Zhu's algebras, $C_2$-algebras and abelian radicals

This paper consists of three parts. In the first part we prove that Zhu’s and $C_2$-algebras in type $A$ have the same dimensions. In the second part we compute the graded decomposition of the $C_2$-algebras in type $A$, thus proving the Gaberdiel-Gannon’s conjecture. Our main tool is the theory of abelian radicals, which we develop in the third part.