Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality

2025/08/14 by Morinelli, Vincenzo, Neeb, Karl-Hermann, Olafsson, Gestur
#22D45 #22E46 #22E70 #81T05 #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #Mathematical Physics (math-ph) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2508.10960

Abstract

The current article continues our project on representation theory, Euler elements, causal homogeneous spaces and Algebraic Quantum Field Theory (AQFT). We call a pair (h,k) of Euler elements orthogonal if e^{πi \ad h} k = -k. We show that, if (h,k) and (k,h) are orthogonal, then they generate a 3-dimensional simple subalgebra. We also classify orthogonal Euler pairs in simple Lie algebras and determine the fundamental groups of adjoint Euler elements in arbitrary finite-dimensional Lie algebras. Causal complements of wedge regions in spacetimes can be related to so-called twisted complements in the space of abstract Euler wedges, defined in purely group theoretic terms. We show that any pair of twisted complements can be connected by a chain of successive complements coming from 3-dimensional subalgebras.

Citations

• Nets of real subspaces on homogeneous spaces and Algebraic Quantum Field Theory
• A geometric perspective on Algebraic Quantum Field Theory
• Modular geodesics and wedge domains in non-compactly causal symmetric spaces
• From Euler elements and 3-gradings to non-compactly causal symmetric spaces
• A family of non-modular covariant AQFTs
• On the geometry of standard subspaces
• Antiunitary representations and modular theory
• Projective completions of Jordan pairs. Part I: The generalized projective geometry of a Lie algebra
• Conformal Nets, Maximal Temperature and Models from Free Probability

Cited by

• Crowned Lie groups and nets of real subspaces

Related

• From Euler elements and 3-gradings to non-compactly causal symmetric spaces
• Algebraic Quantum Field Theory and Causal Symmetric Spaces
• Wedge domains in non-compactly causal symmetric spaces
• From local nets to Euler elements
• Wedge domains in compactly causal symmetric spaces