Abstract: Cluster algebras introduced
a few years ago in a joint work with Sergey Fomin are a new class of
commutative rings designed to provide an algebraic framework for the study
of canonical bases and total positivity in semisimple Lie groups. The original
motivation for this concept comes from the geometry of double Bruhat cells
in a semisimple group (that is, the intersections of cells in two Bruhat
decompositions with respect to a pair of opposite Borel subgroups). I will
explain this motivation and introduce cluster algebras associated with
double Bruhat cells. This is work in progress with Arkady Berenstein and
Sergey Fomin.
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