Deformations of local Lie algebras and an analog of Carles' algebraicity theorem (Q1907180)

Carles' algebraicity theorem [\textit{R. Carles}, Ann. Inst. Fourier 34, No. 3, 65-82 (1984; Zbl 0544.17004)] states that a rigid Lie algebra must be algebraic. In this article, we show that the classical (and geometrical) theory of deformations can be extended to local Lie algebras. By using a suitable definition of the cohomology, we recover all the classical criteria of rigidity linked with cohomology groups. Furthermore, we prove an analog of Carles' characterization and construct deformations coming from central extensions.