No-go theorems for $ r $-matrices in symplectic geometry

Jonas Schnitzer; Jonas Schnitzer

doi:10.3934/cam.2024021

Communications in Analysis and Mechanics

2024, Volume 16, Issue 3: 448-456. doi: 10.3934/cam.2024021

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No-go theorems for $ r $-matrices in symplectic geometry

Jonas Schnitzer ^,

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

Received: 17 November 2023 Revised: 31 January 2024 Accepted: 16 May 2024 Published: 01 July 2024
53D05, 16W25

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 $ \mathbb{C}P^n $ and higher genus Pretzel surfaces cannot carry symplectic structures that are induced by triangular Lie algebras.
- symplectic geometry,
- Lie algebras,
- Yang-Baxter equation
Citation: Jonas Schnitzer. No-go theorems for $ r $-matrices in symplectic geometry[J]. Communications in Analysis and Mechanics, 2024, 16(3): 448-456. doi: 10.3934/cam.2024021

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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 $ \mathbb{C}P^n $ and higher genus Pretzel surfaces cannot carry symplectic structures that are induced by triangular Lie algebras.

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