Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures
type: Item
instance of: scholarly article
title: Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures
zbMATH Open document ID: 1387.22012
published in: Manuscripta Mathematica
publication date: 2018-02-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1611.02089
full work available at URL: http://hdl.handle.net/11336/59983
review text: The authors study left-invariant locally conformally Kähler and locally conformally symplectic structures on almost abelian Lie groups and on their compact quotients. In particular, they focus on the existence of lattices for such Lie groups endowed with such structures. Locally conformally Kähler, respectively symplectic, structures are structures that locally admit a conformal change that is Kähler, respectively symplectic. For some classes of solvable Lie groups, criteria are known that assure the existence of lattices: in this case, left-invariant lcK and lcs structures move to the solvmanifold defined as the compact quotient. For nilpotent Lie groups, a criterion for the existence of lattices is given in [\textit{A. I. Mal'tsev}, Izv. Akad. Nauk SSSR, Ser. Mat. 9, 329--356 (1945; Zbl 0061.05303)], left-invariant lcK structures are studied in [\textit{H. Sawai}, Geom. Dedicata 125, 93--101 (2007; Zbl 1121.53052)], and left-invariant lcs structures are studied in [\textit{G. Bazzoni} and \textit{J. C. Marrero}, Bull. Sci. Math. 143, 1--57 (2018; Zbl 1383.22004)]. For almost abelian Lie groups, (that is, Lie groups whose Lie algebra has a codimension-one abelian ideal) a criterion for the existence of lattices is given in [\textit{C. Bock}, Asian J. Math. 20, No. 2, 199--262 (2016; Zbl 1346.53052)], and left-invariant lcK and lcs structures are the object of the paper under review. Section 3 deals with lcK structures: Theorem 3.3 characterizes the Lie algebras of almost abelian Lie groups admitting left-invariant lcK structures; Theorems 3.7 and 3.9 show that lattices for such Lie groups exist only in dimension \(4\), where Inoue surfaces of type \(S^0\) appear. Section 4 deals with lcs structures: Theorems 4.1 and 4.5 characterize the Lie algebras of almost abelian Lie groups admitting left-invariant lcs structures; Corollary 4.3 and Table 1 specify whether there exist such structures of first kind or second kind (this meaning that the Lee morphism is surjective,…
reviewed by: Daniele Angella
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53D05
zbMATH DE Number: 6845360
zbMATH Keywords: lattice
zbMATH Keywords: locally conformal Kähler metric
zbMATH Keywords: locally conformal symplectic structure
zbMATH Keywords: almost abelian Lie group
author: Adrián Andrada
author: Marcos Origlia
Wikidata QID: Q115388325
MaRDI profile type: Publication
OpenAlex ID: W2554616549
arXiv ID: 1611.02089
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DOI: 10.1007/S00229-017-0938-3
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