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

2021/10/14 by Neeb, Karl-Hermann, Oeh, Daniel
#22E15 #22E60 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2110.07297

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.

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