In this paper, we study gravitational symmetry algebras that live on 2-dimensional cuts S of asymptotic infinity. We define a notion of wedge algebra \(\mathcal {W}(S)\) that depends on the topology of S. For the cylinder \(S={\mathbb {C}}^*\) , we recover the celebrated \(Lw_{1+\infty }\) algebra. For the 2-sphere \(S^2\) , the wedge algebra reduces to a central extension of the anti-self-dual projection of the Poincaré algebra. We then extend \(\mathcal {W}(S)\) outside of the wedge space and build a new Lie algebra \(\mathcal {W}_\sigma (S)\) , which can be viewed as a deformation of the wedge algebra by a spin two field \(\sigma \) playing the role of the shear at a cut of . This algebra represents the gravitational symmetry algebra in the presence of a non-trivial shear and is characterized by a covariantized version of the wedge condition. Finally, we construct a dressing map that provides a Lie algebra isomorphism between the covariant and regular wedge algebras.
