Quiver varieties and tensor products

Abstract

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.

39 pages, no figures; Several references are added

References in corpus (1)

• Quantum Groups and Quantum Cohomology

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