Three-Dimensional Higher-Order Schr\"odinger Algebras and Lie Algebra Expansions

We provide a Lie algebra expansion procedure to construct three-dimensional higher-order Schr\"odinger algebras which relies on a particular subalgebra of the four-dimensional relativistic conformal algebra. In particular, we reproduce the extended Schr\"odinger algebra and provide a new higher-order Schr\"odinger algebra. The structure of this new algebra leads to a discussion on the uniqueness of the higher-order non-relativistic algebras. Especially, we show that the recent d-dimensional symmetry algebra of an action principle for Newtonian gravity is not uniquely defined but can accommodate three discrete parameters. For a particular choice of these parameters, the Bargmann algebra becomes a subalgebra of that extended algebra which allows one to introduce a mass current in a Bargmann-invariant sense to the extended theory.