Parabolic Presentations of the Super Yangian $${Y(\mathfrak{gl}_{M|N})}$$

Abstract

Associated to a composition of M and a composition of N, a new presentation of the super Yangian of the general linear Lie superalgebra Y (gl M |N ) is obtained. IntroductionFor each simple finite-dimensional Lie algebra g over C, the associated Yangian Y (g) was defined by Drinfeld in [D1] as a deformation of the universal enveloping algebra U (g [x]) for the polynomial current Lie algebra g [x]. The Yangians form a family of quantum groups which give rise to rational solutions of the Yang-Baxter equation originating from statistical mechanics; see [CP]. ) associated to each composition λ of N. Roughly speaking, the new presentation corresponds to a block matrix decomposition of gl N of shape λ. In the special case when λ = (1, 1, . . . , 1), the corresponding parabolic presentation is just a variation of Drinfeld's; see [BK1, Remark 5.12]. On the other extreme case when λ = (N ), the corresponding parabolic presentation is exactly the original RTT presentation. The parabolic presentation allows Brundan and Kleshchev to further define the standard Levi and parabolic subalgebras of Y (gl N ), and thus to obtain a Levi decomposition of Y (gl N ). The parabolic presentations have played a crucial role in their subsequent work [BK2], in which they derived generators and relations for the finite W -algebras.The main goal of this article is to obtain the superalgebra generalization of the parabolic presentations of [BK1] for the super Yangian Y (gl M |N ). The super Yangian of the general linear Lie superalgebra Y (gl M |N ) was introduced by Nazarov in [Na], and it shares many properties with the usual Yangian, such as the PBW theorem, the RTT relation and the Hopf algebra structure. The results of this article will be used in a sequel on the connection between Y (gl M |N ) and the super W -algebras.Let λ be a composition of M and ν be a composition of N . We first define some distinguished elements in Y (gl M |N ), denoted by D's, E's and F's, by Gauss decomposition and quasideterminants. We show that these new elements form a set of generators for Y (gl M |N ). The next step is to find the relations among the new generators, where the signs arising from the Z 2 -grading are involved here. However, since the (λ|ν)-block decomposition of gl M |N respects the Z 2 -grading of the superalgebra, the signs in the relations are determined by the block positions only. It is known (cf. [BK1]) that if the elements are from two different blocks and the blocks are not "close", then they commute. This phenomenon remains to be true in our super Yangian setting and it dramatically reduces the number of the nontrivial relations. Hence we only have to focus on the commutation relations of the elements in the same block or when their block-positions are "close". Let m be the number of parts of λ and n be the number of parts of ν. Then the first new non-trivial case will be m = n = 1, and the new ones will be m = 2, n = 1 and m = 1, n = 2 (see Section 4). In these special cases, we determined various relations among D's, E'...

Parabolic Presentations of the Super Yangian $${Y(\mathfrak{gl}_{M|N})}$$ Y ( gl M | N ) Associated with Arbitrary 01-Sequences

“…In [Go], a presentation of Y M |N when µ = (1 M +N ) is obtained, which is a generalization of Drinfeld's presentation. In [Pe1], a similar result when µ is of the form µ = (λ, ν), where λ is a composition of M and ν is a composition of N, is obtained. However, the results only explained the case when s = s st .…”

“…We quickly explain the idea, which is basically generalizing the argument in [BK1] and adapting some techniques in [Go,Pe1] dealing with the sign factors. Fix a composition µ of M + N and fix an arbitrary 01-sequence s of gl M |N .…”

“…As a matter of fact, there are already a few results [Go,Pe1] on such a generalization focusing on the canonical case when s = s st . In [Go], a presentation of Y M |N when µ = (1 M +N ) is obtained, which is a generalization of Drinfeld's presentation.…”

On shifted super Yangians and a class of finite W-superalgebras

“…, µ m+1 ) be a composition of (1|n). By [Pe1,Proposition 3.1], the parabolic generators D (r) a;i,j E (r) a,b;i,j and F (r) b,a;i,j of Y µ = Y 1|n are linear combinations of supermonomials in t (s) i,j of total degree r. On the other hand, if we set D (r) a;i,j , E (r) a,b;i,j and F (r) b,a;i,j all to be of degree r, by multiplying out the matrix equation…”

“…We start with a brief review about parabolic presentations of Y 1|n from [Pe1] in §3.1, and then we extend the notion to Y 1|n (σ) in §3.2.…”