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

TL;DR

The paper tackles the high combinatorial complexity of computing Lie algebra cohomology by introducing a partitioning strategy that splits the full cochain complex into minimal subcomplexes, implemented in the LieCohomology program. It applies this method to the Hamiltonian vector field algebra $H(2|0)$ and its central extension $Po(2|0)$, achieving substantial inefficiency reductions over the naive approach and uncovering new cohomology classes. The results include detailed cohomology data up to grade $8$, with several 1-dimensional classes such as $H^7_8$ and $H^{10}_6$, and a discussion of subcomplex structure and multiplicative relations. The work demonstrates practical feasibility for higher-grade computations and suggests finite-field strategies as a route to further extend computational reach, while noting that arithmetic over $\mathbb{Q}$ dominates runtime.

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)).