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

In {Kawasetsu:2018irs}, Kawasetsu proved that the simple W-algebra associated with a minimal nilpotent element Wₖ(g,f_θ) is rational and C₂-cofinite for g=D₄,E₆,E₇,E₈ with non-admissible level k=-h^/6. In this paper, we study Wₖ(g,f_θ) algebra for g=E₆,E₇,E₈ with non-admissible level k=-h^/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.