Symplectic structures on nilmanifolds: an obstruction for its existence

TL;DR

This paper introduces a cohomological obstruction that determines whether nilpotent Lie algebras, particularly those related to classical simple Lie algebras, can admit symplectic structures, providing both positive and negative results.

Contribution

It presents a necessary cohomological condition for the existence of symplectic structures on nilpotent Lie algebras, especially in the context of nilradicals of minimal parabolic subalgebras.

Findings

Identifies a cohomological obstruction to symplectic structures

Provides examples of nilpotent Lie algebras with and without symplectic structures

Applies the obstruction to classical Lie algebra families

Abstract

In this work we introduce an obstruction for the existence of symplectic structures on nilpotent Lie algebras. Indeed, a necessary condition is presented in terms of the cohomology of the Lie algebra. Using this obstruction we obtain both positive and negative results on the existence of symplectic structures on a large family of nilpotent Lie algebras. Namely the family of nilradicals of minimal parabolic subalgebras associated to the real split Lie algebra of classical complex simple Lie algebras.