Cosets of affine vertex algebras inside larger structures - Creutzig, Thomas; Linshaw, Andrew R. (Journal of Algebra, 2019)
Given a finite-dimensional reductive Lie algebra g equipped with a nondegenerate, invariant, symmetric bilinear form B, let Vk(g,B) denote the universal affine vertex algebra associated to g and B at level k. Let Ak be a vertex (super)algebra admitting a homomorphism Vk(g,B)→Ak. Under some technical conditions on Ak, we characterize the coset Com(Vk(g,B),Ak) for generic values of k. We establish the strong finite generation of this coset in full generality in the following cases: Ak=Vk(g′,B′), Ak=Vk−l(g′,B′)⊗F, and Ak=Vk−l(g′,B′)⊗Vl(g″,B″). Here g′ and g″ are finite-dimensional Lie (super)algebras containing g, equipped with nondegenerate, invariant, (super)symmetric bilinear forms B′ and B″ which extend B, l∈C is fixed, and F is a free field algebra admitting a homomorphism Vl(g,B)→F. Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. As an application, we give a new proof of the rationality of the simple N=2 superconformal algebra with c=[Formula presented] for all positive integers k.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Affine vertex algebra; Commutant construction; Coset construction; Finite generation; Invariant theory; N=2 superconformal algebra; Orbifold construction; W-algebra