## A root space decomposition for finite vertex algebras

### Summary

Summary: Let $L$ be a Lie pseudoalgebra, $a \in L$. We show that, if $a$ generates a (finite) solvable subalgebra $S = \langle a \rangle \subset L$, then one may find a lifting $\bar a \in S$ of $[a] \in S/S'$ such that $\langle \bar a \rangle$ 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 $V = U \sd N$, 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 $\Nil V$.

17B69

### Keywords/Phrases

pseudoalgebra, vertex algebra