Cohomological rigidity of conformal Galilei algebras and their central extensions

In this paper, we study the second adjoint cohomology of the compexification of the real conformal Galilei algebras \(\mathfrak{cga}_\ell(d,\mathbb{R})\) and their central extensions. These algebras are non-semisimple Lie algebras that appear as non-relativistic analogues of conformal Lie algebras. Using cohomological methods, including the Hochschild-Serre factorization theorem, we compute the space \(H^2(\mathfrak{g}, \mathfrak{g})\) and examine conditions under which these algebras are cohomologically rigid. Our main result shows that the conformal Galilei algebras are rigid for all spatial dimensions \(d \neq 2\) when the spin \(\ell\) is a half-integer. This provides a complete characterization of the formal rigidity of these algebras in the specified parameter range.