Conformal embeddings of affine vertex algebras in minimal W-algebras I: structural results

We find all values of k∈C, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra Wk(g, θ)is conformal, where gis a basic simple Lie superalgebra and −θits minimal root. In particular, it turns out that if Wk(g, θ)does not collapse to its affine part, then the possible values of these kare either −2/3h∨ or −(h∨−1)/2, where h∨is the dual Coxeter number of gfor the normalization (θ,θ)=2. As an application of our results, we present a realization of simple affine vertex algebra V−n+12(sl(n +1))inside the tensor product of the vertex algebra Wn−12(sl(2|n), θ)(also called the Bershadsky–Knizhnik algebra) with a lattice vertex algebra.