The Working Group in Algebra, Geometry and Quantization is a weekly meeting of the research team of Prof. Martin Schlichenmaier. Its aim is to present both research works and surveys of mathematical areas of common interest.

### Meetings

The working group currently meets on Tuesday from 14h00 to 15h00.
The room is specified next to the date (Maison du Nombre, 6th floor - Belval).

Occassionally we will have talks by invited guests at other times to be announced here.

### Organizers

• Laurent La Fuente-Gravy
• Francois Petit
• Martin Schlichenmaier

### Upcoming sessions

• 02.03.2018 -- Room S.6B
Alexey Kalugin (Université du Luxembourg, Luxembourg), "Quantization of Lie bialgebras and perverse sheaves"
Abstract: Quantization of Lie bialgebras is a canonical way (up to a choice of Drinfeld associator) to produce a Hopf algebra (quantum group) from an arbitrary (conilpotent) Lie bialgebra. I am going to discuss a new (geometric) approach to the problem of quantization which is based on a Beilinson’s gluing technique of perverse sheaves and a notion of factorization algebras (factorizable sheaves) of Beilinson and Drinfeld. If we will have enough time I will discus a connection with the theory of mixed Tate motives.
• 06.03.2018 -- Room S.6A
Yulia Kuznetsova (University Bourgogne Franche-Comté, Dijon), "Quantum semigroups with involution and their duals"
Abstract: It is well known that locally compact quantum groups admit a duality which generalizes the Pontryagin's duality theorem. This allows to apply the same methods to a quantum group $\mathbb {G}$ and its dual $\mathbb{\widehat G}$, or on the other hand, to pass between the group and its dual as it is done in the classical case with the help of Fourier transform. In this talk I will speak of duality of quantum semigroups which might be not groups. Applying this duality map two times, we do not arrive at an isomorphism between $\mathbb G$ and $\widehat{\widehat {\mathbb G}}$ as in the group case, but rather to a maximal subgroup in the initial quantum semigroup. This is demonstrated on several examples, including semigroup compactifications of (non-quantum) locally compact groups. For locally compact quantum groups, we recover the universal duality of Kustermans, and thus show that one does not need to know the Haar weight (the quantum version of Haar measure) measure to construct it. Based on: ArXiv 1611.04830 [math.OA].
• 09.03.2018 -- Room S.6B
Matteo Felder (University of Zurich, Zurich), "On a homotopy version of the Duflo isomorphism"
Abstract: For a finite dimensional Lie algebra $g$, the Duflo map $S(g)$ \rightarrow $U(g)$ defines an isomorphism of g-modules. On g-invariant elements it gives an isomorphism of algebras. Moreover, it induces an isomorphism of algebras on the level of Lie algebra cohomology $H(g,S(g)) \rightarrow H(g, U(g))$. However, as shown by J. Alm and S. Merkulov, it cannot be extended in a universal way to an $A_\infty$-isomorphism between the corresponding Chevalley-Eilenberg complexes. In this talk, we will try to give an elementary and self-contained proof of this fact using a version of M. Kontsevich's graph complex.
• 20.03.2018 -- Room S.6A
Louis Ioos (IMJ-PRG, Paris), "Asymptotics of isotropic states in holomorphic quantization"
Abstract: In real geometric quantization of a symplectic manifold, quantum states are repre- sented by some specific isotropic submanifolds satisfying the so-called "Bohr-Sommerfeld condition", whereas in holomorphic quantization of a Kähler manifold, quantum states are represented by the holomorphic sections of a holomorphic line bundle. In this talk, I will construct a bridge between the two point of views, giving a natural definition of an "isotropic state" associated to a Bohr-Sommerfeld manifold in holomorphic quanti- zation. I will then study their "semi-classical limit", when the tensor powers of the line bundle tend to infinity, and then show how these results extend to various contexts. If time permits, I will then present some applications to the theory of automorphic forms.

### Archive

For former programs see here.