Laboratory of Information Technologies,

Joint Institute for Nuclear Research,

141980 Dubna,

RUSSIA

E-mail: kornyak@jinr.ru

**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.