Weak mirror symmetry relates a manifold with complex structure to
another manifold equipped with a symplectic structure through a quasiisomorphism
of associated differential Gerstenhaber algebras. The two
manifolds are then mirror partners. In this paper we consider the analogous
problem on Lie algebras. In particular we show that the semi-direct
product of a Lie algebra equipped with a torsion-free flat connection
with itself is a mirror partner of a semi-direct product of the same
Lie algebra with its dual space. For nilpotent algebras this analysis on
Lie algebras can be applied to the compact quotients of the underlying
nilpotent group. We classify mirror pairs among 6-dimensional
nilpotent Lie algebras that have the semi-direct product structure as
well as mirror pairs admitting the more involved special Lagrangian
structure, namely compatible complex and symplectic structures on
the same space.