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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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