Possible contractions of quantum orthogonal groups

Possible contractions of quantum orthogonal groups which correspond to different choices of primitive elements of Hopf algebra are considered and all allowed contractions in Cayley--Klein scheme are obtained. Quantum deformations of kinematical groups have been investigated and have shown that quantum analog of (complex) Galilei group G(1,3) do not exist in our scheme.Comment: 10 pages, Latex. Report given at XXIII Int. Colloquium on Group Theoretical Methods in Physics, July 31- August 5, 2000, Dubna (Russia

Single Boson Images Via an Extended Holstein Primakoff Mapping

The Holstein-Primakoff mapping for pairs of bosons is extended in order to accommodate single boson mapping. The proposed extension allows a variety of applications and especially puts the formalism at finite temperature on firm grounds. The new mapping is applied to the O(N+1) anharmonic oscillator with global symmetry broken down to O(N). It is explicitly demonstrated that N-Goldstone modes appear. This result generalizes the Holstein-Primakoff mapping for interacting boson as developed in ref.[1].Comment: 9 pages, LaTeX. Physical content unchanged. Unnecessary figure remove

The family of quaternionic quasi-unitary Lie algebras and their central extensions

The family of quaternionic quasi-unitary (or quaternionic unitary Cayley--Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N+1) and sp(p,q) in the Cartan series C_{N+1}, as well as many non-semisimple real Lie algebras which can be obtained from these simple algebras by particular contractions. The algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature. This common framework allows to perform the study of many properties for all these Lie algebras simultaneously. In this paper the central extensions for all quasi-simple Lie algebras of the quaternionic unitary Cayley--Klein family are completely determined in arbitrary dimension. It is shown that the second cohomology group is trivial for any Lie algebra of this family no matter of its dimension.Comment: 17 pages, LaTe

Quantization, group contraction and zero point energy

We study algebraic structures underlying 't Hooft's construction relating classical systems with the quantum harmonic oscillator. The role of group contraction is discussed. We propose the use of SU(1,1) for two reasons: because of the isomorphism between its representation Hilbert space and that of the harmonic oscillator and because zero point energy is implied by the representation structure. Finally, we also comment on the relation between dissipation and quantization.Comment: 6 pages, 3 figure

de Sitter group and Einstein-Hilbert Lagrangian

Axial vector torsion in the Einstein-Cartan space $U_{4}$ is considered here. By picking a particular term from the SO(4,1) Pontryagin density and then modifying it in a SO(3,1) invariant way, we get a Lagrangian density with Lagrange multipliers. Then considering torsion and torsion-less connection as independent fields, it has been found that $\kappa$ and $\lambda$ of Einstein-Hilbert Lagrangian, appear as integration constants in such a way that $\kappa$ has been found to be linked with the topological Nieh-Yan density of $U_{4}$ space.Comment: 14 page

Extensions, expansions, Lie algebra cohomology and enlarged superspaces

After briefly reviewing the methods that allow us to derive consistently new Lie (super)algebras from given ones, we consider enlarged superspaces and superalgebras, their relevance and some possible applications.Comment: 9 pages. Invited talk delivered at the EU RTN Workshop, Copenhagen, Sep. 15-19 and at the Argonne Workshop on Branes and Generalized Dynamics, Oct. 20-24, 2003. Only change: wrong number of a reference correcte

Graded contractions of bilinear invariant forms of Lie algebras

We introduce a new construction of bilinear invariant forms on Lie algebras, based on the method of graded contractions. The general method is described and the $\Bbb Z_2$-, $\Bbb Z_3$-, and $\Bbb Z_2\otimes\Bbb Z_2$-contractions are found. The results can be applied to all Lie algebras and superalgebras (finite or infinite dimensional) which admit the chosen gradings. We consider some examples: contractions of the Killing form, toroidal contractions of $su(3)$, and we briefly discuss the limit to new WZW actions.Comment: 15 page

Galilei invariant theories. I. Constructions of indecomposable finite-dimensional representations of the homogeneous Galilei group: directly and via contractions

All indecomposable finite-dimensional representations of the homogeneous Galilei group which when restricted to the rotation subgroup are decomposed to spin 0, 1/2 and 1 representations are constructed and classified. These representations are also obtained via contractions of the corresponding representations of the Lorentz group. Finally the obtained representations are used to derive a general Pauli anomalous interaction term and Darwin and spin-orbit couplings of a Galilean particle interacting with an external electric field.Comment: 23 pages, 2 table

Graded contractions and bicrossproduct structure of deformed inhomogeneous algebras

A family of deformed Hopf algebras corresponding to the classical maximal isometry algebras of zero-curvature N-dimensional spaces (the inhomogeneous algebras iso(p,q), p+q=N, as well as some of their contractions) are shown to have a bicrossproduct structure. This is done for both the algebra and, in a low-dimensional example, for the (dual) group aspects of the deformation.Comment: LaTeX file, 20 pages. Trivial changes. To appear in J. Phys.

Rotations associated with Lorentz boosts

It is possible to associate two angles with two successive non-collinear Lorentz boosts. If one boost is applied after the initial boost, the result is the final boost preceded by a rotation called the Wigner rotation. The other rotation is associated with Wigner's O(3)-like little group. These two angles are shown to be different. However, it is shown that the sum of these two rotation angles is equal to the angle between the initial and final boosts. This relation is studied for both low-speed and high-speed limits. Furthermore, it is noted that the two-by-two matrices which are under the responsibility of other branches of physics can be interpreted in terms of the transformations of the Lorentz group, or vice versa. Classical ray optics is mentioned as a case in point.Comment: LaTeX, 16 Pages, 4 epsfigure