Lie wedges of endomorphism semigroups of standard subspaces in admissible Lie algebras

Abstract

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra containing pointed generating invariant closed convex cones. We determine those derivations $D$ of $\mathfrak{g}$ which induce a 3-grading of the form $\mathfrak{g}={\mathfrak{g}}_{-1}\oplus {\mathfrak{g}}_0\oplus {\mathfrak{g}}_1$ on $\mathfrak{g}$ such that the $(\pm 1)$-eigenspaces ${\mathfrak{g}}_{\pm 1}$ of $D$ are generated by the intersections with generating cones of the form $W={\mathcal{O}}_f^{\ast }$, where ${\mathcal{O}}_f$ is the coadjoint orbit of a linear functional $f\in \mathfrak{z}(\mathfrak{g}{)}^{\ast }$ and ${\mathcal{O}}_f^{\ast }$ is the dual cone of ${\mathcal{O}}_f$. In particular, we show that, if $\mathfrak{g}$ is solvable, no such derivation except the trivial one exists. This continues our classification of Lie algebras generated by Lie wedges of endomorphism semigroups of standard subspaces. The classification is motivated by the relation of nets of standard subspaces to Haag-Kastler nets of von Neumann algebras in Algebraic Quantum Field Theory.