  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.
