Local Derivations on Conformal Galilei Algebras and Their Central Extensions

Abstract

We give a unified study of local derivations on conformal Galilei Lie algebras in three natural settings: the algebra without central extension, the mass central extension, and the exotic central extension. We first determine the structural form of the derivation algebras and isolate the relevant outer derivations. For the mass extension, we complete the one-string rigidity at the two low parameters not covered by the previous one-dimensional treatment, using direct calculations, and combine this with new spatial compatibility arguments to obtain the result in arbitrary spatial dimension. The exotic extension is treated by a self-contained argument from its defining relations. For the algebra without central extension we resolve the remaining algebraic-reflexivity problem completely. If $d\ge2$, every local derivation is again a derivation. In spatial dimension $d=1$, the same conclusion holds when $2\ell$ is even and also for $\ell=\frac12$; however, for odd $2\ell\ge3$ there is an additional $(2\ell-1)$-dimensional space of pure local derivations. This gives a sharp contrast with the mass central extension, where the central Heisenberg pairing removes precisely this extra local freedom. We also determine the Lie algebra structure of the full local-derivation space: in the exceptional one-dimensional non-central case it is an explicit semidirect product with an additional abelian irreducible ideal.