Abstract We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang (arXiv:0708.2520, 2007), in relation to sym- plectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by discrete subgroups. We obtain families of pure and full almost com- plex structures on compact nilmanifolds and solvmanifolds. Some of these families are parametrized by real 2-forms which are anti-invariant with respect to the almost complex structures.
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