Skein-triangulated representations of generalised braids

For zoom details contact Christian Kaiser (kaiser@mpim-bonn.mpg.de)

Ordinary braid group Br_n is a well-known algebraic structure which encodes configurations of n non-touching strands (“braids”) up to continious transformations (“isotopies”). There are many examples where Br_n acts categorically on the derived category of an algebraic variety: the minimal resolutions of Kleinian singularities, the cotangent bundles of flag varieties, etc.
In this talk, I will introduce a new structure: the category GBr_n of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In the context of triangulated categories, it is natural to impose certain relations which result in the notion of a skein-triangulated representation of GBr_n. These relations generalise the famous skein relation used to define oriented link invariants such as Jones polynomial.
We give two examples of skein-triangulated actions of GBr_n: on the cotangent bundles of varieties of full and partial flags in C^n and on categorical nil-Hecke algebras. The latter example, in fact, shows that any categorical action of Br_n can be lifted to a skein-triangulated action of GBr_n, generalising a result of Ed Segal for n=2. This is a joint work with Rina Anno.