Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality
Year: 2025
Type: article
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 ) {(h,k)} of Euler elements orthogonal if e π i ad h k = - k {e^{\pi i\mathop{{\rm ad}}\nolimits h}k=-k} . We show that, if ( h , k ) {(h,k)} and ( k , h ) {(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.
Source: Forum Mathematicum
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
Award: G8220758391
OpenAlex ID: W4415592576