On -representations of the Hopf -algebra associated with the quantum group
On the index of nonlocal elliptic operators for compact Lie groups
We consider a class of nonlocal operators associated with a compact Lie group G acting on a smooth manifold. A notion of symbol of such operators is introduced and an index formula for elliptic elements is obtained. The symbol in this situation is an element of a noncommutative algebra (crossed product by G) and to obtain an index formula, we define the Chern character for this algebra in the framework of noncommutative geometry.
On the problem of classifying simple compact quantum groups
We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups.
Quantum Fibre Bundles. An Introduction
An approach to construction of a quantum group gauge theory based on the quantum group generalisation of fibre bundles is reviewed.
Quantum geometry of noncommutative Bernoulli shifts
We construct an example of a noncommutative dynamical system defined over a two dimensional noncommutative differential manifold with two positive Lyapunov exponents equal to ln d each. This dynamical system is isomorphic to the quantum Bernoulli shift on the half-chain with the quantum dynamical entropy equal to 2 ln d. This result can be interpreted as a noncommutative analog of the isomorphism between the classical one-sided Bernoulli shift and the expanding map of the circle and moreover as...
Quantum permutation groups: a survey
This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks...
Quantum spaces: notes and comments on a lecture by S. L. Woronowicz
Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras.
Restricting the bi-equivariant spectral triple on quantum SU(2) to the Podleś spheres
It is shown that the isospectral bi-equivariant spectral triple on quantum SU(2) and the isospectral equivariant spectral triples on the Podleś spheres are related by restriction. In this approach, the equatorial Podleś 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...
Secondary invariants for Frechet algebras and quasihomomorphisms.
Sobolev theory for non commutative tori
Some remarks on quantum and braided group gauge theory
We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzeziński and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. The bosonisation of any braided group provides us a trivial principal bundle in three ways.
Symmetrized non-commutative tori.
Symplectic models of groups with noncommutative spaces
The -pseudodifferential calculus on Galois coverings and a higher Atiyah-Patodi-Singer index theorem
The closed Friedman world model with the initial and final singularities as a non-commutative space
The most elegant definition of singularities in general relativity as b-boundary points, when applied to the closed Friedman world model, leads to the disastrous situation: both the initial and final singularities form the single point of the b-boundary which is not Hausdorff separated from the rest of space-time. We apply Alain Connes' method of non-commutative geometry, defined in terms of a C*-algebra, to this case. It turns out that both the initial and final singularities can be analysed as...
The cyclic homology of
The Frölicher-Nijenhuis bracket in non-commutative differential geometry.
The Local Index Formula in Noncommutative Geometry.
The quantum spheres and their embedding into quantum Minkowski space-time.