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Finite-dimensional representations of quantum affine algebras. (English) Zbl 0915.17011
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.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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