SDiff(2) and uniqueness of the Pleba\'{n}ski equation

TL;DR

This paper derives the second Plebański equation from the geometry of area-preserving diffeomorphisms, showing it is uniquely characterized by this geometric structure without additional assumptions.

Contribution

It introduces a novel derivation of the Plebański equation based solely on the geometry of the area-preserving diffeomorphism group, emphasizing its Lie remarkable property.

Findings

Derivation of the Plebański equation from Lie algebra extensions and cohomology.

The Plebański equation is shown to be uniquely determined by the geometry of area-preserving transformations.

The approach does not rely on Kähler or other additional geometric structures.

Abstract

The group of area preserving diffeomorphisms showed importance in the problems of self-dual gravity and integrability theory. We discuss how representations of this infinite-dimensional Lie group can arise in mathematical physics from pure local considerations. Then using Lie algebra extensions and cohomology we derive the second Pleba\'{n}ski equation and its geometry. We do not use K\"ahler or other additional structures but obtain the equation solely from the geometry of area preserving transformations group. We conclude that the Pleba\'{n}ski equation is Lie remarkable.