Compact symplectic nilmanifolds associated with graphs
type: Item
instance of: scholarly article
title: Compact symplectic nilmanifolds associated with graphs
zbMATH Open document ID: 1170.53062
author: Hannes Pouseele
author: Paulo Tirao
published in: Journal of Pure and Applied Algebra
publication date: 2009-06-04T00:00:00Z
review text: Associated to any finite graph \(G\) is a 2-step nilpotent Lie algebra \({\mathcal L}_G\) and nilmanifold \(L_G/\Gamma\) obtained as the quotient of the canonical simply connected Lie group \(L_G\) by the lattice generated by the vertices of \(G\). The authors show how to compute the homology groups \(H^2({\mathcal L}_G, {\mathbb R})\) of these 2-step nilpotent Lie algebras directly from the graph, and they use this to characterize which graphs give rise to symplectic or contact nilmanifolds. Interestingly, only one Lie algebra of the form \({\mathcal L}_G\) admits a contact structure, and that is the one associated with the complete graph on two vertices otherwise known as the 3-dimensional Heisenberg Lie algebra.
reviewed by: Hans U. Boden
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 5561730
zbMATH Keywords: finite graph
zbMATH Keywords: Lie algebra
zbMATH Keywords: nilmanifold
zbMATH Keywords: symplectic form
zbMATH Keywords: contact form
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.01.014
OpenAlex ID: W2060191774
cites work: Q4442695
cites work: Symplectic or contact structures on Lie groups
cites work: Constructing Lie algebra homology classes
cites work: On the cohomology of compact homogeneous spaces of nilpotent Lie groups
cites work: Q5850573
cites work: On the homology of free 2-step nilpotent Lie algebras.
cites work: Laplacian and homology of free two-step nilpotent Lie algebras
cites work: Anosov automorphisms on compact nilmanifolds associated with graphs
cites work: Symplectic structures on Heisenberg-type nilmanifolds
DOI: 10.1016/J.JPAA.2009.01.014
description: scientific article; zbMATH DE number 5561730