Compact symplectic nilmanifolds associated with graphs
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.
scientific article; zbMATH DE number 5561730