Using the long exact sequence of nonabelian derived functors, an eight term exact sequence of Lie algebra homology with $\Lambda/q\Lambda$ coefficients is obtained, where $\Lambda$ is a ground ring and $q$ is a nonnegative integer. Hopf formulas for the second and third homology of a Lie algebra are proved. The condition for the existence and the description of the universal $q$-central relative extension of a Lie epimorphism in terms of relative homologies are given.
Keywords: Lie algebra, nonabelian derived functor, exact sequence, homology group.
2000 MSC: 18G10, 18G50.
Theory and Applications of Categories, Vol. 10, 2002, No. 4, pp 113-126.