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Noncommutative Clusters (NAGRT) May 22, 2013 (03:30PM PDT - 04:30PM PDT)
Parent Program: Noncommutative Algebraic Geometry and Representation Theory
Location: MSRI: Simons Auditorium
Speaker(s) Arkady Berenstein (University of Oregon)
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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.

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