  We study generalized complex structures and $T$-duality (in the sense of
Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the
corresponding Cavalcanti and Gualtieri map. Such a construction is called
"Infinitesimal $T$-duality". As an application we deal with the problem of
finding symplectic structures on 2-step nilpotent Lie algebras. We also give a
criteria for the intregability of the infinitesimal $T$-duality of Lie algebras
to topological $T$-duality of the associated nilmanifolds.
