Invariant complex structures on tangent and cotangent Lie groups of dimension six
type: Item
instance of: scholarly article
title: Invariant complex structures on tangent and cotangent Lie groups of dimension six
zbMATH Open document ID: 1256.53024
author: Gabriela Paola Ovando
author: Rutwig Campoamor Stursberg
published in: Osaka Journal of Mathematics
publication date: 2012-07-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/0805.2520
full work available at URL: https://projecteuclid.org/euclid.ojm/1340197936
review text: Tangent and cotangent Lie algebras are semidirect products \(\mathrm{T}_{\pi} \mathfrak{h}=\mathfrak{h} \ltimes_{\pi} V\), where \(\pi\) is the adjoint or the co-adjoint representation, respectively. The present paper deals with left invariant complex structures on simply connected Lie groups, whose Lie algebra is a tangent or cotangent Lie algebra. As a start, the existence of totally real complex structures is studied, i.e., complex structures \(J\) satisfying the constraint \(J\mathfrak{h} = V\). It is shown that, if \(\pi\) is the adjoint representation, this kind of complex structures are associated to non-singular derivations of \(\mathfrak{h}\). Thus, the existence of totally real complex structures on \(\mathrm{T}_{\pi} \mathfrak{h}\) implies that \(\mathfrak{h}\) must be nilpotent. Then, the three dimensional case is studied in detail. For the coadjoint representation, the general form of totally real complex structures is given. Moreover, pseudo-Kähler structures are examined.
reviewed by: Sergio Console
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 6060696
zbMATH Keywords: complex structures
zbMATH Keywords: Lie algebras
zbMATH Keywords: symplectic structures
MaRDI profile type: Publication
arXiv ID: 0805.2520
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description: scientific article; zbMATH DE number 6060696