Ricerc@Sapienza

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 Vk (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 discover a large class of simple affine vertex algebras Vk(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 Vk(g)-modules at an arbitrary collapsing level. We also
determine the generators of the maximal ideal in the universal affine vertex algebra Vk(g) at certain negative integer levels. Considering some conformal embeddings in