A root space decomposition for finite vertex algebras (Q1946051)

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 \(V = U \ltimes N\), where \(U\) is a subalgebra of \(V\) whose underlying Lie conformal algebra \(U^{\text{Lie}}\) is a nilpotent self-normalizing subalgebra of \(V^{\text{Lie}}\), and \(N = V^{[\infty]}\) is a canonically determined ideal contained in the nilradical \(\text{Nil} V\).