  We study the homology and cohomology groups of super Lie algebra of
supersymmetries and of super Poincare Lie algebra in various dimensions. We
give complete answers for (non-extended) supersymmetry in all dimensions $\leq
11$. For dimensions $D=10,11$ we describe also the cohomology of reduction of
supersymmetry Lie algebra to lower dimensions. Our methods can be applied to
extended supersymmetry algebra.
