Free q-Schrodinger Equation from Homogeneous Spaces of the 2-dim Euclidean Quantum Group

After a preliminary review of the definition and the general properties of the homogeneous spaces of quantum groups, the quantum hyperboloid qH and the quantum plane qP are determined as homogeneous spaces of Fq(E(2)). The canonical action of Eq(2) is used to define a natural q-analog of the free Schro"dinger equation, that is studied in the momentum and angular momentum bases. In the first case the eigenfunctions are factorized in terms of products of two q-exponentials. In the second case we determine the eigenstates of the unitary representation, which, in the qP case, are given in terms of Hahn-Exton functions. Introducing the universal T-matrix for Eq(2) we prove that the Hahn-Exton as well as Jackson q-Bessel functions are also obtained as matrix elements of T, thus giving the correct extension to quantum groups of well known methods in harmonic analysis.Comment: 19 pages, plain tex, revised version with added materia

Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2)SU(2) and SO(3)SO(3) groups

We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spaces of quantum SU(2) and SO(3) groups.Comment: 30 pages (with very slight changes

Differential and Twistor Geometry of the Quantum Hopf Fibration

We study a quantum version of the SU(2) Hopf fibration $S^7 \to S^4$ and its associated twistor geometry. Our quantum sphere $S^7_q$ arises as the unit sphere inside a q-deformed quaternion space $\mathbb{H}^2_q$. The resulting four-sphere $S^4_q$ is a quantum analogue of the quaternionic projective space $\mathbb{HP}^1$. The quantum fibration is endowed with compatible non-universal differential calculi. By investigating the quantum symmetries of the fibration, we obtain the geometry of the corresponding twistor space $\mathbb{CP}^3_q$ and use it to study a system of anti-self-duality equations on $S^4_q$, for which we find an `instanton' solution coming from the natural projection defining the tautological bundle over $S^4_q$.Comment: v2: 38 pages; completely rewritten. The crucial difference with respect to the first version is that in the present one the quantum four-sphere, the base space of the fibration, is NOT a quantum homogeneous space. This has important consequences and led to very drastic changes to the paper. To appear in CM

Finitely-Generated Projective Modules over the Theta-deformed 4-sphere

We investigate the "theta-deformed spheres" C(S^{3}_{theta}) and C(S^{4}_{theta}), where theta is any real number. We show that all finitely-generated projective modules over C(S^{3}_{theta}) are free, and that C(S^{4}_{theta}) has the cancellation property. We classify and construct all finitely-generated projective modules over C(S^{4}_{\theta}) up to isomorphism. An interesting feature is that if theta is irrational then there are nontrivial "rank-1" modules over C(S^{4}_{\theta}). In that case, every finitely-generated projective module over C(S^{4}_{\theta}) is a sum of a rank-1 module and a free module. If theta is rational, the situation mirrors that for the commutative case theta=0.Comment: 34 page

Wilson function transforms related to Racah coefficients

The irreducible $*$-representations of the Lie algebra $su(1,1)$ consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch-Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for U_q(\su(1,1)), which turn out to be Askey-Wilson functions and Askey-Wilson polynomials.Comment: 48 page

Noncommutative Spheres and Instantons

We report on some recent work on deformation of spaces, notably deformation of spheres, describing two classes of examples. The first class of examples consists of noncommutative manifolds associated with the so called $\theta$-deformations which were introduced out of a simple analysis in terms of cycles in the $(b,B)$-complex of cyclic homology. These examples have non-trivial global features and can be endowed with a structure of noncommutative manifolds, in terms of a spectral triple (\ca, \ch, D). In particular, noncommutative spheres $S^{N}_{\theta}$ are isospectral deformations of usual spherical geometries. For the corresponding spectral triple (\cinf(S^{N}_\theta), \ch, D), both the Hilbert space of spinors \ch= L^2(S^{N},\cs) and the Dirac operator $D$ are the usual ones on the commutative $N$-dimensional sphere $S^{N}$ and only the algebra and its action on $\ch$ are deformed. The second class of examples is made of the so called quantum spheres $S^{N}_q$ which are homogeneous spaces of quantum orthogonal and quantum unitary groups. For these spheres, there is a complete description of $K$-theory, in terms of nontrivial self-adjoint idempotents (projections) and unitaries, and of the $K$-homology, in term of nontrivial Fredholm modules, as well as of the corresponding Chern characters in cyclic homology and cohomology.Comment: Minor changes, list of references expanded and updated. These notes are based on invited lectures given at the ``International Workshop on Quantum Field Theory and Noncommutative Geometry'', November 26-30 2002, Tohoku University, Sendai, Japan. To be published in the workshop proceedings by Springer-Verlag as Lecture Notes in Physic

Coherent States for Quantum Compact Groups

Coherent states are introduced and their properties are discussed for all simple quantum compact groups. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit and interpret the coherent state as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compact R--matrix formulation (generalizing this way the $q$--deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel--Weil construction) are described using the concept of coherent state. The relation between representation theory and non--commutative differential geometry is suggested.}Comment: 25 page