Dressing operator approach to Moyal algebraic deformation of selfdual gravity

TL;DR

This paper extends the dressing operator method from soliton theory to a Moyal algebraic deformation of selfdual gravity, demonstrating its integrability and algebraic structure.

Contribution

It introduces a novel dressing operator approach for Moyal-deformed selfdual gravity, linking it to loop algebras and integrability.

Findings

Dressing operators satisfy a factorization relation similar to KP and Toda hierarchies.

The algebraic structure is a loop algebra of the Moyal (or star product) algebra.

The nonlinear problem becomes linearized on the loop group, confirming integrability.

Abstract

Recently Strachan introduced a Moyal algebraic deformation of selfdual gravity, replacing a Poisson bracket of the Plebanski equation by a Moyal bracket. The dressing operator method in soliton theory can be extended to this Moyal algebraic deformation of selfdual gravity. Dressing operators are defined as Laurent series with coefficients in the Moyal (or star product) algebra, and turn out to satisfy a factorization relation similar to the case of the KP and Toda hierarchies. It is a loop algebra of the Moyal algebra (i.e., of a $W_\infty$ algebra) and an associated loop group that characterize this factorization relation. The nonlinear problem is linearized on this loop group and turns out to be integrable.