The Duflo isomorphism and its relatives
Workshop on the Duflo isomorphism and related isomorphisms in algebra and geometry
What 


When 
Feb 18, 2008 09:00 AM
to Feb 20, 2008 06:00 PM 
Where  JohannesGutenbergUniversität Mainz 
Contact Name  Marc NieperWißkirchen 
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Introduction
By work of Chevalley, HarishChandra, Dixmier, and Duflo one knows that there exists an isomorphism of algebras between the set H(g, S g) of invariant polynomials on the dual of a finite dimensional Lie algebra g (over C) and the center of its enveloping algebra. This isomophism is called the Duflo isomorphism. It happens to be a composition of the wellknown PoincaréBirkhoffWitt isomorphism (which is only an isomorphism on the level of vector spaces) and a map H(g, Sg) > H(g, Sg) which is induced by the power series sinh(x/2)/(x/2).
The same power series also appears in Kontsevich's claim that the Hochschild cohomology of a complex manifold is isomorphic as an algebra to the cohomology ring of the polyvector fields on this manifold. His claim has been confirmed recently by Van den Bergh and Calaque. Formally, the Hochschild cohomology plays the rôle of the center of an universal enveloping algebra, while the cohomology ring of polyvector fields corresponds to the invariant polynomials from above.
The wheeling theorem by BarNatan, Le, and Thurston, which shows that two spaces of graph homology are isomorphic as algebras, is again of the same form. This is again related to complex geometry by the theory of RozanskyWitten invariants on holomorphic symplectic manifolds.
Furthermore, the power series is related with the Ahat genus and the Todd genus, which show up in various index theorems, e.g. in the RiemannRoch theorems.
Aims
In this workshop, we want to study the Duflo isomorphism, and its generalisations and applications in geometry as described above. Talks will be given by the participants of the workshop on a level of a seminar that can be followed in detail by graduate students.
Programme
 Arrival
 Arrival is possible on Sunday evening or Monday morning
 Monday 10:3013:00
 Reception
The Duflo isomorphism (Ansgar Schneider)
 Monday 13:0014:00
 Lunch
 Monday 14:0018:30
 The Duflo isomorphism, Kontsevich's formality theorem, and the Poisson cohomology of Sg (Christian Blohmann)
Coffee break
The Duflo isomorphism for metrical Lie algebras via noncommutative ChernWeil theory (Timo Schürg)  Tuesday 8:3013:00
 Graph homology and the Wheeling Theorem (Frank Ditsche)
Coffee break
The Wheeling Theorem via noncommutative ChernWeil theory (Marc NieperWißkirchen)  Tuesday 13:0014:00
 Lunch break
 Tuesday 14:0018:30
 The Atiyah class, Hochschild cohomology, and the RiemannRoch theorem (Ulrich Bunke)
Coffee break
The RozanskyWitten weight system (Emanuele Macrì)  Tuesday evening
 Social event (informal dinner)
 Wednesday 8:3013:00
 The Mukai paring and Hochschild (co)homology of compact complex manifolds (Christian Lehn)
Coffee break
The Kontsevich isomorphism (Sergey Mozgovoy)  Wednesday 13:0014:00
 Lunch
 Departure
 The workshop ends on Wednesday after lunch.
The Talks
 Talk 1: The Duflo isomorphism
 Introduction to the classical Duflo isomorphism, idea of proof, the universal enveloping algebra as distributions on G.
Speaker: Ansgar Schneider (Regensburg)
Literature: Duflo.  Talk 2: The Duflo isomorphism, Kontsevich's formality theorem, and the Poisson cohomology of Sg
 The proof of Duflo's theorem via Kontsevich's formality theorem and the extension to the Poisson cohomology.
Speaker: Christian Blohmann (Regensburg)
Literature: Pevzner/Torossian  Talk 3: The Duflo isomorphism for metrical Lie algebras via noncommutative ChernWeil theory
 The ChernWeil map for noncommutative gdifferential algebras with connection and its multiplicativity up to homotopy; the Duflo isomorphism for metrical Lie algebras as a corollary.
Speaker: Timo Schürg (Mainz)
Literature: Alekseev/Meinrenken, Alekseev/Meinrenken  Talk 4: Graph homology and the Wheeling Theorem
 Jacobi diagrams and graph homology; analogy of the PoincaréBirkhoffWitt isomorphism in graph homology; the proof of the Wheeling Theorem via the Kontsevich integral
Speaker: Frank Ditsche (Mainz)
Literature: BarNatan/Le/Thurston  Talk 5: The Wheeling Theorem via noncommutative ChernWeil theory
 Ideas of a combinatorial proof of the Wheeling theorem by drawing analogies to the noncommutative ChernWeil map.
Speaker: Marc NieperWißkirchen (Mainz)
Literature: Kricker  Talk 6: The Atiyah class, Hochschild cohomology and the RiemannRoch theorem
 The Atiyah class and the shifted tangent sheaf T[1] as a Lie algebra; the universal enveloping algebra of T[1] and the Hochschild homology and cohomology complex; the RiemannRoch theorem for Hodge cohomology
Speaker: Ulrich Bunke (Regensburg)
Literature: Markarian/Ramadoss  Talk 7: The RozanskyWitten weight system
 The shifted tangent sheaf of a holomorphic symplectic manifold as a metric Lie algebra and the RozanskyWitten weight system; implications of the Wheeling theorem for holomorphic symplectic manifolds
Speaker: Emanuele Macrì (Bonn)
Literature: Roberts/Willerton  Talk 8: The Mukai pairing and Hochschild (co)homology of compact complex manifolds
 The Hochschild structure of smooth compact spaces, the Mukai pairing and the isomorphisms to the Hodge structure.
Speaker: Christian Lehn (Mainz)
Literature: Caldararu/Willerton, Caldararu  Talk 9: The Kontsevich isomorphism
 The fact that the sheaf of polyvector fields is isomorphic as a Gerstenhaber algebra in the derived category of sheaves of Cmodules to the sheaf of polydifferential operations.
Speaker: Sergey Mozgovoy (Wuppertal)
Literature: Calaque/Van den Bergh
Participants
This workshop is addressed to nonspecialists in the field who want to learn more about this beautiful part of mathematics  either to broaden their mathematical horizon or in relation to their own work.
If you want to participate in this workshop, please email as soon as possible to the address given.
Support
Support (tickets, accommodation expenses) is available for members of the SFB/TR 45 and in addition for all those who volunteer to give a talk.
Organisers
This workshop is organised by Ulrich Bunke (Regensburg) and Marc NieperWißkirchen (Mainz).
Literature
 Alekseev, A.; Meinrenken, E.: The noncommutative Weil algebra
 Alekseev, A.; Meinrenken, E.: Lie theory and the ChernWeil homomorphism
 BarNatan, D.; Le, T.; Thurston, D.: Two applications of elementary knot theory to Lie algebras and Vassiliev invariants
 Van den Bergh, M.; Calaque, D.: Hochschild cohomology and Atiyah classes
 Calaque, D., Rossi, C.: Lectures on Duflo isomorphisms in Lie algebras and complex geometry
 Caldararu, A.: The Mukai pairing II: the HochschildKostantRosenberg isomorphism
 Caldararu, A.; Willerton, S.: The Mukai pairing, I: a categorical approach
 Duflo, M.: Opérateurs différentiels biinvariants sur un groupe de Lie
 Kashiwara, M.; Vergne, M.: The CampbellHausdorff fromula and invariant hyperfunctions
 Kontsevich, M.: Deformation quantization of Poisson manifolds
 Kricker, A.: Noncommutative ChernWeil theory and the combinatorics of wheeling
 Markarian, N.: The Atiyah class, Hochschild cohomology and the RiemannRoch theorem
 Pevzner, M.; Torossian, Ch.: Isomorphisme de Duflo et la cohomologie tangentielle
 Ramadoss, A.: The relative RiemannRoch theorem from Hochschild homology
 Roberts, J.; Willerton, S.: On the RozanskyWitten weight systems
 Shoikhet , B.: Tsygan formality and Duflo formula