Skein-triangulated representations of generalised braids

There are many examples where the braid group Br_n acts on the derived category of an algebraic variety: the minimal resolutions of Kleinian singularities, the cotangent bundles of flag varieties, etc. I will describe a long-running project with Rina Anno (Kansas) where we 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. We consider these strands embedded in a ribbon graph, and allow ribbon twists.

For triangulated categories, it is natural to impose certain relations which result in the notion of a skein-triangulated representation of GBr_n. These relations categorify the polynomial sl_n-web calculus used in the construction of Khovanov-Rozansky homology. We give two examples of skein-triangulated actions of GBr_n: on the cotangent bundles of flag varieties and on categorical nil-Hecke algebras. The latter example 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 that all autoequivalences are spherical twists.