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 mathop {mathrm{ad}}nolimits h defines a 5-grading, and characterize the cases where we even get a 3-grading.
Highlights
Convexity properties of adjoint orbits Ox = Inn(g)x in a finite dimensional real Lie algebra, where Inn(g) = ead g is the group of inner automorphisms, play a role in many contexts
They appear in the theory of invariant convex cones
In the literature pointed generating invariant cones have been studied from the perspective of their interior: If W ⊆ g is pointed and generating, its interior consists of elliptic elements and W is determined by its intersection with a compactly embedded Cartan subalgebra [5,12]
Summary
Convexity properties of adjoint orbits Ox = Inn(g)x in a finite dimensional real Lie algebra, where Inn(g) = ead g is the group of inner automorphisms, play a role in many contexts Most directly, they appear in the theory of invariant convex cones. Beyond reductive Lie algebras, the natural context for our investigation is the class of admissible Lie algebras, i.e., those containing an Inn(g)-invariant pointed generating closed convex subset C (cf [12, Def. VII.3.2]). Since every pointed invariant convex subset spans an admissible ideal, we shall assume throughout that g is admissible It is of vital importance for our arguments, that admissible Lie algebras permit a powerful structure theory.
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