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

Abstract

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 $\mathcal V$ of universal affine vertex algebra $V^k(\mathfrak{g})$ is simple. We present a new method for proving simplicity of quotients of universal affine vertex algebras in the case of affine vertex algebra $L_{k_n}(\mathfrak{sl}_{2n})$ at level $k_n:=-\frac{2n+1}{2}$. In that way we describe the maximal ideal in $V^{k_n}(\mathfrak{sl}_{2n})$. For that purpose, we use the representation theory of minimal affine $W$-algebra $W^{min}_{k_{n+1}}(\mathfrak{sl}_{2n+2})$ developed in [2]. In particular, we use the embedding $L_{k_n}(\mathfrak{sl}_{2n}) \subset W^{min}_{k_{n+1}}(\mathfrak{sl}_{2n+2})$ and fusion rules for $L_{k_n}(\mathfrak{sl}_{2n})$--modules. We apply this result in the cases $n=3,4$ and prove that a maximal ideal is generated by one singular vector of conformal weight $4$. As a byproduct, we classify irreducible modules in the category $\mathcal{O}$ for the simple affine vertex algebra $L_{-7/2}(\mathfrak{sl}_{6})$.