Quantum teardrops

Algebras of functions on quantum weighted projective spaces are introduced, and the structure of quantum weighted projective lines or quantum teardrops are described in detail. In particular the presentation of the coordinate algebra of the quantum teardrop in terms of generators and relations and classification of irreducible *-representations are derived. The algebras are then analysed from the point of view of Hopf-Galois theory or the theory of quantum principal bundles. Fredholm modules and associated traces are constructed. C*-algebras of continuous functions on quantum weighted projective lines are described and their K-groups computed.Comment: 18 page

Local Proof of Algebraic Characterization of Free Actions

Let G be a compact Hausdorff topological group acting on a compact Hausdorff topological space X. Within the C∗-algebra C(X) of all continuous complex-valued functions on X, there is the Peter-Weyl algebra PG(X) which is the (purely algebraic) direct sum of the isotypical components for the action of G on C(X). We prove that the action of G on X is free if and only if the canonical map PG(X)⊗C(X/G)PG(X)→PG(X)⊗O(G) is bijective. Here both tensor products are purely algebraic, and O(G) denotes the Hopf algebra of ''polynomial'' functions on G

Associated noncommutative vector bundles over the Vaksman–Soibelman quantum complex projective spaces

Analysis and Stochastic

On piecewise trivial Hopf—Galois extensions

We discuss a noncommutative generalization of compact principal bundles that can be trivialized relative to the finite covering by closed sets. In this setting we present bundle reconstruction and reduction

An equivariant pullback structure of trimmable graph C^*-algebras

To unravel the structure of fundamental examples studied in noncommutative topology, we prove that the graph C*-algebra C*(E) of a trimmable graph E is U(1)-equivariantly iso-morphic to a pullback C *-algebra of a subgraph C *-algebra C*(E'') and the C *-algebra of func-tions on a circle tensored with another subgraph C*-algebra C*(E'). This allows us to approach the structure and K-theory of the fixed-point subalgebra C*(E)U .1/ through the (typically simpler) C *-algebras C*(E'), C*(E'') and C*(E'')U.1/. As examples of trimmable graphs, we consider one-loop extensions of the standard graphs encoding respectively the Cuntz algebra O2 and the Toeplitz algebra T . Then we analyze equivariant pullback structures of trimmable graphs yielding the C*-algebras of the Vaksman-Soibelman quantum sphere S2n+1 q and the quantum lens space L3 q(l;1, l), respectively.Analysis and Stochastic

Quantized algebras of functions on homogeneous spaces with Poisson stabilizers

Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0<q<1. We study a quantization C(G_q/K_q) of the algebra of continuous functions on G/K. Using results of Soibelman and Dijkhuizen-Stokman we classify the irreducible representations of C(G_q/K_q) and obtain a composition series for C(G_q/K_q). We describe closures of the symplectic leaves of G/K refining the well-known description in the case of flag manifolds in terms of the Bruhat order. We then show that the same rules describe the topology on the spectrum of C(G_q/K_q). Next we show that the family of C*-algebras C(G_q/K_q), 0<q\le1, has a canonical structure of a continuous field of C*-algebras and provides a strict deformation quantization of the Poisson algebra \C[G/K]. Finally, extending a result of Nagy, we show that C(G_q/K_q) is canonically KK-equivalent to C(G/K).Comment: 23 pages; minor changes, typos correcte

Noncommutative Circle Bundles and New Dirac Operators

We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection

Braided join comodule algebras of bi-Galois objects

A bi-Galois object A is a bicomodule algebra for Hopf-Galois coactions with trivial invariants. In the spirit of Milnor's construction, we define the join of noncommutative bi-Galois objects (quantum torsors). To ensure that the diagonal coaction on the join algebra of the right-coacting Hopf algebra is an algebra homomorphism, we braid the tensor product A circle times A with the help of the left-coacting Hopf algebra. Our main result is that the diagonal coaction is principal. Then we show that an anti-Drinfeld double is a symmetric bi-Galois object with the Drinfeld-double Hopf algebra coacting on both left and right. In this setting, we consider a finite quantum covering as an example. Finally, we take the noncommutative torus with the natural free action of the classical torus as an example of a symmetric bi-Galois object equipped with a *-structure. It yields a noncommutative deformation of a nontrivial torus bundle. \ua9 2016, University at Albany. All rights reserved