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On Drinfeld’s realization of quantum affine algebras. (English) Zbl 0784.17022
In Sov. Math., Dokl. 32, 212-216 (1988); translation from Dokl. Akad. Nauk SSSR 269, 13-17 (1987; Zbl 0667.16004), V. G. Drinfel’d gave an alternative presentation by generators and relations of \(U_ q(\widehat {\mathfrak g})\), the quantized enveloping algebra of the Kac-Moody affine, non-twisted, algebra corresponding to a finite dimensional semisimple Lie algebra \({\mathfrak g}\). The authors work out some formulae and notations in the framework of the usual presentation of \(U_ q(\widehat {\mathfrak g})\) by Chevalley generators, by means of the notion of “normalized basis” in the root system. This allows them to relate the above mentioned Drinfel’d’s presentation with the usual one. Then they explain how to get the formulae for the comultiplication in Drinfel’d’s presentation, using the explicit formula for the universal \(R\)-matrix found by them in [Funkts. Anal. Prilozh. 26, 85-88 (1992)].

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