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Quantum Fibre Bundles. An Introduction

Tomasz Brzeziński (1997)

Banach Center Publications

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

Robert Alicki (1998)

Banach Center Publications

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

Teodor Banica, Julien Bichon, Benoît Collins (2007)

Banach Center Publications

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...

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