A Program for computing cohomologies of Lie superalgebras of vector fields

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. At present, because of the need for very tedious algebraic computation, the explicitly computed cohomology for different classes of Lie (super)algebras is known only in a few cases. That is why the application of computer algebra methods is important for this problem. We describe here an algorithm and its C implementation for computing the cohomology of Lie algebras and superalgebras. When elaborating the algorithm we paid primary attention to cohomology in trivial, adjoint and coadjoint modules for Lie algebras and superalgebras of the formal vector fields. These algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics. As an example, we present 3- and 5-cocycles from the cohomology in the trivial module for the Poisson algebra $Po(2)$, as found by computer.