Poisson Geometry, Deformation Quantisation and Group Representations (original) (raw)

Deformation Quantization of Poisson manifolds

2005

I shall focus, in this presentation of Deformation Quantization, on the construction of star products on symplectic and Poisson manifolds. The first lectures will be about the general concept of Deformation Quantisation, with examples, with Fedosov’s construction of a star product on a symplectic manifold and with the classification of star products on a symplectic manifold. The next lectures will introduce the notion of formality and its link with star products, give a flavour of Kontsvich’s construction of a formality for Rd and a sketch of the globalisation of a star product on a Poisson manifold following the approach of Cattaneo, Felder and Tomassini. In the last lecture I shall only briefly mention different aspects of the deformation quantisation programme such as action of a Lie group on a deformed product, reduction procedures in deformation quantisation, states and representations in deformed algebras, convergence of deformations, leaving out many interesting and deep aspe...

Deformation quantisation of Poisson manifolds

2011

In the last lectures I only briefly mentioned different aspects of the deformation quantisation programme such as action of a Lie group on a deformed product, reduction procedures in deformation quantisation, states and representations in deformed algebras, convergence of deformations, leaving out many interesting and deep aspects of the theory (such as traces and index theorems, extension to fields theory); these are not included in these notes and I include a bibliography with many references to those topics.

From local to global deformation quantization of Poisson manifolds

Duke Mathematical Journal, 2002

Dedicated to James Stasheff on the occasion of his 65th birthday ABSTRACT. We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.

SG ] 2 9 A pr 2 01 9 Group Actions in Deformation Quantisation

2019

This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements -with intersectionthe introduction to formal deformation quantization and group actions published in [38], corresponding to a course given in Villa de Leyva in July 2015. After an introduction to the concept of deformation quantization, we briefly recall existence, classification and representation results for formal star products. We come then to results concerning the notion of formal star products with symmetries; one has a Lie group action (or a Lie algebra action) compatible with the Poisson structure, and one wants to consider star products such that the Lie group acts by automorphisms (or the Lie algebra acts by derivations). We recall in particular the link between left invariant star products on Lie groups and Drinfeld twists, and the notion of universal deformation formulas. Classically, symmetries are particularly interesting when they are ...

Deformation quantization and the action of Poisson vector fields

Lobachevskii Journal of Mathematics, 2017

As one knows, for every Poisson manifold M there exists a formal noncommutative deformation of the algebra of functions on it; it is determined in a unique way (up to an equivalence relation) by the given Poisson bivector. Let a Lie algebra g act by derivations on the functions on M. The main question, which we shall address in this paper is whether it is possible to lift this action to the derivations on the deformed algebra. It is easy to see, that when dimension of g is 1, the only necessary and sufficient condition for this is that the given action is by Poisson vector fields. However, when dimension of g is greater than 1, the previous methods do not work. In this paper we show how one can obtain a series of homological obstructions for this problem, which vanish if there exists the necessary extension.

Group Actions in Deformation Quantisation

arXiv (Cornell University), 2019

This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements-with intersection-the introduction to formal deformation quantization and group actions published in [38], corresponding to a course given in Villa de Leyva in July 2015. After an introduction to the concept of deformation quantization, we briefly recall existence, classification and representation results for formal star products. We come then to results concerning the notion of formal star products with symmetries; one has a Lie group action (or a Lie algebra action) compatible with the Poisson structure, and one wants to consider star products such that the Lie group acts by automorphisms (or the Lie algebra acts by derivations). We recall in particular the link between left invariant star products on Lie groups and Drinfeld twists, and the notion of universal deformation formulas. Classically, symmetries are particularly interesting when they are implemented by a moment map and we give indications to build a corresponding quantum moment map. Reduction is a construction in classical mechanics with symmetries which allows to reduce the dimension of the manifold; we describe one of the various quantum analogues which have been considered in the framework of formal deformation quantization. We end up by some considerations about convergence of star products.

Classical covariant Poisson structures and Deformation Quantization

2014

Starting with the well-defined product of quantum fields at two spacetime points, we explore an associated Poisson structure for classical field theories within the deformation quantization formalism. We realize that the induced star-product is naturally related to the standard Moyal product through the causal Green functions connecting points in the space of classical solutions to the equations of motion. Our results resemble the Peierls-DeWitt bracket analyzed in the multisymplectic context. Once our star-product is defined we are able to apply the Wigner-Weyl map in order to introduce a generalized version of Wick's theorem. Finally, we include a couple of examples to explicitly test our method: the real scalar field and the bosonic string. For both models we have encountered causal generalizations of the creation/annihilation relations, and also a causal generalization of the Virasoro algebra in the bosonic string case.

Star products and quantization of poisson-lie groups

Journal of Geometry and Physics, 1992

We prove some theorems by Drinfeld about solutions of the triangular quantum Yang-Baxter equation and corresponding quantum groups. These theorems are to be understood in the natural setting of invariant star products on a Lie group. We also set out and prove another theorem about the invariant Hochschild cohomological meaning of the quantum Yang-Baxter equation, which underlies the others.

Deformation quantization of Poisson algebras

Proceedings of the Japan Academy. Series A, Mathematical sciences, 1992

O. Introduction. Let M be a C Poisson manifOld, and C(M) the set .of all C-valued C functions on M. In what follows, we put a=C-(M) for simplicity. By definition of Poisson manifolds, there exists a bilinear map. { }'a a-a, called the Poisson bracket, with the following properties" For anyf, g, hea,