M. Tarlini

Three‐dimensional quantum groups from contractions of SU(2)q

Contractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one‐dimensional Heisenberg H(1)q and the two‐dimensional Euclidean quantum group E(2)q obtained by contracting SU(2)q.

The quantum Heisenberg group H(1)q

The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R‐matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.

The three‐dimensional Euclidean quantum group E(3)q and its R‐matrix

A contraction procedure starting from SO(4)q is used to determine the quantum analog E(3)q of the three‐dimensional Euclidean group and the structure of its representations. A detailed analysis of the contraction of the R‐matrix is then performed and its explicit expression has been found. The classical limit of R is shown to produce an integrable dynamical system. By means of the R‐matrix the pseudogroup of the noncommutative representative functions is considered. It will finally be shown that a further contraction made on E(3)q produces the two‐dimensional Galilei quantum group and this, in turn, can be used to give a new realization of E(3)q and E(2,1)q.

Onq-tensor operators for quantum groups

The fundamental theorem for tensor operators in quantum groups is proved using an appropriate definition forq-tensor operators. An example is discussed based on theq-boson realization of SUq(2).

Noncommutative Instantons on the 4-Sphere¶from Quantum Groups

Abstract: We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle ?7→?4 making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson–Lie structure of U(4) shows that the diagonal SU(2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SUq(2); it determines a new deformation of the 4-sphere ∑4q as the algebra of coinvariants in ?q7. We show that the quantum vector bundle associated to the fundamental corepresentation of SUq(2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of ∑4q, we define two 0-summable Fredholm modules and we compute the Chern–Connes pairing between the projector and their characters. It comes out that even the zero class in cyclic homology is non-trivial.

The Dirac-Coulomb problem for the κ-Poincaré quantum group

Abstract The recently introduced κ-Poincare-Dirac equation is gauged to treat the κ-Dirac-Coulomb problem. For the resulting equation, we prove that the perturbation to first order in the quantum group parameter vanishes identically. The second order perturbation is singular, but assuming a heuristic cut-off allows a qualitative estimate of the quantum group parameter.

Bijectivity of the canonical map for the non-commutative instanton bundle

Abstract It is shown that the quantum instanton bundle introduced in [Commun. Math. Phys. 226 (2002) 419] has a bijective canonical map and is, therefore, a coalgebra Galois extension.

Freeq-Schrödinger 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 hyperboloidqH and the quantum planeqP are determined as homogeneous spaces ofFq(E(2)). The canonical action ofEq(2) is used to define a naturalq-analog of the free Schrödinger equation, that is studied in the momentum and angular momentum bases. In the first case the eigenfunctions are factorized in terms of products of twoq-exponentials. In the second case we determine the eigenstates of the unitary representation, which, in theqP case, are given in terms of Hahn-Exton functions. Introducing the universalT-matrix forEq(2) we prove that the Hahn-Exton as well as Jacksonq-Bessel functions are also obtained as matrix elements ofT, thus giving the correct extension to quantum groups of well known methods in harmonic analysis.

Exponential mapping for non-semisimple quantum groups

The concept of a universal T matrix, introduced by Fronsdal and Galindo (1993) in the framework of quantum groups, is discussed here as a generalization of the exponential mapping. New examples related to inhomogeneous quantum groups of physical interest are developed, the duality calculations are explicitly presented and it is found that in some cases the universal T matrix, as for Lie groups, is expressed in terms of usual exponential series.

An R-matrix approach to the quantization of the Euclidean group E(2)

The R-matrices for two different deformations of the Euclidean group E(2), calculated in a two-dimensional representation, are used to determine the deformed Hopf algebra of the representative functions. The duality of the latter with the initial quantum algebras is explicitly proved and the relationship between the two quantum groups is discussed and clarified.