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 Wk(g,θ) as a module for its maximal affine subalgebra k( g♮) at a conformal level k, that is, whenever the Virasoro vectors of Wk(g,θ) and k(g♮) 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, Wk(g,θ) is isomorphic to an extension of k(g♮) by its simple module. We are able to prove that in certain cases Wk(g,θ) is a simple current extension of k(g♮) . In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra Wk(sl(4),θ) at k = −8/3. We prove, as conjectured in [3], that Wk(sl(4),θ) is isomorphic to the vertex algebra ℛ(3), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra Vk(sl(n)) at certain admissible levels and for Vk(sl(m|n)),m≠n,m,n≥1 at arbitrary levels.