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47
Variations on deformation quantization
, 2000
"... I was asked by the organisers to present some aspects of Deformation Quantization. Moshé has pursued, for more than 25 years, a research program based on the idea that physics progresses in stages, and one goes from one level of the theory to the next one by a deformation, in the mathematical sense ..."
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I was asked by the organisers to present some aspects of Deformation Quantization. Moshé has pursued, for more than 25 years, a research program based on the idea that physics progresses in stages, and one goes from one level of the theory to the next one by a deformation, in the mathematical sense of the word, to be defined in an appropriate category. His study of deformation theory applied to mechanics started in 1974 and led to spectacular developments with the deformation quantization programme. I first met Moshé at a conference in Liège in 1977. A few months later he became my thesis “codirecteur”. Since then he has been one of my closest friends, present at all stages of my personal and mathematical life. I miss him.... 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.
The Hidden Group Structure Of Quantum Groups: Strong Duality, Rigidity And Preferred Deformations.
, 1993
"... : A notion of wellbehaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeldtype and their duals, algebras of coefficients of compact semisimple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantu ..."
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: A notion of wellbehaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeldtype and their duals, algebras of coefficients of compact semisimple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyalproductlike deformations are naturally found for all FRTmodels on coefficients and C 1 functions. Strong rigidity (H 2 bi = f0g) under deformations in the category of bialgebras is proved and consequences are deduced. AMS classification: Primary 17B37, 16W30, 22C05 46H99, 81R50. Running title : Topological quantum groups. (In press in Communications in Mathematical Physics, end of 1993) 1 Universit'e de Bourgogne  Laboratoire de Physique Math'ematique B.P. 138, 21004 DIJON Cedex  FRANCE, email: flato@satie.ubourgogne.fr 2 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 191046395 U.S.A. email: mgersten@mail.sas.upenn.edu and murray@math...
Natural and projectively equivariant quantizations by means of Cartan connections
 Lett. Math. Phys
, 2005
"... Abstract. The existence of a natural and projectively equivariant quantization in the sense of Lecomte [20] was proved recently by M. Bordemann [4], using the framework of ThomasWhitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain a ..."
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Abstract. The existence of a natural and projectively equivariant quantization in the sense of Lecomte [20] was proved recently by M. Bordemann [4], using the framework of ThomasWhitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain an explicit formula in terms of these connections. Our method yields the existence of a projectively equivariant quantization if and only if an sl(m + 1, R)equivariant quantization exists in the flat situation in the sense of [18], thus solving one of the problems left open by M. Bordemann.
Symplectic connections
"... This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Riccitype connections (for which the curvature is entirely determined by the Ricci tensor) i ..."
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This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Riccitype connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching generalization to special connections. A twistorial construction shows a relation between Riccitype connections and complex geometry. We give a construction of Ricciflat symplectic connections. We end up by presenting, through an explicit example, an approach to noncommutative symplectic symmetric spaces.
Formality and Star Products
 Poisson Geometry, Deformation Quantisation and Group Representations, vol. 323 in London Mathematical Society Lecture Note Series, 79–144
, 2005
"... These notes, based on the minicourse 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 ..."
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These notes, based on the minicourse 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.
Quantum surfaces, special functions, and the tunneling effect
, 2001
"... The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with nonLie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace ..."
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The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with nonLie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace formulas over leaves with complex polarization are obtained. The noncommutative product on the leaves incorporates a closed 2form and a measure which (in general) are different from the classical symplectic form and the Liouville measure. The quantum objects are related to some generalized special functions. The difference between classical and quantum geometrical structures could even occur to be exponentially small with respect to the deformation parameter. That is interpreted as a tunneling effect in the quantum geometry. Dedicated to the memory of Professor M. Flato 1
Irreducible highestweight modules and equivariant quantization
, 2005
"... The notion of deformation quantization, motivated by ideas coming from both physics and mathematics, was introduced in classical papers [2, 7, 8]. Roughly speaking, a deformation quantization of a Poisson manifold (P, { ,}) is a formal ..."
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The notion of deformation quantization, motivated by ideas coming from both physics and mathematics, was introduced in classical papers [2, 7, 8]. Roughly speaking, a deformation quantization of a Poisson manifold (P, { ,}) is a formal