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Final Activity Report Summary - DYNQUANTGR (Dynamical quantum groups, deformation quantisation of Lie bialgebroids and integrable systems in mathematical physics)

Deformation quantisation is a mathematically rigorous formulation of the passage from classical to quantum mechanics. The main goal of the present project was to explore various aspects of deformation quantisation, and relations with Hopf algebras and dynamical Hopf algebroids. In particular, the leading question of the project was the following: starting with a classical system having a certain number of nice properties, can one quantise this system in a way compatible (in a certain sens) with the mentioned properties.

1) as a first main result of this project we gave, together with Carlo Rossi, a positive answer to a conjecture in deformation quantisation about the compatibility of Tsygan's formality (which plays a crucial role in finding quantum trace functionals out of classical trace functionals) with so-called cap products.
a) as a first consequence of this result we obtained an analog of the famous Duflo isomorphism (from representation theory) for coinvariants.
b) as a second consequence of this result we plan to prove another famous conjecture, due to Caldararu, in the field of complex geometry (this is a work in progress with Carlo Rossi and Michel Van den Bergh).

These results are particularly spectacular as they make the bridge between different subjects and disciplines: mathematical physics and deformation quantisation on the one side, pure mathematics with Lie theory and complex geometry on the other side.

2) together with Thomas Willwacher we also proved the cyclic formality conjecture of Maxim Kontsevich (Field medalist), which itself implies the existence and classification of closed star-products. This situation is a typical example of a quantisation procedure that respects a given property.

3) the proof of both above results make a crucial use of Feynman diagrams methods, coming from theoretical physics, and that are so important for renormalisation. Together with Dominique Manchon and Kurusch Ebrahimi-Fard we noticed that the diagram techniques used by numerical analysts can be understood via a Hopf algebra and a coaction from it on the Connes-Kreimer algebra in renormalisation ... and precisely give an example of a dynamical Hopf algebroid.

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