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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>h</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(h,k)} of Euler elements orthogonal if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mi>π</m:mi> <m:mo>⁢</m:mo> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>ad</m:mo> <m:mi>h</m:mi> </m:mrow> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi>k</m:mi> </m:mrow> </m:mrow> </m:math> {e^{\pi i\mathop{{\rm ad}}\nolimits h}k=-k} . We show that, if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>h</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(h,k)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>h</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(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 orbits of 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.