Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

In this note, we study in a finite dimensional Lie algebra g the set of all those elements x forwhich 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 Cx . Assuming that 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 Cx pointed. Given x, we show that such elements h can be constructed in such a way that ad h defines a 5-grading, and characterize the cases where we even get a 3-grading.