2024/09/03 by Dražen Adamović, Adamovic, Drazen, Andrei Babichenko +1 · 1 citation
Computer Science · Mathematics · #17B69 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2409.02093
openalex publication_date 2024/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The representation theory of the Nappi-Witten VOA was initiated in arXiv:1104.3921 and arXiv:2011.14453. In this paper we use the technique of inverse quantum hamiltonian reduction to investigate the representation theory of the Nappi-Witten VOA V1(\mathfrak h4). We first prove that the quantum hamiltonian reduction of V1(\mathfrak h4) is the Heisenberg-Virasoro VOA LHVir of level zero investigated in arXiv:math/0201314 and arXiv:1405.1707. We invert the quantum hamiltonian reduction in this case and prove that V1(\mathfrak h4) is realized as a vertex subalgebra of LHVir ⊗ Π, where Π is a certain lattice-like vertex algebra. Using such an approach we shall realize all relaxed highest weight modules which were classified in arXiv:2011.14453. We show that every relaxed highest weight module, whose top components is neither highest nor lowest weight \mathfrak h4-module, has the form M1 ⊗ Π1 (λ) where M1 is an irreducible, highest weight LHVir-module and Π1 (λ) is an irreducible weight Π-module. Using the fusion rules for LHVir-modules and the previously developed methods of constructing logarithmic modules we are able to construct a family of logarithmic V1(\mathfrak h4)-modules. The Loewy diagrams of these logarithmic modules are completely analogous to the Loewy diagrams of projective modules of weight Lk(\mathfraksl(2))-modules, so we expect that our logarithmic modules are also projective in a certain category of weight V1(\mathfrak h4)-modules.