We study the category O of representations of the rational Cherednik algebra A W attached to a complex reflection group W . We construct an exact functor, called Knizhnik-Zamolodchikov functor: O → H W -mod, where H W is the (finite) Iwahori-Hecke algebra associated to W . We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between O/O tor , the quotient of O by the subcategory of A W -modules supported on the discriminant, and the category of finite-dimensional H W -modules. The standard A W -modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of "cells", provided W is a Weyl group and the Hecke algebra H W has equal parameters. We prove that the category O is equivalent to the module category over a finite dimensional algebra, a generalized "q-Schur algebra" associated to W .

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Given an affine Kac-Moody algebra g and its associated Yangian Y (g), we explain how to construct a coproduct for Y (g). In order to prove that this coproduct is an algebra homomorphism, we obtain, in the first half of this paper, a minimalistic presentation of Y (g) when g is, more generally, a symmetrizable Kac-Moody algebra.

The main purpose of this note is to give a proof of a statement of Drinfeld in ['Quantum groups', Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, CA, 1986) (American Mathematical Society, Providence, RI, 1987) 798-820.] regarding Yangians and quantum loop algebras, namely how the former can be constructed as limit forms of the latter. We also apply the same ideas to twisted quantum loop algebras to recover the (non-twisted) Yangians.

We prove the equivalence of two presentations of the Yangian Y(g) of a simple Lie algebra g and we also show the equivalence with a third presentation when g is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians. ContentsYangian of gl n : this was accomplished in [BrKl] where the authors obtained so-called parabolic presentations of that Yangian depending on a partition of n, one extreme case being the RT T -presentation, the other extreme case being the current presentation. As a consequence, an isomorphism between the RT T and current presentations of the Yangian of sl n is obtained; it is in agreement with the formulas provided in [Dr3]: see Remarks 5.12 and 8.8 of [BrKl]. The results of [BrKl] are also explained in Section 3.1 of [Mo3].The formulas for the equivalence between the RT T and current presentation in [BrKl] came from the Gauss decomposition of the matrix of generators in the RT T -presentation. Very recently, this approach has been successfully extended to Yangians of orthogonal and symplectic Lie algebras: see [JLM]. (The so 3 -case was treated previously in [JL].)In this paper, we give a proof of Theorem 6 in [Dr1] for orthogonal and symplectic Lie algebras (see Theorem 3.16) and a proof of Theorem 1 in [Dr3] for any g (see Theorem 2.6). These are two of the three main contributions of this paper. The RT T -presentation for symplectic and orthogonal Yangians was first treated in [AACFR, AMR], and later received more attention in the mathematical literature in such papers as [Mo4, MM1, MM2, GR, GRW2]. The initial motivation for this paper came from a desire to better understand why this presentation of the Yangian is equivalent to the others. This led us to consider the analogous question regarding the J and current presentations. Another motivation came from representation theory. It has been known since [Dr3] that finite-dimensional irreducible representations of Yangians are classified by certain polynomials, usually called Drinfeld polynomials in the literature: to prove this classification result, Drinfeld used the current presentation. When g = sl n , such a classification theorem can also be proved using the RT T -presentation (see, for instance, Corollary 3.4.8 in [Mo3]) and the resulting classification is also given in terms of certain polynomials. This raises the question of how these two families of polynomials are related: the answer is provided in the proof of Corollary 3.4.9 in loc. cit. and uses the Gauss decomposition. When g = so N or g = sp N , the classification theorem was reproved in [AMR] using the RT T -presentation: see Corollary 5.19 in loc. cit. It was explained by the authors of [AMR] that, without an explicit isomorphism between the RT T and current presentations available, it was not clear how to translate Drinfeld's classification result into one compatible with the RT T -presenta...