Abstract. We find all values of k ∈ C, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra W k (g, θ) is conformal, where g is a basic simple Lie superalgebra and −θ its minimal root. In particular, it turns out that if W k (g, θ) does not collapse to its affine part, then the possible values of these k are either − 2 3, where h ∨ is the dual Coxeter number of g for the normalization (θ, θ) = 2. As an application of our results, we present a realization of simple affine vertex algebra V − n+1 2

We discover a large class of simple affine vertex algebras V k (g), associated to basic Lie superalgebras g at non-admissible collapsing levels k, having exactly one irreducible g-locally finite module in the category O. In the case when g is a Lie algebra, we prove a complete reducibility result for V k (g)-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra V k (g) at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras V −1/2 (Cn) and V −4 (E 7 ), we surprisingly obtain the realization of non-simple affine vertex algebras of types B and D having exactly one non-trivial ideal., are non-simple, with a unique non-trivial ideal.The decompositions of the embeddings above is still an open problem, and will be a subject of our forthcoming papers.

Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras Vk(g0) ⊂ Vk(g) , corresponding to an embedding of a maximal equal rank reductive subalgebra g0 into a simple Lie algebra g, is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when Vk(g) decomposes finitely as a Vk(g0) -module

We present methods for computing the explicit decomposition of the minimal simple affine W-algebra Wk(g,θ) as a module for its maximal affine subalgebra k( g♮) at a conformal level k, that is, whenever the Virasoro vectors of Wk(g,θ) and k(g♮) coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when ♮ is a semisimple Lie algebra, we show that, for a suitable conformal level k, Wk(g,θ) is isomorphic to an extension of k(g♮) by its simple module. We are able to prove that in certain cases Wk(g,θ)\ud is a simple current extension of k(g♮) . In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra Wk(sl(4),θ) at k = −8/3. We prove, as conjectured in [3], that Wk(sl(4),θ) is isomorphic to the vertex algebra ℛ(3), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra Vk(sl(n)) at certain admissible levels and for Vk(sl(m|n)),m≠n,m,n≥1 at arbitrary levels

We complete the classification of conformal embeddings of a maximally reductive subalgebra k into a simple Lie algebra g at non-integrable non-critical levels k by dealing with the case when k has rank less than that of g. We describe some remarkable instances of decomposition of the vertex algebra V k (g) as a module for the vertex subalgebra generated by k. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings A1 × A1 ֒→ C3 at level k = −1/2, and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding A1 × D4 ֒→ C8 at level k = −1/2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.