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1.
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.  相似文献   

2.
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.  相似文献   

3.
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  相似文献   

4.
For transcendental values of the deformation parameter q all bicovariant first order differential calculi on the Hopf algebras are classified.  相似文献   

5.
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.  相似文献   

6.
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.  相似文献   

7.
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).  相似文献   

8.
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  相似文献   

9.
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.  相似文献   

10.
We present a general method to deform the inhomogeneous algebras of theB n,Cn,Dn type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromB n+1,Cn+1,Dn+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.  相似文献   

11.
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.  相似文献   

12.
13.
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 Bi6Cu2Se3.2Br0.8O6 and Bi6Cu2Se3.6Cl0.4O6, respectively, indicating the promising thermoelectric performance in n-type Bi6Cu2Se4O6 oxyselenides.  相似文献   

14.
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 C2v symmetry group and two other (18O18O16O and 16O16O18O) to the Cs 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 C2v, and hybrid (both anhar-monic and Coriolis) for Cs 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.  相似文献   

15.
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.  相似文献   

16.
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.  相似文献   

17.
(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+.  相似文献   

18.
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 .  相似文献   

19.
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  相似文献   

20.
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.  相似文献   

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