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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 31, Issue 3

# Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds

Marcos Origlia
• Corresponding author
• KU Leuven Kulak, E. Sabbelaan 53, BE-8500 Kortrijk, Belgium; and FaMAF-UNC, CIEM-CONICET, Ciudad Universitaria, 5000 Córdoba, Argentina
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Published Online: 2018-11-23 | DOI: https://doi.org/10.1515/forum-2018-0200

## Abstract

We study Lie algebras of type I, that is, a Lie algebra $𝔤$ where all the eigenvalues of the operator ${\mathrm{ad}}_{X}$ are imaginary for all $X\in 𝔤$. We prove that the Morse–Novikov cohomology of a Lie algebra of type I is trivial for any closed 1-form. We focus on locally conformal symplectic structures (LCS) on Lie algebras of type I. In particular, we show that for a Lie algebra of type I any LCS structure is of the first kind. We also exhibit lattices for some 6-dimensional Lie groups of type I admitting left invariant LCS structures in order to produce compact solvmanifolds equipped with an invariant LCS structure.

MSC 2010: 22E25; 53C15; 53D05; 53C55; 22E40

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Revised: 2018-10-31

Published Online: 2018-11-23

Published in Print: 2019-05-01

This work was partially supported by CONICET (PIP 11220120100451), ANPCyT (PICT 2014 NÂ° 2706), SECyT-UNC (Proyecto “A” 30720150100731CB) and the Research Foundation Flanders (Project G.0F93.17N).

Citation Information: Forum Mathematicum, Volume 31, Issue 3, Pages 563–578, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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