Quiver Varieties and Tensor Products
نویسنده
چکیده
In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety Z̃ in a quiver variety, and show the following results: (1) The homology group of Z̃ is a representation of a symmetric Kac-Moody Lie algebra g, isomorphic to the tensor product V (λ1)⊗· · ·⊗V (λN ) of integrable highest weight modules. (2) The set of irreducible components of Z̃ has a structure of a crystal, isomorphic to that of the q-analogue of V (λ1)⊗ · · ·⊗V (λN ). (3) The equivariant K-homology group of Z̃ is isomorphic to the tensor product of universal standard modules of the quantum loop algebraUq(Lg), when 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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