**Authors: Dražen Adamović, Pierluigi Möseneder Frajria, Paolo Papi, Ozren Perše**

Advances in Mathematics, Volume 360, 22 January 2020, 106918

**Abstract:** 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 where is a basic classical simple Lie superalgebra. Let be the subalgebra of generated by . We first classify all levels for which the embedding in is conformal. Next we prove that, for a large family of such conformal levels, is a completely reducible–module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of as a finite, non simple current extension of . This decomposition uses our previous work [10] on the representation theory of .

We also study conformal embeddings and in most cases we obtain decomposition rules.