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Rationality of admissible affine vertex algebras in the category \(\mathcal O\). (English) Zbl 1395.17057

Summary: We study the vertex algebras associated with modular invariant representations of affine Kac-Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph’s characteristic varieties. We show that an irreducible highest weight representation of a nontwisted affine Kac-Moody algebra at an admissible level \(k\) is a module over the associated simple affine vertex algebra if and only if it is an admissible representation whose integral root system is isomorphic to that of the vertex algebra itself. This in particular proves the conjecture of Adamović and Milas on the rationality of admissible affine vertex algebras in the category \(\mathcal{O}\).

MSC:

17B69 Vertex operators; vertex operator algebras and related structures
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B35 Universal enveloping (super)algebras
17B56 Cohomology of Lie (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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References:

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