##### Department of Mathematics,

University of California San Diego

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### Algebra Colloquium

## Andrei Zelevinsky

#### Northeastern University

## Cluster algebras and their quantizations

##### Abstract:

Cluster algebras, introduced jointly with S. Fomin, are a class of axiomatically defined commutative rings equipped with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of the same finite cardinality. The original motivation for this theory was to create an algebraic framework for total positivity and canonical bases in semisimple algebraic groups. Since their introduction, cluster algebras have found applications in a diverse variety of settings which include quiver representations, Teichmuller theory, Poisson geometry, discrete dynamical systems, tropical geometry, and algebraic combinatorics. We discuss the basics of the theory of cluster algebras and their quantum analogues constructed jointly with A. Berenstein.

Host: Efim Zelmanov

### February 6, 2006

### 2:00 PM

### AP&M 6218

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