Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 068, 28 pages      arXiv:2306.15582      https://doi.org/10.3842/SIGMA.2024.068
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces

Hans Z. Munthe-Kaas a and Jonatan Stava b
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

Received December 20, 2023, in final form July 03, 2024; Published online July 25, 2024

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.

Key words: Lie admissible triple algebra; connection algebra; symmetric spaces.

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