A root space decomposition for finite vertex algebras

Abstract

Let $L$ be a Lie pseudoalgebra, $a\in L$. We show that, if $a$ generates a (finite) solvable subalgebra $S=⟨a⟩\subset L$, then one may find a lifting $\bar{a}\in S$ of $[a]\in S/S^{\prime }$ such that $⟨\bar{a}⟩$ 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 $N=V^{[\infty ]}$ is a canonically determined ideal contained in the nilradical $\mathrm{Nil}V$.