Twenty Years of Bialowieza: A Mathematical Anthology

The following sections are included:

• Preface

• Contents

The following sections are included:

• Diffeomorphism Groups and Physical Space

• Diffeomorphism Group Representations and Quantum Mechanics

• Causal Diffeomorphisms and Space-Time

• Diffeomorphism Groups and Particle Configurations

• Unitary Representations: Quasi-invariant Measures and Cocycles

• Comments on Quantization

• Deriving Brackets for Creation and Annihilation Fields

• Infinite Dimensional Configuration Spaces

• Bibliography

We propose that geometric quantization of symplectic manifolds is the arrow part of a functor, whose object part is deformation quantization of Poisson manifolds. The ‘quantization commutes with reduction’ conjecture of Guillemin and Sternberg then becomes a special case of the functoriality of quantization. In fact, our formulation yields almost unlimited generalizations of the Guillemin–Sternberg conjecture, extending it, forexample, to arbitrary Lie groups or even Lie groupoids. Technically, this involves symplectic reduction and Weinstein's dual pairs on the classical side, and Kasparov's bivariant K-theory for C*-algebras (KK-theory) on the quantum side.

A notion of a mechanical system is introduced. On the basis of this idea the relationship of the coherent state method with the Kostant–Souriau geometric quantization and path integral quantization is described.

We survey various results about the Lie algebra of unimodular vector fields and the corresponding group, such as computations of cohomology, extensions, and rigidity properties. This subject is closely related to recent problems in mathematical physics, e.g quantization of branes.

The following sections are included:

• Introduction

• Some Properties of Ricci Type Connections

• Examples of Manifolds with Ricci Type Connections

• Complex Projective Space

• Bibliography

This paper gives a brief review of some of the results of the authors regarding Banach Lie- Poisson spaces. The category of these spaces is described and the properties of the morphisms is discussed. Its relationship to W*-algebras and Banach Lie algebras is also presented. Several examples are given.

This contribution is an overview of the ergodic, the entropy and the stochastic properties of weighted shift operators associated with their spectral characteristics. The presented material forms in essence an overview of a series of lectures given by the authors at several Białowieża conferences.

The following sections are included:

• Invariant Vector Fields on Homogeneous Spaces

• Tangent Bundles of Symmetric Spaces

• Commuting Vector Fields and Flows

• The Ergodic Arnold–Liouville Theorem

• Bibliography

The following sections are included:

• Introduction

• Percolation

• The Ising Model

• Possible Construction of a Conformal Field Theory

• Calculations for the Symmetric Group

• Final Remarks

• Bibliography

We describe the geometrical ladder of equations for Abelian bundles and gerbes, as well as higher generalisations, in terms of the cohomology of an operator that combines de Rham and Čech cohomology.

The concept of a quantum group of unitary operators is relevant for the theory of non-compact locally compact quantum groups. It plays a similar rôle as the concept of a quantum matrix group in the compact case. To show the usefulness of this notion we present an approach to a construction of the quantum ‘az + b’ group based on this idea. A brief survey of the present status of quantum group theory is also included.