From Poisson Brackets to Universal Quantum Symmetries

From Poisson Brackets to Universal Quantum Symmetries

18.08.2014 -  22.08.2014 | Warsaw



The subject of the conference is at the intersection of differential geometry, abstract algebra, functional analysis (in particular operator algebras), topology and mathematical physics.

The goal is to present studies of the quantization of classical dynamical systems and their symmetries with special emphasis on relation between the Hamiltonian reduction in the classical and quantum context. The subject has two main strands. The first is deformation quantization of Poisson manifolds and related symmetries, the topic that has been of great interest for mathematics and physics communities for many years. This theory has a well developed "toolbox" centered on the formality type theorems and related methods. While deformation quantization of Poisson manifolds and stacks is by now well understood, the associated deformation quantization of symmetries of the Poisson structure is not really understood, but of major importance in the applications of the theory, both in the mathematical physics (string theory) and in the analysis (semiclassical limit and its applications). The second strand centers around C*-algebraic version of deformation quantization, a counterpart of geometric quantization in classical quantum mechanics and representation theory. This involves, on one hand, understanding of C*-algebraic quantum groups and their actions, and on the other hand, study of appropriate notion of Hamiltonian actions and Hamiltonian reduction, which in the context of C*-algebras and their symmetry (quantum-) groups is at the moment very poorly understood. Thus the goal is to bring together specialists in both subjects (formal vs. C*-algebraic quantization and symmetries) to help resolve problems in both fields. For instance, formal deformation quantization and its methods can be used in certain cases to construct non-formal deformations of algebras and their symmetries while, from the other side, the notions and structures uncovered in C*-algebraic quantum groups can be used to better understand phenomena involved in the formal deformation quantization of symmetries of Poisson manifolds.