In this work a systematic study of finite-dimensional nonrelativistic conformal groups is carried out under two complementary points of view. First, the conformal Killing equation is solved to obtain a whole family of finite-dimensional conformal algebras corresponding to each of the Galilei and Newton–Hooke kinematical groups. Some of their algebraic and geometrical properties are studied in a second step. Among the groups included in these families one can identify, for example, the contraction of the Minkowski conformal group, the analog for a nonrelativistic de Sitter space, or the nonextended Schrödinger group.

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Lie bialgebra contractions are introduced and classified. A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras so(p, q) starting from the one corresponding to so(N + 1). It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. This construction is explicitly given for the cases N = 2, 3, 4. All Lie bialgebra contractions studied in this paper define Hopf algebra contractions for the Drinfel'd-Jimbo deformations U z so(p, q). They are explicitly used to generate new non-semisimple quantum algebras as it is the case for the Euclidean, Poincaré and Galilean algebras.

A new quantum deformation, which we call null-plane, of the (3+1) Poincaré algebra is obtained. The algebraic properties of the classical null-plane description are generalized to this quantum deformation. In particular, the classical isotopy subalgebra of the null-plane is deformed into a Hopf subalgebra, and deformed spin operators having classical commutation rules can be defined. Quantum Hamiltonian, mass and position operators are studied, and the null-plane evolution is expressed in terms of a deformed Schrödinger equation.

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For chains of regular injections A p ⊂ A p−1 ⊂ . . . ⊂ A 1 ⊂ A 0 of Hopf algebras the sets of maximal extended Jordanian twists {F E k } are considered. We prove that under certain conditions there exists for A 0 the twist F B k≺0 composed by the factors F E k . The general construction of a chain of twists is applied to the universal envelopings U (g) of classical Lie algebras g . We study the chains for the infinite series A n , B n and D n . The properties of the deformation produced by a chain U (g) F B k≺0 are explicitly demonstrated for the case of g = so(9).

The finite-dimensional conformal groups associated with the Galilei and (oscillating or expanding) Newton–Hooke space–time manifolds was characterized by the present authors in a recent work. Three isomorphic group families, one for each nonrelativistic kinematics, were obtained, whose members are labeled by a half-integer number l. Since the action of these groups on their corresponding space–time manifolds is only local, a linearization is introduced here such that the corresponding action is well defined everywhere. In particular, the (l=1)-conformal cases that can be obtained by contraction from the well-known Minkowskian conformal group are treated in more detail. As an application of physical interest, the conformal invariance of the Galilean electromagnetism is studied. In order to achieve it, the pertinent local representations of the Galilean conformal algebras are derived.