Solutions of Klein--Gordon and Dirac equations on quantum Minkowski spaces

Covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified. In the case of quantum Minkowski spaces they have the same dimensions as in the classical case. Formal solutions of the corresponding Klein--Gordon and Dirac equations are found. The Fock space construction is sketched.Comment: 21 pages, LaTeX file, minor change

Twisted cyclic homology of all Podles quantum spheres

We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of all Podles quantum spheres relative to arbitary automorphisms. Our calculations are based on a free resolution due to Masuda, Nakagami and Watanabe. The dimension drop in Hochschild homology can be overcome by twisting by automorphisms induced from the canonical modular automorphism associated to the Haar state on quantum SU(2). We specialize our results to the standard quantum sphere, and identify the class in twisted cyclic cohomology of the 2-cocycle discovered by Schmuedgen and Wagner corresponding to the distinguished covariant differential calculus found by Podles.Comment: 17 pages, no figures, uses the amscd package. v6: final version accepted for publicatio

The coisotropic subgroup structure of SL_q(2,R)

We study the coisotropic subgroup structure of standard SL_q(2,R) and the corresponding embeddable quantum homogeneous spaces. While the subgroups S^1 and R_+ survive undeformed in the quantization as coalgebras, we show that R is deformed to a family of quantum coisotropic subgroups whose coalgebra can not be extended to an Hopf algebra. We explicitly describe the quantum homogeneous spaces and their double cosets.Comment: LaTex2e, 10pg, no figure

Fredholm Modules for Quantum Euclidean Spheres

The quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, \SO_q(N). The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres $S_q^{N-1}$. We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i. e., for N even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra $A(S^{N-1}_q)$.Comment: LaTeX, Euler package, a few improvements and added reference

On the Construction of Covariant Differential Calculi on Quantum Homogeneous Spaces

Let A be a coquasitriangular Hopf algebra and X the subalgebra of A generated by a row of a matrix corepresentation u or by a row of u and a row of the contragredient representation u^c. In the paper left-covariant first order differential calculi on the quantum group A are constructed and the corresponding induced calculi on the left quantum space X are described. The main tool for these constructions are the L-functionals associated with u. The results are applied to the quantum homogeneous space GL_q(N)/GL_q(N-1).Comment: 25 pages, Late

Geometry of Quantum Spheres

Spectral triples on the q-deformed spheres of dimension two and three are reviewed.Comment: 23 pages, revie

Twisted Configurations over Quantum Euclidean Spheres

We show that the relations which define the algebras of the quantum Euclidean planes \b{R}^N_q can be expressed in terms of projections provided that the unique central element, the radial distance from the origin, is fixed. The resulting reduced algebras without center are the quantum Euclidean spheres $S^{N-1}_q$. The projections $e=e^2=e^*$ are elements in \Mat_{2^n}(S^{N-1}_q), with N=2n+1 or N=2n, and can be regarded as defining modules of sections of q-generalizations of monopoles, instantons or more general twisted bundles over the spheres. We also give the algebraic definition of normal and cotangent bundles over the spheres in terms of canonically defined projections in \Mat_{N}(S^{N-1}_q).Comment: 14 pages, latex. Additional minor changes; final version for the journa

Quasitriangularity and enveloping algebras for inhomogeneous quantum groups

Coquasitriangular universal ${\cal R}$ matrices on quantum Lorentz and quantum Poincar\'e groups are classified. The results extend (under certain assumptions) to inhomogeneous quantum groups of [10]. Enveloping algebras on those objects are described.Comment: 18 pages, LaTeX file, minor change

The Dirac operator and gamma matrices for quantum Minkowski spaces

Gamma matrices for quantum Minkowski spaces are found. The invariance of the corresponding Dirac operator is proven. We introduce momenta for spin 1/2 particles and get (in certain cases) formal solutions of the Dirac equation.Comment: 25 pages, LaTeX fil

Global Symmetries of Noncommutative Space-time

The global counterpart of infinitesimal symmetries of noncommutative space-time is discussed.Comment: 7 pages, no figures; minor changes in the bibliography; final version accepted for publication in Phys. Rev.