Vertex operator algebra

Affine transformation

Pure mathematics

Mathematical physics

Discrete mathematics

Lie algebra

Vertex (graph theory)

Bilinear form

Current algebra

Graph

Mathematics

Invariant (physics)

Combinatorics

Homomorphism

Coset

Algebra over a field

Algebra homomorphism

Jordan algebra

Journal of Algebra

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=3kk+2 for all positive integers k.

Cosets of affine vertex algebras inside larger structures