  In this article, we give geometric constructions of tensor products in
various categories using quiver varieties. More precisely, we introduce a
lagrangian subvariety $\Zl$ in a quiver variety, and show the following
results:
  (1) The homology group of $\Zl$ is a representation of a symmetric Kac-Moody
Lie algebra $\mathfrak g$, isomorphic to the tensor product
$V(\lambda_1)\otimes...\otimes V(\lambda_N)$ of integrable highest weight
modules.
  (2) The set of irreducible components of $\Zl$ has a structure of a crystal,
isomorphic to that of the $q$-analogue of $V(\lambda_1)\otimes...\otimes
V(\lambda_N)$.
  (3) The equivariant $K$-homology group of $\Zl$ is isomorphic to the tensor
product of universal standard modules of the quantum loop algebra $\Ul$, when
$\mathfrak g$ is of type $ADE$.
  We also give a purely combinatorial description of the crystal of (2). This
result is new even when N=1.
