Bicovariant differential calculus on quantum groupsSU q (N) andSO q (N)

Following Woronowicz's proposal the bicovariant differential calculus on the quantum groupsSU _q (N) andSO _q (N) is constructed. A systematic construction of bicovariant bimodules by using the matrix is presented. The relation between the Hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules, and theq-deformed universal enveloping algebras is given. Imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined. As an application the Maurer-Cartan equations and theq-analogue of the structure constants are formulated.Address after 1 Dec. 1990, Institute of Theoretical Physics, University of München.     

Remarks on bicovariant differential calculi and exterior Hopf algebras

We show that every bicovariant differential calculus over the quantum groupA defines a bialgebra structure on its exterior algebra. Conversely, every exterior bialgebra ofA defines bicovariant bimodule overA. We also study a quasitriangular structure on exterior Hopf algebras in some detail.     

Levi-Civita Connections on the Quantum Groups SLq(N), Oq(N) and Spq(N)

For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, Levi-Civita connections) are introduced and studied. It is proved that for the bicovariant differential calculi on SL _q (N), O _q (N) and Sp _q (N) from the classification in [18] there exist unique Levi-Civita connections. Received: Received: 28 February 1996 / Accepted: 1 October 1996     

Bicovariant differential calculi on SL q(N) and Sp q(N)

For transcendental values of the deformation parameter q all bicovariant first order differential calculi on the Hopf algebras are classified.     

Quantum double and differential calculi

We show that bicovariant bimodules as defined by Woronowicz are in one-to-one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is connected to the existence of a particular (n+1)-dimensional representation of the double. An example of bicovariant differential calculus on the nonquasitriangular quantum group E _q (2) is developed. The construction is studied in terms of Hochschild cohomology and a correspondence between differential calculi and 1-cocycles is proved. Some differences of calculi on quantum and finite groups with respect to Lie groups are stressed.     

Bicovariant differential calculus on quantum groups and wave mechanics

The bicovariant differential calculus on quantum groups being defined by Woronowicz and later worked out explicitly by Carow-Watamura et at. and Juro for the real quantum groupsSU _q (N) andSO _q (N) through a systematic construction of the bicovariant bimodules of these quantum groups is reviewed forSU _q (2) andSO _q (N). The resulting vector fields build representations of the quantized universal enveloping algebras acting as covariant differential operators on the quantum groups and their associated quantum spaces. As an application a free particle stationary wave equation on quantum space is formulated and solved in terms of a complete set of energy eigenfunctions.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.     

Classification of bicovariant differential calculi on quantum groups

Suppose thatq is not a root of unity. We classify all bicovariant differential calculi of dimension greater than one on the quantum groupsGL _q (N),O _q (N) andSp _q (N) for which the differentials du _j ⁱ of the matrix entriesu _j ⁱ generate the left module of first order forms. Our first classification theorem asserts that there are precisely two one-parameter families of such calculi onGL _q (N) forN3. In the limitq1 only two of these calculi give the ordinary differential calculus onGL(N). Our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties onO _q (N) andSp _q (N) forN4. This strengthens the corresponding result proved in our previous paper [SS2]. There are four such calculi onO _q (3). We introduce two new 4-dimensional bicovariant differential calculi onO _q (3).     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Differential calculus on the inhomogeneous quantum groups IGL q (n)

We obtain the inhomogeneousq-groups IGL _q (n) via a projection from GL _q (n + 1). The bicovariant differential calculus of IGL _q (n) is constructed, and the corresponding quantum Lie algebra is given explicitly.     

Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity

We present a general method to deform the inhomogeneous algebras of theB _n,C_n,D_n type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromB _n+1,C_n+1,D_n+1. For example we obtain the (bicovariant) inhomogeneousq-algebraISO _q(N) as a consistent projection of the (bicovariant)q-algebraSO _q(N=2). This projection works for particular multiparametric deformations ofSO(N+2), the so-called minimal deformations. The case ofISO _q(4) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameterq. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains theclassical Lorentz algebra. Only the commutation relations involving the momenta depend onq. Finally, we discuss aq-deformation of gravity based on the gauging of thisq-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.     

Complex quantum group,dual algebra and bicovariant differential calculus

The method used to construct the bicovariant bimodule in ref. [CSWW] is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group. The complex quantum group Fun _q (SL(N, C)) is defined by requiring that it contains Fun _q (SU(N)) as a subalgebra analogously to the quantum Lorentz group. Analyzing the properties of the fundamental bimodule, we show that the dual algebra has the structure of the twisted product Fun _q (SU(N))Fun _q (SU(N)) _reg ^* . Then the bicovariant differential calculi on the complex quantum group are constructed.     

Electrical and Thermal Transport Properties of n-type Bi6Cu2Se4O6 (2BiCuSeO + 2Bi2O2Se)

New thermoelectric materials, n-type Bi₆Cu₂Se₄O₆ oxyselenides, composed of well-known BiCuSeO and Bi₂O₂Se oxyselenides, are synthesized with a simple solid-state reaction. Electrical transport properties, microstructures, and elastic properties are investigated with an emphasis on thermal transport properties. Similar to Bi₂O₂Se, it is found that the halogen-doped Bi₆Cu₂Se₄O₆ possesses n-type conducting transports, which can be improved via Br/Cl doping. Compared with BiCuSeO and Bi₂O₂Se, an extremely low thermal conductivity can be observed in Bi₆Cu₂Se₄O₆. To reveal the origin of low thermal conductivity, elastic properties, sound velocity, Grüneisen parameter, and Debye temperature are evaluated. Importantly, the calculated phonon mean free path of Bi₆Cu₂Se₄O₆ is comparable to the interlayer distance for BiO─CuSe and BiO─Se layers, which is ascribed to the strong interlayer phonon scattering. Contributing from the outstanding low thermal conductivity and improved electrical transport properties, the maximum ZT ≈0.15 at 823 K and ≈0.11 at 873K are realized in n-type Bi₆Cu₂Se_3.2Br_0.8O₆ and Bi₆Cu₂Se_3.6Cl_0.4O₆, respectively, indicating the promising thermoelectric performance in n-type Bi₆Cu₂Se₄O₆ oxyselenides.     

The bicovariant differential caculus on the κ-Poincaré and κ-Weyl groups

The bicovariant differential calculus on four-dimensional -Poincare group and corresponding Lie-algebra like structure for any metric tensor are described. The bicovariant differential calculus on four-dimensional -Weyl group and corresponding Lie-algebra like structure for any metric tensor in the reference frame in which g ₀₀ = 0 are considered.     

The 5v 3 bands of 18O enriched ozone: line positions of 16O16O18O, 16O18O16O, 16O18O18O and 18O16O18O

The four 5v ₃ bands of ¹⁸O enriched ozone have been observed and analysed for the first time. Two species (¹⁶O¹⁸O¹⁶O and ¹⁸O¹⁶O¹⁸O) belong to the C_2v symmetry group and two other (¹⁸O¹⁸O¹⁶O and ¹⁶O¹⁶O¹⁸O) to the C_s symmetry group. They have been recorded at a resolution of 0.008 cm^?1 with a pathlength of 32.16 m. Despite the very weak absorptions observed, almost 250 energy levels have been derived for each of the 4 species, with J ? 35 and K _a ? 13, and suitable sets of Hamiltonian parameters have been determined. For 3 species it has been necessary to account for the resonance between the (005) and (311) states to correctly reproduce the spectra observed. These resonances, anharmonic for C_2v, and hybrid (both anhar-monic and Coriolis) for C_s symmetry confirm the accidentally extremely strong coupling between the (005) and (311) states for ¹⁶O₃, due in that case to the very close distance between unperturbed energy levels. This work also confirms the excellent prediction of band centres of these four species derived from the recently determined isotopically invariant molecular potential function.     

Electric field gradients of111Cd in the hexagonal (A-phase) La2O3 and Nd2O3 sesquioxides

Perturbed angular correlation measurements of¹¹¹In in hexagonal La₂O₃ and Nd₂O₃ yield well defined symmetric electric field gradients with coupling constantv _Q =280 MHz (extrapolated) in La₂O₃ andv _Q =269 MHz for Nd₂O₃ at room temperature. The coupling constant in La₂O₃ linearly decreases with temperature. The spectra of La₂O₃, strongly damped at RT, rise to full amplitude at 870 K indicating dynamical interactions.     

Glass Transition (T g) And Ftir Study Of Na2O-CuO-Bi2O3-P2O5 Glasses

(40?x)Na₂O-xCuO-10Bi₂O₃-50P₂O₅ glasses (0 h x h 40) were prepared and studied. Their density, molar volume, glass transition temperature and IR spectroscopy have been investigated in order to understand the structural role of Bi₂O₃ and CuO in these glasses. In the sodium bismo-phosphate glass, the structures consist of some phosphate chains linked together through P-O-Bi bonds. When copper oxide is added to sodium bismo-phosphate glass, phosphate chains are depolymerised by the incorporation of Cu through P-O-Cu bonds. The former bonds are the origin of the partial glass forming ability of Bi³⁺ and Cu²⁺.     

Antiferromagnetic order in PrBa2Cu3O x (x≈6, 7)

Zero and weak transverse fieldμ ⁺ SR measurements on PrBa₂Cu₃O _x samples withx≈6 andx≈7 show ordered magnetism in both oxygen concentration limits. As expected for equivalent doping (nominally PrBa₂Cu₃O _x ∼YBa₂Cu₃O _x −0.5), neither sample is superconducting. Two muon signals are observed in thex≈6 sample, as in YBa₂Cu₃O₆, but only one (the weaker) has the same local magnetic field as in YBa₂Cu₃O₆. In thex≈7 sample, only one site is observed; its local field is reduced with respect to that of the primary site in YBa₂Cu₃O₆ by a factor roughly consistent with the carrier-density dependence of the Cu ion moment in antiferromagnetic YBa₂Cu₃O _x .     

Gravity on Finite Groups

Gravity theories are constructed on finite groups G. A self-consistent review of the differential calculi on finite G is given, with some new developments. The example of a bicovariant differential calculus on the nonabelian finite group S ₃ is treated in detail, and used to build a gravity-like field theory on S ₃. Received: 11 November 1999 / Accepted: 11 December 2000     

The classification of differential structures on quantum 2-spheres

Exterior algebras of differential forms on quantum 2-spheresS _qc ² ,q[–1, 1]/{0},c[0, ] (c=0 forq=±1), are classified. In the definition of exterior algebras we assume the invariance w.r.t. the action of the quantumSU(2) group and dimensionality conditions (which imply that we deal with two-dimensional manifolds). The exterior algebras exist only forc=0 and are unique in that case. The corresponding generalized directional derivatives are provided.