On the homology of graded Lie algebras

We consider the homology of finite-dimensional-graded Lie algebras with coefficients in a finite-dimensional-graded module. By a combinatorial approach we give a lower bound for their total homology. Our result extends a result of Deninger and Singhof for the case of trivial coefficients. Applications for 2-step and free nilpotent Lie algebras are given.