Hilbert space representations of the cross product *-algebras of the Hopf *-algebra U_q(su_2) and its module *-algebras O(S^2_{qr}) of Podles spheres are investigated and classified by describing the action of generators. The representations are analyzed within two approaches. It is shown that the Hopf *-algebra O(SU_q(2)) of the quantum group SU_q(2) decomposes into an orthogonal sum of projective Hopf modules corresponding to irreducible integrable *-representations of the cross product algebras and that each irreducible integrable *-representation appears with multiplicity one. The projections of these projective modules are computed. The decompositions of tensor products of irreducible integrable *-representations with spin l representations of U_q(su_2) are given. The invariant state h on O(S^2_{qr}) is studied in detail. By passing to function algebras over the quantum spheres S^2_{qr}, we give chart descriptions of quantum line bundles and describe the representations from the first approach by means of the second approach.
Hilbert space representations of the cross product *-algebra of the Hopf *-algebra U_q(su_2) and its module *-algebras O(S^2_{qc}) of Podles' spheres are studied. Two classes of representations are described by explicit formulas for the actions of the generators.
A Dirac operator D on the standard Podles sphere is defined and investigated. It yields a spectral triple such that |D|^{-z} is of trace class for Re z>0. Commutators with the Dirac operator give the distinguished 2-dimensional covariant differential calculus on the standard Podles sphere. The twisted cyclic cocycle associated with the volume form of the differential calculus is expressed by means of the Dirac operator.
For $μ\in (0,1), c> 0,$ we identify the quantum group $SO_μ(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries' in the sense of \cite{goswami2} on the natural spectral triple on the Podles sphere $S^2_{μ, c}$ constructed by Dabrowski, D'Andrea, Landi and Wagner in \cite{Dabrowski_et_al}.
Starting with the braided quantum group $\operatorname{SU}_q(2)$ for a complex deformation parameter $q$ we perform the construction of the quotient $\operatorname{SU}_q(2)/\mathbb{T}$ which serves as a model of a quantum sphere. Then we follow the reasoning of Podleś who for real $q$ classified quantum spaces with the action of $\operatorname{SU}_q(2)$ with appropriate spectral properties. These properties can also be expressed in the context of the braided quantum $\operatorname{SU}_q(2)$ (with complex $q$) and we find that they lead to precisely the same family of quantum spaces as found by…
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 …
Building on the now established presentation of the standard Podles sphere as an example of a noncommutative complex structure, we investigate how its classical Kahler geometry behaves under $q$-deformation. Discussed are noncommutative versions of Hodge decomposition, Lefschetz decomposition, the Kahler identities, and the refinement of de Rham cohomology by Dolbeault cohomology
The dual coalgebra of Podleś' quantum sphere O_q(S^2_c) is determined explicitly. This result is used to classify all finite dimensional covariant first order differential calculi over O_q(S^2_c) for all but exceptional values of the parameter c.
The fundamental Hochschild cohomology class of the standard Podles quantum sphere is expressed in terms of the spectral triple of Dabrowski and Sitarz by means of a residue formula.
We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of `modular' spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podles sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class w…
The cup and cap product in twisted Hochschild (co)homology is computed for the standard quantum 2-sphere and used to construct a cyclic 2-cocycle that represents the fundamental Hochschild class.
It is shown that the isospectral bi-equivariant spectral triple on quantum SU(2) and the isospectral equivariant spectral triples on the Podles spheres are related by restriction. In this approach, the equatorial Podles sphere is distinguished because only in this case the restricted spectral triple admits an equivariant grading operator together with a real structure (up to infinitesimals of arbitrary high order). The real structure is expressed by the Tomita operator on quantum SU(2) and it is shown that the failure of the real structure to satisfy the commutant property is related to the fa…
We give a new definition of dimension spectrum for non-regular spectral triples and compute the exact (i.e. non only the asymptotics) heat-trace of standard Podles spheres $S^2_q$ for $0<q<1$, study its behavior when $q\to 1$ and fully compute its exact spectral action for an explicit class of cut-off functions.
We show that the family of Podles' spheres is complete under equivariant Morita equivalence (with respect to the action of quantum SU(2)), and determine the associated orbits. We also give explicit formulas for the actions which are equivariantly Morita equivalent with the quantum projective plane. In both cases, the computations are made by examining the localized spectral decomposition of a generalized Casimir element.
A wide class of noncommutative spaces, including 4-spheres based on all the quantum 2-spheres and suspensions of matrix quantum groups is described. For each such space a noncommutative vector bundle is constructed. This generalises and clarifies various recent constructions of noncommutative 4-spheres.
Using principles of quantum symmetries we derive the algebraic part of the real spectral triple data for the standard Podleś quantum sphere: equivariant representation, chiral grading $γ$, reality structure $J$ and the Dirac operator $D$, which has bounded commutators with the elements of the algebra and satisfies the first order condition.
We construct spectral triples on all Podles quantum spheres. These noncommutative geometries are equivariant for a left action of $U_q(su(2))$ and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the 2-sphere. There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.
We propose a slight modification of the properties of a spectral geometry a la Connes, which allows for some of the algebraic relations to be satisfied only modulo compact operators. On the equatorial Podles sphere we construct suq2-equivariant Dirac operator and real structure which satisfy these modified properties.
We discuss spectral properties of the equatorial Podles sphere. As a preparation we also study the `degenerate' (i.e. $q=0$) case (related to the quantum disk). We consider two different spectral triples: one related to the Fock representation of the Toeplitz algebra and the isopectral one. After the identification of the smooth pre-$C^*$-algebra we compute the dimension spectrum and residues. We check the nontriviality of the (noncommutative) Chern character of the associated Fredholm modules by computing the pairing with the fundamental projector of the $C^*$-algebra (the nontrivial generato…
The purpose of the paper is twofold: First, known results of the noncommutative spin geometry of the standard Podles sphere are extended by discussing Poincare duality and orientability. In the discussion of orientability, Hochschild homology is replaced by a twisted version which avoids the dimension drop. The twisted Hochschild cycle representing an orientation is related to the volume form of the distinguished covariant differential calculus. Integration over the volume form defines a twisted cyclic 2-cocycle which computes the q-winding numbers of quantum line bundles. Second, a "twisted" …
We give an explicit form of the symplectic groupoid that integrates the semiclassical standard Podles sphere. We show that Sheu's groupoid, whose convolution C*-algebra quantizes the sphere, appears as the groupoid of the Bohr-Sommerfeld leaves of a (singular) real polarization of the symplectic groupoid. By using a complex polarization we recover the convolution algebra on the space of polarized sections. We stress the role of the modular class in the definition of the scalar product in order to get the correct quantum space.
We define the $C^*$-algebra of quantum real projective space $\R P_q^2$, classify its irreducible representations and compute its $K$-theory. We also show that the $q$-disc of Klimek-Lesniewski can be obtained as a non-Galois $\Z_2$-quotient of the equator Podleś quantum sphere. On the way, we provide the Cartesian coordinates for all Podleś quantum spheres and determine an explicit form of isomorphisms between the $C^*$-algebras of the equilateral spheres and the $C^*$-algebra of the equator one.
We define even dimensional quantum spheres Sigma_q^2n that generalize to higher dimension the standard quantum two-sphere of Podle's and the four-sphere Sigma_q^4 obtained in the quantization of the Hopf bundle. The construction relies on an iterated Poisson double suspension of the standard Podle's two-sphere. The Poisson spheres that we get have the same symplectic foliation consisting of a degenerate point and a symplectic plane and, after quantization, have the same C^*-algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.
We construct new families of spectral triples over quantum spheres, with a particular attention focused on the standard Podles quantum sphere and twisted Dirac operators.
The quantum flag manifold ${SU_q(3)/\mathbb{T}^2}$ is interpreted as a noncommutative bundle over the quantum complex projective plane with the quantum or Podleś sphere as a fibre. A connection arising from the (associated) quantum principal $U_q(2)$-bundle is described.
Whereas the classical sphere $C P^1$ can be defined as the coordinate algebra generated by the matrix entries of a projector $e$ with $\trace(e)=1$, the fuzzy-sphere is defined in the same way by $\trace(e)=1+λ$. We show that the standard $q$-sphere is similarly defined by $\trace_q(e)=1$ and the Podles 2-spheres by $\trace_q(e)=1+λ$, thereby giving a unified point of view in which the 2-parameter Podles spheres are $q$-fuzzy spheres. We show further that they arise geometrically as `constant time slices' of the unit hyperboloid in $q$-Minkowski space viewed as the braided group $B_q[SU_2]$. T…
The Noncommutative Index Theorem is used to prove that the Chern character of quantum Hopf line bundles over the standard Podles quantum sphere equals the winding number of the representations defining these bundles. This result gives an estimate of the positive cone of the algebraic $K_0$ of the standard quantum sphere.
The paper presents applications of Toeplitz algebras in Noncommutative Geometry. As an example, a quantum Hopf fibration is given by gluing trivial U(1) bundles over quantum discs (or, synonymously, Toeplitz algebras) along their boundaries. The construction yields associated quantum line bundles over the generic Podles spheres which are isomorphic to those from the well-known Hopf fibration of quantum SU(2). The relation between these two versions of quantum Hopf fibrations is made precise by giving an isomorphism in the category of right U(1)-comodules and left modules over the C*-algebra of…
We consider two Z/2Z-actions on the Podles generic quantum spheres. They yield, as noncommutative quotient spaces, the Klimek-Lesniewski q-disc and the quantum real projective space, respectively. The C*-algebras of all these quantum spaces are described as graph C*-algebras. The K-groups of the thus presented C*-algebras are then easily determined from the general theory of graph C*-algebras. For the quantum real projective space, we also recall the classification of the classes of irreducible *-representations of its algebra and give a linear basis for this algebra.
Using non canonical braidings, we first introduce a notion of symmetric tensors and corresponding Hodge operators on a class of left covariant 3d differential calculi over the quantum SU(2) group, then we induce Hodge operators on the left covariant 2d exterior algebras over the standard Podles quantum sphere.
We describe the quantum sphere of Podleś for $c=0$ by means of a stereographic projection which is analogous to that which exhibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential calculus on the sphere are covariant under the coaction of fractional transformations with \SU coefficients as well as under the action of \SU vector fields. Going to the classical limit we obtain the Poisson sphere. Finally, we study the invariant integration of functions on the sphere and find its relation with the translationally invariant integration on the c…
A Dirac operator D on quantized irreducible generalized flag manifolds is defined. This yields a Hilbert space realization of the covariant first-order differential calculi constructed by I. Heckenberger and S. Kolb. All differentials df=i[D,f] are bounded operators. In the simplest case of Podles' quantum sphere one obtains the spectral triple found by L. Dabrowski and A. Sitarz.
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.
The recent result of Brown and Zhang establishing Poincare duality in the Hochschild (co)homology of a large class of Hopf algebras is extended to right coideal subalgebras over which the Hopf algebra is faithfully flat, and applied to the standard Podles quantum 2-sphere.
The Podles quantum sphere S^2_q admits a natural commutative C*-subalgebra I_q with spectrum {0} \cup {q^{2k}: k = 0,1,2,...}, which may therefore be considered as a quantised version of a classical interval. We study here the compact quantum metric space structure on I_q inherited from the corresponding structure on S^2_q, and provide an explicit formula for the metric induced on the spectrum. Moreover, we show that the resulting metric spaces vary continuously in the deformation parameter q with respect to the Gromov-Hausdorff distance, and that they converge to a classical interval of lengt…
We construct the action of the quantum double of $\uq$ on the standard Podleś sphere and interpret it as the quantum projective formula generalizing to the q-deformed setting the action of the Lorentz group of global conformal transformations on the ordinary Riemann sphere.
An algebraic framework for noncommutative bundles with (quantum) homogeneous fibres is proposed. The framework relies on the use of principal coalgebra extensions which play the role of principal bundles in noncommutative geometry which might be additionally equipped with a Hopf algebra symmetry. The proposed framework is supported by two examples of noncommutative $\mathbb{C} P_q^1$-bundles: the quantum flag manifold viewed as a bundle with a generic Podleś sphere as a fibre, and the quantum twistor bundle viewed as a bundle over the quantum 4-sphere of Bonechi, Ciccoli and Tarlini.
We show that if $G$ is a compact Lie group and $\mathfrak{g}$ is its Lie algebra, then there is a map from the Hopf-cyclic cohomology of the quantum enveloping algebra $U_q(\mathfrak{g})$ to the twisted cyclic cohomology of quantum group algebra $\mathcal{O}(G_q)$. We also show that the Schmüdgen-Wagner index cocycle associated with the volume form of the differential calculus on the standard Podleś sphere $\mathcal{O}(S^2_q)$ is in the image of this map.
We study covariant differential calculus on the quantum spheres S_q^{N-1} which are quantum homogeneous spaces with coactions of the quantum groups O_q(N). The first part of the paper is devoted to first order differential calculus. A classification result is proved which says that for N>=6 there exist exactly two covariant first order differential calculi on S_q^{N-1} which satisfy the classification constraint that the bimodule of one-forms is generated as a free left module by the differentials of the generators of S_q^{N-1}. Both calculi exist also for 3<=N<=5. The same calculi can be cons…
Using the corepresentation of the quantum group $ SL_q(2)$ a general method for constructing noncommutative spaces covariant under its coaction is developed. The method allows us to treat the quantum plane and Podleś' quantum spheres in a unified way and to construct higher dimensional noncommutative spaces systematically. Furthermore, we extend the method to the quantum supergroup $ OSp_q(1|2).$ In particular, a one-parameter family of covariant algebras, which may be interpreted as noncommutative superspheres, is constructed.
We consider noncommutative line bundles associated with the Hopf fibrations of SUq(2) over all Podles spheres and with a locally trivial Hopf fibration of S^3_{pq}. These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U(1) with Galois-type extensions encoding the principal fibrations of SUq(2) and S^3_{pq}. We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them.
We consider three quantum algebras: the q-oscillator algebra, the Podles' sphere and the q-deformed enveloping algebra of $su(2).$ To each of these *-algebras we associate certain partial dynamical system and perform the "Mackey analysis" of *-representations developed in [SS]. As a result we get the description of "standard" irreducible *-representations. Further, for each of these examples we show the existence of a "$C^*$-envelope" which is canonically isomorphic to the covariance $C^*$-algebra of the partial dynamical system. Finally, for the q-oscillator algebra and the q-deformed $\cU(su…
We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices $M_2(\C)=\C\Z_2\cdot\C\Z_2$. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct $q$-monopoles on all the Podleś quantum spheres $S^2_{q,s}$.
Let (Γ,d) be the 3D-calculus or the 4D_{\pm}-calculus on the quantum group SU_q(2). We describe all pairs (π, F) of a *-representation πof O(SU_q(2)) and of a symmetric operator F on the representation space satisfying a technical condition concerning its domain such that there exist a homomorphism of first order differential calculi which maps dx into the commutator [iF,π(x)] for x\in O(SU_q(2)). As an application commutator representations of the 2-dimensional left-covariant calculus on Podles' quantum 2-sphere S^2_{qc} with c=0 are given.
This paper presents an analysis of the set of connections and covariant derivatives on a U(1) quantum Hopf bundle on the standard Podles sphere, whose total space quantum SU(2) is equipped with the 3d left covariant differential calculus by Woronowicz. The introduction of a Hodge duality on the exterior algebras on both total and base space of the bundle allows for the study of Laplacians and of gauged Laplacians.
A fibre product construction is used to give a description of quantum line bundles over the generic Podles spheres by gluing two quantum discs along their boundaries. Representatives of the corresponding $K_0$-classes are given in terms of 1-dimensional projections belonging to the C*-algebra, and in terms of analogues of the classical Bott projections. The $K_0$-classes of quantum line bundles derived from the generic Hopf fibration of quantum SU(2) are determined and the index pairing is computed. It is argued that taking the projections obtained from the fibre product construction yields a …
We find a substantial class of pairs of $*$-homomorphisms between graph C*-algebras of the form $C^*(E)\hookrightarrow C^*(G)\twoheadleftarrow C^*(F)$ whose pullback C*-algebra is an AF graph C*-algebra. Our result can be interpreted as a recipe for determining the quantum space obtained by shrinking a quantum subspace. There is a variety of examples from noncommutative topology, such as quantum complex projective spaces (including the standard Podleś quantum sphere) or quantum teardrops, that instantiate the result. Furthermore, to go beyond AF graph C*-algebras, we consider extensions of gra…
A method of constructing covariant differential calculi on a quantum homogeneous space is devised. The function algebra X of the quantum homogeneous space is assumed to be a left coideal of a coquasitriangular Hopf algebra H and to contain the coefficients of any matrix over H which is the two-sided inverse of one with entries in X. The method is based on partial derivatives. For the quantum sphere of Podles and the quantizations of symmetric spaces due to Noumi, Dijkhuizen and Sugitani the construction produces the subcalculi of the standard bicovariant calculus on the quantum group.
We construct covariant $q$-deformed holomorphic structures for all finitely-generated relative Hopf modules over the irreducible quantum flag manifolds endowed with their Heckenberger--Kolb calculi. In the classical limit these reduce to modules of sections of holomorphic homogeneous vector bundles over irreducible flag manifolds. For the case of simple relative Hopf modules, we show that this covariant holomorphic structure is unique. This generalises earlier work of Majid, Khalkhali, Landi, and van Suijlekom for line modules of the Podleś sphere, and subsequent work of Khalkhali and Moatadel…
We construct a $U_q(\mathrm{so}(2n+1))$-equivariant local star-product on the complex sphere $\mathbb{S}^{2n}$ as a Non-Levi conjugacy class $SO(2n+1)/SO(2n)$.
We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded Kasparov cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on $\mathbb R^{n+1}$. We find that the resulting spectral triple for the algebra $C(\mathbb S^n)$ differs from the Dirac operator on the round sphere by a so-called index cycle, whose class in $KK_0(\mathbb C, \mathbb C)$ represents the multiplicative unit. At all points we check that our construction involving the unbounded Kasparov product is compa…
We analyze the expansion of the fuzzy sphere non-commutative product in powers of the non-commutativity parameter. To analyze this expansion we develop a graphical technique that uses spin networks. This technique is potentially interesting in its own right as introducing spin networks of Penrose into non-commutative geometry. Our analysis leads to a clarification of the link between the fuzzy sphere non-commutative product and the usual deformation quantization of the sphere in terms of the star-product.
