(In)finite extensions of algebras from their İnönü–Wigner contractions

Abstract

The method to obtain massive non-relativistic states from the Poincaré algebra is twofold. First, following İnönü and Wigner, the Poincaré 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 İnönü–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-zero mass) 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 Poincaré algebra, the exotic Newton–Hooke algebra and some others. This paper is dedicated to the memory of Laurent Houart (1967–2011).

Relating non-relativistic string theories

“…and also the recent applications[34,35,68] 19. This is analogous to the way in which the Bargmann algebra can be obtained from the Poincaré algebra using Lie algebra expansion[67], where in that case one decomposes with only one longitudinal direction (i.e. time) and the transverse ones being the spatial directions.…”

Infinite S-expansion with ideal subtraction and some applications

“…Moreover, it gives further connections between the contraction processes and the expansion methods, which was an open question already mentioned in Ref. [58].…”

Three-dimensional (higher-spin) gravities with extended Schrödinger and l-conformal Galilean symmetries

“…These issues will be considered elsewhere. [11][12][13][14][15][16][17][18][19][20][21][22]2019) and at the School of Physics and Astrophysics, University of Western Australia where part of this work was done. Work of D.C. was supported by the Russian Science Foundation, grant No 19-11-00005.…”

“…2 It might be of interest to see whether the extended Schrödinger algebra (9) with the addition of (18) can be alternatively viewed as a certain algebra expansion, a technique considered e.g. in [17,18,19] and references there in. 3 Note that, instead of the contraction of su(1, 2) ⊕ su(1, 2) we could also consider the contraction of sl(3, R) ⊕ sl(3, R) by simply assuming that the vector indices i, j in (21) be transformed under the SO(1, 1) group instead of SO(2) (see Appendix B).…”