Level 0 modules of affine Lie algebras
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Abstract: The favourite R-matrices and transfer matrices (which give the 6 vertex model) arise from the evaluation representation of the standard representation of the quantum group of sl(n). This is a level 0 representation of quantum affine sl(n) and the lattice models are based on tensor products of this representation. With Finn McGlade and Yaping Yang we have written a survey about the classification of these modules (by dominant weights for extremal weight modules and by Drinfeld polynomials for finite dimensional modules), their characters (which are q-Whittaker functions) and their crystals. I will try to sketch how I think this category (of level 0 modules) is the controlling structure for vertex operators, Fock spaces (Kyoto path model) and the Algebraic Bethe ansatz.