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 V(λ1)⊗...⊗V(λ 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 V(λ1)⊗...⊗V(λ 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.
KeywordsTensor Product Irreducible Component Homology Group Geometric Construction Weight Module
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