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Symmetry, Integrability and Geometry: Methods and Applications, 2024, Volume 20, 068, 28 pp.
DOI: https://doi.org/10.3842/SIGMA.2024.068
(Mi sigma2070)
 

This article is cited in 1 scientific paper (total in 1 paper)

Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces

Hans Z. Munthe-Kaasa, Jonatan Stavab

a Department of Mathematics and Statistics, UiT The Arctic University of Norway, P.O. Box 6050, Stakkevollan, 9037 Tromsø, Norway
b Department of Mathematics, University of Bergen, P.O. Box 7803, 5020 Bergen, Norway
Full-text PDF (497 kB) Citations (1)
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Abstract: Associated to a symmetric space there is a canonical connection with zero torsion and parallel curvature. This connection acts as a binary operator on the vector space of smooth sections of the tangent bundle, and it is linear with respect to the real numbers. Thus the smooth section of the tangent bundle together with the connection form an algebra we call the connection algebra. The constraints of zero torsion and constant curvature makes the connection algebra into a Lie admissible triple algebra. This is a type of algebra that generalises pre-Lie algebras, and it can be embedded into a post-Lie algebra in a canonical way that generalises the canonical embedding of Lie triple systems into Lie algebras. The free Lie admissible triple algebra can be described by incorporating triple-brackets into the leaves of rooted (non-planar) trees.
Keywords: Lie admissible triple algebra, connection algebra, symmetric spaces.
Funding agency Grant number
Research Council of Norway 302831
The authors are supported by the Research Council of Norway through project 302831 Computational Dynamics and Stochastics on Manifolds (CODYSMA).
Received: December 20, 2023; in final form July 3, 2024; Published online July 25, 2024
Document Type: Article
Language: English
Citation: Hans Z. Munthe-Kaas, Jonatan Stava, “Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces”, SIGMA, 20 (2024), 068, 28 pp.
Citation in format AMSBIB
\Bibitem{MunSta24}
\by Hans~Z.~Munthe-Kaas, Jonatan~Stava
\paper Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces
\jour SIGMA
\yr 2024
\vol 20
\papernumber 068
\totalpages 28
\mathnet{http://mi.mathnet.ru/sigma2070}
\crossref{https://doi.org/10.3842/SIGMA.2024.068}
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    Symmetry, Integrability and Geometry: Methods and Applications
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