Cluster algebras were introduced by Fomin and Zelevinsky in 2001 in order to model the algebraic properties of the dual canonical basis of a quantum group associated to a simple Lie algebra and to study total positivity in algebraic groups. There are connections with Y-systems, the Bethe ansatz, recurrence problems, Poisson geometry, Teichmueller spaces, stacks, braid monoids, representation theory of algebras, Lie algebras, toric varieties, and other areas. Cluster algebras of finite type are classified by the Dynkin diagrams.
Cluster algebras are combinatorially defined subalgebras of the field of rational functions in finitely many indeterminates. I will give an introduction to this theory and I will also talk about the Laurent phenomenon, which is the fact that the generators of the cluster algebra, defined via a recurrence relation from the indeterminates, are all Laurent polynomials. I will also discuss the Grassmannian from a cluster algebra perspective.
Contact A.P. Fordy (email: allan@maths.leeds.ac.uk ) for further details.
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Last updated January 25th, 2007.