2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160035The existence of a flat torsion-free connection, or left symmetric algebra structure on a Lie algebra g gives rise to a canonically defined complex structure on g+g and a symplectic structure on g+g^*. We verify that the associated differential Gerstenhaber algebras controlling the deformation theories of the complex and symplectic form are isomorphic. This provides a class of examples of "weak mirror symmetry" as suggested by Merkulov. For nilpotent algebras in dimension 4 and 6 the isomorphism classes of the semi-direct products g+g and g+g^* are listed. A one-parameter family of inequivalent pseudo-Kähler structures is given.36 pages, 3 tablesAlgebraic GeometryMathematical PhysicsDifferential Geometry17B81 (Primary); 32G81, 16E45, 53D05 (Secondary)text