Papers by Martin Bordemann
A gentle introduction to Drinfel'd associators
arXiv (Cornell University), Apr 14, 2023
Http Www Theses Fr, Nov 12, 2012
Je voudrais remercier dans ces lignes les différentes personnes m'ayant soutenu et encouragé dura... more 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 Strasbourg assister à des cours ou des conférences. Je remercie chaleureusement M me 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 récents échanges, partages et découvertes. Je remercie vivement M me Fricker, pour la gestion et résolution d'innombrables détails administratifs, M me 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 retraites 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.
(Bi)Modules, morphisms, and reduction of star-products: the symplectic case, foliations, and obstructions
eprint arXiv:q-alg/9605038, May 1, 1996
We show that for every vector bundle E over any given symplectic manifold there exists an explici... more We show that for every vector bundle E over any given symplectic manifold there exists an explicitly given super Poisson bracket on the space of sections of the dual Grassmann bundle associated to E built out of the symplectic structure of M , a fibre metric on E and a connection in E compatible with the given fibre metric. Moreover, we construct a deformation quantization for this space of sections by means of a Fedosov type procedure.
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Rack-Bialgebras and Star-Products on duals of Leibniz algebras
Journal für die reine und angewandte Mathematik (Crelles Journal), 2010
In this paper we establish a notion of deformation quantization of a surjective submersion which ... more In this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a principal fibre bundle: the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold. In case of a principal fibre bundle we require in addition invariance under the principal action. We prove existence and uniqueness of such deformations. The commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operators. Applications to noncommutative gauge field theories and phase space reduction of star products are discussed.
Communications in Mathematical Physics, 1993
The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, whic... more The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into the r-s-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matrices r and s are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrix c. It is proposed that all these Poisson brackets taken together are representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.
2-dimensional nonlinear sigma models: zero curvature and Poisson structure
Aspects of Mathematics, 1994
ABSTRACT In this chapter some aspects of two-dimensional nonlinear sigma models are discussed. Th... more ABSTRACT In this chapter some aspects of two-dimensional nonlinear sigma models are discussed. The solutions to the field equations are (pseudo-)harmonic maps of twodimensional Minkowski space into some homogeneous Riemannian manifolds. In Section 2 we shall give a brief introduction to the physics of these models and discuss the conditions under which a zero curvature ansatz built out of the conserved currents is possible: Theorem (2.1) which is an extended version of old work of the physicists K. Pohlmeyer [29] and H. Eichenherr & M. Forger [11] shows under weak technical assumptions that such an ansatz works if and only if the target manifold is a (pseudo-)Riemannian symmetric space. We have included the rather technical, mainly Lie algebraic proof because the result is new (although not very surprising). The use of symmetric spaces in this context had recently been rediscovered in the mathematics literature (see, for example [31]) and applied to harmonic tori, see for...
arXiv: Differential Geometry, 2016
In this article, we present an integration of any real finite-dimensional Leibniz algebra as a Li... more In this article, we present an integration of any real finite-dimensional Leibniz algebra as a Lie rack which reduces in the particular case of a Lie algebra to the ordinary connected simply connected Lie group. The construction is not functorial.
In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. Firs... more In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the -invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.
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Papers by Martin Bordemann