Acceleration-enlarged symmetries in nonrelativistic space-time with a cosmological constant

Abstract

By considering the nonrelativistic limit of de-Sitter geometry one obtains the nonrelativistic space-time with a cosmological constant and Newton-Hooke (NH) symmetries. We show that the NH symmetry algebra can be enlarged by the addition of the constant acceleration generators and endowed with central extensions (one in any dimension (D) and three in D=(2+1)). We present a classical Lagrangian and Hamiltonian framework for constructing models quasi-invariant under enlarged NH symmetries which depend on three parameters * Supported by the KBN grant 1P03B01828 1 described by three nonvanishing central charges. The Hamiltonian dynamics then splits into external and internal sectors with new noncommutative structures of external and internal phase spaces. We show that in the limit of vanishing cosmological constant the system reduces to the one presented in [1] which possesses accelarationenlarged Galilean symmetries.

Conformal mechanics in Newton–Hooke spacetime

“…Whereas in arbitrary dimension the Newton-Hooke algebra admits only one central charge, in (2 + 1)-dimensions the second central charge is allowed which characterizes the exotic Newton-Hooke symmetry [38,41,42,43]. Extensions of the Newton-Hooke algebra by extra vector generators and their dynamical realizations were discussed recently in [44,45,46]. In the context of non-relativistic strings and branes generalizations of the Newton-Hooke algebra were studied in [19,47,48,49].…”

Conformal Galilei groups, Veronese curves and Newton–Hooke spacetimes

“…., a conformal Newton-Cartan structure together with a distinguished, flat, normal Cartan connection associated with CGal N (d). 5 The general construction of the normal Cartan connection associated with a Schrödingerconformal Newton-Cartan structure has been performed in [21]. , guarantee having zero torsion.…”

Twist Deformation of Doubly Enlarged Newton–hooke Hopf Algebra

“…with u · = f (1) S 0 (f (2) ) (we use Sweedler's notation F · = f (1) ⊗f (2) ). Besides, it should be noted, that the twist factor F · ∈ U · ( NH ± ) ⊗ U · ( NH ± ) satisfies the classical cocycle condition…”

“…It should be noted, that the Hopf structure (1), (2) is the largest known explicitly symmetry (quantum) group at nonrelativistic level. By its different contraction schemes we get respectively: i) For R i → 0 -the acceleration-enlarged Newton-Hooke Hopf algebra U 0 ( NH ± ) proposed in [1], [2], ii) In the case of R i and F i generators approaching zero -the (usual) Newton-Hooke quantum group U 0 (NH ± ) [3], iii) For R i → 0 and τ → ∞ -the acceleration-enlarged Galilei Hopf structure U 0 ( G) provided in [4], iv) For both R i and F i operators approaching zero as well as for parameter τ running to infinity -the (usual) Galilei quantum group U 0 (G), and, finally v) In the case of τ → ∞ -the (new) doubly enlarged Galilei Hopf algebra U 0 ( G) .…”