Minimal W-algebras with non-admissible levels and intermediate Lie algebras

TL;DR

This paper explores specific W-algebras associated with exceptional Lie algebras at non-admissible levels, identifying modules, characters, and connections to intermediate Lie algebras, extending previous rationality results.

Contribution

It extends the study of minimal W-algebras at non-admissible levels for E6, E7, E8, determining modules, characters, and coset constructions, linking to intermediate Lie algebras.

Findings

All irreducible modules are classified.

Characters of modules are explicitly computed.

Connections to intermediate Lie algebras are established.

Abstract

In \cite{Kawasetsu:2018irs}, Kawasetsu proved that the simple W-algebra associated with a minimal nilpotent element $W_{k}(\mathfrak{g},f_\theta)$ is rational and $C_2$-cofinite for $\mathfrak{g}=D_4,E_6,E_7,E_8$ with non-admissible level $k=-h^\vee/6$. In this paper, we study ${W}_{k}(\mathfrak{g},f_\theta)$ algebra for $\mathfrak{g}=E_6,E_7,E_8$ with non-admissible level $k=-h^\vee/6+1$. We determine all irreducible (Ramond twisted) modules, compute their characters and find coset constructions and Hecke operator interpretations. These W-algebras are closely related to intermediate Lie algebras and intermediate vertex subalgebras.