2-Local Derivations on Conformal Galilei Lie Algebras and Their Central Extensions

Abstract

We classify 2-local derivations on the finite-dimensional conformal Galilei Lie algebras in three natural settings: the algebra without central extension, the mass central extension, and the exotic central extension. No additivity or linearity is assumed in the definition of a 2-local derivation. We prove that in every case each 2-local derivation is an ordinary derivation. For the unextended algebra, the proof combines characteristic-quotient normalization with separating vectors, invariant bilinear forms, rank-two mixed zero anchors, and a semisimple-perturbation argument for the standard module. For the mass central extension with $2\ell\ge3$, the residual map is first reduced to the center and then eliminated by a symplectic-anchor construction on the Heisenberg radical; a dimension argument in the centralizer of a regular semisimple element simultaneously handles singular reductive actions and the additional central shear that occurs in spatial dimension two. The endpoint $\ell=\tfrac12$ is the Schrödinger algebra; a self-contained symplectic-anchor proof includes all spatial dimensions, with explicit treatment of dimensions one and two. For the exotic extension, after center normalization we use conjugate middle-weight positions and an affine middle anchor to eliminate every central translate, including the one-sided middle-weight configurations. Consequently the set of 2-local derivations coincides with the derivation algebra for all admissible parameters, and hence every 2-local derivation in these families is automatically linear.