Piotr M. Hajac

Strong connections on quantum principal bundles

A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the two-sphere fibrationS2→RP2. A certain class of strongUq(2)-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq-dependent hermitian metric. A particular form of the Yang-Mills action on a trivialUq(2)-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent ofq.

Projective Module Description of the q-Monopole

The Dirac q-monopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The Chern-Connes pairing of cyclic cohomology and K-theory is computed for the winding number -1. The non-triviality of this pairing is used to conclude that the quantum principal Hopf fibration is non-cleft. Among general results, we provide a left-right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (Hopf-Galois extensions) their associated covariant derivatives on projective modules.

Piecewise principal comodule algebras

A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality is a piecewise property: given N comodule-algebra surjections P->Pi whose kernels intersect to zero, P is principal if and only if all Pis are principal. Furthermore, assuming the principality of P, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with B. Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N-families of surjections P->Pi and such that the comodule algebra of global sections is P.

Bundles over Quantum Sphere and Noncommutative Index Theorem

The Noncommutative Index Theorem is used to prove that the Chern numbers of quantum Hopf line bundles over the standard Podles quantum sphere equal the winding numbers of the repres- entations defining these bundles. This result gives an estimate of the positive cone of the algebraic K0 of the standard quantum sphere.

Metrics and pairs of left and right connections on bimodules

Properties of metrics and pairs consisting of left and right connections are studied on the bimodules of differential 1‐forms. Those bimodules are obtained from the derivation based calculus of an algebra of matrix valued functions, and an SLq(2,C)‐covariant calculus of the quantum plane at a generic q and the cubic root of unity. It is shown that, in the aforementioned examples, giving up the middle‐linearity of metrics significantly enlarges the space of metrics. A metric compatibility condition for the pairs of left and right connections is defined. Also, a compatibility condition between a left and right connection is discussed. Consequences entailed by reducing to the center of a bimodule the domain of those conditions are investigated in detail. Alternative ways of relating left and right connections are considered.

Quantum double-torus

Abstract A symmetry extending the T 2 -symmetry of the noncommutative torus T q 2 is studied in the category of quantum groups. This extended symmetry is given by the quantum double-torus DT q 2 defined as a compact matrix quantum group consisting of the disjoint union of T 2 and T q 2 2 . The bicross-product structure of the polynomial Hopf algebra A(DT q 2 ) is computed. The Haar measure and the complete list of unitary irreducible representations of T q 2 2 are determined explicitly.

Equivariant Join and Fusion of Noncommutative Algebras

We translate the concept of the join of topological spaces to the language of C -algebras, replace the C -algebra of functions on the interval (0; 1) with evaluation maps at 0 and 1 by a unital C -algebra C with appropriate two surjections, and introduce the notion of the fusion of unital C -algebras. An appropriate modification of this construction yields the fusion comodule algebra of a comodule algebra P with the coacting Hopf alge- bra H. We prove that, if the comodule algebra P is principal, then so is the fusion comodule algebra. When C = C((0; 1)) and the two surjections are evaluation maps at 0 and 1, this result is a noncommutative-algebraic incarnation of the fact that, for a compact Hausdorff principal G-bundle X, the diagonal action of G on the join X G is free.

The K -theory of Heegaard quantum lens spaces

Representing Z/NZ as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/NZ, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/NZ to construct an associated complex loine bundle. This paper proves the stable non-triviality of these line bundles over any of the quantum lens spaces we consider. We use the pullback structure of the C*-algebra of the lens space to compute its K-theory via the Mayer-Vietoris sequence, and an explicit form of the Bass connecting homomorphism to prove the stable non-triviality of the bundles. On the algebraic side we prove the universality of the coordinate algebra of such a lens space for aparticular set of generators and relations. We also prove the non-existence of non-trivial invertibles in the coordinate algebra of a lens space. Finally, we prolongate the Z/NZ-fibres of the Heegaard quantum sphere to U (1), and determine the algebraic structure of such U (1) prolongation.

Quantum complex projective spaces from Toeplitz cubes

From N -tensor powers of the Toeplitz algebra, we construct a multi-pullback C*-algebra that is a noncommutative deformation of the complex projective space P.C/. Using Birkhoff’s Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multi-pullback C*-algebra is free. This shows that our deformation preserves the freeness of the lattice of subsets generated by the affine covering of the complex projective space. Mathematics Subject Classification (2010). 06B10, 06B25.

The Einstein action for algebras of matrix valued functions—Toy models

Two toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi–Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical‐like, a quantum‐like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for M4(R). It is pointed out that a solution to the Palatini variational problem is not necessarily a Levi–Civita connection. In this model, no additional assumptions regarding metrics are made.