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

Abstract

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.

arXiv comment19 pages

References in corpus (5)

• Vertex Algebras and Algebraic Curves
• Nonmeromorphic operator product expansion and C_2-cofiniteness for a family of W-algebras
• A theory of tensor products for vertex operator algebra satsifying C_2-cofiniteness
• Zhu's algebra, the C_2 algebra, and twisted modules
• Integrability of C_2-cofinite vertex operator algebras

Cited by in corpus (4)

• ${\mathbb{G}}_a^M$ degeneration of flag varieties
• The structure of Zhu's algebras for certain W-algebras
• PBW filtration and bases for irreducible modules in type $A_n$
• PBW--filtration over $\bz$ and compatible bases for $V_\bz(\la)$ in type ${\mathtt{A}}_n$ and ${\mathtt{C}}_n$