Vertex algebras and TKK algebras

2023/07/03 by Chen, Fulin, Ding, Lingen, Wang, Qing
#FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2307.00707

Abstract

In this paper, we associate the TKK algebra \widehatG(J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ, the category of restricted \widehatG(J)-modules of level ℓ is canonically isomorphic to the category of σ-twisted V_\widehatC_{\mathfrak g}(ℓ,0)-modules, where V_\widehatC_{\mathfrak g}(ℓ,0) is a vertex algebra arising from the toroidal Lie algebra of type C₂ and σ is an isomorphism of V_\widehatC_{\mathfrak g}(ℓ,0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ, the integrable restricted \widehatG(J)-modules of level ℓ are exactly the σ-twisted modules for the quotient vertex algebra L_\widehatC_{\mathfrak g}(ℓ,0) of V_\widehatC_{\mathfrak g}(ℓ,0). Finally, we classify the irreducible (1)/(2)ℕ-graded σ-twisted L_\widehatC_{\mathfrak g}(ℓ,0)-modules.

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