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

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)


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


Zbl 0667.16004
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