Computer Algebra Study of Cohomology of Lie (Super)algebras

Abstract:
Cohomology and its mirror variant homology play a central role in the algebraic topology and its numerous applications. Unfortunately computation of cohomology is a problem of intrinsically high complexity. Recently we developed a new algorithm for computing cohomology based on combination of two ideas: 1) splitting the whole cochain complex into minimal subcomplexes and 2) modular search within these subcomplexes. This algorithm increases considerably the efficiency of computation. It can be applied to computation of cohomology and homology of different nature. Writing the C implementation of the algorithm we concentrate here our efforts on cohomology of Lie algebras and superalgebras. In this report, we explain main features of the algorithm and present new results on the structure of cohomology of the restricted Lie algebra of Hamiltonian vector fields. We reveal these results first with the help of the C program and then prove them rigorously. We present also results of application of the program for computing cohomology of some Lie superalgebras of vector fields preserving odd-symplectic (periplectic) structures. These algebras --- known in physics as Lie superalgebras of vector fields with antibrackets --- play an important role in the Batalin--Vilkovisky formalism for quantizing gauge fields.