A root space decomposition for finite vertex algebras

TL;DR

This paper establishes a root space decomposition for finite vertex algebras, revealing a structural breakdown into a nilpotent part and a complementary subalgebra, advancing understanding of their internal algebraic structure.

Contribution

It introduces a root space decomposition for finite vertex algebras, connecting Lie pseudoalgebra properties with vertex algebra structure, and identifies a canonical decomposition into nilpotent and semi-simple parts.

Findings

Finite vertex algebras decompose into a semi-direct product of a nilpotent subalgebra and an ideal.

Existence of a lifting of elements to nilpotent generators in solvable subalgebras.

Structural insight into the internal organization of finite vertex algebras.

Abstract

Let L be a Lie pseudoalgebra, a in L. We show that, if a generates a (finite) solvable subalgebra S=<a>, then one may find a lifting a' in S of [a] in S/S' such that <a'> is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra V admits a decomposition into a semi-direct product V = U + N, where U is a subalgebra of V whose underlying Lie conformal algebra U^lie is a nilpotent self-normalizing subalgebra of V^lie, and N is a canonically determined ideal contained in the nilradical Nil V.