|Location:||MSRI: Simons Auditorium|
Cluster algebras were introduced by Fomin and Zelevinsky in 2001 and have become an important tool in representation theory, higher category theory, and algebraic/Poisson geometry.
The goal of my talk (based on a joint paper with V. Retakh) is to introduce totally noncommutative clusters and their mutations, which can be viewed as generalizations of both ``classical" and quantum cluster structures.
Each noncommutative cluster X is built on a torsion-free group G and a certain collection of its automorphisms. We assign to X a noncommutative algebra A(X) related to the group algebra of G, which is an analogue of the cluster algebra, and expect a Noncommutative Laurent Phenomenon to hold in the most of algebras A(X).
Our main examples of "cluster groups" G include principal noncommutative tori which we define for any initial exchange matrix B and noncommutative triangulated groups which we define for all oriented surfaces.No Notes/Supplements Uploaded No Video Files Uploaded