M. A. del Olmo

Nonrelativistic conformal groups

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 Schrodinger group.

Moyal quantization of 2+1‐dimensional Galilean systems

Stratonovich–Weyl kernels are constructed for some of the coadjoint orbits of the two‐dimensional extended Galilean group G(2+1). As an intermediate step, the unitary irreducible representations associated with a given group orbit are obtained by using the Kirillov–Mackey theory. Star products are defined, in the sense of Moyal, for functions on each of these orbits. The central extension of G(2+1) with parameter k is also analyzed, which results from the commutator between the generators of boosts, to conclude that it originates a sort of nonrelativistic remainder of the Thomas precession.

Nonrelativistic conformal groups. II. Further developments and physical applications

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.

Quantum structure of the motion groups of the two-dimensional Cayley-Klein geometries

A simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two-dimensional Cayley-Klein geometries. Their central extensions are also considered under this unified pattern. In both cases some fundamental properties characterizing the classical CK geometries (as the existence of a set of commuting involutions, contractions and dualities relationships), remain in the quantum version.

A new “null-plane” quantum Poincaré algebra

Abstract A new quantum deformation, which we call null-plate, of the (3+1) Poincare 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 Schrodinger equation.

Four‐dimensional quantum affine algebras and space–time q‐symmetries

A global model of the q deformation for the quasiorthogonal Lie algebras generating the groups of motions of the four‐dimensional affine Cayley–Klein (CK) geometries is obtained starting from the three‐dimensional deformations. It is shown how the main algebraic classical properties of the CK systems can be implemented in the quantum case. Quantum deformed versions either of the space–time or space symmetry algebras [Poincare (3+1), Galilei (3+1), 4‐D Euclidean as well as others] appear in this context as particular cases. For some of these classical algebras several q deformations are directly obtained.

Anyons, group theory and planar physics

Relativistic and nonrelativistic anyons are described in a unified formalism by means of the coadjoint orbits of the symmetry groups in the free case as well as when there is an interaction with a constant electromagnetic field. To deal with interactions we introduce the extended Poincare and Galilei Maxwell groups.

Cayley-Klein algebras as graded contractions of so(N+1)

We study Z2(X)N graded contractions of the real compact simple Lie algebra so(N+1), and we identify within them the Cayley-Klein algebras as a naturally distinguished subset.

Integrable systems based on SU(p,q) homogeneous manifolds

The general theory of the separation of variables in Hamilton–Jacobi and Laplace–Beltrami equations on the SU(p,q) hyperboloid is used to introduce completely integrable Hamiltonian systems on O(p,q) hyperboloids. Each of the q+1 different Cartan subalgebras of su(p,q) leads to a different integrable O(p,q) potential. Different complete sets of integrals of motion are obtained for each of the integrable systems.

Maximal Abelian Subalgebras of Pseudounitary Lie Algebras

Abstract The task of classifying and constructing all maximal Abelian subalgebras of su( p , q ) ( p ⩾ q ⩾1) is reduced to that of classifying orthogonally indecomposable MASAs. These are either maximal Abelian nilpotent subalgebras (represented by nilpotent matrices in any finite-dimensional representation), or for p = q they can be (nonorthogonally) decomposable and their study can be reduced to a construction of MANSs of sl( p , C ). Two types of MANSs of su( p , q ) are shown to exist (“one-rowed” and “non-one-rowed”). Numerous classification theorems are proven and applied to obtain all MASAs of su( p ,1), su( p ,2), and su( p , q ) with p + q ⩽6. Physical applications are discussed.