#### DOCUMENTA MATHEMATICA, Vol. 17 (2012), 783-805

A. D'Andrea and G. Marchei

A Root Space Decomposition for Finite Vertex Algebras

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 $\Nil V$.

2010 Mathematics Subject Classification: 17B69

Keywords and Phrases: Pseudoalgebra, vertex algebra

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