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,

This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra V_k(g) where g=g_0+g_1 is a basic classical simple Lie superalgebra. Let VV_k(g_0) be the subalgebra of V_k(g) generated by g_0. We first classify all levels k for which the embedding VV_k(g_0) in V_k(g) is conformal. Next we prove that, for a large family of such conformal levels, V_k(g) is a completely reducible VV_k(g_0)–module and obtain decomposition rules.