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