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On level-zero representation of quantized affine algebras. (English) Zbl 1033.17017
An extremal weight vector is a generalization of highest weight vectors. The first part of this long paper studies modules of a quantized affine algebra generated by an extremal vector of weight $$\lambda$$. It is shown that all the weights are contained in the convex hull of the Weyl group orbit of $$\lambda$$. Moreover, if the extremal vector has a level-zero fundamental weight then the module is irreducible and isomorphic to the affinization of an irreducible finite dimensional module. In the second part, the author proves a conjecture set forth in T. Akasaka and M. Kashiwara [ Publ. Res. Inst. Math. Sci. 33, 839–867 (1997; Zbl 0915.17011)], about the irreducibility of the tensor product of certain fundamental representations, with the use of canonical bases.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations
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##### References:
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