On Classification of Deformations of the Gardner Bracket Over Local Infinitesimal Transformations

TL;DR

This paper investigates the classification of deformations of the Gardner bracket Lie algebra over local infinitesimal transformations, focusing on computing relevant cohomology groups to understand their structure.

Contribution

It provides a detailed analysis of the cohomology associated with the Gardner bracket, identifying key classes of deformations and advancing understanding of their algebraic properties.

Findings

Identified the cohomology groups relevant to the deformations.

Classified a significant class of deformations of the Gardner bracket.

Enhanced understanding of the algebraic structure of local functionals.

Abstract

The Lie algebra specified by space of local functionals with commutator determined by the Gardner bracket was under survey. Problem of classification of deformations of this bracket over local infinitesimal transformations of functionals was interpreted as a problem of computing the appropriate cohomology group of order 2. The most interesting class of deformations was investigated.