| Summary: | © 2018, Springer International Publishing AG, part of Springer Nature. 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.
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