New Approaches for Studying Conformal Embeddings and Collapsing Levels for <i>W</i>–Algebras

Abstract In this paper, we prove a general result saying that under certain hypothesis an embedding of affine vertex algebra into $W$–algebra is conformal if and only their central charges coincide. This extends our previous obtained in the case minimal $W$-algebras [ 3]. We also find sufficient condition showing levels are collapsing. new enables us to some $k$ where $W_{k}(sl(N), x, f )$ collapses its part when $f$ hook or rectangular type. Our methods can be applied non-admissible levels. particular, Creutzig’s conjecture 18] on type $W$-algebra $W_{k}(sl(n+m), f_{m,n})$ subalgebra. Quite surprisingly, problem not collapsing turns out very difficult. cases admissible conformal, Then, by generalizing results semi-simplicity embeddings from 2], 5], many which semi-simple as module for subalgebra at level provide explicit decompositions.