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.
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...
Let ${\mathfrak{g}}$ be a symmetrizable Kac–Moody algebra with associated Yangian $Y_\hbar{\mathfrak{g}}$ and Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$. An elementary result of fundamental importance to the theory of Yangians is that, for each $c\in{\mathbb{C}}$, there is an automorphism $\tau _c$ of $Y_\hbar{\mathfrak{g}}$ corresponding to the translation $t\mapsto t+c$ of the complex plane. Replacing $c$ by a formal parameter $z$ yields the so-called formal shift homomorphism $\tau _z$ from $Y_\hbar{\mathfrak{g}}$ to the polynomial algebra $Y_\hbar{\mathfrak{g}}[z]$. We prove that $\tau _z$ uniquely extends to an algebra homomorphism $\Phi _z$ from the Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$ into the $\hbar $-adic closure of the algebra of Laurent series in $z^{-1}$ with coefficients in the Yangian $Y_\hbar{\mathfrak{g}}$. This induces, via evaluation at any point $c\in{\mathbb{C}}^\times $, a homomorphism from $\textrm{D}Y_\hbar{\mathfrak{g}}$ into the completion of the Yangian with respect to its grading. We show that each such homomorphism gives rise to an isomorphism between completions of $\textrm{D}Y_\hbar{\mathfrak{g}}$ and $Y_\hbar{\mathfrak{g}}$ and, as a corollary, we find that the Yangian $Y_\hbar{\mathfrak{g}}$ can be realized as a degeneration of the Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$. Using these results, we obtain a Poincaré–Birkhoff–Witt theorem for $\textrm{D}Y_\hbar{\mathfrak{g}}$ applicable when ${\mathfrak{g}}$ is of finite type or of simply laced affine type.
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