M-Seminar: Nil Hecke algebra in the category of bimodules
Speaker: Rina Anno (KSU)
Abstract: Let A be a DG algebra or a small DG category. Let H_1, ... H_n be objects in the derived category D(A-A) of A-A bimodules that satisfy the braid-like relation $H_i \otimes H_j \otimes H_i \simeq H_j\otimes H_i \otimes H_j$ for |i-j|=1. Then we can construct a certain algebra object in D(A-A). When A is commutative and each H_i is isomorphic to A, this gives the usual nil Hecke algebra over A; overall, this construction is best thought of as the nil Hecke algebra where the generators don't necessarily commute with the coefficients. This algebra has a collection of subalgebras enumerated by ordered partitions of $n$. If H_i's are invertible in the monoidal category D(A-A), the derived categories of representations of these subalgebras are equivalent whenever the corresponding partitions differ by a permutation, and those equivalences generate (colored) braid group actions. We use this construction to show that any (DG-enhanceable) braid group action on a triangulated category can be completed to an action of a bigger diagrammatic category that we call the category of generalized braids. This talk is based on an ongoing project with Timothy Logvinenko.
Thursday, April 9 at 3:30pm to 5:00pm