Conformal embeddings of affine vertex algebras in minimal W-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine W-algebra as a module for its maximal affine subalgebra at a conformal level k, that is, whenever the Virasoro vectors of and coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when is a semisimple Lie algebra, we show that, for a suitable conformal level k, is isomorphic to an extension of by its simple module. We are able to prove that in certain cases is a simple current extension of . In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra at k = -8/3. We prove, as conjectured in [3], that is isomorphic to the vertex algebra , and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra at certain admissible levels and for at arbitrary levels.