Cohomology in algebra, geometry, physicsand statistics

Modules of quantized Lie algebras and their braiding from homology of configuration spaces

From any semi-simple Lie algebra, Drinfel'd has defined an associated quantized version called a quantum group. The theory of modules over quantum groups has been widely used to produce topological invariants in low dimension such as: braid groups representations, the famous Jones polynomial for knots or topological quantum field theories à la Witten--Reshetikhin--Turaev (providing representations of mapping class groups of surfaces expected to have rich properties and 3-manifold invariants).
All these constructions rely on the algebraic background surrounding quantum groups so that their topological content is often mysterious in the end, and finally the subject of many conjectures in this field called quantum topology.
We are able to recover quantum groups modules from homology of configuration spaces, and it gives a homological model for quantum braid group representations and knot invariants such as the ones arising from the Jones family.

In this talk I'll present in details how to recover the sl_2 case: quantum Verma modules and their braiding, from homology of configuration spaces.
If I have time I'll say a few words on how to generalize this to every semi-simple Lie algebra (which is a joint work in progress with S. Bigelow), and how it sheds light on the topological content of Jones invariants of knots.

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Our ZOOM meeting shall be open at 11.15 at

https://cesnet.zoom.us/j/99598413922?pwd=czl5K2hPSm0zakx0SllzMklZc2JkQT09
Meeting ID
995 9841 3922
Passcode Drinfel'd

and closed at 13.00