On the Construction of Covariant Differential Calculi on Quantum Homogeneous Spaces

Let A be a coquasitriangular Hopf algebra and X the subalgebra of A generated by a row of a matrix corepresentation u or by a row of u and a row of the contragredient representation u^c. In the paper left-covariant first order differential calculi on the quantum group A are constructed and the corresponding induced calculi on the left quantum space X are described. The main tool for these constructions are the L-functionals associated with u. The results are applied to the quantum homogeneous space GL_q(N)/GL_q(N-1).Comment: 25 pages, Late

The 3D Spin Geometry of the Quantum Two-Sphere

We study a three-dimensional differential calculus on the standard Podles quantum two-sphere S^2_q, coming from the Woronowicz 4D+ differential calculus on the quantum group SU_q(2). We use a frame bundle approach to give an explicit description of the space of forms on S^2_q and its associated spin geometry in terms of a natural spectral triple over S^2_q. We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.Comment: v2: 25 pages; minor change

Unbounded Operators on Hilbert C∗C^*-Modules

Let $E$ and $F$ be Hilbert $C^*$-modules over a $C^*$-algebra \CAlg{A}. New classes of (possibly unbounded) operators $t:E\to F$ are introduced and investigated. Instead of the density of the domain \Def(t) we only assume that $t$ is essentially defined, that is, \Def(t)^\bot=\{0\}. Then $t$ has a well-defined adjoint. We call an essentially defined operator $t$ graph regular if its graph \Graph(t) is orthogonally complemented in $E\oplus F$ and orthogonally closed if \Graph(t)^{\bot\bot}=\Graph(t). A theory of these operators is developed. Various characterizations of graph regular operators are given. A number of examples of graph regular operators are presented ($E=C_0(X)$, a fraction algebra related to the Weyl algebra, Toeplitz algebra, Heisenberg group). A new characterization of affiliated operators with a $C^*$-algebra in terms of resolvents is given

Twisted cyclic homology of all Podles quantum spheres

We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of all Podles quantum spheres relative to arbitary automorphisms. Our calculations are based on a free resolution due to Masuda, Nakagami and Watanabe. The dimension drop in Hochschild homology can be overcome by twisting by automorphisms induced from the canonical modular automorphism associated to the Haar state on quantum SU(2). We specialize our results to the standard quantum sphere, and identify the class in twisted cyclic cohomology of the 2-cocycle discovered by Schmuedgen and Wagner corresponding to the distinguished covariant differential calculus found by Podles.Comment: 17 pages, no figures, uses the amscd package. v6: final version accepted for publicatio

Operator Representations of a q-Deformed Heisenberg Algebra

A class of well-behaved *-representations of a q-deformed Heisenberg algebra is studied and classified.Comment: 17 pages; Plain Tex; no figure