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K-State Today Student Edition

March 27, 2012

University of Oregon professor to present 47th William J. Spencer Lecture today

By Reta McDermott

Arkady Berenstein, associate professor of mathematics at the University of Oregon, will present the 47th William J. Spencer Lecture at 2:30 p.m. today in 102 Cardwell Hall. The title of Berenstein's lecture is "Quantum Cluster Algebras."

Cluster algebras have been introduced by Fomin and Zelevinsky in 2001 as an algebraic framework for total positivity and canonical bases in representations of reductive groups. Now the theory of cluster algebras is connected to many different areas of mathematics, for example, representation theory of finite dimensional algebras, Poisson geometry and Teichmuller Theory. 

The goal of Berenstein's talk (based on joint work with A. Zelevinsky) is to introduce quantum deformations of cluster algebras. While a cluster algebra corresponds to an integer skew-symmetrizable matrix B, its quantum version corresponds to a compatible pair of B with a skew-symmetric matrix responsible for the q-commutation relations. 

It turns out that all classical cluster structures can be carried over (sometimes conjecturally) to the quantum world. For instance, we expect that each quantum reductive group admits a quantum cluster structure. Same is expected for each double Bruhat cell in quantum Kac-Moody groups. 

Ultimately, Berenstein will explain how to fulfill one of the original goals of the cluster project: construction of the canonical basis in acyclic case.