On Lie groups with left-invariant symplectic or Kählerian structures
type: Item
instance of: scholarly article
title: On Lie groups with left-invariant symplectic or Kählerian structures
zbMATH Open document ID: 0665.53046
DOI: 10.1007/BF00398959
author: André Lichnerowicz
author: A. Medina
published in: Letters in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: In this paper the authors search for conditions, on a connected Lie group, which ensure the existence of invariant symplectic or Kählerian structures. Every Kählerian manifold has a canonical symplectic structure but the converse does not hold in general. There has been recent interest in examples of compact symplectic manifolds which do not admit Kählerian structures. The main results of the present paper are: Theorem 5. A simply connected Kählerian group is a direct product of a unimodular Kählerian group which is metabelian and flat by a Kählerian group with non vanishing Ricci curvature which either admits a holomorphic fibration with flat Kählerian fibers of admits an exact symplectic structure given by its Ricci 2-form. Theorem 6. Let G be a symplectic group admitting a left invariant metric such that its Ricci curvature is \(\geq 0\). The group G is necessarily unimodular, metabelian and admits a left invariant Kählerian structure which is flat.
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53B35
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 4088448
zbMATH Keywords: Lie group
zbMATH Keywords: symplectic structure
zbMATH Keywords: Kählerian structures
zbMATH Keywords: Kählerian group
zbMATH Keywords: symplectic group
zbMATH Keywords: Ricci curvature
Wikidata QID: Q115395058
reviewed by: Isabel Dotti Miatello
MaRDI profile type: Publication
cites work: On Kaehlerian Homogeneous Spaces of Unimodular Lie Groups
cites work: Q4728948
full work available at URL: https://doi.org/10.1007/bf00398959
OpenAlex ID: W1981514453
description: scientific article; zbMATH DE number 4088448