Abstract
We prove the rationality of all the minimal series principal $W$-algebras discovered by Frenkel, Kac and Wakimoto, thereby giving a new family of rational and $C_2$-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu’s algebra of simple $W$-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu’s algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu’s algebras of vertex operator algebras associated with admissible representations of affine Kac-Moody algebras as well.
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