Zhu's algebras, C2-algebras and abelian radicals

This paper consists of three parts. In the first part we prove that Zhu's and C2-algebras in type A have the same dimensions. In the second part we compute the graded decomposition of the C2-algebras in type A, thus proving the Gaberdiel–Gannon conjecture. Our main tool is the theory of abelian radicals, which we develop in the third part.