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Crystal base for the basic representation of \(U_ q({\mathfrak sl}^\wedge (n))\). (English) Zbl 0724.17010

It is proved that the basic representation of \(U_ q({\mathfrak sl}^{\wedge}(n))\) has a crystal base (cf. the preceding review). An explicit description in terms of Young diagrams is given. In a recent preprint [Res. Inst. Math. Sci., Kyoto. Univ., 767 (1991)], M. Kashiwara and T. Nakashima have given further explicit descriptions of crystal bases.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)

Citations:

Zbl 0724.17009
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References:

[1] Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: One-dimensional configuration sums in vertex models and affine Lie algebra characters. Lett. Math. Phys.17, 69–77 (1989). · Zbl 0681.17016 · doi:10.1007/BF00420017
[2] Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Paths, Maya diagrams and representations of \(\widehat{\mathfrak{s}\mathfrak{l}}(r,C)\) . In: Integrable systems in quantum field theory and statistical mechanics. Adv. Stud. Pure Math.19, 149–191 (1989). · Zbl 0704.17013
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[5] Kashiwara, M.: Crystalizing theq-analogue of universal enveloping algebras. RIMS, Kyoto University, preprint RIMS-676 (1989)
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