  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.
