Equivalences between three presentations of orthogonal and symplectic Yangians

Abstract

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...

Spinorial R operator and Algebraic Bethe Ansatz

“…The linear ansatz for L-operator (2.5) implies the so(n) or sp(2m) Lie algebra relations, 8) and the symmetry condition,…”

“…According to (5.20) this R-matrix has the following non-vanishing entries: In analogy to the so(4) case, considering (α 1 , α 2 ) = (1, 1) and (γ 1 , γ 2 ) = (1, 2), (1, 3), (1,5), (1,8), in (5.13), one obtains for the arbitrary 8 × 8 monodromy matrix T (u):…”

“…(v) = 0. Finally, considering say (α 1 , α 2 ) = (2, 1), (2,4), (2,6), (γ 1 , γ 2 ) = (2,8) and (α 1 , α 2 ) = (3, 7), (γ 1 , γ 2 ) = (3, 8) one obtains:…”

“…so (8) is linear in the spectral parameter and contains two invariant tensors, while R (2) so (8) is quadratic in u and contains three tensors.…”

Orthogonal and symplectic Yangians: Linear and quadratic evaluations

“…In the cases G = so(n) and G = sp(n) the center has been analyzed in [22]. Proofs of the equivalence of the Yangian definition via (1.3) and the center factorization to the definitions by Drinfeld are given in [23,24].…”