Computation of Cohomology of Lie Superalgebras of Vector Fields

TL;DR

This paper addresses the computational bottleneck in Lie (super)algebra cohomology by introducing KornProg, a C implementation capable of computing H^k(A;M) for finite-dimensional and certain infinite-dimensional graded Lie superalgebras of vector fields. The method leverages internal grading to reduce problems to grade zero and solves cohomology equations by constructing cochains, coboundaries, and performing linear-algebraic reductions to obtain Z^k/B^k, with separate treatment of even and odd components. The author reports explicit computations for Buttin-related algebras B(1), Le(1), M(1) and graded variants SB(1), SLe(1), SM(1), including concrete cocycles such as a^3, a^2, and a^1_-1, and discusses central extensions and BV-formalism relevance. The work demonstrates the feasibility of nontrivial cohomology computations in supersymmetric contexts and provides a foundation for exploring deformation theory and cohomology ring structures in mathematical physics.

Abstract

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. It carries most fundamental ("topological") information about algebra under consideration. 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 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. The program can proceed finite-dimensional algebras and infinite-dimensional graded algebras with finite-dimensional homogeneous components. Among the last algebras Lie algebras and superalgebras of formal vector fields are most important. We present some results of computation of cohomology for Lie superalgebras of Buttin vector fields and related algebras. These algebras being super-analogs of Poisson and Hamiltonian algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics.