Lie wedges of endomorphism semigroups of standard subspaces in admissible Lie algebras

2020/07/27 by Oeh, Daniel
#22E45 (Primary) 81R05 #81T05 (Secondary) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2007.13445

Abstract

Let \mathfrakg be a real finite-dimensional Lie algebra containing pointed generating invariant closed convex cones. We determine those derivations D of \mathfrakg which induce a 3-grading of the form \mathfrakg = \mathfrakg_-1 ⊕ \mathfrakg₀ ⊕ \mathfrakg₁ on \mathfrakg such that the (± 1)-eigenspaces \mathfrakg_{± 1} of D are generated by the intersections with generating cones of the form W = O_f^*, where O_f is the coadjoint orbit of a linear functional f ∈ \mathfrakz(\mathfrakg)^* and O_f^* is the dual cone of O_f. In particular, we show that, if \mathfrakg is solvable, no such derivation except the trivial one exists. This continues our classification of Lie algebras generated by Lie wedges of endomorphism semigroups of standard subspaces. The classification is motivated by the relation of nets of standard subspaces to Haag-Kastler nets of von Neumann algebras in Algebraic Quantum Field Theory.

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