2021/10/28 by Dražen Adamović, Adamovic, Drazen, Thomas Creutzig +3
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2110.15203
openalex publication_date 2021/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In [8], the affine vertex algebra Lk(\mathfraksl2) is realized as a subalgebra of the vertex algebra Virc ⊗ Π(0), where Virc is a simple Virasoro vertex algebra and Π(0) is a half-lattice vertex algebra. Moreover, all Lk(\mathfraksl2)--modules (including, modules in the category KLk, relaxed highest weight modules and logarithmic modules) are realized as Virc ⊗ Π(0)--modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case \mathfrakg= \mathfraksl3 and present realization of the VOA Lk(\mathfrak g) for k ∉ \mathbb Z≥ 0 as a vertex subalgebra of Wk ⊗ \mathcal S ⊗ Π(0), where Wk is a simple Breshadsky Polykov vertex algebra and \mathcal S is the βγ vertex algebra. We use this realization to study ordinary modules, relaxed highest weight modules and logarithmic modules. We prove the irreducibility of all our relaxed highest weight modules having finite-dimensional weight spaces (whose top components are Gelfand-Tsetlin modules). The irreducibility of relaxed highest weight modules with infinite-dimensional weight spaces is proved up to a conjecture on the irreducibility of certain \mathfrak g--modules which are not Gelfand-Tsetlin modules. The next problem that we consider is the realization of logarithmic modules. We first analyse the free-field realization of Wk from [11] and obtain a realization of logarithmic modules for Wk of nilpotent rank two at most admissible levels. Using logarithmic modules for the βγ VOA, we are able to construct logarithmic Lk(\mathfrak g)--modules of rank three.