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Research Article | Open Access

No-go theorems for r-matrices in symplectic geometry

Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Str. 1, 79104 Freiburg i. Br., Germany
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Abstract

If a triangular Lie algebra acts on a smooth manifold, it induces a Poisson bracket on it. In case this Poisson structure is actually symplectic, we show that this already implies the existence of a flat connection on any vector bundle over the manifold the Lie algebra acts on, in particular the tangent bundle. This implies, among other things, that CPn and higher genus Pretzel surfaces cannot carry symplectic structures that are induced by triangular Lie algebras.

Keywords

symplectic geometryLie algebrasYang-Baxter equation
CLC number: 53D05, 16W25

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Communications in Analysis and Mechanics
Volume 16 Issue 3,
September 2024
Pages 448-456
DOI: 10.3934/cam.2024021

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Schnitzer J. No-go theorems for r-matrices in symplectic geometry. Communications in Analysis and Mechanics, 2024, 16(3): 448-456. https://doi.org/10.3934/cam.2024021

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Received: 17 November 2023
Revised: 31 January 2024
Accepted: 16 May 2024
Published: 15 September 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)