Inventiones mathematicae

, Volume 146, Issue 2, pp 399–449 | Cite as

Quiver varieties and tensor products

  • Hiraku Nakajima


In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety &?tilde; in a quiver variety, and show the following results: (1) The homology group of &?tilde; is a representation of a symmetric Kac-Moody Lie algebra ?, isomorphic to the tensor product V1)⊗...⊗V N ) of integrable highest weight modules. (2) The set of irreducible components of &?tilde; has a structure of a crystal, isomorphic to that of the q-analogue of V1)⊗...⊗V N ). (3) The equivariant K-homology group of &?tilde; is isomorphic to the tensor product of universal standard modules of the quantum loop algebra U q (L?), when ? is of type ADE. We also give a purely combinatorial description of the crystal of (2). This result is new even when N=1.


Tensor Product Irreducible Component Homology Group Geometric Construction Weight Module 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Hiraku Nakajima
    • 1
  1. 1.Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan (e-mail:

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