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Beck, Jonathan

Braid group action and quantum affine algebras. (English) Zbl 0807.17013

Commun. Math. Phys. 165, No. 3, 555-568 (1994).
The purpose of this paper is to establish explicitly the isomorphism between the quantum enveloping algebra \(U_ q (\widehat{g})\) of Drinfeld and Jimbo (\(\widehat{g}\) an untwisted affine Kac-Moody algebra) and the “new realization” of V. G. Drinfeld [Sov. Math., Dokl. 36, 212- 216 (1988); translation from Dokl. Akad. Nauk SSSR 269, 13-17 (1987; Zbl 0667.16004)]. (The last form makes the study of finite dimensional representations easier.)
The author uses the action on \(U_ q (\widehat{g})\) of the braid group associated with the extended affine Weyl group of \(\widehat{g}\). This action fixes the Heisenberg subalgebra pointwise. Loop-like generators of the algebra are obtained as translations of the usual Drinfeld-Jimbo generators. They satisfy the relations of Drinfeld’s new realization. Coproduct formulas are given and a PBW type basis is constructed.
Reviewer: Yu.Bespalov (Kiev)

Cited in 5 Reviews
Cited in 127 Documents

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
20F36 Braid groups; Artin groups

Keywords:

quantum affine algebras; quantum enveloping algebra; braid group; Drinfeld’s new realization

Citations:

Zbl 0667.16004
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\textit{J. Beck}, Commun. Math. Phys. 165, No. 3, 555--568 (1994; Zbl 0807.17013)
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References:

[1] [B] Bourbaki, N.: Groupes et algèbres de Lie. Ch. 4, 5, 6, Paris: Hermann, 1968 · Zbl 0186.33001
[2] [Da] Damiani, I.: A basis of type PBW for the quantum algebra of \(\widehat{\mathfrak{s}\mathfrak{l}_2 }\) . Journal of Algebra161, 291–310 (1993) · Zbl 0803.17003
[3] [D] Drinfeld, V.G.: Quantum groups. Proc. ICM Berkeley1 (1986), pp. 789–820
[4] [D2] Drinfeld, V.G.: A new realization of Yangians and Quantized Affine Algebras. Sov. Math. Dokl.36, 212–216 (1988)
[5] [DC-K] De Concini, C., Kac, V.G.: Representations of quantum groups at roots of 1. Progress in Math.92, Boston: Birkhäuser, 1990, pp. 471–506 · Zbl 0738.17008
[6] [Dc-K-P] De Concini, C., Kac, V.G., Procesi, C.: Quantum coadjoint action. J. AMS5, 151–190 (1992) · Zbl 0747.17018
[7] [DC-K-P2] De Concini, C., Kac, V.G., Procesi, C.: Some remarkable degenerations of quantum groups. Commun. Math. Phys.157, 405–427 (1993) · Zbl 0795.17006
[8] [G-K] Gabber, O., Kac, V.G.: On defining relations of certain infinite dimensional Lie algebras. Bull. AMS5, 185–189 (1981) · Zbl 0474.17007
[9] [G] Garland, H.: The Arithmetic Theory of Loop Algebras. J. Algebra53, 480–551 (1978) · Zbl 0383.17012
[10] [H] Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge: Cambridge University Press, 1990 · Zbl 0725.20028
[11] [J] Jimbo, M.: Aq-difference analog ofU(g) and the Yang-Baxter equation. Lett. Math. Physics10, 63–69 (1985) · Zbl 0587.17004
[12] [K] Kac, V.G.: Infinite Dimensional Lie Algebras. 3rd. edn. Cambridge: Cambridge University Press, 1990 · Zbl 0716.17022
[13] [K-R] Kirillov, A.N., Reshetikhin, N.:q-Weyl group and a multiplicative formula for universalR-matrices. Commun. Math. Phys.134, 421–431 (1990) · Zbl 0723.17014
[14] [L-S] Levendorskii, S., Soibelman, Y.: Some applications of the quantum Weyl groups. J. Geom. Phys.7, 241–254 (1990) · Zbl 0729.17009
[15] [Lor] Levendorskii, S., Soibelman, Y., Stukopin, V.: Quantum Weyl Group and Universal QuantumR-matrix for Affine Lie AlgebraA 1 (1) . Lett. in Math. Physics27, 253–264 (1993) · Zbl 0776.17011
[16] [L] Lusztig, G.: Introduction to Quantum Groups. Boston: Birkhäuser, 1993 · Zbl 0788.17010
[17] [L2] Lusztig, G.: Affine Hecke algebras and their graded version. AMS2, 599–625 (1989) · Zbl 0715.22020
[18] [L3] Lusztig, G.: Some examples of square integrable representations of semisimplep-adic groups. Trans. AMS277, 623–653 (1983) · Zbl 0526.22015
[19] [L4] Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. in Math.70, 237–249 (1988) · Zbl 0651.17007
[20] [L5] Lusztig, G.: Finite dimensional Hopf algebras arising from quantized universal enveloping algebras. J. AMS3, 257–296 (1990) · Zbl 0695.16006
[21] [Ro] Rosso, M.: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra. Commun. Math. Phys.117, 581–593 (1988) · Zbl 0651.17008
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