A Program for Computing the Cohomology of Lie Superalgebras of Vector Fields

Abstract

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. At present, since the required algebraic computations are very tedious, the cohomology is explicitly computed only in a few cases for different classes of Lie (super)algebras. That is why application of computer algebra methods is important for this problem. We describe an algorithm (and its C implementation) for computing the cohomology of Lie algebras and superalgebras. In elaborating the algorithm, we focused mainly on the cohomology with coefficients in trivial, adjoint, and coadjoint modules for Lie (super)algebras of the formal vector fields. These algebras have many applications to modern supersymmetric models of theoretical and mathematical physics. As an example, we consider the cohomology of the Poisson algebra Po(2) with coefficients in the trivial module and present 3- and 5-cocycles found by a computer. Bibliography: 6 titles.