On the representation theory of the vertex algebra \(L_{-5/2}(sl(4))\)

• Published in: arXiv.org 2021-12
• Main Authors: Adamovic, Drazen, Perse, Ozren, Vukorepa, Ivana
• Format: Article
• Language: English
• Subjects: Algebra; Homology; Modules; Quotients; Reduction; Tensors
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2103.02985

We study the representation theory of non-admissible simple affine vertex algebra \(L_{-5/2} (sl(4))\). We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra \(V^{-5/2} (sl(4))\), and show that it generates the maximal ideal in \(V^{-5/2} (sl(4))\). We classify irreducible \(L_{-5/2} (sl(4))\)--modules in the category \({\mathcal O}\), and determine the fusion rules between irreducible modules in the category of ordinary modules \(KL_{-5/2}\). It turns out that this fusion algebra is isomorphic to the fusion algebra of \(KL_{-1}\). We also prove that \(KL_{-5/2}\) is a semi-simple, rigid braided tensor category. In our proofs we use the notion of collapsing level for the affine \(\mathcal{W}\)--algebra, and the properties of conformal embedding \(gl(4) \hookrightarrow sl(5)\) at level \(k=-5/2\) from arXiv:1509.06512. We show that \(k=-5/2\) is a collapsing level with respect to the subregular nilpotent element \(f_{subreg}\), meaning that the simple quotient of the affine \(\mathcal{W}\)--algebra \(W^{-5/2}(sl(4), f_{subreg})\) is isomorphic to the Heisenberg vertex algebra \(M_J(1)\). We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor \(H_{f_{subreg}}\). It turns out that the properties of \(H_{f_{subreg}}\) are more subtle than in the case of minimal reducition.