EMS abstract

18 June 2010, Professor H Nakajima (LMS Hardy Lecturer)

Quiver varieties and cluster algebras

We realize a Fomin-Zelevinsky cluster algebra (when the underlying graph is bi-partite) as a subquotient of the dual of the upper part of the quantized enveloping algebra. Then we show that Lusztig dual canonical base induces a base of the cluster algebra which includes all the cluster monomials. In particular, the positivity conjecture on cluster monomials is a consequence.