Abstract
AbstractIn 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.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Anesthesiology and Pain Medicine
Reference20 articles.
1. Bourbaki, N.: Groupes et algèbres de Lie, Chap. VII–VIII. Masson, Paris (1990)
2. Brunetti, R., Guido, D., Longo, R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–785 (2002)
3. Harish-Chandra (1956) Representations of semi-simple Lie groups, V, VI. Am. J. Math. 78 (1–41), 564–628
4. Hilgert, J., Neeb, K.-H.: Structure and Geometry of Lie Groups. Springer (2012)
5. Hilgert, J., Hofmann, K.H., Lawson, J.D.: Lie Groups, Convex Cones, and Semigroups. Oxford University Press, Oxford (1989)