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Formality Related to Universal Enveloping Algebras and Study of Hom-(Co)Poisson Algebras Olivier Elchinger

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Formality Related to Universal Enveloping Algebras and Study of Hom-(Co)Poisson Algebras Olivier Elchinger Formality related to universal enveloping algebras and study of Hom-(co)Poisson algebras Olivier Elchinger To cite this version: Olivier Elchinger. Formality related to universal enveloping algebras and study of Hom-(co)Poisson algebras. Mathematics [math]. Université de Haute Alsace - Mulhouse, 2012. English. tel-01225555 HAL Id: tel-01225555 https://tel.archives-ouvertes.fr/tel-01225555 Submitted on 6 Nov 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Public Domain Université de Haute Alsace École Doctorale Jean-Henri Lambert Laboratoire de Mathématiques, Informatique et Applications Formality related to universal enveloping algebras and study of Hom-(co)Poisson algebras presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics proposed by Olivier Elchinger Thesis directed by Martin Bordemann and Abdenacer Makhlouf defended the 12th of November 2012 in front of the jury composed by: M. Benjamin Enriquez Université de Strasbourg (Referee) M. Joakim Arnlind University of Linköping (Referee) M. Camille Laurent-Gengoux Université de Metz (Referee) M. Martin Schlichenmaier Université de Luxembourg M. Martin Bordemann Université de Haute Alsace (Advisor) M. Abdenacer Makhlouf Université de Haute Alsace (Advisor) M. Augustin Fruchard Université de Haute Alsace (Invited member) Remerciements Je voudrais remercier dans ces lignes les différentes personnes m’ayant soutenu et encouragé durant ces trois années de thèse. Tout d’abord mes parents, qui m’ont suivi toute ma scolarité jusque dans les études supérieures. Mon épouse, qui s’est investie pour moi dans diverses démarches administratives, trajets, et au quotidien ; et me signalait tous les progrès successifs faits par notre fils. Fabien ensuite, qui m’a fait part de l’opportunité d’une place en thèse sur Mulhouse. Takashi et Elodie, ainsi que Philippe, pour leur hospitalité lorsque j’avais besoin de retourner sur Stras- bourg assister à des cours ou des conférences. Je remercie chaleureuse- ment Mme Wurth pour sa disponibilité et son accueil à mon égard. Je tiens aussi à exprimer ma gratitude à Celui qui m’a protégé sur la route plusieurs fois plus longue que prévue. Je souhaite également remercier les membres du laboratoire de m’avoir bien accueilli parmi eux, pour les discussions, énigmes et réflexions menées lors des repas et poursuivies quelques fois plus avancé dans l’après-midi. Merci à Abdennour pour les ré- cents échanges, partages et découvertes. Je remercie vivement Mme Fricker, pour la gestion et résolution d’innombrables détails ad- ministratifs, Mme Robert pour les échanges prompts avec l’école doctorale, ainsi que les personnes des ressources humaines pour leurs conseils. Merci aussi à mes directeurs de thèse pour avoir cru en moi, m’avoir introduit dans leurs recherches, expliqué et ré-expliqué les points moins évidents, fait et refait des calculs gigantesques, envoyé suivre des conférences et des écoles d’été en diverses re- traites et conseillé durant tout ces travaux. Merci de m’avoir offert à de nombreuses reprises le couvert et/ou le café. Je tiens enfin à remercier les rapporteurs et membres du jury d’avoir accepté de prendre le temps de regarder mon travail, et de leurs remarques et critiques pour l’améliorer. iii iv Contents Remerciements iii Contents v General introduction ix I Formality and L1 structures 1 1 Preliminaries 5 1.1 Graded structures ................... 6 1.1.1 Graded vector spaces ............... 6 1.1.2 Shifted spaces ................... 8 1.1.3 Tensor bialgebra .................. 8 1.1.4 Symmetric bialgebra . 14 1.1.5 Universal enveloping algebras . 17 1.2 Cohomology and deformations . 18 1.2.1 Hochschild cohomology . 18 1.2.2 Chevalley-Eilenberg cohomology . 19 1.2.3 Properties ..................... 20 1.2.4 Link with deformations . 20 2 Kontsevich formality 23 2.1 Definitions ........................ 24 2.2 Case of associative algebras . 25 2.2.1 Differential graded Lie algebras . 25 2.2.2 Sections ...................... 27 2.2.3 Formality ..................... 27 2.2.4 Application to deformation . 31 2.3 Case of Lie algebras . 32 2.3.1 Polyvectors fields and polynomials functions . 32 2.3.2 Linear Poisson structure . 33 2.3.3 Formality ..................... 34 2.4 Universal enveloping algebras . 35 2.5 Perturbation Lemma . 36 2.5.1 Two-degree cohomology . 39 2.5.2 General result ................... 42 v 3 Study of the formality for the free algebras 45 3.1 Description of the spaces . 46 3.1.1 Definitions ..................... 46 3.1.2 Examples for spaces of dimension 0 and 1 . 47 3.1.3 Results for spaces of dimension greater than 2 . 48 3.1.4 Case of a finite dimensional space . 55 3.2 Perturbed formality . 57 3.2.1 Computations for the cocycle part in the case 1 . 58 3.2.2 Computations for the cocycle part in the case 2 . 58 3.2.3 Computations for the coboundary part in the case 1 .......................... 59 3.2.4 Computations for the coboundary part in the case 2 .......................... 62 4 Study of the formality for the Lie algebra so(3) 65 4.1 Description of the Lie algebra so(3) . 66 4.2 Subalgebra of the Chevalley-Eilenberg complex . 67 4.3 Deformation retract of the complex . 70 4.4 Computation of the L structure . 71 1 II Hom-algebraic structures 75 5 Twisting of Hom-(co)algebras 79 5.1 Hom-associative algebras and Hom-Lie algebras . 80 5.1.1 Definitions ..................... 80 5.1.2 Twisting principle . 82 5.1.3 Construction of Hom-Lie algebras . 83 5.2 Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras ......................... 87 5.2.1 Hom-coalgebras and duality . 87 5.2.2 Hom-bialgebra and Hom-Hopf algebra . 90 5.3 Hom-Lie coalgebras and Hom-Lie bialgebras . 92 6 Hom-(co)Poisson structures 95 6.1 Hom-Poisson algebras . 96 6.1.1 Definitions and examples . 96 6.1.2 Twisting principle . 97 6.1.3 Application to Sklyanin algebra . 97 6.1.4 Constructing Hom-Poisson algebras from Hom- Lie algebras .................... 98 6.2 1-operation structures . 100 6.2.1 Flexibles Hom-algebras . 100 6.2.2 Link with Hom-Poisson algebras . 101 6.3 Hom-coPoisson algebras and duality . 103 vi 6.3.1 Link with Hom-Lie bialgebras . 103 6.3.2 Duality . 105 7 Deformation and quantization of Hom-algebras109 7.1 Formal Hom-deformation . 110 7.1.1 Formal deformation of Hom-associative algebras . 110 7.1.2 Deformations of Hom-coalgebras and Hom-Bialgebras112 7.2 Quantization and twisting of ?-products . 112 7.2.1 Twists of Moyal-Weyl ?-product . 114 7.2.2 Twists of the Poisson bracket . 116 7.2.3 Quantization of the Poisson automorphisms . 118 A Computations with Mathematica 125 A.1 Computation of the morphisms of sl(2) . 126 A.2 Hom-Lie structures associated to the Jackson sl(2) bracket . 129 Bibliography 133 vii viii General introduction his thesis aims to study some algebraic aspects of structures T linked to the problem of deformation quantization. At first, we examine the formality for the case of free algebras and for the Lie algebra so(3) and then, we consider deformation quantization for Hom-algebraic structures. The following is about the historic of these subjects, the results are exposed with more details at the beginning of each part. Deformation of structure theories is useful to formalize quan- tum physics. If one has a quantum description of a physical sys- tem, then the passage to the classical description is done letting “Planck’s constant ~ tend to zero”. The reverse operation, i.e. pro- ducing a quantum description from a classical one is called quanti- zation. The algebraic structure considered by classical mechanics is the associative commutative algebra of smooth functions over a symplectic, or more generally, a Poisson manifold. Deformation quantization consists to construct an associative non-commutative multiplication (more precisely a ?-product) on the formal series in ~ with coefficients in this algebra, which encodes the Poisson bracket in the first order. The Poisson structure is then called the quasi-classical limit and the deformation is the ?-product. This point of view, initiated in 1978 in [BFF+78], tries to consider quan- tum mechanics as a deformation from classical mechanics, and the Poincaré group as a deformation of the Galileo group. This pioneering work raises the fundamental questions about the ex- istence and the uniqueness of a deformation quantization for a given Poisson manifold. The first results treated the case of symplectics manifolds. The general case of Poisson manifolds was solved by Kontsevich in 1997 in [Kon03]. He deduced the result by proving a much more general statement, which he called “formality conjecture”. En- ix dowed with the Gerstenhaber bracket, the Hochschild complex of the algebra of smooth functions over a Poisson manifold admits a graded Lie algebraic structure by shift, which controls the defor- mations of the Poisson bracket. Kontsevich shows that this com- plex is linked with its cohomology — which therefore controls the same deformations — by a L -quasi-isomorphism, called a for- mality map. This boils down1 the problem of deformation to the previously solved case. Konstevich shows in particular that the formality criterion is true for symmetric algebras over a finite-dimensional vector space are formal. Bordemann and Makhlouf have examined in [BM08] a slight generalization to the case of universal enveloping algebras.
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