Discrete Series for Loop Groups.I. An algebraic Realization of Standard Modules

In this paper we consider the category $C (\tilde k, \tilde H)$ of the $(\tilde k, \tilde H)$-modules, including all the Verma modules, where $k$ is some compact Lie algebra and H some Cartan subgroup, $\tilde k$ and $\tilde H$ are the corresponding affine Lie algebra and the affine Cartan group, respectively. To this category we apply the Zuckerman functor and its derivatives. By using the determinant bundle structure, we prove the natural duality of the Zuckerman derived functors, and deduce a Borel-Weil-Bott type theorem on decomposition of the nilpotent part cohomology.Comment: 16 pages, LaTeX2e file, This paper is a revised version of 92-015(1992),IV.1-IV.16, SFB 343, Uni Bielefel

Quantum Strata of Coadjoint Orbits

In this paper we construct quantum analogs of strata of coadjoint orbits and describe their representations. This kind objects play an important role in describing quantum groups as repeated extensions of quantum strata.Comment: 26 pages, AMS-LaTeX2e, no figur

The Noncommutative Chern-Connes Character of the Locally Compact Quantum Normalizer of SU(1,1) in SL(2,C)

We observe that the von Neumann envelope of the quantum algebra of functions on the normalizer of thegroup \SU(1,1)\cong \SL(2,\mathbb R) in \SL(2,\mathbb C) via deformation quantization contains the von Neumann algebraic quantum normalizer of \SU(1,1) in the frame work of Waronowicz-Korogodsky. We then use the technique of reduction to the maximal subgroup to compute the K-theory, the periodic cyclic homology and the corresponding Chern-Connes character.Comment: 9 pages, LaTeX, no figur

Multiparty Quantum Telecommunication Using Quantum Fourier Transforms

Consider the problem: Alice wishes to send the same key to $n-1$ users (Bob, Carol,. . . , Nathan), while preventing eavesdropper Eve from acquiring information without being detected. The problem has no solution in the classical cryptography but in quantum telecommunication there are some codes to solve the problem. In the paper \cite{zengetall}, Guo-Jyun Zeng, Kuan-Hung Chen, Zhe-Hua Chang, Yu-Shan Yang, and Yao-Hsin Chou from one side and Cabello in \cite{cabello} from other side, used Hadamard gates, Pauli gates in providing the quantum communication code for two-partity telecommunication with 3 persons and then generalized it to the case of arbitrary number of participants, indicating the position of measurements of participants. We remark that the Hadamard gate with precising the position of measurement is the same as Fourier transform for two qubits and hence use the general Fourier transform for $n$ entangled qubits, in place of Hadamard gates. The result is more natural for arbitrary $n$ qudits.Comment: LaTeX2e, 11 page

On the Twisted KK-Theory and Positive Scalar Curvature Problem

Positiveness of scalar curvature and Ricci curvature requires vanishing the obstruction $\theta(M)$ which is computed in some KK-theory of C*-algebras index as a pairing of spin Dirac operator and Mishchenko bundle associated to the manifold. U. Pennig had proved that the obstruction $\theta(M)$ does not vanish if $M$ is an enlargeable closed oriented smooth manifold of even dimension larger than or equals to 3, the universal cover of which admits a spin structure. Using the equivariant cohomology of holonomy groupoids we prove the theorem in the general case without restriction of evenness of dimension. Our groupoid method is different from the method used by B. Hanke and T. Schick in reduction to the case of even dimension.Comment: 8 pages, LaTeX2

Category of Noncommutative CW complexes. III

We prove in this paper a noncommutative version of Leray Spectral Sequence Theorem and then Leray-Serre Spectral Theorem for noncommutative Serre fibrations: for NC Serre fibration there are converging spectral sequences with \E^2 terms as \E^2_{p,q} = \HP_p(A; \HP_q(B,A)) \Longrightarrow \HP_{p+q}(B) and \E^2_{p,q} = \HP_p(A;\K_q(B,A)) \Longrightarrow \K_{p+q}(B).Comment: LaTeX2e, 10 pages, no figur

Noncommutative Spherical Tight Frames in finitely generated Hilbert C*-modules

In the paper we describe the C*-algebras of noncommutative spherical tight frames over some C*-algebras and then apply to study the noncommutative version of the universal classifying space.Comment: LaTeX2e, no figur

Jeffrey-Kirwan-Witten Localization Formula for Reductions at Regular Co-adjoint Orbits

For Marsden-Weinstein reductions at the point 0 in the vector space dual to the Lie algebra, the well-known Jeffrey-Kirwan-Witten localization formula was proven and lastly modified by M. Vergne. We prove in this paper the same kind formula for the reductions at regular coadjoint orbits by using the universal orbital formula of characters.Comment: 17 pages, AMSLaTeX file, accepted for publication in Matimyas Matemati

Category of Noncommutative CW Complexes

We expose the notion of noncommutative CW (NCCW) complexes, define noncommutative (NC) mapping cylinder and NC mapping cone, and prove the noncommutative Approximation Theorem. The long exact homotopy sequences associated with arbitrary morphisms are also deduced.Comment: LaTeX2e, 12 pages, no figure

Quantum Communication and Quantum Multivariate Polynomial Interpolation

The paper is devoted to the problem of multivariate polynomial interpolation and its application to quantum secret sharing. We show that using quantum Fourier transform one can produce the protocol for quantum secret sharing distribution.Comment: 7 pages, no figure, LaTeX2