## Duke Mathematical Journal

### Rationality of admissible affine vertex algebras in the category ${\mathcal{O}}$

Tomoyuki Arakawa

#### Abstract

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 ${\mathcal{O}}$.

#### Article information

Source
Duke Math. J., Volume 165, Number 1 (2016), 67-93.

Dates
Revised: 17 November 2014
First available in Project Euclid: 4 November 2015

https://projecteuclid.org/euclid.dmj/1446648393

Digital Object Identifier
doi:10.1215/00127094-3165113

Mathematical Reviews number (MathSciNet)
MR3450742

Zentralblatt MATH identifier
06543258

#### Citation

Arakawa, Tomoyuki. Rationality of admissible affine vertex algebras in the category ${\mathcal{O}}$. Duke Math. J. 165 (2016), no. 1, 67--93. doi:10.1215/00127094-3165113. https://projecteuclid.org/euclid.dmj/1446648393

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