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

TL;DR

This paper characterizes elements in finite-dimensional Lie algebras whose adjoint orbits generate pointed convex cones, and explores pairs of elements with specific commutation relations related to invariant cones and gradings.

Contribution

It provides a natural characterization of elements with pointed convex cones in admissible Lie algebras and studies pairs satisfying certain commutation relations, connecting to invariant cone structures.

Findings

Characterization of elements with pointed convex cones in admissible Lie algebras

Construction of element pairs $(x,h)$ with $[h,x]=x$ where $C_x$ is pointed

Identification of conditions for 3- and 5-gradings in Lie algebra 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.