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

We find all values of $k\in \mathbb{C}$, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra $W_k(\mathfrak g,\theta)$isconformal,where$\mathfrak g$ is a basic simple Lie superalgebra and $-\theta$ its minimal root. In particular, it turns out that if $W_k(\mathfrak{g},\theta )$ does not collapse to its affine part, then the possible values of these $k$ are either $-\frac{2}{3}{h}^{\vee }$ or $-\frac{h^{\vee }-1}{2}$, where $h^{\vee }$ is the dual Coxeter number of $\mathfrak g$forthenormalization$(\theta,\theta)=2$. As an application of our results, we present a realization of simple affine vertex algebra $V_{-\tfrac{n+1}{2} } (sl(n+1))$insideofthetensorproductofthevertexalgebra$W_{\tfrac{n-1}{2}} (sl(2| n), \theta)$ (also called the Bershadsky-Knizhnik algebra) with a lattice vertex algebra.