I will show that self-dual gravity in Euclidean four-dimensional Anti-de Sitter space (AdS4 ) can be described by a minimally coupled scalar field with a cubic interaction written in terms of a deformed Poisson bracket, providing a remarkably simple generalisation of the Plebanski action for self-dual gravity in flat space. This implies a novel symmetry algebra in self-dual gravity, notably an AdS4 version of the so-called kinematic algebra. This provides a concrete starting point for defining the double copy for Einstein gravity in AdS4 by expanding around the self-dual sector. Moreover, I will show that the new kinematic Lie algebra can be lifted to a deformed version of the w1+∞ algebra, which plays a prominent role in celestial holography.

The idea of writing supergravity as a double copy of super Yang-Mills theories has proved a fruitful one, most notably in the context of scattering amplitudes. I will explore this idea of "squaring" at a fundamental level to explain how the symmetries of the former(both local and global) can be written as a double copy of the latter. I will show how the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry are obtained from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincare. We give a gravity-gauge dictionary by convoluting fields and parameters. At the global level, I will explain how the coset groups of supergravity can be derived from the global R-symmetries of the SYM multiplets, via the 4 division algebras:reals, complexes, quaternions and octonions. Finally I will explore some intriguing possible applications of our dictonary.