Quiver varieties and cluster algebras

TL;DR

This paper establishes a deep connection between cluster algebras and the representation theory of quantum loop algebras, showing how cluster monomials relate to simple modules and canonical bases, with implications for positivity and independence conjectures.

Contribution

It embeds cluster algebras into the Grothendieck ring of quantum loop algebra representations, linking cluster monomials to simple modules and canonical bases, and proves related conjectures.

Findings

Cluster monomials correspond to classes of simple modules.

Positivity and linear independence of cluster monomials are confirmed.

Cluster monomials form a subset of Lusztig's dual canonical base.

Abstract

Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig's dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials [arXiv:math/0104151] follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of `prime' simple ones according to the cluster expansion.