On deformations and extensions of $\text{Diff}(S^2)$

We investigate the algebra of vector fields on the sphere. First, we find that linear deformations of this algebra are obstructed under reasonable conditions. In particular, we show that $hs[\lambda]$, the one-parameter deformation of the algebra of area-preserving vector fields, does not extend to the entire algebra. Next, we study some non-central extensions through the embedding of $\mathfrak{v}\mathfrak{e}\mathfrak{c}\mathfrak{t}(S^2)$ into $\mathfrak{v}\mathfrak{e}\mathfrak{c}\mathfrak{t}(\mathbb{C}^*)$. For the latter, we discuss a three parameter family of non-central extensions which contains the symmetry algebra of asymptotically flat and asymptotically Friedmann spacetimes at future null infinity, admitting a simple free field realization.