Deformations of Dolbeault cohomology classes for Lie algebra with complex structures

In this paper, we study deformations of complex structures on Lie algebras and its associated Dolbeault cohomology classes. A complete deformation is constructed in a way similar to the Kuranishi family. The extension isomorphism shown be valid case. As an application, prove that given family left invariant $\{M_t\}_{t\in B}$ compact manifold $M=(\Gamma\setminus G, J)$ where $G$ group, $\Gamma$ sublattice $J$ structure, set all $t\in B$ such $M_t$ may computed by tensor fields analytic open subset $B$.