Cotangent bundles of 4-dimensional hypercomplex Lie groups
type: Item
instance of: scholarly article
title: Cotangent bundles of 4-dimensional hypercomplex Lie groups
zbMATH Open document ID: 1035.53062
author: Anna Fino
published in: Manuscripta Mathematica
publication date: 2003-03-27T00:00:00Z
review text: This paper deals with Lie groups \(\text{Aff}(\mathbb{C})\) (the group of affine motions of \(\mathbb{C}\)), \(S_{\mathbb{R}}\) (the solvable Lie group that acts simply transitively and isometrically on the 4-dimensional and hyperbolic space) and \(\mathbb{H}^*\) (the multiplicative group of non-zero quaternions). The author proves the existence of invariant conformally hyper-Kähler metrics on the cotangent bundles \(T^*\text{Aff}(\mathbb{C})\) and \(T^* S_{\mathbb{R}}\) endowed with a Lie group structure determined by the Obata connection on \(\text{Aff}(\mathbb{C})\) and \(S_{\mathbb{R}}\). Furthermore, she proves that \(T^*\text{Aff}(\mathbb{C})\) and \(T^* \mathbb{H}^*\) admits a hyper-Kähler with torsion structure which is invariant with respect to the Lie group structure determined by the affine structure associated to the left invariant symplectic form on \(\text{Aff}(\mathbb{C})\) and Obata connection on \(\mathbb{H}^*\).
Mathematics Subject Classification ID: 53C26
zbMATH DE Number: 1889173
zbMATH Keywords: hyper-Kähler metric
zbMATH Keywords: Lie group
zbMATH Keywords: cotangent bundle
Wikidata QID: Q115388455
reviewed by: S. Ivashkovich
MaRDI profile type: Publication
full work available at URL: s00229-002-0324-6
OpenAlex ID: W1991758102
DOI: 10.1007/S00229-002-0324-6
description: scientific article; zbMATH DE number 1889173