  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.
