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 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.