody algebra \(\widehat{\mathfrak{sl}}_3\). Recall that \(\mathcal{L}(k,0)\) possesses a vertex operator algebra structure and that the Cartan subalgebra \(\mathfrak{h}\) of \(\mathfrak{sl}_3\) generates the Heisenberg vertex operator subalgebra \(M(k)\) of \(\mathcal{L}(k,0)\). The parafermion vertex operator algebra \(K(\mathfrak{sl}_3,k)\) associated with \(\mathcal{L}(k,0)\) is defined as the commutant of \(M(k)\) in \(\mathcal{L}(k,0)\), i.e. \[ K(\mathfrak{sl}_3,k) =\left\{v\in \mathcal{L}(k,0)\,:\,h(n)v=0\text{ for all }h\in\mathfrak{h},\,n\geq 0\right\}. \] In this paper, the automorphism group \(\Aut(K(\mathfrak{sl}_3,k))\) of the parafermion vertex operator algebra \(K(\mathfrak{sl}_3,k)\) is determined. More precisely, it is proved that \(\Aut(K(\mathfrak{sl}_3,k))\) is isomorphic to the automorphism group of the root system of the simple Lie algebra \(\mathfrak{sl}_3\), i.e. \[ \text{Aut}(K(\mathfrak{sl}_3,k))\cong S_3\rtimes \mathbb{Z}/2\mathbb{Z}\quad \text{for all }k\geq 3. \] The proof relies on the integrable representation theory of the affine Kac-Moody algebras.
scientific article; zbMATH DE number 6499129
scientific article