Zhu's algebra and the $C_2$-algebra in the symplectic and the orthogonal cases

TL;DR

This paper investigates the relationship between Zhu's algebra and the $C_2$-algebra in symplectic and orthogonal Lie algebra types, establishing their dimension equality in type ${ t C}_m$ and computing the graded character.

Contribution

It proves the dimension equality of Zhu's algebra and the $C_2$-algebra for type ${ t C}_m$ and provides a graded character computation, utilizing maximal parabolic subalgebras.

Findings

Zhu's algebra and the $C_2$-algebra of type ${ t C}_m$ have the same dimension.

The graded character of the $C_2$-algebra is computed.

For orthogonal algebras, only a quotient of the $C_2$-algebra is described.

Abstract

We prove that Zhu's algebra and the $C_2$-algebra of type ${\tt C}_m$ have the same dimension, and we compute the graded character of the latter. Maximal parabolic subalgebras of the symplectic algebra play a central role in our construction. For the orthogonal algebras our methods do not allow to describe the whole $C_2$-algebras, we get only a description of a certain quotient of the algebra.