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

In this note, we study in a finite dimensional Lie algebra $${\mathfrak g}$$ 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 $$C_x$$ C x . Assuming that $${\mathfrak g}$$ 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 $$[h,x]=x$$ [ h , x ] = x for which $$C_x$$ C x pointed. Given x, we show that such elements h can be constructed in such a way that $$\mathop {\mathrm{ad}}\nolimits h$$ ad h defines a 5-grading, and characterize the cases where we even get a 3-grading.