Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912538192510976 |
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| author | Morinelli, Vincenzo Neeb, Karl-Hermann Olafsson, Gestur |
| author_facet | Morinelli, Vincenzo Neeb, Karl-Hermann Olafsson, Gestur |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10960 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality Morinelli, Vincenzo Neeb, Karl-Hermann Olafsson, Gestur Representation Theory Mathematical Physics Group Theory 81T05, 22D45, 22E46, 22E70 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. |
| title | Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality |
| topic | Representation Theory Mathematical Physics Group Theory 81T05, 22D45, 22E46, 22E70 |
| url | https://arxiv.org/abs/2508.10960 |