Bulletin (new Series) of the American Mathematical Society

In his fundamental 1985 paper [6], Drinfeld attached a certain Hopf algebra, which he called a Yangian, to each finite dimensional simple Lie algebra over the ground field C. These Hopf algebras can be regarded as a tool for producing rational solutions of the quantum Yang-Baxter equation and are one of the main families of examples in Drinfeld's seminal ICM address [8] which marked the beginning of the era of quantum groups. For sl n (C), Drinfeld's Yangian embeds into a slightly larger Hopf algebra Y (gl n), the Yangian of gl n (C), which was discovered a few years earlier in the work on the quantum inverse scattering method by the St. Petersburg school. A few years later, Olshanski [14] introduced the twisted Yangians Y (so n) and Y (sp n) (assuming n is even in the latter case). These are certain subalgebras of Y (gl n) defined by " folding " the generators with respect to an approriate invo-lution. The terminology here is confusing, as Olshanski's twisted Yangians Y (so n) and Y (sp n) are quite different from Drinfeld's Yangians associated to so n (C) and sp n (C); in particular, the former are not Hopf algebras. For the rest of this review, we are interested not with Drinfeld's Yangians in general, but just with the three families Y (gl n), Y (so n), and Y (sp n). The study of these algebras has revealed some hidden structure in the underlying classical Lie algebras, in the spirit of the sort of invariant theory to be found in Weyl's book [16]. Formally, the Yangian Y (gl n) can be defined as the associative algebra over C with generators {t (r) ij | 1 ≤ i, j ≤ n, r ≥ 1} subject to the relations (1) [t (r) ij , t (s) kl ] = min(r,s) a=1 (t (a−1) kj t (r+s−a) il − t (r+s−a) kj t (a−1) il), where [x, y] = xy − yx is the commutator and t (0) ij should be interpreted as the Kronecker δ ij. The motivation behind these relations will be explained shortly, but first the reader should compare them to the familiar relations satisfied by the matrix units {e ij | 1 ≤ i, j ≤ n} which generate the universal enveloping algebra U (gl n) of the Lie algebra gl n (C). It follows that there are algebra …