SDiff(2) and uniqueness of the Pleba\'nski 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 Pleba\'nski 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\'nski equation is Lie remarkable.
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