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Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds charges) and a fusion structure (braided, comes from G-bundles on curves, preserves central charge). In the quantum case, there is the usual tensor product (braided meromorphic-braided$^\ast$), but all I see in the literature about fusion is vague comments that it can't exist. I guess my question should be "what is the major malfunction?"

$^\ast$ Edit: The meromorphic property (in the sense of Soibelman's Meromorphic tensor categories) seems to be a first hint at problems, and I should have paid better attention to it.

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I don't really think ordinary tensor products are boring. I should change it to a more judgment-neutral word. – S. Carnahan Oct 12 '09 at 6:38
up vote 4 down vote accepted

The fusion product for affine Lie algebras is closely related to the existence of "evaluation homomorphisms" from the loop algebra to the finite-dimensional semisimple Lie algebra g, which split the natural inclusion of g as the subalgebra of constant loops. In the quantum case there is no evaluation map from the quantum affine algebra to the finite-type quantum algebra outside of type A (this is proved - at least for Yangians - in Drinfeld's original paper I'm pretty sure).

You see consequences of this in lots of places: e.g. for representations of g, the evaluation homomorphisms mean any irreducible representation for g can be lifted to an irreducible representation of the affine Lie algebra Lg. On the other hand, irreducible representations of the associated quantum groups do not (necessarily) lift to representations of quantum affine algebras, and so one asks about "minimal affinizations" -- irreducible finite dimensional representations of the quantum affine algebra which have the given irreducible as a constituent when restricted to the finite-type quantum group.

That said, the "ordinary" tensor product for finite dimensional representations of quantum affine algebras is pretty interesting -- it's not braided any more for example.

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The poles of the R-matrices for quantum affine algebras are the price to pay for the abovementioned simplification - the braiding becomes symmetric under q-deformation. If there were no poles, the category would have been symmetric and hence would be a representation category of a (usual) group. In fact, the poles of the R-matrices contain much of the information about the structure of this category.

Also, I'd like to add that there is an interesting paper by Hernandez, math/0504269, where he discusses a new "fusion" tensor product on representations of quantum affine algebras.

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