The cohomology groups of Lie superalgebras and, more generally, of color Lie
algebras, are introduced and investigated. The main emphasis is on the case
where the module of coefficients is non-trivial. Two general propositions are
proved, which help to calculate the cohomology groups. Several examples are
included to show the peculiarities of the super case. For L = sl(1|2), the
cohomology groups H^1(L,V) and H^2(L,V), with V a finite-dimensional simple
graded L-module, are determined, and the result is used to show that
H^2(L,U(L)) (with U(L) the enveloping algebra of L) is trivial. This implies
that the superalgebra U(L) does not admit of any non-trivial formal
deformations (in the sense of Gerstenhaber). Garland's theory of universal
central extensions of Lie algebras is generalized to the case of color Lie
algebras.Comment: 50 pages, Latex, no figures. In the revised version the proof of
Lemma 5.1 is greatly simplified, some references are added, and a pertinent
result on sl(m|1) is announced. To appear in the Journal of Mathematical
Physic