Finite-dimensional representations of quantum affine algebras
Let \(\widehat{\mathfrak g}\) be an untwisted affine Lie algebra and \(U_q(\widehat{\mathfrak g})\) the associated quantum group over the field \(\mathbb{C} ({\mathfrak q})\) of rational functions of an indeterminate \(q\). Let \(\lambda_1, \dots, \lambda_n\) be the fundamental weights of the underlying finite-dimensional complex simple Lie algebra \({\mathfrak g}\), and let \(V(\lambda_i)_a\) be the evaluation representation with parameter \(a\in\mathbb{C} (q)^\times\). The authors propose a conjecture on the condition for irreducibility of a tensor product \(V(\lambda_{i_1})_{a_1} \otimes \cdots \otimes V(\lambda_{i_N})_{a_N}\). This has the following consequence: this tensor product is irreducible iff the \(R\)-matrix \(V(\lambda_{i_r})_x\otimes V(\lambda_{i_s})_y\to V(\lambda_{i_s})_y \otimes V(\lambda_{i_r})_x\) has no pole at \((x,y)= (a_r,a_s)\) for all \(r<s\). Moreover, when this \(R\)-matrix has no pole at \((x,y)= (a_r,a_s)\) for all \(r>s\), the submodule of \(v(\lambda_{i_1})_{a_1} \otimes\cdots \otimes V(\lambda_{i_N})_{a_N}\) generated by the tensor product of highest weight vectors is irreducible, and every finite-dimensional irreducible integrable module is obtained in this way. The main conjecture is proved when \({\mathfrak g}\) is of type \(A_n\) and \(C_n\), using crystal basis techniques.
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