Computation of Cohomology of Lie Algebra of Hamiltonian Vector Fields by Splitting Cochain Complex into Minimal Subcomplexes

TL;DR

This paper introduces an efficient algorithm for computing the cohomology of Lie algebras, demonstrated on Hamiltonian vector fields, revealing new cohomological classes and outperforming traditional methods.

Contribution

The paper presents a novel algorithm that partitions cochain complexes into minimal subcomplexes, significantly improving computational efficiency for Lie algebra cohomology.

Findings

The new algorithm is more efficient than traditional methods.

Application to H(2|0) reveals new cohomological classes.

The approach is effective for Hamiltonian and Poisson algebras.

Abstract

Computation of homology or cohomology is intrinsically a problem of high combinatorial complexity. Recently we proposed a new efficient algorithm for computing cohomologies of Lie algebras and superalgebras. This algorithm is based on partition of the full cochain complex into minimal subcomplexes. The algorithm was implemented as a C program LieCohomology. In this paper we present results of applying the program LieCohomology to the algebra of hamiltonian vector fields H(2|0). We demonstrate that the new approach is much more efficient comparing with the straightforward one. In particular, our computation reveals some new cohomological classes for the algebra H(2|0) (and also for the Poisson algebra Po(2|0)).