We study rigidity and stability of infinite dimensional algebras which are not subject to the HochschildSerre factorization theorem. In particular, we consider algebras appearing as asymptotic symmetries of three dimensional spacetimes, the BMS3, u(1) KacMoody and Virasoro algebras. We construct and classify the family of algebras which appear as deformations of BMS3, u(1) KacMoody 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 InÃ¶nÃ¼Wigner 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.
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