On the Comultiplication in Quantum Affine Algebras

In [8], V.G. Drinfeld gave a new set of generators and relations for the quantum affine algebra Uq(ĝ) (and also for the Yangian). He also gave an isomorphism between the two realizations, but there was no proof in that article. In [1], J. Beck found these new generators inside Uq(ĝ), and proved that they satisfy the relations given by Drinfeld. He also proved that the two realizations were isomorphic as Hopf algebras, and an explicit isomorphism was given in that article. Beck also gave new formulas satisfied by the comultiplication, but no explicit expressions for the comultiplication of the generators were found. Another comultiplication was found by Drinfeld in an unpublished paper, see [5], [6], but this new comultiplication has values in an extension of Uq(ĝ)⊗Uq(ĝ), and do not coincide with the comultiplication induced by the isomorphism given by Beck in [1]. Its advantage over the induced comultiplication is that it is much easier to work with (cf. [10], [6]). In this paper we find the comultiplication of the generators in Drinfelds new realization of the quantum affine algebra Uq(ŝl2), using Beck’s isomorphism (see [12] for the corresponding case for the Yangian associated to ŝl2). The paper is organized as follows: Section 1 contains the definition of the algebras Uq(ŝl2) and Ûq. We define certain generating functions, X ± 0 (z), Y ± 0 (z), and some further relations involving the generators of Ûq are found. In Section 2, the main theorem is Theorem 2.1, where we give relations satisfied by the comultiplication of the generators of the algebra Ûq. The result is written in terms of the generating functions defined in Section 1.