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