# Locally conformally symplectic structures on Lie algebras and solvmanifolds

## Marcos Origlia

created by daniele on 19 Mar 2019
modified on 21 Mar 2019

27 mar 2019 -- 10:00

Sala S, Dipartimento di Matematica, Università di Torino

Abstract.

A locally conformal symplectic structure (LCS for short) on the manifold $M$ is a non degenerate $2$-form $\omega$ such that there exists an open cover $\{U_i\}$ and smooth functions $f_i$ on $U_i$ such that $\omega_i=\exp(-f_i)\omega$ is a symplectic form on $U_i$. This condition is equivalent to requiring that $d\omega=\theta\wedge\omega$ for some closed $1$-form $\theta$, called the Lee form. The pair $(\omega, \theta)$ is called an LCS structure on $M$. According to Vaisman a LCS structure can be of the first or of the second kind. In this talk we will focus on left invariant LCS structures on Lie group or equivalently these structures on their Lie algebras. We will study LCS structures on Lie algebras of type I, and we will show a method to build examples of Lie algebras admitting LCS structures of the second kind. We will also discuss about the existence of lattices in the associated simply connected Lie groups.

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