Duke Mathematical Journal

Rationality of admissible affine vertex algebras in the category O

Tomoyuki Arakawa

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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 O.

Article information

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

Received: 28 May 2013
Revised: 17 November 2014
First available in Project Euclid: 4 November 2015

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Zentralblatt MATH identifier

Primary: 17B69: Vertex operators; vertex operator algebras and related structures 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Secondary: 17B08: Coadjoint orbits; nilpotent varieties 17B55: Homological methods in Lie (super)algebras

affine Kac–Moody algebras Kac–Wakimoto admissible representations vertex operator algebras Joseph’s characteristic variety


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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