A method for describing the maximal ideal in universal affine vertex algebras at non-admissible levels

The problem of determining maximal ideals in universal affine vertex algebras is difficult for levels beyond admissible, since there are no simple character formulas which can be applied. Here we investigate when certain quotient [Formula: see text] of universal affine vertex algebra [Formula: see text] is simple. We present a new method for proving simplicity of quotients of universal affine vertex algebras in the case of affine vertex algebra [Formula: see text] at level [Formula: see text]. In that way we describe the maximal ideal in [Formula: see text]. For that purpose, we use the representation theory of minimal affine [Formula: see text]-algebra [Formula: see text] developed in [D. Adamović, T. Creutzig, O. Perše and I. Vukorepa, Tensor category [Formula: see text] via minimal affine W-algebras at the non-admissible level [Formula: see text], J. Pure Appl. Algebra 228(5) (2024) 107565]. In particular, we use the embedding [Formula: see text] and fusion rules for [Formula: see text]-modules. We apply this result in the cases [Formula: see text] and prove that a maximal ideal is generated by one singular vector of conformal weight [Formula: see text]. As a byproduct, we classify irreducible modules in the category [Formula: see text] for the simple affine vertex algebra [Formula: see text].