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.
It is known that, under strong combinatorial axioms, $O_N\subset O_N^*\subset O_N^+$ are the only orthogonal quantum groups. We prove here similar results for the noncommutative spheres $S^{N-1}_\mathbb R\subset S^{N-1}_{\mathbb R,*}\subset S^{N-1}_{\mathbb R,+}$, the noncommutative projective spaces $P^{N-1}_\mathbb R\subset P^{N-1}_\mathbb C\subset P^{N-1}_+$, and the projective orthogonal quantum groups $PO_N\subset PO_N^*\subset PO_N^+$.
We investigate the "theta-deformed spheres" C(S^{3}_{theta}) and C(S^{4}_{theta}), where theta is any real number. We show that all finitely-generated projective modules over C(S^{3}_{theta}) are free, and that C(S^{4}_{theta}) has the cancellation property. We classify and construct all finitely-generated projective modules over C(S^{4}_θ) up to isomorphism. An interesting feature is that if theta is irrational then there are nontrivial "rank-1" modules over C(S^{4}_θ). In that case, every finitely-generated projective module over C(S^{4}_θ) is a sum of a rank-1 module and a free module. If t…
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.
We describe the quantum subspaces of Banica-Goswami's half-liberated real-spheres, showing in particular that there is a bijection between the symmetric ones and the conjugation stable closed subspaces of the complex projective spaces.
We show that much of the structure of the 2-sphere as a complex curve survives the q-deformation and has natural generalizations to the quantum 2-sphere - which, with additional structures, we identify with the quantum projective line. Notably among these is the identification of a quantum homogeneous coordinate ring with the coordinate ring of the quantum plane. In parallel with the fact that positive Hochschild cocycles on the algebra of smooth functions on a compact oriented 2-dimensional manifold encode the information for complex structures on the surface, we formulate a notion of twisted…
We show that the standard SU(n)-covariant Poisson sphere $S^{2n-1}$ is embedded in the nonstandard $SU(n+1)$-covariant Poisson complex projective spaces $CP^{n}$.
Noncommutative (NC) sphere is introduced as a quotient of the enveloping algebra of the Lie algebra su(2). Using the Cayley-Hamilton identities we introduce projective modules which are analogues of line bundles on the usual sphere (we call them quantum line bundles) and define a multiplicative structure in their family. Also, we compute a pairing between certain quantum line bundles and finite dimensional representations of the NC sphere in the spirit of the NC index theorem. A new approach to constructing the differential calculus on a NC sphere is suggested. The approach makes use of the pr…
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 projecti…
We introduce $q$-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with $q$-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics. We also give metric connec…
By a diagonal embedding of $U(1)$ in $SU_q(m)$, we prolongate the diagonal circle action on the Vaksman-Soibelman quantum sphere $S^{2n+1}_q$ to the $SU_q(m)$-action on the prolongated bundle. Then we prove that the noncommutative vector bundles associated via the fundamental representation of $SU_q(m)$, for $m\in\{2,\ldots,n\}$, yield generators of the even K-theory group of the C*-algebra of the Vaksman-Soibelman quantum complex projective space $\mathbb{C}{\rm P}^n_q$.
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.
Associated to the standard $SU_{q}(n)$ R-matrices, we introduce quantum spheres $S_{q}^{2n-1}$, projective quantum spaces $CP_{q}^{n-1}$, and quantum Grassmann manifolds $G_{k}(C_{q}^{n})$. These algebras are shown to be homogeneous quantum spaces of standard quantum groups and are also quantum principle bundles in the sense of T Brzezinski and S. Majid (Comm. Math. Phys. 157,591 (1993)).
We explore the possibility of introducing q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. For the module of 1-forms on the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a g…
We quantize homogeneous vector bundles over an even complex sphere $\mathbb{S}^{2n}$ as one-sided projective modules over its quantized coordinate ring. We realize them in two different ways: as locally finite $\mathbb{C}$-homs between pseudo-parabolic Verma modules and as induced modules of the quantum orthogonal group. Based on this alternative, we study representations of a quantum symmetric pair related to $\mathbb{S}^{2n}_q$ and prove their complete reducibility.
Noncommutative geometry of quantised contact spheres introduced by Omori, Maeda, Miyazaki and Yoshioka is studied. In particular it is proven that these spheres form a noncommutative Hopf fibration in the sense of Hopf-Galois extensions. The monopole (strong) connection is constructed, and projectors describing projective modules of all monopole charges are computed.
The (left coalgebra) line bundle associated to the quantum Hopf fibration of any quantum two-sphere is shown to be a finitely generated projective module. The corresponding projector is constructed and its monopole charge is computed. It is shown that the Dirac $q$-monopole connection on any quantum two-sphere induces the Grassmannian connection built with this projector.
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.
We determine the Chern characters of two projective modules over the fuzzy sphere and calculate the corresponding topological charges (Chern numbers). These turn out to have corrections - compared to the commutative limit - induced by the noncommutative structure of the three coordinates.
The quantum deformation $CP_q(N)$ of complex projective space is discussed. Many of the features present in the case of the quantum sphere can be extended. The differential and integral calculus is studied and $CP_q(N)$ appears as a quantum Kähler manifold. The braiding of several copies of $CP_q(N)$ is introduced and the anharmonic ratios of four collinear points are shown to be invariant under quantum projective transformations. They provide the building blocks of all projective invariants. The Poisson limit is also described.
We derive an explicit expression for an associative *-product on fuzzy complex projective spaces. This generalises previous results for the fuzzy 2-sphere and gives a discrete non-commutative algebra of functions on fuzzy complex projective spaces, represented by matrix multiplication. The matrices are restricted to ones whose dimension is that of the totally symmetric representations of SU(N). In the limit of infinite dimensional matrices we recover the commutative algebra of functions on ordinary projective space. Derivatives on the fuzzy projective space are also expressed as matrix commuta…
We investigate covariant first order differential calculi on the quantum complex projective spaces CP_q^{N-1} which are quantum homogeneous spaces for the quantum group SU_q(N). Hereby, one more well-studied example of covariant first order differential calculus on a quantum homogeneous space is given. Since the complex projective spaces are subalgebras of the quantum spheres S_q^{2N-1} introduced by Vaksman and Soibelman, we get also an example of the relations between covariant differential calculus on two closely related quantum spaces. Two approaches are combined in obtaining covariant fir…
When considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connec…
