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Isomorphism of two realizations of quantum affine algebra \(U_ q(\widehat{{\mathfrak g}{\mathfrak l}(n)})\). (English) Zbl 0786.17008
There are known at least four different descriptions of the quantum affine algebra \(U_ q(\widehat {\mathfrak g})\): (1) in terms of Chevalley generators by Drinfeld-Jimbo; (2) the so called “new” realization by Drinfeld where the whole currents are quantized; (3) in terms of \({\mathcal L}\)-operator by Semenov-Tian-Shansky, Reshetikhin and Frenkel; and (4) in terms of Cartan-Weyl generators by Khoroshkin-Tolstoj.
Here an explicit isomorphism of the forms (2) and (3) is established for the case of \(U_ q(\widehat {{\mathfrak {gl}}_ n})\). More precisely, it is proved that Drinfeld generators are the entries of the Gauss decomposition of the corresponding \({\mathcal L}\)-operators.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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