Non-relativistic conformal symmetries and Newton-Cartan structures

LaTeX, 47 pages. Bibliographical improvements. To appear in J. Phys. AInternational audienceThis article provides us with a unifying clas\-sification of the conformal infini\-tesimal sym\-metries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational ``dyna\-mical exponent'', $z$. The Schrödinger-Virasoro algebra of Henkel et al. corresponds to $z=2$. Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schrödinger Lie algebra, for which~$z=2$. For lightlike geodesics, they yield, in turn, the Conformal Galilean Algebra (CGA) and Lukierski, Stichel and Zakrzewski [alias ``$\alt$" of Henkel], with $z=1$. Physical systems realizing these symmetries include, e.g., classical systems of massive, and massless non-relativistic particles, and also hydrodynamics, as well as Galilean electromagnetism