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

TL;DR

This paper explores the algebraic structure of vector fields on the sphere, demonstrating the rigidity of linear deformations and examining non-central extensions relevant to spacetime symmetries.

Contribution

It shows linear deformations of the sphere's vector field algebra are obstructed and introduces a three-parameter family of non-central extensions linked to spacetime symmetries.

Findings

Linear deformations are obstructed under reasonable conditions.

The algebra $hs[bb]b4$ does not extend to the entire algebra.

A three-parameter family of non-central extensions is constructed.

Abstract

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.