Deformation Quantization of Poisson manifolds (original) (raw)

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.

Deformation Quantisation and Connections

European Women in Mathematics - Proceedings of the 13th General Meeting, 2010

After a brief introduction to the concept of formal Deformation Quantisation, we shall focus on general konwn constructions of star products, enhancing links with linear connections. We first consider the symplectic context: we recall how any natural star product on a symplectic manifold determines a unique symplectic connection and we recall Fedosov's construction which yields a star product, given a symplectic connection. In the more general context, we consider universal star products, which are defined by bidifferential operators expressed by universal formulas for any choice of a linear torsionfree connection and of a Poisson structure. We recall how formality implies the existence (and classification) of star products on a Poisson manifold. We present Kontsevich formality on R d and we recall how Cattaneo-Felder-Tomassini globalisation of this result proves the existence of a universal star product.

Deformation Quantization : an introduction

2005

I shall focus, in this presentation of Deformation Quantization, to some mathematical aspects of the theory of star products. 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. We will then introduce the notion of formality and its link with star products, give a flavour of Kontsvich’s construction of a formality for R and a sketch of the globalisation of a star product on a Poisson manifold following the approach of Cattaneo, Felder and Tomassini. The last lectures will be devoted to the action of a Lie group on a deformed product. The notes here are a brief summary of those lectures; I start with a Further Reading section which includes expository papers with details of what is presented. I shall not mention many very important aspects of the deformation quantisation programme such as reduction p...

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.

Poisson Geometry, Deformation Quantisation and Group Representations

2005

These notes, based on the mini-course given at the PQR2003 Euroschool held in Brussels in 2003, aim to review Kontsevich's formality theorem together with his formula for the star product on a given Poisson manifold. A brief introduction to the employed mathematical tools and physical motivations is also given.

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.

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 ...

Variations on deformation quantization

2000

I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness --up to equivalence-- of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manifold and the construction of some convergent star products on Hermitian symmetric spaces. Those subjects will appear in a promenade through the history of existence and equivalence in deformation quantization.