zbMATH — the first resource for mathematics

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)].

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
Full Text: DOI
[1] Alekseev, A.; Faddeev, L.D.; Semenov-Tian-Shansky, M.A., Hidden quantum groups inside Kac-Moody algebras, (1991), LOMI preprint E-5-91
[2] Asherova, R.M.; Smirnov, Yu.F.; Tolstoy, V.N., A description of some class of projection operators for semisimple complex Lie algebras, Matem. zametki, 26, 15-25, (1979) · Zbl 0414.17005
[3] Drinfeld, V.G., A new realization of Yangians and quantized affine algebras, Soviet math. dokl., 32, 212-216, (1988), (see also preprint FTINT USSR AS, 1986, No. 30) · Zbl 0667.16004
[4] Drinfeld, V.G., Quantum groups, (), 798-820
[5] Feigin, B.; Frenkel, E., Free field resolutions in affine Toda field theories, Phys. lett. B, 276, 79-86, (1992)
[6] Jimbo, M., Quantum R-matrix for the generalized Toda system, Commun. math. phys., 102, 537-547, (1986) · Zbl 0604.58013
[7] Khoroshkin, S.M.; Tolstoy, V.N., The universal R-matrix for quantum nontwisted affine Lie algebras, Funkt. anal. pril., 26, 85-88, (1992) · Zbl 0834.17013
[8] Khoroshkin, S.M.; Tolstoy, V.N., The uniqueness theorem for the universal R-matrix, Lett. math. phys., 24, 231-244, (1992) · Zbl 0761.17012
[9] Khoroshkin, S.M.; Tolstoy, V.N., Universal R-matrix for quantized (super)algebras, Commun. math. phys., 141, 599-617, (1991) · Zbl 0744.17015
[10] Levendorshii, S.Z.; Soibelman, Ya.S., Some applications of quantum Weyl groups, the multiplicative formula for universal R-matrix for simple Lie algebras, Geom. phys., 7, 4, 1-14, (1990)
[11] Reshetikhin, N.Yu.; Semenov-Tian-Shansky, M.A., Central extensions of quantum current groups, Lett. math. phys., 19, 133-142, (1990) · Zbl 0692.22011
[12] Tolstoy, V.N., Extremal projectors for contragredient Lie algebras and superalgebras of finite growth, Usp. math. nauk., 44, 211-212, (1989) · Zbl 0674.17008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.