Yangians and Classical Lie Algebras

Introduction The term `Yangian’ was introduced by V. G. Drinfeld to specify quantum groups related to rational solutions of the classical Yang–Baxter equation; see Belavin– Drinfeld [BD1,BD2] for the description of these solutions. In Drinfeld [D1] for each simple finite-dimensional Lie algebra a, a certain Hopf algebra Y(a) was constructed so that Y(a) is a deformation of the universal enveloping algebra for the polynomial current Lie algebra a[x]. An alternative description of the algebra Y(a) was given in Drinfeld [D3]; see Theorem 1 therein. Prior to the intruduction of the Hopf algebra Y(a) in Drinfeld [D1], the algebra which may be called the Yangian for the reductive Lie algebra gl(N) and may be denoted by Y , was considered in the works of mathematical physicists from St.-Petersburg; see for instance Takhtajan–Faddeev [TF]. The latter algebra is a deformation of the universal enveloping algebra U . Representations of were studied in Kulish–Reshetikhin–Sklya