Invariant subalgebras of affine vertex algebras

TL;DR

This paper proves that invariant subalgebras of affine vertex algebras under reductive automorphism groups are strongly finitely generated for generic levels, leading to new families of deformable W-algebras applicable for most levels.

Contribution

It establishes strong finite generation of invariant subalgebras of affine vertex algebras under reductive automorphisms for generic levels, introducing new deformable W-algebras.

Findings

Invariant subalgebras are strongly finitely generated for generic levels.

Existence of new deformable W-algebras W(g,B,G)_k for all but finitely many levels.

Provides structural insights into automorphism-invariant subalgebras of affine vertex algebras.

Abstract

Given a finite-dimensional complex Lie algebra g equipped with a nondegenerate, symmetric, invariant bilinear form B, let V_k(g,B) denote the universal affine vertex algebra associated to g and B at level k. For any reductive group G of automorphisms of V_k(g,B), we show that the invariant subalgebra V_k(g,B)^G is strongly finitely generated for generic values of k. This implies the existence of a new family of deformable W-algebras W(g,B,G)_k which exist for all but finitely many values of k.