  We study the connection of Moyal deformations of self-dual gravity and
self-dual Yang-Mills theory to chiral higher-spin theories, and also to
deformations of operator algebras in celestial holography. The relation to
Moyal deformations illuminates various aspects of the structure of chiral
higher-spin theories. For instance, the appearance of the self-dual kinematic
algebra in all the theories considered here leads via the double copy to
vanishing tree-level scattering amplitudes. Regarding celestial holography, the
Moyal deformation of self-dual gravity was recently shown to lead to the loop
algebra of $W_{\wedge}$, and we obtain here the analogous deformation of a
Kac-Moody algebra corresponding to Moyal-deformed self-dual Yang-Mills theory.
We also introduce the celestial algebras for various chiral higher-spin
theories.
