Elements in pointed invariant cones in Lie algebras and corresponding affine pairs

Abstract

In this note we study in a finite dimensional Lie algebra ${\mathfrak g}$ the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone~$C_x$. Assuming that ${\mathfrak g}$ is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs $(x,h)$ of Lie algebra elements satisfying $[h,x]=x$ for which $C_x$ pointed. Given $x$, we show that such elements $h$ can be constructed in such a way that ${\rm ad} h$ defines a $5$-grading, and characterize the cases where we even get a $3$-grading.