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

TL;DR

This paper classifies orthogonal pairs of Euler elements in Lie algebras, explores their algebraic properties, and relates these structures to causal complements in spacetime models within algebraic quantum field theory.

Contribution

It provides a classification of orthogonal Euler pairs in simple Lie algebras and analyzes their fundamental groups and geometric implications.

Findings

Orthogonal Euler pairs generate 3D simple subalgebras.

Classification of orthogonal Euler pairs in simple Lie algebras.

Connection between twisted complements and 3D subalgebras in spacetime models.

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 $e^{\pi i \ad h} k = -k$. 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.