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 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 $B$.