Cohomological properties of unimodular six dimensional solvable Lie algebras

TL;DR

This paper investigates the cohomological properties of six-dimensional unimodular solvable Lie algebras, focusing on symplectic structures, Betti numbers, and the Hard Lefschetz property, confirming Guan's conjecture in this setting.

Contribution

It provides a complete classification of symplectic structures on these Lie algebras and verifies Guan's conjecture for six-dimensional unimodular solvable cases.

Findings

All symplectic structures on six-dimensional unimodular solvable Lie algebras are listed.

Guan's conjecture about the nilpotency step is confirmed for these algebras.

Some solvmanifolds satisfy the Hard Lefschetz property based on Tseng and Yau's cohomologies.

Abstract

In the present paper we study six dimensional solvable Lie algebras with special emphasis on those admitting a symplectic structure. We list all the symplectic structures that they admit and we compute their Betti numbers finding some properties about the codimension of the nilradical. Next, we consider the conjecture of Guan about step of nilpotency of a symplectic solvmanifold finding that it is true for all six dimensional unimodular solvable Lie algebras. Finally, we consider some cohomologies for symplectic manifolds introduced by Tseng and Yau in the context of symplectic Hogde theory and we use them to determine some six dimensional solvmanifolds for which the Hard Lefschetz property holds.