Markus Engeli - Academia.edu (original) (raw)
Uploads
Papers by Markus Engeli
The text consists of two parts. In the first two chapters, we construct a trace on the deformed a... more The text consists of two parts. In the first two chapters, we construct a trace on the deformed algebra of functions on a supersymplectic manifold. It is a generalization of the trace that has been given by Feigin, Felder and Shoikhet [16] for symplectic manifolds. The main ingredients are a generalization of Fedosov’s connection to supersymplectic manifolds found by Bordemann [9], which is explained in chapter 1, and a Hochschild cocycle for the Weyl-Clifford algebra, which is constructed in chapter 2. Using then a general construction for an isomorphism between the Hochschild homologies of a differential graded algebra and the corresponding twisted algebra, we find a chain map from the Hochschild homology of the algebra of deformed functions to the de Rham cohomology of the symplectic manifold. In Hochschild degree 0, we get a volume form on the symplectic manifold depending on a deformed function. Its integral over the manifold defines a unique normalized trace. In the chapters 3...
Annales scientifiques de l'École normale supérieure, 2008
Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on... more Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology HH 2n (Dn, D * n) of the algebra of differential operators on a formal neighbourhood of a point. If D is the identity, the formula reduces to the Riemann-Roch-Hirzebruch formula. Résumé. Soit D un opérateur différentiel holomorphe opérant sur les sections d'un fibré vectoriel holomorphe sur une variété complexe de dimension n. Nous démontrons une formule, conjecturée par Feigin et Shoikhet, donnant le nombre de Lefschetz de D comme intégrale d'une forme différentielle sur la variété. La classe de cette forme différentielle est obtenue, via la géométrie différentielle formelle, du générateur canonique de la cohomologie de Hochschild HH 2n (Dn, D * n) de l'algèbre des opérateurs différentiels sur un entourage formel d'un point. Si D est l'identité, la formule se réduità la formule de Riemann-Roch-Hirzebruch. Contents 1. Introduction 1 2. Hochschild homology of the algebra of differential operators 3 3. The third trace 10 4. The first trace is proportional to the third.. . 11 5.. .. and so is the second 14 6. Asymptotic topological quantum mechanics 15 Appendix A. Triangulations and signs 20 Appendix B. Heat kernel estimates and asymptotic expansion 20 References 30
We quantize the Poisson-Lie group SL(2,R)^* as a bialgebra using the product of Kontsevich. The c... more We quantize the Poisson-Lie group SL(2,R)^* as a bialgebra using the product of Kontsevich. The coproduct is a deformation of the coproduct that comes from the group structure. The resulting bialgebra structure is isomorphic to the quantum universal enveloping algebra U_hsl(2,R).
The text consists of two parts. In the first two chapters, we construct a trace on the deformed a... more The text consists of two parts. In the first two chapters, we construct a trace on the deformed algebra of functions on a supersymplectic manifold. It is a generalization of the trace that has been given by Feigin, Felder and Shoikhet [16] for symplectic manifolds. The main ingredients are a generalization of Fedosov’s connection to supersymplectic manifolds found by Bordemann [9], which is explained in chapter 1, and a Hochschild cocycle for the Weyl-Clifford algebra, which is constructed in chapter 2. Using then a general construction for an isomorphism between the Hochschild homologies of a differential graded algebra and the corresponding twisted algebra, we find a chain map from the Hochschild homology of the algebra of deformed functions to the de Rham cohomology of the symplectic manifold. In Hochschild degree 0, we get a volume form on the symplectic manifold depending on a deformed function. Its integral over the manifold defines a unique normalized trace. In the chapters 3...
Annales scientifiques de l'École normale supérieure, 2008
Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on... more Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology HH 2n (Dn, D * n) of the algebra of differential operators on a formal neighbourhood of a point. If D is the identity, the formula reduces to the Riemann-Roch-Hirzebruch formula. Résumé. Soit D un opérateur différentiel holomorphe opérant sur les sections d'un fibré vectoriel holomorphe sur une variété complexe de dimension n. Nous démontrons une formule, conjecturée par Feigin et Shoikhet, donnant le nombre de Lefschetz de D comme intégrale d'une forme différentielle sur la variété. La classe de cette forme différentielle est obtenue, via la géométrie différentielle formelle, du générateur canonique de la cohomologie de Hochschild HH 2n (Dn, D * n) de l'algèbre des opérateurs différentiels sur un entourage formel d'un point. Si D est l'identité, la formule se réduità la formule de Riemann-Roch-Hirzebruch. Contents 1. Introduction 1 2. Hochschild homology of the algebra of differential operators 3 3. The third trace 10 4. The first trace is proportional to the third.. . 11 5.. .. and so is the second 14 6. Asymptotic topological quantum mechanics 15 Appendix A. Triangulations and signs 20 Appendix B. Heat kernel estimates and asymptotic expansion 20 References 30
We quantize the Poisson-Lie group SL(2,R)^* as a bialgebra using the product of Kontsevich. The c... more We quantize the Poisson-Lie group SL(2,R)^* as a bialgebra using the product of Kontsevich. The coproduct is a deformation of the coproduct that comes from the group structure. The resulting bialgebra structure is isomorphic to the quantum universal enveloping algebra U_hsl(2,R).