(In)finite extensions of algebras from their Inonu-Wigner contractions

The method to obtain massive non-relativistic states from the Poincare algebra is twofold. First, following. Inonu and Wigner, the Poincare algebra has to be contracted to the Galilean one. Second, the Galilean algebra has to be extended to include the central mass operator. We show that the central extension might be properly encoded in the non-relativistic contraction. In fact, any. Inonu-Wigner contraction of one algebra to another corresponds to an infinite tower of Abelian extensions of the latter. The proposed method is straightforward and holds for both central and non-central extensions. Apart from the Bargmann (non-zeromass) extension of the Galilean algebra, our list of examples includes the Weyl algebra obtained from an extension of the contracted SO(3) algebra, the Carrollian (ultrarelativistic) contraction of the Poincare algebra, the exotic Newton-Hooke algebra and some others.