Kostant Pairs of Lie Type and Conformal Embeddings

© 2019, Springer Nature Switzerland AG. We deal with some aspects of the theory of conformal embeddings of affine vertex algebras, providing a new proof of the Symmetric Space Theorem and a criterion for conformal embeddings of equal rank subalgebras. We study some examples of embeddings at the critical level. We prove a criterion for embeddings at the critical level which enables us to prove equality of certain central elements

On the classification of non-equal rank affine conformal embeddings and applications

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