  We study rigidity and stability of infinite dimensional algebras which are
not subject to the Hochschild-Serre factorization theorem. In particular, we
consider algebras appearing as asymptotic symmetries of three dimensional
spacetimes, the BMS3, u(1) Kac-Moody and Virasoro algebras. We construct and
classify the family of algebras which appear as deformations of BMS3, u(1)
Kac-Moody and their central extensions by direct computations and also by
cohomological analysis. The Virasoro algebra appears as a specific member in
this family of rigid algebras; for this case stabilization procedure is inverse
of the In\"on\"u-Wigner contraction relating Virasoro to BMS3 algebra. We
comment on the physical meaning of deformation and stabilization of these
algebras and relevance of the family of rigid algebras we obtain
