Zhu's algebra, C2C2-algebra and C2C2-cofiniteness of parafermion vertex operator algebras

We study Zhu's algebra, C2C2-algebra and C2C2-cofiniteness of parafermion vertex operator algebras. We first give a detailed study of Zhu's algebra and C2C2-algebra of parafermion vertex operator algebras associated with the affine Kac–Moody Lie algebra slˆ2. We show that they have the same dimension and Zhu's algebra is semisimple. The classification of irreducible modules is also established. Finally, we prove that the parafermion vertex operator algebras for any finite dimensional simple Lie algebras are C2C2-cofinite.