The Duflo isomorphism and its relatives

Introduction

By work of Chevalley, Harish-Chandra, 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 well-known Poincaré-Birkhoff-Witt 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 Bar-Natan, 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 Rozansky-Witten invariants on holomorphic symplectic manifolds.

Furthermore, the power series is related with the A-hat genus and the Todd genus, which show up in various index theorems, e.g. in the Riemann-Roch 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 10:30-13:00

Reception
The Duflo isomorphism (Ansgar Schneider)

Monday 13:00-14:00

Lunch

Monday 14:00-18:30

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 non-commutative Chern-Weil theory (Timo Schürg)

Tuesday 8:30-13:00

Graph homology and the Wheeling Theorem (Frank Ditsche)
Coffee break
The Wheeling Theorem via non-commutative Chern-Weil theory (Marc Nieper-Wißkirchen)

Tuesday 13:00-14:00

Lunch break

Tuesday 14:00-18:30

The Atiyah class, Hochschild cohomology, and the Riemann-Roch theorem (Ulrich Bunke)
Coffee break
The Rozansky-Witten weight system (Emanuele Macrì)

Tuesday evening

Social event (informal dinner)

Wednesday 8:30-13:00 

The Mukai paring and Hochschild (co-)homology of compact complex manifolds (Christian Lehn)
Coffee break
The Kontsevich isomorphism (Sergey Mozgovoy)

Wednesday 13:00-14:00

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 non-commutative Chern-Weil theory

The Chern-Weil map for non-commutative g-differential 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 Poincaré-Birkhoff-Witt isomorphism in graph homology; the proof of the Wheeling Theorem via the Kontsevich integral
Speaker: Frank Ditsche (Mainz)
Literature: Bar-Natan/Le/Thurston

Talk 5: The Wheeling Theorem via non-commutative Chern-Weil theory

Ideas of a combinatorial proof of the Wheeling theorem by drawing analogies to the non-commutative Chern-Weil map.
Speaker: Marc Nieper-Wißkirchen (Mainz)
Literature: Kricker

Talk 6: The Atiyah class, Hochschild cohomology and the Riemann-Roch 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 Riemann-Roch theorem for Hodge cohomology
Speaker: Ulrich Bunke (Regensburg)
Literature: Markarian/Ramadoss

Talk 7: The Rozansky-Witten weight system

The shifted tangent sheaf of a holomorphic symplectic manifold as a metric Lie algebra and the Rozansky-Witten 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 C-modules to the sheaf of polydifferential operations.
Speaker: Sergey Mozgovoy (Wuppertal)
Literature: Calaque/Van den Bergh

Participants

This workshop is addressed to non-specialists 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 e-mail 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 Nieper-Wißkirchen (Mainz).

Literature

• Alekseev, A.; Meinrenken, E.: The non-commutative Weil algebra
• Alekseev, A.; Meinrenken, E.: Lie theory and the Chern-Weil homomorphism
  
• Bar-Natan, 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 Hochschild-Kostant-Rosenberg isomorphism
• Caldararu, A.; Willerton, S.: The Mukai pairing, I: a categorical approach
  
• Duflo, M.: Opérateurs différentiels bi-invariants sur un groupe de Lie
• Kashiwara, M.; Vergne, M.: The Campbell-Hausdorff fromula and invariant hyperfunctions
  
• Kontsevich, M.: Deformation quantization of Poisson manifolds
• Kricker, A.: Non-commutative Chern-Weil theory and the combinatorics of wheeling
• Markarian, N.: The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem
  
• Pevzner, M.; Torossian, Ch.: Isomorphisme de Duflo et la cohomologie tangentielle
• Ramadoss, A.: The relative Riemann-Roch theorem from Hochschild homology
• Roberts, J.; Willerton, S.: On the Rozansky--Witten weight systems
  
• Shoikhet , B.: Tsygan formality and Duflo formula