Quiver varieties and tensor products

Details Quiver varieties and tensor products Quiver varieties and tensor products

2001-03-02

arxiv.org/abs/math/0103008v2

Mathematics

Quantum Algebra

39 pages, no figures; Several references are added

Scientific paper

10.1007/PL00005810

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.

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