SDiff(2) and uniqueness of the Plebański equation

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 Plebański equation and its geometry. We do not use Kähler or other additional structures but obtain the equation solely from the geometry of area preserving transformations group. We conclude that the Plebański equation is Lie remarkable.

References in corpus (4)

• Hyper-K{ä}hler Hierarchies and their twistor theory
• Partner symmetries and non-invariant solutions of four-dimensional heavenly equations
• Structure of Symmetry Groups via Cartan's Method: Survey of Four Approaches
• Method of group foliation, hodograph transformation and non-invariant solutions of the Boyer-Finley equation

Cited by in corpus (4)

• Integrable dispersionless PDE in 4D, their symmetry pseudogroups and deformations
• Ordinary differential equations described by their Lie symmetry algebra
• Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations
• Webs and the Plebański equation