From quantum affine Kac-Moody algebras to Drinfeldians and Yangians

Abstract

A general scheme of construction of Drinfeldians and Yangians from quantum non-twisted affine Kac-Moody algebras is presented. Explicit description of Drinfeldians and Yangians for all Lie algebras of the classical series A, B, C, D is given in terms of a Chevalley basis. Rational and trigonometric deformations of a universal enveloping algebra U(g(u)) (g is a finite-dimensional complex simple Lie algebra) - Yangian Yη(g) and a quantum (q-deformed) affine algebraUq(g(u)) - are playing increasingly a role in the theory of integrable systems and the quantum field theory. For applications it is useful to know various realizations (or bases) of these deformations. Among various realizations the Chevalley basis plays a special role due to its compactness and simplicity. In the case of the quantum affine algebras the Chevalley basis was originally introduced in the defining relations while the Yangians were initially de- fined in terms of a finite Cartesian basis (the first Drinfeld realization (2)) as well as in terms of a special infinite basis (the second Drinfeld realization (3)). It was not clear whether does exist a Chevalley basis for the Yangians. Recently in (12, 13) it was shown that the Yangian can be realized in terms of Chevalley basis and, moreover, there was found a simple connection of this realization with the corresponding quantum affine algebra (also see (14, 15)). This connection allows to introduce a new Hopf algebra called Drinfeldian which depends on two deformation parameters q and η. The Drinfeldian Dqη(g) can be considered as quantization of U(g(u)) in the direction of a classical r-matrix which is a sum of simplest rational and trigonometric r-matrices. When the parameter q goes to 1 we obtain the Yangian Yη(g) in terms of the Chevalley basis. In this paper we remind once more the procedure of construction of the Drin- feldian Dqη(g) and the Yangian Yη(g) in terms of the Chevalley basis starting from