Workshop: Kontsevich formality theory and the Duflo isomorphism
Banach Center, 06-09 April 2009

Program

Introduction (15 min)(D. Calaque)

1. presentation and goals of the workshop

Lecture 1 (45 min): Introduction to deformation quantization after Kontsevich I (P. Witkowski)

1. Definition of a differential graded Lie algebra (DGLA)
2. Definition of T_{poly} : vector fields, polyvector fields, Schouten bracket
3. Poisson bracket \Leftrightarrow Maurer-Cartan element (MCE) in T_{poly}
4. Definition of D_{poly} : differential operators, polydifferential operators, Gerstenhaber bracket, Hochschild coboundary operator
5. Associative product \Leftrightarrow MCE in D_{poly}

Lecture 2 (45 min): Introduction to deformation quantization after Kontsevich II (J. Milles)

1. Formal deformation \Rightarrow Poisson structure (semi-classical limit)
2. Statement of the deformation quantization problem
3. Statement of the Hochschild-Kostant-Rosenberg theorem
4. Lie algebra morphisms between DGLAs induces a map between MC elements
5. Consequence: if H-K-R was a Lie morphism, then we would obtain a solution to the deformation quantization problem
6. Default of being a Lie morphism (example)

Lecture 3 (45 min) : Introduction to the formality theorem (A. Kaygun)

1. Reminder on cofree coalgebras
2. Definition of L_\infty -algebras (example: DGLAs are L_\infty -algebras) and L_\infty -morphisms
3. MCE elements are sent to MCE via a L_\infty -morphism
4. Statement of the formality theorem (existence of a L_\infty -morphism)
5. Consequences for deformation quantization

Lecture 4 (1h) : Kontsevich's graphs and formality (E. Hoffbeck)

1. Definition of an admissible graph
2. Polydifferential operator associated to an admissible graph (examples)
3. Sign ambiguity in the previous definition
4. Construction of what-would-be-a-formality-map in terms of graphs (weights of graphs not yet specified)
5. Quadratic identities on weights of graphs \Rightarrow L_\infty -condition

Lecture 5 (1h30) : Angle forms on compactified configuration spaces (J.-M. Oudom)

1. Configuration spaces C_{n,m}^+ and their compactifications
2. Combinatorics of the boundary of the compactified configuration spaces
3. Angle forms on configuration spaces

Lecture 6 (45min) : Proof of Kontsevich's formality theorem (Y. Frégier)

1. Weights of graphs as integrals of angle forms
2. Sign ambiguity in the definition (cancel with the sign ambiguity of Lecture 4)
3. Stokes theorem
4. Vanishing lemmata (without proofs!)
5. Examples of two graphs and their weights
6. Proof of the appropriate identities on weights (i.e proof of the L_\infty condition)

Lecture 7 (45min) : Compatibility between cup products I (E. Burgunder)

1. The homotopy equation
2. Expression of the cup-products in terms of weights and graphs
3. A path in \bar{C}_{2,0}
4. Expression of the homotopy in terms of weights and graphs

Lecture 8 (45min) : Compatibility between cup products II (M. Livernet)

1. Stokes formula
2. Proof of the homotopy equation

Lecture 9 (1h30) : Duflo isomorphism (C. Blohmann)

1. Statement (without proof!) of the Poincar\'e-Birkhoff-Witt theorem (isomorphism of vector spaces between S( g) and U( g) )
2. Statement of the Duflo isomorphism (isomorphism of algebras between S( g)^{ g} and U( g)^{ g} )
3. Proof of the statement via the compatibility between cup-products (after Kontsevich)

Lecture 10 (1h30): The combinatorial Kashiwara-Vergne conjecture and the Duflo isomorphism (M. Ronco)

1. Baker-Campbell-Hausdorff formula
2. Presentation of the KV conjecture
3. Statement of the combinatorial KV conjecture
4. KV implies Duflo (algebraically)

Lecture 11 (1h30): Proof of the KV conjecture (J.-L. Loday)

1. Lie- and wheel-type graphs
2. Reformulation of the KV conjecture via deformations (equivalence)
3. A deformation of the Baker-Campbell-Hausdorff formula
4. Alekseev-Meinrenken proof (sketch)

Lecture 12 (1h30) : Introduction to Drinfeld associators (A. Brochier)

1. Universal Lie algebra t_n , relation with semi-simple Lie algebras
2. Definition of associators
3. Categorical interpretation
4. Torsor structure
5. Existence/C = existence/Q

Lecture 13 (1h30) : KV associators, after Alekseev-Torossian (D. Calaque)

1. tder, sder, tr, kv_2 , kv_3 , etc ...
2. Definition of KV-associators
3. Relation with Drinfeld associators
4. Relation with the KV problem

Lecture 14 (1h30 min): Existence of Drinfeld associators (U. Kraehmer)

1. Knizhnik-Zamolodchikov equations
2. Construction of the KZ associator
3. Proof that it satisfies the axioms of associators

Lecture 15 (45 min): Drinfeld double (P. M. Hajac)

1. The Drinfeld double and its quasi-triangular structure

Lecture 16 (45 min): Quantization of Lie bialgebras (T. Maszczyk)

1. Lie bialgebras, quantization problem
2. Strategy of Etingof-Kazhdan proof

Lecture 17: Alekseev-Enriquez-Torossian explicit proof of the KV conjecture (A. Alekseev)

References :