aa r X i v : . [ m a t h . QA ] F e b FINITE W -SUPERALGEBRAS VIA SUPER YANGIANS YUNG-NING PENG
Abstract.
Let e be an arbitrary even nilpotent element in the general linear Lie super-algebra gl M | N and let W e be the associated finite W -superalgebra. Let Y m | n be the superYangian associated to the Lie superalgebra gl m | n . A subalgebra of Y m | n , called the shiftedsuper Yangian and denoted by Y m | n ( σ ), is defined and studied. Moreover, an explicit iso-morphism between W e and a quotient of Y m | n ( σ ) is established. Contents
1. Introduction 12. Finite W -superalgebras and pyramids 53. The super Yangian Y m | n Introduction
A finite W -algebra is an associative algebra determined by a pair ( g , e ), where g is afinite dimensional semisimple or reductive Lie algebra and e is a nilpotent element in g . Inthe extreme case when e = 0, the corresponding finite W -algebra is the universal envelopingalgebra U ( g ). In the other extreme case when e is the principal (also called regular ) nilpotentelement, Kostant [Ko] proved that the associated finite W -algebra is isomorphic to the centerof the universal enveloping algebra.The study of finite W -algebra for a general e was firstly developed systematically byPremet [Pr1], in which the modern terminologies were given and a proof of the long-standingKac-Weisfeiler conjecture [WK] was established. Moreover, finite W -algebras can be under-stood as quantizations of Slodowy slices [GG, Pr2]. Since then, finite W -algebras have appeared in many branches of mathematics so that their behavior and properties can beexplained from different viewpoints. In recent years, the finite W -algebras have been inten-sively studied by various approaches; see the survey articles [Ar, Lo, Wa] for details.On the other hand, Yangians are certain non-commutative Hopf algebras that are impor-tant examples of quantum groups. They first appeared in physics in the work of Faddeevand his school around 80’s concerning the quantum inverse scattering method. The termYangian was given by Drinfeld [Dr1] in honor of C.N. Yang and had been commonly usedsince then. They were used to provide rational solutions of the Yang-Baxter equation; seethe book [Mo] for related topics and further applications of Yangians.The connection between Yangians and finite W -algebras was firstly noticed by Ragoucyand Sorba [RS] for type A Lie algebras. Suppose that the nilpotent element e is rectangular ,which means that all the Jordan blocks of e are of the same size, say ℓ . They showed thatthe associated finite W -algebra is isomorphic to the Yangian of level ℓ , which is a certainquotient of the Yangian, considered by Cherednik [C1, C2].This observation is further generalized by Brundan and Kleshchev [BK2] to an arbitrarynilpotent e ∈ gl N . The main result [BK2, Theorem 10.1] can be shortly described as follows:the finite W -algebra associated to a nilpotent e ∈ gl N is isomorphic to a quotient of somesubalgebra of the Yangian (called the shifted Yangian ) associated to gl n , where n is thenumber of Jordan blocks of e . Moreover, an explicit realization of type A finite W -algebraby generators and relations is obtained. This provides a powerful tool for the study offinite W -algebras, their representations and further applications [BGK, BK3, BK4]. It isalso observed recently that the shifted Yangian can also be defined by different approachestogether with generalizations and applications; see [BFN, FKPRW, FPT, KWWY].The finite W -superalgebras are defined in a very similar way as the Lie algebra case exceptthat the nilpotent element e ∈ g is assumed to be even (with respect to the Z -grading ofthe Lie superalgebra) with other modifications. In recent years, finite W -superalgebras andtheir representations have been extensively studied [BBG, BGK, WZ1, WZ2, ZS1, ZS2, Zh]with different emphases.The super Yangian associated to gl m | n , denoted by Y m | n , was defined by Nazarov [Na1]in terms of the RTT presentation . It is natural to seek for connections between finite W -superalgebras and super Yangians. The very first result is obtained by Briot and Ragoucy[BR], saying that if the nilpotent element e ∈ gl M | N is rectangular, then the associatedfinite W -superalgebra is isomorphic to a certain quotient of Y m | n called the truncated superYangian , where m and n are the numbers of Jordan blocks of e restricted to the even andodd spaces, respectively. In recent years, there have been some results [BBG, Pe2, Pe3]generalizing the above observation when the nilpotent element e satisfies some assumptions,but for a general e the problem remains to be open. INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 3 The goal of this article is to give a solution to this open problem, generally establishingthe connection between the finite W -superalgebras and super Yangians for type A. That is,we explicitly give a superalgebra isomorphism between the finite W -superalgebra associatedto an arbitrary even nilpotent element e ∈ gl M | N and a quotient of a certain subalgebra of Y m | n , obtaining a super analogue of the main result of [BK2] for type A Lie superalgebras infull generality.We shortly explain our approach, which is basically generalizing the arguments in [BK2]to the general linear Lie superalgebras with suitable modifications and try to overcome allof the difficulties along the way. Although there are similarities between gl N and gl M | N and similarities between the associated super Yangians, some of the earlier approaches areno longer available in the case of Lie superalgebras. Moreover, there are other technical orconceptual obstacles that did not appear in the Lie algebra case.Our first step is to define a subalgebra of Y m | n which we call the shifted super Yangian and denote by Y m | n ( σ ). To obtain this subalgebra, we need to use certain presentations of Y m | n called the parabolic presentations . Similar to the Lie algebra case [BK1, Dr2], the RTTpresentation and the Drinfeld’s presentation can be treated as special cases of the parabolicpresentations. There have been some results [Go, Pe1] giving suitable presentations of Y m | n ,where the results [BBG, Pe3] are in fact based on them. However, as noticed in [BBG, Pe3],they are no longer suitable presentations for the general case. What we need is a kindof “more” generalized parabolic presentation which works for any Y m | n . Such a presentationwas recently obtained by the author in [Pe4]. As a consequence, the shifted super Yangian Y m | n ( σ ) can be defined as a subalgebra of Y m | n generated by a certain subset of the generatingset for the whole Y m | n .However, to establish the desired connection, we need not only the subalgebra but also its presentation . By suitably modifying the defining relations for Y m | n found in [Pe4], we obtaina set of defining relations and hence a presentation of the shifted super Yangian Y m | n ( σ ). Itshould be emphasized that there are a few extra series of defining relations for Y m | n that didnot appear in [BK2]. Although we are able to guess the suitable modifications, it is highlynon-trivial to check that our proposed relations actually hold in Y m | n ( σ ). With some effort,one can eventually overcome this difficulty and a presentation of Y m | n ( σ ) is obtained, whichallows one to define some homomorphisms called baby comultiplications , see §
6, that willplay important roles in the desired connection.We further define the shifted super Yangian of level ℓ , denoted by Y ℓm | n ( σ ), as a quotient of Y m | n ( σ ) over some 2-sided ideal. Roughly speaking, σ is a matrix recording the generatingset for Y m | n ( σ ), while ℓ is an integer recording the size of the ideal in the quotient. It turnsout that the data σ and ℓ can be recorded by a diagram called pyramid [EK, Ho], which we YUNG-NING PENG denote by π , and it makes sense to set the notation Y π := Y ℓm | n ( σ ). On the other hand, thediagram π also determines a finite W -superalgebra which we denote by W π .In §
9, we introduce the notion of super column height so that one may explicitly write downsome distinguished elements in W π according to the diagram π by modifying the descriptionin [BK2, § Y π into these distinguished elements in W π is an isomorphism of (filtered) superalgebras,obtaining a presentation of the finite W -superalgebra W π .It is an interesting question to generalize the results in this article to other types ofLie superalgebras. In particular, there have been some results in the case of queer Liesuperalgebras and their associated Yangians [Na2] when the even nilpotent element is regular[PS1] or rectangular [PS2], but it is still open in general. We expect that the approachesin this article can be suitably modified to deal with the queer Lie superalgebra case for ageneral nilpotent element.This article is organized as follows. In §
2, we set up our notations and recall some necessarybackground knowledge about finite W -superalgebras. In particular, the notion of pyramidwith respect to a 01-sequence is recalled. In §
3, we recall some well-known facts about Y m | n .The shifted super Yangian Y m | n ( σ ) is defined in § Y m | n , where some computations are relatively easier in thissetting. Then we show that Y m | n ( σ ) can be identified as a subalgebra of Y m | n . Some basicproperties of Y m | n ( σ ) are also provided.In § Y m | n , todefine Y m | n ( σ ) and establish the corresponding properties obtained in § § § §
7, we introduce the canonical filtration of Y m | n ( σ ), which eventually corresponds tothe Kazhdan filtration of finite W -superalgebras. The shifted super Yangian of level ℓ isdefined in § Y m | n ( σ ).In §
9, we explicitly define some distinguished elements in the universal enveloping algebra U ( gl M | N ) that will eventually be identified as generators of our finite W -superalgebra. Ourmain result is stated and proved in § C , which can be replaced byany algebraically closed field of characteristic zero. The term subalgebra always means a sub-superalgebra . For homogeneous elements x and y in an associated superalgebra L , the INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 5 supercommutator of x and y is defined by (cid:2) x, y (cid:3) = xy − ( − | x || y | yx, where | x | is the Z -grading of x in L , called the parity of x . By convention, a homogeneouselement x is called even (resp. odd ) if | x | = 0 (resp. 1). L and L denote the set of evenand odd elements in L , respectively. Acknowledgements.
The author is grateful to Shun-Jen Cheng and Weiqiang Wang forcountless discussions and encouragement. A part of this article was finished during theauthor’s visit to RIMS (Kyoto, Japan) in 2016. The author would like to thank the RIMS forproviding an excellent working environment, and also thank Naoki Genra, Ryosuke Koderaand Hiraku Nakajima for stimulating discussions during the visit. The visit is supportedby the NCTS (Taipei, Taiwan), which is greatly acknowledged. The author would also liketo thank Lucy Gow, Alexander Molev and Alexander Tsymbaliuk for communication. Thiswork is partially supported by MOST grant 105-2628-M-008-004-MY4.2.
Finite W -superalgebras and pyramids In this section, we recall the definition of a finite W -superalgebra, which is determinedby an even nilpotent element e and a semisimple element h of gl M | N . Also, a combinatorialobject called pyramid is introduced so that we may encode e and h simultaneously by adiagram π .Throughout this section, g = gl M | N is identified with the set of ( M + N ) × ( M + N )matrices with the standard Z -grading g = g ⊕ g and ( · , · ) means the non-degenerateeven supersymmetric g -invariant bilinear form on g defined by( x, y ) := str( xy )for all x, y ∈ g , where xy stands for the usual matrix product and str means the supertrace.Every elements of g appearing in any equations are considered homogeneous with respect tothe Z -grading unless specifically mentioned.2.1. Finite W -superalgebras of gl M | N . Let e be an even nilpotent element in g . It is well-known [Ho, Wa] that there exists (not uniquely in general) a semisimple element h ∈ g suchthat ad h : g → g gives a good Z -grading of g for e , which means the following conditionsare satisfied:(1) ad h ( e ) = 2 e ,(2) g = L j ∈ Z g ( j ), where g ( j ) := { x ∈ g | ad h ( x ) = jx } ,(3) the center of g is contained in g (0),(4) ad e : g ( j ) → g ( j + 2) is injective for all j ≤ − e : g ( j ) → g ( j + 2) is surjective for all j ≥ − YUNG-NING PENG
In order to simplify the definition of finite W -superalgebras, throughout this article, weassume in addition that the Z -grading is even ; that is, g ( i ) = 0 for all i / ∈ Z . We say h e, h i is a good pair if ad h gives an even good Z -grading of g for e . Remark 2.1.
In general, a good pair may fail to exist in other types of classical Lie super-algebras [Ho] . But for any even nilpotent e ∈ gl M | N we can always find some h such that h e, h i is a good pair; see Theorem . Fix a good pair h e, h i in g . Define the following subalgebras of g by p := M j ≥ g ( j ) , m := M j< g ( j ) . (2.1)Define χ ∈ g ∗ by χ ( y ) := ( y, e ) ∀ y ∈ g . The restriction of χ on m extends to a one dimensional U ( m )-module. Let I χ be the leftideal of U ( g ) generated by { a − χ ( a ) | a ∈ m } . As a consequence of the PBW theorem for U ( g ), we have U ( g ) = I χ ⊕ U ( p ) together withthe following identification U ( g ) /I χ ∼ = U ( p )by the natural projection pr χ : U ( g ) → U ( p ). One defines the following χ -twisted action of m on U ( p ) by a · y := pr χ ([ a, y ]) , for all a ∈ m , y ∈ U ( p ).The finite W-superalgebra , which we will usually omit the prefix “finite” from now on, isdefined to be the space of m -invariants in U ( p ) under the χ -twisted action; to be explicit, W e,h := U ( p ) m = { y ∈ U ( p ) | pr χ ([ a, y ]) = 0 , ∀ a ∈ m } = { y ∈ U ( p ) | (cid:0) a − χ ( a ) (cid:1) y ∈ I χ , ∀ a ∈ m } . For example, if e = 0, then χ = 0, g = g (0) = p and m = 0. Thus the associated W -superalgebra is exactly U ( g ).At this point, it seems that the definition of a W -superalgebra depends on both of e and h in the good pair. In fact, the definition is independent of the choices of h up to isomorphisms;see Remark 10.12. INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 7 Pyramids and W -superalgebras. We recall the notion of pyramid [EK, Ho] as aconvenient tool to present a good pair h e, h i . We will identify a partition λ = ( λ , λ , . . . )with its corresponding Young diagram in French style, which means that the diagrams areleft-justified and the longest row is located in the bottom. Definition 2.2.
Let λ be a Young diagram. A pyramid is a diagram obtained by horizontallyshifting the rows of λ such that for each box not in the bottom row, there is exactly one boxbelow it. For example, only the left-most diagram is a pyramid obtained from λ = (3 , , V = V ⊕ V be a Z -graded vector space with dim V = M and dim V = N . Weidentify g = gl M | N with End V and one has the following identification for g g ∼ = End( V ) ⊕ End( V ) . As a result, an even nilpotent element e ∈ gl M | N can be thought as a sum of two nilpotentelement e = e + e , where e i ∈ End V i for i ∈ { , } . Thus we may describe e by two Youngdiagrams µ and ν corresponding to the Jordan types of e and e , respectively.For example, the diagram + ++ + + ⊕ − −− − − − represents an even nilpotent element in gl | , which is a sum of a nilpotent element in End C with Jordan type µ = (3 ,
2) and a nilpotent element in End C with Jordan type ν = (4 , − in the boxes because we now stack the two diagrams together to obtain anew Young diagram, and we need to track from which diagram the boxes originally are.For example, there are two possibilities if we stack the above two Young diagrams togetherto obtain one Young diagram: + + − − + + + − − − − − − + ++ + + − − − − (2.2) Remark 2.3.
The pyramids in this article correspond to certain even nilpotent elements in gl M | N , hence the following condition always holds:every boxes in a row have the same + or − labeling. YUNG-NING PENG
As one may expect, we shift the rows of the stacked Young diagram to obtain a pyramid.For example, we take the right diagram in (2.2) and list all possibilities below: − − − − + + ++ + − − − − − − + + ++ + − − − − − − + + ++ + − − − − − − + + ++ + − −
Soon we will see (Theorem 2.4) that each of these pyramids represents a good pair h e, h i in gl | . Moreover, these are all good pairs we could have for that given e ∈ gl | .Now we do the other way around: obtaining a good pair h e, h i from a given pyramid π satisfying the condition described in Remark 2.3. Assume that we have M (resp. N ) boxeslabeled with + (resp. − ) in π , where they came from the Young diagram of e ∈ gl M | (resp. e ∈ gl | N ). We enumerate those “ + ” boxes by 1 , , . . . , M down columns from left to right,and enumerate those “ − ” boxes by 1 , , . . . , N by the same rule.Next we imagine that each box of π is of size 2 × x -axis,where the center of π is exactly located above the origin. For instance: π = x -coordinates: 1 3 5 62 4 51 32 4 • − − I = { < . . . < M < < . . . < N } be an ordered index set and let { v i | i ∈ I } be thestandard basis of C M | N with respect to the following order v i < v j if i < j in I. Let { e i,j | i, j ∈ I } denote the elementary matrices in gl M | N . Define the element e π := X i j ∈ π e i,j ∈ g , (2.4)where the sum is taken over all adjacent pairs i j appeared in π. Let col x ( i ) denote the x -coordinate of the center of the box numbered with i ∈ I , whichmust be an integer by our construction. Define the following diagonal matrix h π := − diag (cid:0) col x (1) , . . . , col x ( M ) , col x (1) , . . . , col x ( N ) (cid:1) (2.5) INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 9 For example, the elements e π and h π associated to the pyramid π in (2.3) are e π = e + e + e + e + e + e + e ,h π = diag(1 , , − , − , − , , , , − , − , − . It is easy to check that h e π , h π i forms a good pair.Note that if we horizontally shift the rows of π to obtain another pyramid ~π , then e π = e ~π but h π = h ~π . The following theorem implies that every even good Z -gradings for e π can beobtained by shifting the rows of π . Theorem 2.4. [Ho, Theorem 7.2]
Let π be a pyramid. Let e = e π and h = h π be theelements in gl M | N defined by (2.4) and (2.5) , respectively. Then h e, h i forms a good pair for e . Moreover, any good pair for e is of the form h e, h ~π i where ~π is some pyramid obtained byshifting rows of π horizontally. In other words, Theorem 2.4 classifies all of the even good Z -gradings of gl M | N for any even nilpotent e . (In fact, [Ho, Theorem 7.2] classifies all good Z -gradings, not just thoseeven good Z -gradings considered in this article.) As a consequence, for a given pyramid π ,it makes sense to denote the W -superalgebra associated to the good pair h e π , h π i simply by W π := W e π ,h π . Remark 2.5.
If we permute the rows with the same length of π to obtain a new pyramid π ′ ,then we have e π = e π ′ and h π = h π ′ . For example, the two Young diagrams in (2.2) give usexactly the same list of good pairs by shifting their rows. We label the columns of π from left to right by 1 , . . . , ℓ . For any i ∈ I , let col( i ) denotethe column where i appear. The Kazhdan filtration of U ( g ) · · · ⊆ F d U ( g ) ⊆ F d +1 U ( g ) ⊆ · · · is defined by setting deg( e i,j ) := col( j ) − col( i ) + 1 (2.6)for each i, j ∈ I , where F d U ( g ) denotes the span of all supermonomials e i ,j · · · e i s ,j s for s ≥ P sk =1 deg ( e i k ,j k ) ≤ d . Let gr U ( g ) denote the graded superalgebra associated tothe Kazhdan filtration. A natural grading on W π is induced from the projection g ։ p andwe denote by gr W π the associated graded superalgebra.Let g e denote the centralizer of e in g and let S ( g e ) denote the associated supersymmetricsuperalgebra. The same setting (2.6) defines the Kazhdan filtration on S ( g e ). The followingresult still holds in our case since our pyramid π satisfies the condition in Remark 2.3. Proposition 2.6. [Zh, Remark 3.11] S ( g e ) and gr W π are isomorphic as graded superalge-bras. Shift matrix.
We give an alternative way to describe a pyramid. An ( m + n ) × ( m + n )matrix σ = ( s i,j ) ≤ i,j ≤ m + n is called a shift matrix if its entries are non-negative integerssatisfying the following condition s i,j + s j,k = s i,k , (2.7)whenever | i − j | + | j − k | = | i − k | . For example, the following matrix is a shift matrix: σ = (2.8) Lemma 2.7.
The follow facts hold for a shift matrix σ = ( s i,j ) ≤ i,j ≤ m + n . (1) If the entries in the last column { s i,m + n | ≤ i ≤ m + n } are known, then the wholeupper-triangular part of σ is determined. (2) If the entries in the upper-diagonal { s i,i +1 | ≤ i < m + n } are known, then the wholeupper-triangular part of σ is determined. (3) If the entries in the last row { s m + n,i | ≤ i ≤ m + n } are known, then the wholelower-triangular part of σ is determined. (4) If the entries in the lower-diagonal { s i +1 ,i | ≤ i < m + n } are known, then the wholelower-triangular part of σ is determined.Proof. By (2 . (cid:3) In our superalgebra setting, we need to record the ± -labeling of each row in our pyramid,so we introduce the following terminology. Let m, n ∈ Z ≥ . A 0 m n -sequence , or 01 -sequence for short, is an ordered sequence Υ consisting of m n ≤ i ≤ m + n , thei-th digit of Υ is denoted by | i | .Suppose that σ ∈ M m + n ( Z ≥ ) is a shift matrix. Let ℓ be an integer such that ℓ >s ,m + n + s m + n, and let Υ be a fixed 0 m n -sequence. Then one can obtain a pyramid π , with m (resp. n ) rows labeled by “+” (resp. “ − ”) and the bottom row consisting of ℓ boxes, fromthe triple ( σ, ℓ, Υ) by the following fashion.Start with a rectangular Young diagram consisting of m + n rows and ℓ columns, whichwe denote by Ξ. We number the rows of Ξ from top to bottom by 1 , , . . . , m + n . For each1 ≤ i ≤ m + n , we label every boxes in the i -th row of Ξ by “ + ” if | i | = 0, and by “ − ” if | i | = 1.Next we obtain our pyramid from this rectangle. Consider the entries in the last row andthe last column of σ : { s m + n,i | ≤ i ≤ m + n } and { s i,m + n | ≤ i ≤ m + n } . For each1 ≤ j ≤ m + n , we erase the leftmost s m + n,j boxes and the rightmost s j,m + n boxes in the INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 11 j -th row of Ξ. By (2.7), the resulted diagram is a pyramid which has ℓ boxes in the bottomrow and ℓ − s m + n, − s ,m + n boxes in the top row. For example, take ℓ = 8 and let σ be theone given in (2.8) with Υ = 100010, the resulted pyramid π is − + ++ ++ ++ ++ + − −− −− − + ++ ++ ++ +Conversely, given a pyramid π which represents a good pair. Let ℓ be the number ofboxes in the bottom of π and let m and n be the numbers of rows of π labeled by + and − , respectively. We number the rows of π from top to bottom by 1 , , . . . , m + n as before.Since π satisfies the condition in Remark 2.3, we may obtain a 0 m n -sequence Υ by assigningthe i -th digit of Υ to be 0 (resp. 1) if the boxes in the i -th row are labeled by “ + ” (resp.“ − ”).For each 1 ≤ i ≤ m + n , define the number s m + n,i (resp. s i,m + n ) to be the numberof missing boxes on the left-hand side (resp. right-hand side) of the i -th row of π in arectangular diagram Ξ of size ( m + n ) × ℓ . This gives us the entries of the last row and thelast column of σ and hence we are able to recover the whole σ by Lemma 2.7. The discussionabove is summarized in the following proposition. Proposition 2.8.
Let S be the set of triples ( σ, ℓ, Υ) where σ is a shift matrix of size m + n , ℓ > s m + n, + s ,m + n is an integer and Υ is a m n -sequence. Let P be the set of all pyramids π such that π has m (resp. n ) rows labeled by + (resp. − ) and ℓ columns. Then there existsa bijection between S and P . Roughly speaking, σ determines the shape and height , ℓ determines the width and Υdetermines the ± -labeling of π and vise versa.The following proposition is a super analogue of a well-known result about g e . Since ourpyramid π satisfies the condition described in Remark 2.3, it is similar to the Lie algebracase as remarked in [BBG]. Proposition 2.9.
Let π be a pyramid with row lengths { p i | ≤ i ≤ m + n } , where the rowsare labeled from top to bottom. Let σ = ( s i,j ) ≤ i,j ≤ m + n be the associated shift matrix of π inthe triple ( σ, ℓ, Υ) . Let e = e π be the nilpotent element defined by (2.4) . Let M (resp. N ) be the number of boxes of π labeled in + (resp. − ). For all ≤ i, j ≤ m + n and r > , define c ( r ) i,j := X h,k ∈ Irow ( h )= i, row ( k )= j col ( k ) − col ( h )= r − e h,k ∈ g = gl M | N . Then { c ( r ) i,j | ≤ i, j ≤ m + n, s i,j < r ≤ s i,j + p min ( i,j ) } forms a linear basis for g e . The super Yangian Y m | n In this section, we recall some well-known facts about the super Yangian associated to thegeneral linear Lie superalgebra.3.1.
RTT presentations of Y m | n .Definition 3.1. [Na1] For a given 01-sequence Υ , the Yangian associated to the generallinear Lie superalgebra gl m | n , denoted by Y m | n , is the associative Z -graded algebra with unitygenerated over C by the RTT generators n t ( r ) i,j | ≤ i, j ≤ m + n ; r ≥ o , (3.1) subject to following RTT relations: (cid:2) t ( r ) i,j , t ( s ) h,k (cid:3) = ( − | i | | j | + | i | | h | + | j | | h | min( r,s ) − X g =0 (cid:16) t ( g ) h,j t ( r + s − − g ) i,k − t ( r + s − − g ) h,j t ( g ) i,k (cid:17) , (3.2) where the parity of t ( r ) i,j is defined by | i | + | j | (mod 2) . By convention, we set t (0) i,j := δ ij . The original definition in [Na1] corresponds to the case when Υ is the standard st := m z }| { . . . n z }| { . . . . As observed in [Pe2, Ts], up to isomorphism, the definition of Y m | n is independent of thechoices of Υ so we often omit it in our notation when appropriate.For each 1 ≤ i, j ≤ m + n , define the formal series t i,j ( u ) := X r ≥ t ( r ) i,j u − r ∈ Y m | n [[ u − ]] . It is well-known [Na1] that Y m | n is a Hopf-superalgebra. In particular, the comultiplication∆ : Y m | n → Y m | n ⊗ Y m | n can be nicely described as∆( t ( r ) i,j ) = r X s =0 m + n X k =1 t ( r − s ) i,k ⊗ t ( s ) k,j . (3.3)Moreover, there exists a surjective homomorphismev : Y m | n → U ( gl m | n ) INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 13 called the evaluation homomorphism , defined byev (cid:0) t i,j ( u ) (cid:1) := δ ij + ( − | i | e ij u − , (3.4)where e ij ∈ gl m | n means the elementary matrix.The following proposition gives a PBW basis for Y m | n in terms of the RTT generators,where the proof in [Go] works perfectly for any fixed Υ. Proposition 3.2. [Go, Theorem 1]
The set of supermonomials in the following elements n t ( r ) i,j | ≤ i, j ≤ m + n, r ≥ o taken in some fixed order forms a linear basis for Y m | n . Define the loop f iltration on Y m | n L Y m | n ⊆ L Y m | n ⊆ L Y m | n ⊆ · · · by setting deg t ( r ) ij = r − r ≥ L k Y m | n be the span of all supermonomialsof the form t ( r ) i j t ( r ) i j · · · t ( r s ) i s j s with total degree not greater than k . We denote by gr L Y m | n the associated graded superal-gebra.Let gl m | n [ x ] denote the loop superalgebra gl m | n ⊗ C [ x ], where a basis is given by { e ij x r | ≤ i, j ≤ m + n, r ≥ } . Let U ( gl m | n [ x ]) denote its universal enveloping algebra with the natural filtration and gradinggiven by deg e ij x r := r. The following corollary is a consequence of Proposition 3.2.
Corollary 3.3. [Go, Corollary 1]
The function Y m | n → U ( gl m | n [ x ]) given by t ( r ) ij ( − | i | e ij x r − induces an isomorphism gr L Y m | n ∼ = U ( gl m | n [ x ]) of graded superalgebras. Parabolic generators of Y m | n . In this subsection, we give another generating set for Y m | n . Eventually it will allow us to define a certain subalgebra of Y m | n which can not beobserved by the earlier RTT-presentation except for some special cases.Firstly we introduce a convenient shorthand notation. Let µ = ( µ , . . . , µ z ) be a givencomposition of m + n with length z and let Υ be a fixed 0 m n -sequence. We break Υ into z subsequences according to µ ; that is, Υ = Υ Υ . . . Υ z , where Υ is the subsequence consisting of the first µ digits of Υ, Υ is the subsequenceconsisting of the next µ digits of Υ, and so on. For example, if we have Υ = 011100011 and µ = (2 , , Υ z}|{ Υ z}|{ Υ z}|{ . For each 1 ≤ a ≤ z , let p a and q a denote the number of 0’s and 1’s in Υ a , respectively. Fora fixed 1 ≤ a ≤ z and each value of i = 1 , , . . . , µ a , we define the restricted parity | i | a by | i | a := the i -th digits of Υ a ,or equivalently | i | a = | a − X j =1 µ j + i | . (3.5)Define the ( m + n ) × ( m + n ) matrix with entries in Y m | n [[ u − ]] by T ( u ) := (cid:16) t i,j ( u ) (cid:17) ≤ i,j ≤ m + n Note that the leading minors of the matrix T ( u ) are always invertible and hence the matrix T ( u ) possesses a Gauss decomposition with respect to µ ; that is, T ( u ) = F ( u ) D ( u ) E ( u ) (3.6)for unique block matrices D ( u ), E ( u ) and F ( u ) of the form D ( u ) = D ( u ) 0 · · · D ( u ) · · · · · · D z ( u ) ,E ( u ) = I µ E , ( u ) · · · E ,z ( u )0 I µ · · · E ,z ( u )... ... . . . ...0 0 · · · I µ z ,F ( u ) = I µ · · · F , ( u ) I µ · · · F z, ( u ) F z, ( u ) · · · I µ z , INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 15 where D a ( u ) = (cid:0) D a ; i,j ( u ) (cid:1) ≤ i,j ≤ µ a , (3.7) E a,b ( u ) = (cid:0) E a,b ; h,k ( u ) (cid:1) ≤ h ≤ µ a , ≤ k ≤ µ b , (3.8) F b,a ( u ) = (cid:0) F b,a ; k,h ( u ) (cid:1) ≤ k ≤ µ b , ≤ h ≤ µ a , (3.9)are µ a × µ a , µ a × µ b and µ b × µ a matrices, respectively, for all 1 ≤ a ≤ z in (3.7) and all1 ≤ a < b ≤ z in (3.8) and (3.9). In fact, these matrices can be explicitly obtained by quasideterminants (cf. [GR]).Since all of the submatrices D a ( u )’s are invertible, it allows one to define the µ a × µ a matrix D ′ a ( u ) = (cid:0) D ′ a ; i,j ( u ) (cid:1) ≤ i,j ≤ µ a by D ′ a ( u ) := (cid:0) D a ( u ) (cid:1) − . The entries of these matrices give us some formal series with coefficients in Y m | n : D a ; i,j ( u ) = P r ≥ D ( r ) a ; i,j u − r , D ′ a ; i,j ( u ) = X r ≥ D ′ ( r ) a ; i,j u − r , (3.10) E a,b ; h,k ( u ) = P r ≥ E ( r ) a,b ; h,k u − r , F b,a ; k,h ( u ) = X r ≥ F ( r ) b,a ; k,h u − r . (3.11)Actually we only need the diagonal, upper-diagonal and lower-diagonal blocks. Hence we set E b ; i,j ( u ) := E b,b +1; h,k ( u ) = X r ≥ E ( r ) b ; h,k u − r , F b ; i,j ( u ) := F b +1 ,b ; k,h ( u ) = X r ≥ F ( r ) b ; k,h u − r , (3.12)for 1 ≤ b ≤ z −
1. As proved in [Pe4], these coefficients { D ( r ) a ; i,j , D ′ ( r ) a ; i,j | ≤ a ≤ z, ≤ i, j ≤ µ a , r ≥ }{ E ( r ) b ; h,k | ≤ b < z, ≤ h ≤ µ b , ≤ k ≤ µ b +1 , r ≥ }{ F ( r ) b ; k,h | ≤ b < z, ≤ k ≤ µ b +1 , ≤ h ≤ µ b , r ≥ } form a generating set for Y m | n , called the parabolic generators of Y m | n , which will be denotedby P µ . Moreover, by [Pe4, Lemma 4.2], their parities can be explicitly determined by thefollowing rule: parity of D ( r ) a ; i,j = | i | a + | j | a (mod 2) , (3.13)parity of E ( r ) b ; h,k = | h | b + | k | b +1 (mod 2) , (3.14)parity of F ( r ) b ; k,h = | k | b +1 + | h | b (mod 2) . (3.15)In the special case when µ = (1 m + n ) := ( m + n z }| { , . . . , P D , appeared in an analogue of the Drinfeld presentation for Y m | n [BK1, Dr2, Go, Pe4, St, Ts]. We list P D explicitly here since it will be used right away: { D ( r ) a , D ′ ( r ) a | ≤ a ≤ m + n, r ≥ } , (3.16) { E ( r ) b | ≤ b < m + n, r ≥ } , (3.17) { F ( r ) b | ≤ b < m + n, r ≥ } , (3.18)and their parities are given by | D ( r ) a | = | D ′ ( r ) a | = 0 , | E ( r ) b | = | F ( r ) b | = | b | + | b + 1 | (mod 2) . (3.19)4. Shifted super Yangian: Drinfeld’s presentation
Recall from § π can be uniquely recorded by a triple ( σ, ℓ, Υ) where σ is ashift matrix of size m + n , ℓ is a positive integer and Υ is a 01-sequence. Following [BK2, § σ and Υ to define the following structure, which is one of the main objects studiedin this article. Definition 4.1.
Let m, n ∈ Z ≥ , σ = ( s i,j ) be a shift matrix of size m + n with a fixed m n -sequence Υ . The shifted super Yangian of gl m | n associated to σ , denoted by Y m | n ( σ ) , isthe superalgebra over C generated by following symbols (cid:8) D ( r ) a , D ′ ( r ) a | ≤ a ≤ m + n, r ≥ (cid:9) , (cid:8) E ( r ) b | ≤ b < m + n, r > s b,b +1 (cid:9) , (cid:8) F ( r ) b | ≤ b < m + n, r > s b +1 ,b (cid:9) , where their parities are defined by (3.19) , subject to the following relations: D (0) a = D ′ (0) a = 1 , (4.1) r X t =0 D ( t ) a D ′ ( r − t ) a = δ r , (4.2) (cid:2) D ( r ) a , D ( s ) b (cid:3) = 0 , (4.3)[ D ( r ) a , E ( s ) b ] = ( − | a | (cid:0) δ a,b − δ a,b +1 (cid:1) r − X t =0 D ( t ) a E ( r + s − − t ) b , (4.4)[ D ( r ) a , F ( s ) b ] = ( − | a | (cid:0) δ a,b +1 − δ a,b (cid:1) r − X t =0 F ( r + s − − t ) b D ( t ) a , (4.5)[ E ( r ) a , F ( s ) b ] = δ a,b ( − | a +1 | +1 r + s − X t =0 D ′ ( r + s − − t ) a D ( t ) a +1 , (4.6) INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 17 [ E ( r ) a , E ( s ) a ] = ( − | a +1 | (cid:0) s − X t = s a,a +1 +1 E ( r + s − − t ) a E ( t ) a − r − X t = s a,a +1 +1 E ( r + s − − t ) a E ( t ) a (cid:1) , (4.7)[ F ( r ) a , F ( s ) a ] = ( − | a | (cid:0) r − X t = s a +1 ,a +1 F ( r + s − − t ) a F ( t ) a − s − X t = s a +1 ,a +1 F ( r + s − − t ) a F ( t ) a (cid:1) , (4.8)[ E ( r +1) a , E ( s ) a +1 ] − [ E ( r ) a , E ( s +1) a +1 ] = ( − | a +1 | E ( r ) a E ( s ) a +1 , (4.9)[ F ( r +1) a , F ( s ) a +1 ] − [ F ( r ) a , F ( s +1) a +1 ] = ( − | a || a +1 | + | a +1 || a +2 | + | a || a +2 | F ( s ) a F ( r ) a , (4.10)[ E ( r ) a , E ( s ) b ] = 0 if | b − a | > , (4.11)[ F ( r ) a , F ( s ) b ] = 0 if | b − a | > , (4.12) (cid:2) E ( r ) a , [ E ( s ) a , E ( t ) b ] (cid:3) + (cid:2) E ( s ) a , [ E ( r ) a , E ( t ) b ] (cid:3) = 0 if | a − b | = 1 , (4.13) (cid:2) F ( r ) a , [ F ( s ) a , F ( t ) b ] (cid:3) + (cid:2) F ( s ) a , [ F ( r ) a , F ( t ) b ] (cid:3) = 0 if | a − b | = 1 , (4.14) (cid:2) [ E ( r ) a − , E ( s a,a +1 +1) a ] , [ E ( s a,a +1 +1) a , E ( s ) a +1 ] (cid:3) = 0 when m + n ≥ and | a | + | a + 1 | = 1 , (4.15) (cid:2) [ F ( r ) a − , F ( s a +1 ,a +1) a ] , [ F ( s a +1 ,a +1) a , F ( s ) a +1 ] (cid:3) = 0 when m + n ≥ and | a | + | a + 1 | = 1 , (4.16) for all admissible indices a, b, r, s, t . For example, (4.4) is meant to hold for all r ≥ , s > s b,b +1 , ≤ a ≤ m + n and ≤ b < m + n . Note that when σ is the zero matrix, the presentation above coincides with the presentationof Y m | n given in [Pe4] by taking µ = (1 m + n ) therein (this special case is also obtained in [Ts]).As a result, we may identify Y m | n (0) = Y m | n . In the remaining part of this section, we will show that Y m | n ( σ ) can be identified as asubalgebra of Y m | n in general (Crorllary 4.5). Let P D,σ be the generating set of Y m | n ( σ ) inDefinition 4.1. Let Γ : Y m | n ( σ ) → Y m | n be the map sending P D,σ to the elements with thesame name (3.16)–(3.18) in Y m | n obtained by Gauss decomposition. Proposition 4.2.
The canonical map
Γ : Y m | n ( σ ) → Y m | n is a homomorphism.Proof. By setting µ = (1 m + n ) in [Pe4, Proposition 7.1], or simply by [Ts, (2.2)–(2.10)], therelations (4.1)–(4.14) are preserved by Γ. Setting k = l in the generalized quartic Serrerelations in [Ts, (2.14), (2.15)], we see that (4.15) and (4.16) are preserved by Γ as well. (cid:3) It remains to show that Γ is injective. We introduce the loop f iltration on Y m | n ( σ ) L Y m | n ( σ ) ⊆ L Y m | n ( σ ) ⊆ L Y m | n ( σ ) ⊆ · · · by setting the degrees of the generators D ( r ) a , E ( r ) b , and F ( r ) b to be ( r −
1) and setting L k Y m | n ( σ )to be the span of all supermonomials in the generators of total degree not greater than k .Let gr L Y m | n ( σ ) denote the associated graded superalgebra.For 1 ≤ a < b ≤ m + n , r > s a,b and t > s b,a , define the following higher root elements E ( r ) a,b , F ( t ) b,a ∈ Y m | n ( σ ) recursively by E ( r ) a,a +1 := E ( r ) a , E ( r ) a,b := ( − | b − | [ E ( r − s b − ,b ) a,b − , E ( s b − ,b +1) b − ] , (4.17) F ( t ) a +1 ,a := F ( t ) a , F ( t ) b,a := ( − | b − | [ F ( s b,b − +1) b − , F ( t − s b,b − ) b − ,a ] . (4.18)By definition, we have E ( r ) a,b ∈ L r − Y m | n ( σ ) and F ( t ) b,a ∈ L t − Y m | n ( σ ).Define the elements { e ( r ) a,b | ≤ a, b ≤ m + n, r ≥ s a,b } ⊆ gr L Y m | n ( σ ) by e ( r ) a,b := gr Lr D ( r +1) a if a = b ,gr Lr E ( r +1) a,b if a < b ,gr Lr F ( r +1) a,b if a > b . (4.19)Using the same argument in [Pe4, Lemma 7.5], except that one uses the defining relationsof Y m | n ( σ ) listed in Definition 4.1, we deduce the following result. Proposition 4.3. [BK2, (2.21)][Go, (51)]
For all ≤ a, b, c, d ≤ m + n , r ≥ s a,b , t ≥ s c,d ,the following identity holds in gr L Y m | n ( σ ) : [ e ( r ) a,b , e ( t ) c,d ] = ( − | b | δ b,c e ( r + t ) a,d − ( − | a || b | + | a || c | + | b || c | δ a,d e ( r + t ) c,b (4.20)Let gl m | n [ x ]( σ ) be the subalgebra of the loop superalgebra gl m | n [ x ] generated by the fol-lowing elements { e ij x r | ≤ i, j ≤ m + n, r ≥ s i,j } . By (2.7), gl m | n [ x ]( σ ) is indeed a subalgebra of gl m | n [ x ]. Let the universal enveloping algebra U (cid:0) gl m | n [ x ]( σ ) (cid:1) be equipped with the natural grading induced by the grading on gl m | n [ x ]. Theorem 4.4. [BK2, Theorem 2.1]
The map γ : U (cid:0) gl m | n [ x ]( σ ) (cid:1) −→ gr L Y m | n ( σ ) defined by γ ( e a,b x r ) = ( − | a | e ( r ) a,b , for all ≤ a, b ≤ m + n , r ≥ s a,b , is an isomorphism of graded superalgebras.Proof. γ is a homomorphism by (4.20). Since the image of γ contains the image of P D,σ ingr L Y m | n ( σ ), γ is surjective.It remains to show the injectivity. Consider firstly the special case when σ = 0, where wecan identify Y m | n (0) = Y m | n . By [Pe4, Proposition 7.9], the ordered supermonomials in the INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 19 elements { e ( r ) a,b | ≤ a, b ≤ m + n, r ≥ } are linearly independent in gr L Y m | n . It follows that γ is injective.For the general case, observe that the canonical map Γ : Y m | n ( σ ) → Y m | n is a homo-morphism of filtered superalgebras. It induces a map gr L Y m | n ( σ ) → gr L Y m | n , sending e ( r ) a,b ∈ gr L Y m | n ( σ ) to e ( r ) a,b ∈ gr L Y m | n . By the previous paragraph, the ordered supermonomi-als in the elements { e ( r ) a,b | ≤ a, b ≤ m + n, r ≥ s a,b } are linearly independent in gr L Y m | n ( σ )as well, which implies that γ is injective by the PBW theorem for U (cid:0) gl m | n [ x ]( σ ) (cid:1) . (cid:3) Corollary 4.5.
The canonical map
Γ : Y m | n ( σ ) → Y m | n is injective. As a consequence, thestructure Y m | n ( σ ) defined in Definition can be identified as a subalgebra of Y m | n . Shifted super Yangian: Parabolic presentations
In this section, we provide a more sophisticated definition for Y m | n ( σ ) together with cor-responding results mentioned in §
4. For the sake of the purpose, we introduce some termi-nologies and notations.Let σ = ( s i,j ) be a shift matrix of size m + n . We say a composition µ = ( µ , . . . , µ z ) of m + n of length z is admissible to σ if s µ + µ + ··· + µ a − + i,µ + µ + ··· + µ a − + j = 0for all 1 ≤ a ≤ z , 1 ≤ i, j ≤ µ a . In addition, µ is called minimal admissible if it is admissibleto σ and its length is minimal among all compositions admissible to σ . Clearly, for a shiftmatrix σ , its minimal admissible shape uniquely exists. Moreover, (1 m + n ) is admissible forany σ of size m + n . Remark 5.1.
The notion of admissibility can be intuitively explained in terms of pyramid.Note that one can decompose a pyramid horizontally into a number of rectangles. An admis-sible shape µ records the heights of these rectangles from top to bottom, while the minimaladmissible shape records such a decomposition with the least number of rectangles. When µ = ( µ , µ , . . . , µ z ) is admissible to σ , we will use a shorthand notation s µa,b := s µ + ... + µ a ,µ + ... + µ b , ∀ ≤ a, b ≤ z. (5.1)Note that one can recover the original matrix σ if an admissible shape µ and the numbers { s µa,b | ≤ a, b ≤ z } are known. Moreover, the admissible condition (2.7) implies that for any1 ≤ a, b ≤ z , we have s µ + ··· + µ a − + i,µ + ··· + µ b − + j = s µa,b , ∀ ≤ i ≤ µ a , ≤ j ≤ µ b . (5.2)Let Υ be a fixed 0 m n -sequence. We decompose Υ into z subsequences according to µ Υ = Υ Υ · · · Υ z , and define the restricted parity | i | a as in (3.5). Now we give the following presentation for Y m | n ( σ ), a super analogue of shifted Yangian given in [BK2, § Definition 5.2.
Let σ = ( s i,j ) be a shift matrix of size m + n with a fixed m n -sequence Υ . Let µ = ( µ , . . . , µ z ) be an admissible shape to σ . The shifted super Yangian of gl m | n associated to σ and µ , denoted by Y µ ( σ ) , is the superalgebra over C generated by the followingsymbols (cid:8) D ( r ) a ; i,j , D ′ ( r ) a ; i,j | ≤ a ≤ z, ≤ i, j ≤ µ a , r ≥ (cid:9) , (cid:8) E ( r ) b ; h,k | ≤ b < z, ≤ h ≤ µ b , ≤ k ≤ µ b +1 , r > s µb,b +1 (cid:9) , (cid:8) F ( r ) b ; k,h | ≤ b < z, ≤ h ≤ µ b , ≤ k ≤ µ b +1 , r > s µb +1 ,b (cid:9) , where their parities are defined by (3.13)–(3.15) , subject to the following relations: D (0) a ; i,j = D ′ (0) a ; i,j = δ ij , (5.3) µ a X p =1 r X t =0 D ( t ) a ; i,p D ′ ( r − t ) a ; p,j = δ r δ ij , (5.4) (cid:2) D ( r ) a ; i,j , D ( s ) b ; h,k (cid:3) = δ ab ( − | i | a | j | a + | i | a | h | a + | j | a | h | a × min ( r,s ) − X t =0 (cid:0) D ( t ) a ; h,j D ( r + s − − t ) a ; i,k − D ( r + s − − t ) a ; h,j D ( t ) a ; i,k (cid:1) , (5.5)[ D ( r ) a ; i,j , E ( s ) b ; h,k ] = δ a,b δ hj ( − | h | a | j | a µ a X p =1 r − X t =0 D ( t ) a ; i,p E ( r + s − − t ) b ; p,k − δ a,b +1 ( − | h | b | k | a + | h | b | j | a + | j | a | k | a r − X t =0 D ( t ) a ; i,k E ( r + s − − t ) b ; h,j , (5.6)[ D ( r ) a ; i,j , F ( s ) b ; h,k ] = − δ a,b ( − | i | a | j | a + | h | a +1 | i | a + | h | a +1 | j | a µ a X p =1 r − X t =0 F ( r + s − − t ) b ; h,p D ( t ) a ; p,j + δ a,b +1 ( − | h | a | k | b + | h | a | j | a + | j | a | k | b r − X t =0 F ( r + s − − t ) b ; i,k D ( t ) a ; h,j , (5.7)[ E ( r ) a ; i,j , F ( s ) b ; h,k ] = δ a,b ( − | h | a +1 | k | a + | j | a +1 | k | a + | h | a +1 | j | a +1 +1 r + s − X t =0 D ′ ( r + s − − t ) a ; i,k D ( t ) a +1; h,j , (5.8)[ E ( r ) a ; i,j , E ( s ) a ; h,k ] = ( − | h | a | j | a +1 + | j | a +1 | k | a +1 + | h | a | k | a +1 × INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 21 (cid:0) s − X t = s µa,a +1 +1 E ( r + s − − t ) a ; i,k E ( t ) a ; h,j − r − X t = s µa,a +1 +1 E ( r + s − − t ) a ; i,k E ( t ) a ; h,j (cid:1) , (5.9)[ F ( r ) a ; i,j , F ( s ) a ; h,k ] = ( − | h | a +1 | j | a + | j | a | k | a + | h | a +1 | k | a × (cid:0) r − X t = s µa +1 ,a +1 F ( r + s − − t ) a ; i,k F ( t ) a ; h,j − s − X t = s µa +1 ,a +1 F ( r + s − − t ) a ; i,k F ( t ) a ; h,j (cid:1) , (5.10)[ E ( r +1) a ; i,j , E ( s ) a +1; h,k ] − [ E ( r ) a ; i,j , E ( s +1) a +1; h,k ] = ( − | j | a +1 | h | a +1 δ h,j µ a +1 X q =1 E ( r ) a ; i,q E ( s ) a +1; q,k , (5.11)[ F ( r +1) a ; i,j , F ( s ) a +1; h,k ] − [ F ( r ) a ; i,j , F ( s +1) a +1; h,k ] =( − | i | a +1 ( | j | a + | h | a +2 )+ | j | a | h | a +2 +1 δ i,k µ a +1 X q =1 F ( s ) a +1; h,q F ( r ) a ; q,j , (5.12)[ E ( r ) a ; i,j , E ( s ) b ; h,k ] = 0 if | b − a | > or if b = a + 1 and h = j, (5.13)[ F ( r ) a ; i,j , F ( s ) b ; h,k ] = 0 if | b − a | > or if b = a + 1 and i = k, (5.14) (cid:2) E ( r ) a ; i,j , [ E ( s ) a ; h,k , E ( t ) b ; f,g ] (cid:3) + (cid:2) E ( s ) a ; i,j , [ E ( r ) a ; h,k , E ( t ) b ; f,g ] (cid:3) = 0 if | a − b | ≥ , (5.15) (cid:2) F ( r ) a ; i,j , [ F ( s ) a ; h,k , F ( t ) b ; f,g ] (cid:3) + (cid:2) F ( s ) a ; i,j , [ F ( r ) a ; h,k , F ( t ) b ; f,g ] (cid:3) = 0 if | a − b | ≥ , (5.16) (cid:2) [ E ( r ) a − i,f , E ( s µa,a +1 +1) a ; f ,j ] , [ E ( s µa,a +1 +1) a ; h,g , E ( s ) a +1; g ,k ] (cid:3) = 0 if z ≥ and | h | a + | j | a +1 = 1 , (5.17) (cid:2) [ F ( r ) a − i,f , F ( s µa +1 ,a +1) a ; f ,j ] , [ F ( s µa +1 ,a +1) a ; h,g , F ( s ) a +1; g ,k ] (cid:3) = 0 if z ≥ and | j | a + | h | a +1 = 1 , (5.18) for all indices a, b, f, f , f , g, g , g , h, i, j, k, r, s, t that make sense. For example, (5.11) issupposed to hold for all ≤ a ≤ z − , ≤ i ≤ µ a , ≤ h, j ≤ µ a +1 , ≤ k ≤ µ a +2 , r ≥ s µa,a +1 + 1 , s ≥ s µa +1 ,a +2 + 1 . In the special case where σ is the zero matrix, the above relations are precisely the definingrelations of Y m | n with respect to the parabolic generators P µ given in §
3. We shall write Y µ instead of Y m | n to emphasize that we are using the parabolic presentation in [Pe4] to define Y m | n . The generators of Y µ ( σ ), denoted by P µ,σ , will be called the parabolic generators of Y µ ( σ ). Later we will identify P µ,σ as as subset of P µ . Remark 5.3.
As noticed in [Pe2, Ts] , the definition of Y µ is independent from the choiceof the 01-sequence Υ since the RTT presentation of Y m | n is. For Y µ ( σ ) , we have a similar but slightly weaker phenomenon. Write Y µ ( σ, Υ) for the shifted super Yangian to emphasizethe choice of Υ . Let S m + n be the symmetric group on m + n objects, which acts on Υ bypermutation, and let S µ denote its Young subgroup associated to µ . Then we have Y µ ( σ, Υ) ∼ = Y µ ( σ, ρ · Υ) ∀ ρ ∈ S µ . Fix an admissible shape µ . Similar to §
3, we will show that Y µ ( σ ) can be identified asa subalgebra of Y µ . Let Γ : Y µ ( σ ) → Y µ be the map sending elements in P µ,σ to elements(3.10) and (3.12) with the same name in Y µ obtained by Gauss decomposition with respectto µ . Proposition 5.4.
The canonical map
Γ : Y µ ( σ ) → Y µ is a homomorphism.Proof. By [Pe4], the relations (5.3)–(5.16) hold in Y µ whenever the indices make sense. Itremains to show that (5.17) and (5.18) also hold in Y µ . These relations are crucial differencesfrom the non-super case in [BK2] and checking them turns out to be very technical andinvolved. As a result, we postpone this part to the end of this section; see Proposition 5.15. (cid:3) For 1 ≤ a < b ≤ z , 1 ≤ i ≤ µ a , 1 ≤ j ≤ µ b , r > s µa,b and a fixed 1 ≤ k ≤ µ b − , we definethe higher root elements E ( r ) a,b ; i,j ∈ Y µ ( σ ) recursively by E ( r ) a,a +1; i,j := E ( r ) a ; i,j , E ( r ) a,b ; i,j := ( − | k | b − [ E ( r − s µb − ,b ) a,b − i,k , E ( s µb − ,b +1) b − k,j ] . (5.19)Similarly, using the same indices except that for r > s µb,a , we define F ( r ) b,a ; j,i ∈ Y µ ( σ ) by F ( r ) a +1 ,a ; j,i := F ( r ) a ; j,i , F ( r ) b,a ; j,i := ( − | k | b − [ F ( s µb,b − +1) b − j,k , F ( r − s µb,b − ) b − ,a ; k,i ] . (5.20)It turns out that the above definitions are independent of the choice of k ; see Remark 5.8.We introduce the loop f iltration on Y µ ( σ ) L Y µ ( σ ) ⊆ L Y µ ( σ ) ⊆ L Y µ ( σ ) ⊆ · · · by setting the degrees of the generators D ( r ) a ; i,j , E ( r ) a ; i,j , and F ( r ) a ; i,j to be r − L k Y µ ( σ ) to be the span of all supermonomials in the generators of total degree not greaterthan k . We let gr L Y µ ( σ ) denote the associated graded superalgebra and define the elements { e ( r ) a,b ; i,j | ≤ a, b ≤ z, ≤ i ≤ µ a , ≤ j ≤ µ b , r ≥ s µa,b } ⊆ gr L Y µ ( σ ) by e ( r ) a,b ; i,j := gr Lr D ( r +1) a ; i,j if a = b ,gr Lr E ( r +1) a,b ; i,j if a < b ,gr Lr F ( r +1) a,b ; i,j if a > b .The following is a parabolic version of Proposition 4.3, which can be derived by the sameargument in [Pe4, Lemma 7.5] with the defining relations in Definition 5.2. INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 23 Proposition 5.5. [BK1, Lemma 6.7][Pe4, Lemma 7.5]
For all ≤ a, b, c, d ≤ z , ≤ i ≤ µ a , ≤ j ≤ µ b , r ≥ s µa,b , t ≥ s µc,d , the following identity holds in gr L Y µ ( σ ) : [ e ( r ) a,b ; i,j , e ( t ) c,d ; h,k ] = ( − | j | b | h | c δ b,c δ h,j e ( r + t ) a,d ; i,k − ( − | i | a | j | b + | i | a | h | c + | j | b | h | c δ a,d δ i,k e ( r + t ) c,b ; h,j . (5.21) Theorem 5.6.
The map γ : U (cid:0) gl m | n [ x ]( σ ) (cid:1) −→ gr L Y µ ( σ ) defined by γ ( e µ + ··· + µ a − + i,µ + ··· + µ b − + j x r ) = ( − | i | a e ( r ) a,b ; i,j , for all ≤ a, b ≤ z , ≤ i ≤ µ a , ≤ j ≤ µ b , r ≥ s µa,b , is an isomorphism of gradedsuperalgebras.Proof. γ is a surjective homomorphism by (5.21). For injectivity, we start with the case σ = 0,where we already know that Y µ (0) = Y µ , and the statement follows from Corollary 3.3. Forthe general case, observe that the canonical map Γ : Y µ ( σ ) → Y µ is a homomorphism offiltered superalgebras (under loop filtration), and its induced map gr L Y µ ( σ ) → gr L Y µ sends e ( r ) a,b ; i,j ∈ gr L Y µ ( σ ) to e ( r ) a,b ; i,j ∈ gr L Y µ . By the previous paragraph, the ordered supermonomi-als in the elements { e ( r ) a,b ; i,j | ≤ a, b ≤ m + n, r ≥ s µa,b } are linearly independent in gr L Y µ ( σ ),hence γ is injective by the PBW theorem for U (cid:0) gl m | n [ x ]( σ ) (cid:1) . (cid:3) Theorem 5.7.
Let Y µ ( σ ) be the subalgebra of Y µ generated by the union of the followingsubsets of P µ : (cid:8) D ( r ) a ; i,j , D ′ ( r ) a ; i,j | ≤ a ≤ z, ≤ i, j ≤ µ a , r ≥ (cid:9) , (cid:8) E ( r ) b ; h,k | ≤ b < z, ≤ h ≤ µ b , ≤ k ≤ µ b +1 , r > s µb,b +1 (cid:9) , (cid:8) F ( r ) b ; k,h | ≤ b < z, ≤ k ≤ µ b +1 , ≤ h ≤ µ b , r > s µb +1 ,b (cid:9) . Then the relations (5.3)–(5.18) form a set of defining relations for Y µ ( σ ) . In other words, Y µ ( σ ) defined in Definition can be realized as a subalgebra of the super Yangian Y µ .Proof. We slightly change the notation in this proof to avoid possible confusion. Let ˜ Y µ ( σ )denote the abstract superalgebra generated by elements in P µ,σ with defining relations givenin Definition 5.2 and let Y µ ( σ ) denote the concrete subalgebra of Y µ as stated in the theorem.Let Γ : ˜ Y µ ( σ ) → Y µ ( σ ) be the map sending elements of P µ,σ to the corresponding elementsof Y µ denoted by the same notations. By Proposition 5.4, Γ is a surjective homomorphism.Its injectivity follows from Theorem 5.6. (cid:3) Remark 5.8.
By Theorem , E ( r ) b ; h,k and F ( r ) b ; k,h are now concrete elements in Y µ . Using thesame argument as in [BK1, (6.9)] together with the admissible condition (5.2) , one can showthat the higher root elements defined recursively by (5.19) and (5.20) are independent of thechoices of k . Let Y µ denote the subalgebra of Y µ ( σ ) generated by all of the D ( r ) a ; i,j ’s , Y + µ ( σ ) denote thesubalgebra generated by all of the E ( r ) b ; h,k ’s and Y − µ ( σ ) denote the subalgebra generated by allof the F ( r ) b ; k,h ’s. The following corollary give PBW bases for these subalgebras. Corollary 5.9. [BK2, Theorem 3.2](1)
The set of supermonomials in the elements { D ( r ) a ; i,j | ≤ a ≤ z, ≤ i, j ≤ µ a , r > } taken in some fixed order forms a basis for Y µ . (2) The set of supermonomials in the elements { E ( r ) a,b ; h,k | ≤ a < b ≤ z, ≤ h ≤ µ a , ≤ k ≤ µ b , r > s µa,b } taken in some fixed order forms a basis for Y + µ ( σ ) . (3) The set of supermonomials in the elements { F ( r ) b,a ; k,h | ≤ a < b ≤ z, ≤ k ≤ µ b , ≤ h ≤ µ a , r > s µb,a } taken in some fixed order forms a basis for Y − µ ( σ ) . (4) The set of supermonomials in the union of the elements listed in (1)–(3) taken insome fixed order forms a basis for Y µ ( σ ) .Proof. (4) follows from Theorem 5.6 and the PBW theorem for U (cid:0) gl m | n [ x ]( σ ) (cid:1) , while theothers follow from (5.21). (cid:3) Corollary 5.10. [BK2, Corollary 3.4]
The multiplicative map Y − µ ( σ ) ⊗ Y µ ⊗ Y + µ ( σ ) −→ Y µ ( σ ) is an isomorphism of superspaces. Now we show that the definition of Y µ ( σ ) is independent of the choice of the admissibleshape µ . It suffices to show that Y µ ( σ ) = Y (1 m + n ) ( σ ). Assume that µ = ( µ , . . . , µ z ) isadmissible to σ . If µ j = 1 for all j , then we have done. Otherwise, suppose that µ p > ≤ p ≤ z and we decompose µ p = x + y for some positive integers x, y .Define a finer composition ν of length z + 1 by setting ν i = µ i for all 1 ≤ i ≤ p − ν p = x , ν p +1 = y , ν j +1 = µ j for all p + 1 ≤ j ≤ z , that is, ν = ( µ , . . . , µ p − , x, y, µ p +2 , . . . , µ z ) , which is also admissible to σ . We claim that Y µ ( σ ) = Y ν ( σ ) . INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 25 Consider the Gauss decomposition of the matrix T ( u ) with respect to the two compositions µ and ν , respectively: T ( u ) = µ E ( u ) µ D ( u ) µ F ( u ) = ν E ( u ) ν D ( u ) ν F ( u ) , where the matrices are block matrices as described in § µ D a and ν D a the a -th diagonal matrices in µ D ( u ) and ν D ( u ) with respect tothe compositions µ and ν , respectively. Similarly, let µ E a and µ F a denote the matrices inthe a -th upper and the a -th lower diagonal of µ E ( u ) and µ F ( u ), respectively; ν E a and ν F a are defined to be the matrices in the a -th upper and the a -th lower diagonal of ν E ( u ) and ν F ( u ), respectively. Lemma 5.11.
Using the above notation, define an ( x × x ) -matrix A , an ( x × y ) -matrix B ,a ( y × x ) -matrix C and a ( y × y ) -matrix D from the equation µ D p = I x C I y ! A D ! I x B I y ! . Then (i) ν D a = µ D a for a < p , ν D p = A , ν D p +1 = D , and ν D c = µ D c − for c > p + 1 ; (ii) ν E a = µ E a for a < p − , ν E p − is the submatrix consisting of the first x columnsof µ E p − , ν E p = B , ν E p +1 is the submatrix consisting of the last p rows of µ E p , and ν E c = µ E c − for c > p + 1 ; (iii) ν F a = µ F a for a < p − , ν F p − is the submatrix consisting of the first x rows of µ F p − , ν F p = C , µ F p +1 is the submatrix consisting of the last y columns of µ F p , and ν F c = µ F c − for c > p + 1 .Proof. Matrix multiplication. (cid:3)
As a consequence of Lemma 5.11, one has that Y ν ( σ ) ⊆ Y µ ( σ ). Now the equality followsfrom the fact that the isomorphism U (cid:0) gl m | n [ x ]( σ ) (cid:1) ∼ = gr L Y µ ( σ ) is independent of the choiceof µ . Applying induction on the length of the admissible shape µ , we have deduced thedesired result. Corollary 5.12. Y µ ( σ ) is independent of the choice of the admissible shape µ . Let σ be a shift matrix with an admissible shape µ . Note that the transpose matrix σ t is again a shift matrix while µ is still admissible for σ t . On the other hand, suppose that ~σ = ( ~s i,j ) ≤ i,j ≤ m + n is another shift matrix satisfying (2.7) and the condition ~s i,i +1 + ~s i +1 ,i = s i,i +1 + s i +1 ,i holds for all 1 ≤ i ≤ m + n −
1. As a result, if µ is an admissible shape for σ then it isalso admissible for ~σ . Denote by ~D ( r ) a ; i,j , ~E ( r ) b ; h,k and ~F ( r ) b ; k,h the parabolic generators of Y µ ( ~σ ) toavoid confusion. The following results can be easily deduced from the presentation of Y µ ( σ ). Proposition 5.13.
The map τ : Y µ ( σ ) → Y µ ( σ t ) defined by τ ( D ( r ) a ; i,j ) = D ( r ) a ; j,i , τ ( E ( r ) b ; h,k ) = F ( r ) b ; k,h , τ ( F ( r ) b ; k,h ) = E ( r ) b ; h,k . (5.22) is a superalgebra anti-isomorphism of order 2. Proposition 5.14.
The map ι : Y µ ( σ ) → Y µ ( ~σ ) defined by ι ( D ( r ) a ; i,j ) = ~D ( r ) a ; i,j , ι ( E ( r ) b ; h,k ) = ~E ( r − s µb,b +1 + ~s µb,b +1 ) b ; h,k , ι ( F ( r ) b ; k,h ) = ~F ( r − s µb +1 ,b + ~s µb +1 ,b ) b ; k,h , (5.23) is a superalgebra isomorphism. Now we prove the missing piece in the proof of Proposition 5.4.
Proposition 5.15.
The relations (5.17) and (5.18) hold in Y µ , where E ( r ) b ; h,k and F ( r ) b ; k,h arethe elements in Y µ defined by (3.12) .Proof. We prove (5.17) where (5.18) is similar. Inspired by [BK3, § µ . Our initial step is the case µ = (1 m + n ), where (5.17) reduces to (4.15), which was proved in Proposition 4.2.Assume now the length of µ = ( µ , . . . , µ z ) is strictly less than m + n . Following the samenotations given in the proof of Corollary 5.12, we may choose some 1 ≤ p ≤ z and decompose µ p = x + y to obtain a new composition ν = ( µ , . . . , µ p − , x, y, µ p +2 , . . . , µ z ) . The key ideais to describe the relations between the elements µ E ( r ) b ; h,k and ν E ( r ) b ; h,k .Recall the set P µ,σ consisting of the following elements in Y µ (cid:8) µ D ( r ) a ; i,j | ≤ a ≤ z, ≤ i, j ≤ µ a , r ≥ (cid:9)(cid:8) µ E ( r ) b ; h,k | ≤ b < z, ≤ h ≤ µ b , ≤ k ≤ µ b +1 , r > s µb,b +1 (cid:9)(cid:8) µ F ( r ) b ; k,h | ≤ b < z, ≤ h ≤ µ b , ≤ k ≤ µ b +1 , r > s µb +1 ,b (cid:9) obtained by the Gauss decomposition of T ( u ) with respect to µ . Similarly, replacing µ by ν , we have the following elements in Y ν as well (cid:8) ν D ( r ) a ; i,j | ≤ a ≤ z + 1 , ≤ i, j ≤ ν a , r ≥ (cid:9)(cid:8) ν E ( r ) b ; h,k | ≤ b ≤ z, ≤ h ≤ ν b , ≤ k ≤ ν b +1 , r > s νb,b +1 (cid:9)(cid:8) ν F ( r ) b ; k,h | ≤ b ≤ z, ≤ h ≤ ν b , ≤ k ≤ ν b +1 , r > s νb +1 ,b (cid:9) For every 1 ≤ a < b ≤ z + 1, 1 ≤ i ≤ ν a , 1 ≤ j ≤ ν b , we inductively define higher rootelements ν E ( r ) a,b ; i,j for r > s νa,b by equation (5.19) and similarly define ν F ( r ) b,a ; j,i for r > s νb,a byequation (5.20). We further define the formal series in Y ν ( σ )[[ u − ]]: ν E a,b ; i,j ( u ) := X r>s νa,b ν E ( r ) a,b ; i,j u − r , ν F b,a ; j,i ( u ) := X r>s νb,a ν F ( r ) b,a ; j,i u − r . (5.24) INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 27 Note that the value of k in (5.19) and (5.20) can be arbitrarily chosen between 1 and ν b − due to Remark 5.8. Moreover, one should be careful that they are in general different fromthose series in Y ν [[ u − ]] given by (3.11) so that we have to slightly modify the argument inthe proof of Corollary 5.12. Finally, let ν D a ; i,j ( u ) be given as in (3.7) with respect to ν .Using these series, one defines the following matrices ν D a ( u ) = (cid:0) ν D a ; i,j ( u ) (cid:1) ≤ i,j ≤ ν a ν E a,b ( u ) = (cid:0) ν E a,b ; h,k ( u ) (cid:1) ≤ h ≤ ν a , ≤ k ≤ ν b ν F b,a ( u ) = (cid:0) ν F b,a ; k,h ( u ) (cid:1) ≤ k ≤ ν b , ≤ h ≤ ν a One further defines the block matrices ν D ( u ), ν E ( u ) and ν F ( u ) exactly the same way as(3.6)–(3.9), except that we use their product to define the matrix ν G ( u ): ν G ( u ) := ν F ( u ) ν D ( u ) ν E ( u )By exactly the same way, one defines the higher root elements µ E ( r ) a,b ; i,j , µ F ( r ) b,a ; j,i , formalseries µ E a,b ; i,j ( u ), µ F b,a ; j,i ( u ), µ D a ; i,j ( u ), block matrices µ D ( u ), µ E ( u ) and µ F ( u ) and hencetheir product µ G ( u ) := µ F ( u ) µ D ( u ) µ E ( u ). A key observation from [BK3, § ν G ( u ) = µ G ( u ) and hence we have ν F ( u ) ν D ( u ) ν E ( u ) = µ F ( u ) µ D ( u ) µ E ( u )As a consequence of Lemma 5.11, for each 1 ≤ a < b ≤ z , 1 ≤ i ≤ µ a and 1 ≤ j ≤ µ b , wehave the following relation µ E a,b ; i,j ( u ) = ν E a,b ; i,j ( u ) if b < p ; ν E a,b ; i,j ( u ) if b = p, j ≤ x ; ν E a,b +1; i,j − x ( u ) if b = p, j > x ; ν E a,b +1; i,j ( u ) if a < p < b ; ν E a,b +1; i,j ( u ) − P yq =1 ν E a,a +1; i,q ( u ) ν E a +1 ,b +1; q,j ( u ) if a = p, i ≤ x ; ν E a +1 ,b +1; i − x,j ( u ) if a = p, i > x ; ν E a +1 ,b +1; i,j ( u ) if a > p. (5.25)Back to (5.17), we may assume that f = f = f and g = g = g by (5.11). Moreover, by(5.25), µ E ( r ) a ; i,j = ν E ( r ) a ; i,j except for a ∈ { p − , p, p + 1 } so the general case is further reducedto the special case µ = ( µ , µ , µ , µ ) since (5.17) holds for ν by induction. Therefore, itsuffices to check the following relation holds in Y µ for any t > s µ , : (cid:2) [ µ E ( r )1; i,f , µ E ( t )2; f,j ] , [ µ E ( t )2; h,g , µ E ( s )3; g,k ] (cid:3) = 0 (5.26)This can be checked by a case-by-case discussion. We list all possibilities below: p = 1 , ≤ i ≤ x (5.27) p = 1 , ≤ i − x ≤ y (5.28) p = 2 , ≤ f ≤ x, ≤ h ≤ x (5.29) p = 2 , ≤ f − x ≤ y, ≤ h ≤ x (5.30) p = 2 , ≤ f ≤ x, ≤ h − x ≤ y (5.31) p = 2 , ≤ f − x ≤ y, ≤ h − x ≤ y (5.32) p = 3 , ≤ g ≤ x, ≤ j ≤ x (5.33) p = 3 , ≤ g − x ≤ y, ≤ j ≤ x (5.34) p = 3 , ≤ g ≤ x, ≤ j − x ≤ y (5.35) p = 3 , ≤ g − x ≤ y, ≤ j − x ≤ y (5.36) p = 4 , ≤ k ≤ x (5.37) p = 4 , ≤ k − x ≤ y (5.38)We will check some of them in detail here and the remaining ones can be deduced similarly.Suppose that (5.27) holds. By (5.25), we have µ E ( r )1; i,f = ν E ( r )1 , i,f − X s ν , Although Y m | n is a hopf-superalgebra, the shifted super Yangian Y µ ( σ ) is not closed underthe comultiplication defined by (3.3) in general; that is,∆( Y µ ( σ )) * Y µ ( σ ) ⊗ Y µ ( σ ) . However, one can define some comultiplication-like maps on Y µ ( σ ) as in [BK2, § σ be a non-zero shift matrix of size m + n with minimal admissible shape µ = ( µ , . . . , µ z ). Let Υ be afixed 0 m n -sequence and let Y µ ( σ ) be the shifted super Yangian defined in § 5. Suppose thatthere are p q µ z digits of Υ; that is, Υ z is a 0 p q -sequence and µ z = p + q . Since µ is minimal admissible and σ = 0, we have that 1 ≤ µ z < m + n andeither s m + n − µ z ,m + n +1 − µ z = 0 or s m + n +1 − µ z ,m + n − µ z = 0. Theorem 6.1. Let µ = ( µ , µ , . . . , µ z ) be minimal admissible to σ . For ≤ i, j ≤ µ z ,define ˜ e i,j := e i,j + δ i,j (( m − p ) − ( n − q )) ∈ U ( gl p | q ) . Here e i,j is the elementary matrix identified with the element in gl p | q and its parity is deter-mined by the p q -sequence Υ z . (1) Suppose that s m + n − µ z ,m + n +1 − µ z = 0 . Define ˙ σ = ( ˙ s i,j ) ≤ i,j ≤ m + n by ˙ s i,j = ( s i,j − if i ≤ m + n − µ z < j , s i,j otherwise. (6.1) INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 31 Then the map ∆ R : Y m | n ( σ ) → Y m | n ( ˙ σ ) ⊗ U ( gl p | q ) defined by D ( r ) a ; i,j ˙ D ( r ) a ; i,j ⊗ δ a,z µ z X f =1 ( − | f | z ˙ D ( r − a ; i,f ⊗ ˜ e f,j ,E ( r ) b ; h,k ˙ E ( r ) b ; h,k ⊗ δ b,z − µ z X f =1 ( − | f | z ˙ E ( r − b ; h,f ⊗ ˜ e f,k ,F ( r ) b ; k,h ˙ F ( r ) b ; k,h ⊗ , is a superalgebra homomorphism. (2) Suppose that s m + n +1 − µ z ,m + n − µ z = 0 . Define ˙ σ = ( ˙ s i,j ) ≤ i,j ≤ m + n by ˙ s i,j = ( s i,j − if j ≤ m + n − µ z < i , s i,j otherwise. (6.2) Then the map ∆ L : Y m | n ( σ ) → U ( gl p | q ) ⊗ Y m | n ( ˙ σ ) defined by D ( r ) a ; i,j ⊗ ˙ D ( r ) a ; i,j + δ a,z ( − | i | z µ z X k =1 ˜ e i,k ⊗ ˙ D ( r − a ; k,j ,E ( r ) b ; h,k ⊗ ˙ E ( r ) b ; h,k ,F ( r ) b ; k,h ⊗ ˙ F ( r ) b ; k,h + δ b,z − ( − | i | z µ z X f =1 ˜ e k,f ⊗ ˙ F ( r − b ; f,h , is a superalgebra homomorphism.To avoid possible confusion, in the above description and hereafter, the parabolic generatorsof Y m | n ( ˙ σ ) are denoted by ˙ D ( r ) a ; i,j , ˙ E ( r ) a ; i,j , and ˙ F ( r ) a ; i,j , where ˙ σ is the shift matrix defined by either (6.1) or (6.2) , with respect to the same shape µ which is also admissible to ˙ σ .Proof. Check that ∆ R and ∆ L preserve the defining relations in Definition 5.2. Similar to[BK2, Theorem 4.2], to check (5.15) and (5.16), one needs to use (5.9), (5.10), (5.11) and(5.12) multiple times. Note that it suffices to check the special case when z = 4 since thenon-trivial situations only happen in the very last block.We check (5.18) here as an illustrating example since it is a super phenomenon which doesnot appear in [BK2]. Assume z = 4 and (6.2) holds. Applying ∆ L to the left-hand-side of(5.18), we have (cid:2) [1 ⊗ ˙ F ( r )1; i,f , ⊗ ˙ F ( t )2; f ,j ] , [1 ⊗ ˙ F ( t )2; h,g , ⊗ ˙ F ( s )3; g ,k + ( − | g | µ X x =1 ˜ e g ,x ⊗ ˙ F ( s − x,k ] (cid:3) . It equals to1 ⊗ (cid:2) [ ˙ F ( r )1; i,f , ˙ F ( t )2; f ,j ] , [ ˙ F ( t )2; h,g , ˙ F ( s )3; g ,k ] (cid:3) + θ µ X x =1 ˜ e g ,x ⊗ (cid:2) [ ˙ F ( r )1; i,f , ˙ F ( t )2; f ,j ] , [ ˙ F ( t )2; h,g , ˙ F ( s − x,k ] (cid:3) , where θ = ± Y m | n ( ˙ σ ). (cid:3) The next lemma computes the images of higher root elements E ( r ) a,b ; i,j and F ( r ) b,a ; i,j under ∆ R and ∆ L . Lemma 6.2. ( 1) Suppose the assumption of Theorem holds. For all admissibleindices i, j, r and ≤ a < b − < z , we have ∆ R ( F ( r ) b,a ; i,j ) = ˙ F ( r ) b,a ; i,j ⊗ , ∆ R ( E ( r ) a,b ; i,j ) = ˙ E ( r ) a,b ; i,j ⊗ if b < z, and ∆ R ( E ( r ) a,z ; i,j ) = ( − | h | z − [ ˙ E ( r − s µz − ,z ) a,z − i,h , ˙ E ( s µz − ,z +1) z − h,j ] ⊗ µ z X k =1 ( − | k | z ˙ E ( r − a,z ; i,k ⊗ ˜ e k,j , for any ≤ h ≤ µ z − . ( 2) Suppose the assumption of Theorem holds. For all admissible indices i, j, r and ≤ a < b − < z , we have ∆ L ( E ( r ) a,b ; i,j ) = 1 ⊗ ˙ E ( r ) a,b ; i,j , ∆ L ( F ( r ) b,a ; i,j ) = 1 ⊗ ˙ F ( r ) b,a ; i,j if b < z, and ∆ L ( F ( r ) z,a ; i,j ) = ( − | h | z − (cid:0) ⊗ [ ˙ F ( s µz,z − +1) z − i,h , ˙ F ( r − s µz,z − ) z − ,a ; h,j ] (cid:1) + ( − | i | z µ z X k =1 ˜ e i,k ⊗ ˙ F ( r − z − ,a ; k,j , for any ≤ h ≤ µ z − .Proof. We compute ∆ R ( E ( r ) a,z ; i,j ) for 1 ≤ a ≤ z − ≤ h ≤ µ z − , we have E ( r ) a,z ; i,j = ( − | h | z − [ E ( r − s µz − ,z ) a,z − i,h , E ( s µz − ,z +1) z − h,j ] . INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 33 Also, ∆ R ( E ( r − s µz − ,z ) a,z − i,h ) = ˙ E ( r − s µz − ,z ) a,z − i,h ⊗ 1. Hence∆ R ( E ( r ) a,z ; i,j ) = ( − | h | z − h ˙ E ( r − s µz − ,z ) a,z − i,h ⊗ , ˙ E ( s µz − ,z +1) z − h,j ⊗ i + ( − | h | z − " ˙ E ( r − s µz − ,z ) a,z − i,h ⊗ , β X k =1 ( − | k | z ˙ E ( s µz − ,z ) z − h,k ⊗ ˜ e k,j = ( − | h | z − h ˙ E ( r − s µz − ,z ) a,z − i,h , ˙ E ( s µz − ,z +1) z − h,j i ⊗ µ z X k =1 ( − | k | z ˙ E ( r − a,z ; i,k ⊗ ˜ e k,j . (cid:3) Proposition 6.3. If the assumption of Theorem holds, then ∆ R is injective. Similarly,if the assumption of Theorem holds, then ∆ L is injective.Proof. Let ǫ : U ( gl p | q ) → C be the homomorphism such that ǫ (˜ e i,j ) = 0for 1 ≤ i, j ≤ µ z . By definition, Y µ ( σ ) ⊆ Y µ ( ˙ σ ) ⊆ Y µ is a chain of subalgebras. Note thatthe compositions m ◦ (id ⊗ ǫ ) ◦ ∆ R and m ◦ ( ǫ ⊗ id) ◦ ∆ L coincide with the natural embedding Y µ ( σ ) ֒ → Y µ ( ˙ σ ), where m ( a ⊗ b ) := ab is the usual multiplication map. This deduces thatthe maps ∆ R and ∆ L are injective whenever they are defined. (cid:3) Canonical filtration There is another filtration on Y m | n , called the canonical filtration F Y m | n ⊆ F Y m | n ⊂ F Y m | n ⊆ · · · defined by deg t ( r ) ij := r where F d Y m | n is defined to be the span of all supermonomials in t ( r ) ij of total degree not greater than d . Let gr Y m | n denote the associated superalgebra, which issupercommutative by (3.2).Now we describe the canonical filtration using parabolic presentations. Let µ = ( µ , . . . , µ z )be a composition of m + n . By [Pe4, 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 m | 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 thematrix equation T ( u ) = F ( u ) D ( u ) E ( u ), each t ( r ) ij is a linear combination of supermonomialsin the parabolic generators of total degree r as well. Thus F d Y m | n can be alternatively definedas the span of all supermonomials in the parabolic generators D ( r ) a ; i,j E ( r ) a,b ; i,j and F ( r ) b,a ; i,j of totaldegree ≤ d . For 1 ≤ a, b ≤ z , 1 ≤ i ≤ µ a , 1 ≤ j ≤ µ b and r > 0, define the following elements in gr Y µ by e ( r ) a,b ; i,j := gr r D ( r ) a ; i,j if a = b ,gr r E ( r ) a,b ; i,j if a < b ,gr r F ( r ) a,b ; i,j if a > b . (7.1)Since gr Y µ is supercommutative, together with Corollary 5.9 (4), the following result can bededuced immediately. Proposition 7.1. [BK2, Theorem 5.1] For any shape µ = ( µ , . . . , µ z ) , gr Y µ is the free su-percommutative superalgebra on generators { e ( r ) a,b ; i,j | ≤ a, b ≤ z, ≤ i ≤ µ a , ≤ j ≤ µ b , r > } . Suppose now σ is a shift matrix of size m + n and µ = ( µ , . . . , µ z ) is an admissible shapeto σ . We induce the canonical filtration of Y µ to the subalgebra Y µ ( σ ) by defining F d Y µ ( σ ) := F d Y µ ∩ Y µ ( σ ) . The natural embedding Y µ ( σ ) ֒ → Y µ is a filtered map and the induced map gr Y µ ( σ ) → gr Y µ is injective as well, so that we may identify gr Y µ ( σ ) as a subalgebra of gr Y µ . The nexttheorem gives a set of generators of gr Y µ ( σ ). Theorem 7.2. [BK2, Theorem 5.2] For an admissible shape µ = ( µ , . . . , µ z ) , gr Y µ ( σ ) isthe subalgebra of gr Y µ generated by the elements { e ( r ) a,b ; i,j | ≤ a, b ≤ z, ≤ i ≤ µ a , ≤ j ≤ µ b , r > s µa,b } . Proof. By relations (5.11) and (5.12), the elements e ( r ) a,b ; i,j of gr Y µ ( σ ) can be identified as theelements of the same notation in gr Y µ defined in (7.1) by the embedding gr Y µ ( σ ) → gr Y µ .Now the statement follows from Corollary 5.9 (4) and Proposition 7.1. (cid:3) One consequence of Theorem 7.2 is that we may define the canonical filtration on Y µ ( σ )intrinsically by setting the degree of the elements D ( r ) a ; i,j , E ( r ) a,b ; i,j and F ( r ) a,b ; i,j in Y µ ( σ ) to be r .By Corollary 5.12, this definition is independent of the choice of admissible shape µ .By definition, the comultiplication ∆ : Y µ → Y µ ⊗ Y µ is a filtered map with respect tothe canonical filtration. If we extend the canonical filtration of Y µ ( ˙ σ ) to Y µ ( ˙ σ ) ⊗ U ( gl p | q ) bydeclaring the degree of the matrix unit e ij ∈ gl p | q to be 1, then the baby comultiplications∆ R and ∆ L defined in Theorem 6.1, as long as they are defined, are filtered maps as well.Moreover, the same argument in Proposition 6.3 implies that the associated graded mapsgr ∆ L : gr Y µ ( ˙ σ ) → gr (cid:0) Y µ ( ˙ σ ) ⊗ U ( gl p | q ) (cid:1) gr ∆ R : Y µ ( ˙ σ ) → gr (cid:0) U ( gl p | q ) ⊗ Y µ ( ˙ σ ) (cid:1) are injective as well. We state this fact as a proposition. Proposition 7.3. [BK2, Remark 5.4] The induced maps gr ∆ R and gr ∆ L are injective when-ever they are defined, INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 35 Truncation Let σ be a fixed shift matrix of size m + n . Choose an integer ℓ > s ,m + n + s m + n, , whichwill be called level later. For each 1 ≤ i ≤ m + n , set p i := ℓ − s i,m + n − s m + n,i . (8.1)This defines a tuple ( p , . . . , p m + n ) of integers such that 0 < p ≤ · · · ≤ p m + n = ℓ . Let µ = ( µ , . . . , µ z ) be an admissible shape for σ . For each 1 ≤ a ≤ z , set p µa := p µ + ... + µ a . (8.2)Since µ is admissible, together with (2.7), for any 1 ≤ a ≤ z , we have p i = p µa for any valueof i such that 1 ≤ i − a − X k =1 µ k ≤ µ a .Following [BK2, § shifted super Yangian of level ℓ , denoted by Y ℓµ ( σ ), tobe the quotient of Y µ ( σ ) by the two-side ideal of Y µ ( σ ) generated by the elements { D ( r )1; i,j | ≤ i, j ≤ µ , r > p } . We claim that the definition of Y ℓµ ( σ ) is independent of the choice of the admissible shape µ so that we may simply write Y ℓm | n ( σ ) when appropriate. Let I µ denote the two-sided idealassociated to µ as in the definition. Since ν = (1 m + n ) is admissible for any σ , it suffices toprove that I µ = I ν .By definition, we have ν D ( r )1 = t ( r )1 , . Assume µ is an arbitrary admissible shape. By [Pe4,(3.10)], we have µ D ( r )1;1 , = t ( r )1 , and hence I ν ⊆ I µ . On the other hand, by relation (5.5), wehave µ D ( r )1; i,j ∈ I ν for all 1 ≤ i, j ≤ µ , r > p and hence I µ = I ν .When σ = 0, the two-sided ideal is generated by { t ( r ) i,j | ≤ i, j ≤ m + n, r > ℓ } . In thisspecial case, the quotient is exactly the truncated super Yangian in [BR, Pe2], which is asuper analogy of Yangian of level ℓ due to Cherednik [C1, C2]. It should be clear from thecontext that we are dealing with Y µ ( σ ) or the quotient Y ℓµ ( σ ) and hence, by abusing notation,we will use the same symbols D ( r ) a ; i,j , E ( r ) a,b ; i,j and F ( r ) b,a ; i,j to denote the elements in Y µ ( σ ) andtheir images in the quotient Y ℓµ ( σ ).It is obvious that the anti-isomorphism τ defined in (5.22) factors through the quotientand induces an anti-isomorphism τ : Y ℓµ ( σ ) → Y ℓµ ( σ t ) . (8.3)Similarly, let ~σ be another shift matrix satisfying that ~s i,i +1 + ~s i +1 ,i = s i,i +1 + s i +1 ,i for all1 ≤ i ≤ m + n − 1. Then the isomorphism ι defined by (5.23) also induces an isomorphism ι : Y ℓµ ( σ ) → Y ℓµ ( ~σ ) . (8.4) Recall the canonical filtration defined in § 7. We obtain a filtration F Y ℓµ ( σ ) ⊆ F Y ℓµ ( σ ) ⊆ · · · induced from the quotient map Y µ ( σ ) → Y ℓµ ( σ ), where we define the elements D ( r ) a ; i,j , E ( r ) a,b ; i,j and F ( r ) b,a ; i,j of Y ℓµ ( σ ) to be of degree r and F d Y ℓµ ( σ ) is the span of all supermonomials in theseelements of total degree ≤ d .For 1 ≤ a, b ≤ z , 1 ≤ i ≤ µ a , 1 ≤ j ≤ µ b and r > s µa,b , define element e ( r ) a,b ; i,j (by abusingnotation again) in the associative graded superalgebra gr Y ℓµ ( σ ) according to exactly thesame formula (7.1), except that now our D ’s, E ’s and F ’s here are in the quotient. ByProposition 7.1 and Theorem 7.2, gr Y ℓµ ( σ ) is also supercommutative and is generated by theelements { e ( r ) a,b ; i,j ∈ gr Y ℓµ ( σ ) | ≤ a, b ≤ z, ≤ i ≤ µ a , ≤ j ≤ µ b , r > s µa,b } Following the same argument in [BK2, Lemma 6.1], one may deduce that gr Y ℓµ ( σ ) is in fact finitely generated . Lemma 8.1. For any admissible shape µ = ( µ , . . . , µ z ) , gr Y ℓµ ( σ ) is generated only by theelements { e ( r ) a,b ; i,j | ≤ a, b ≤ z, ≤ i ≤ µ a , ≤ j ≤ µ b , s µa,b < r ≤ s µa,b + p µmin ( a,b ) } . Let σ = ( s ij ) ≤ i,j ≤ m + n be a non-zero shift matrix with minimal admissible shape µ =( µ , . . . , µ z ) and let Υ be a 0 m n -sequence. Then µ z equals to the size of the largest zerosquare matrix in the southeastern corner of σ . Hence we have 1 ≤ µ < m + n and either s m + n − µ z ,m + n +1 − µ z = 0 or s m + n +1 − µ z ,m + n − µ z = 0. Let p and q denote the the number of 0’sand 1’s respectively in the last µ z digits of the 0 m n -sequence Υ.If s m + n − µ z ,m + n +1 − µ z = 0, then the baby comultiplication ∆ R defined in Theorem 6.1 factorsthrough the quotient and we obtain an induced map∆ R : Y ℓµ ( σ ) → Y ℓ − µ ( ˙ σ ) ⊗ U ( gl p | q ) (8.5)where ˙ σ is given by (6.1).Similarly, if s m + n +1 − µ z ,m + n − µ z = 0, then ∆ L induces a map∆ L : Y ℓµ ( σ ) → U ( gl p | q ) ⊗ Y ℓ − µ ( ˙ σ ) (8.6)where ˙ σ is given by (6.2).Recall that ∆ R and ∆ L are filtered maps with respect to the canonical filtration, so theyinduce the following homomorphisms of graded superalgebrasgr ∆ R : gr Y ℓµ ( σ ) → gr (cid:0) Y ℓ − µ ( ˙ σ ) ⊗ U ( gl p | q ) (cid:1) , (8.7)gr ∆ L : gr Y ℓµ ( σ ) → gr (cid:0) U ( gl p | q ) ⊗ Y ℓ − µ ( ˙ σ ) (cid:1) . (8.8) INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 37 Theorem 8.2. For any admissible shape µ = ( µ , . . . , µ z ) , gr Y ℓµ ( σ ) is the free supercommu-tative superalgebra on generators { e ( r ) a,b ; i,j | ≤ a, b ≤ z, ≤ i ≤ µ a , ≤ j ≤ µ b , s µa,b < r ≤ s µa,b + p µmin ( a,b ) } . Also, the maps gr ∆ R and gr ∆ L in (8.7) and (8.8) are injective whenever they are defined,and so are the maps ∆ R and ∆ L in (8.5) and (8.6) .Proof. Similar to the argument in [BK2, Theorem 6.2], except that our induction starts from ℓ = 1. In that case, the assertion follows from [Pe2, Proposition 2.3]. (cid:3) As a corollary, we obtain a PBW basis for Y ℓm | n ( σ ). Corollary 8.3. For any admissible shape µ = ( µ , . . . , µ z ) , the supermonomials in the ele-ments { D ( r ) a ; i,j | ≤ a ≤ z, ≤ i, j ≤ µ a , < r ≤ p µa } , { E ( r ) a,b ; i,j | ≤ a < b ≤ z, ≤ i ≤ µ a , ≤ j ≤ µ b , s µa,b < r ≤ s µa,b + p µa } , { F ( r ) b,a ; i,j | ≤ a < b ≤ z, ≤ i ≤ µ b , ≤ j ≤ µ a , s µb,a < r ≤ s µb,a + p µa } , taken in any fixed order forms a basis for Y ℓm | n ( σ ) . Another corollary is obtained by counting. Corollary 8.4. Consider Y ℓm | n ( σ ) together with the canonical filtration and some fixed Υ .Let S ( g e ) be the supersymmetric superalgebra of g e with the Kazhdan filtration, where e is thenilpotent element corresponding to the triple ( σ, ℓ, Υ) as explained in § . Denote by F d Y ℓm | n ( σ ) and F d S ( g e ) the superspaces with total degree not greater than d in the associated filteredsuperalgebras respectively. Then for each d ≥ , we have dim F d Y ℓm | n ( σ ) = dim F d S ( g e ) .Proof. Take µ = (1 m + n ) in Theorem 8.2. Then the statement follows from Proposition 2.9and induction on d . (cid:3) Remark 8.5. Consider the following inverse system Y ℓm | n ( σ ) և Y ℓ +1 m | n ( σ ) և Y ℓ +2 m | n ( σ ) և · · · where the maps are homomorphisms of filtered superalgebras with respect to the canonicalfiltration. As an observation from Corollary and Corollary , we have Y m | n ( σ ) = lim ← Y ℓm | n ( σ ) where the inverse limit is taken in the category of filtered superalgebras. Similar to [BK2,Remark 6.4] , we may view Y m | n ( σ ) as the inverse limit ℓ → ∞ of the shifted super Yangianof level ℓ . Invariants Let π be a given pyramid of height m + n associated to a 0 m n -sequence Υ. Let M and N be the number of boxes in π labeled by “ + ” and “ − ”, respectively. Let p and m be thesubalgebras of gl M | N associated to the good pair ( e π , h π ). Generalizing [BK2, § m -invariant (under the χ -twisted action) elements in U ( p ); thatis, some elements in W π .We number the columns of π from left to right by 1 , . . . , ℓ . Let h = m − n and let(ˇ q , . . . , ˇ q ℓ ) denote the super column heights of π , where each ˇ q i is defined to be the numberof boxes in the i -th column of π labeled with “ + ” subtract the number of boxes labeledwith “ − ” in the same column.Define ρ = ( ρ , . . . , ρ ℓ ), where ρ r is given by ρ r := h − ˇ q r − ˇ q r +1 − · · · − ˇ q ℓ (9.1)for each r = 1 , . . . , ℓ .Give an order on the index set I := { < . . . < M < < . . . < N } . For all i, j ∈ I , define˜ e i,j := ( − col( j ) − col( i ) ( e i,j + δ i,j ( − pa ( i ) ρ col( i ) ) , (9.2)where pa ( i ) := 0 if i ∈ { , . . . , M } and pa ( i ) := 1 otherwise. Note that the parity notationpa ( i ) used here is for gl M | N , while another parity notation | i | defined in § Y m | n .Calculation shows that[˜ e i,j , ˜ e h,k ] = (˜ e i,k − δ i,k ( − pa( i ) ρ col( i ) ) δ h,j − ( − (pa( i )+pa( j ))(pa( h )+pa( k )) δ i,k (˜ e h,j − δ h,j ( − pa( j ) ρ col( j ) ) . (9.3)The effect of the homomorphism U ( m ) → C induced by the character χ can be obtainedeasily by definition. We explicitly give the result here since it will be frequently use later.For any i, j ∈ I , we have χ (˜ e i,j ) = ( − pa( i )+1 if row( i ) = row( j ) and col( i ) = col( j ) + 1;0 otherwise. (9.4)Now we are going to define certain crucial elements in the universal enveloping algebra U ( gl M | N ). For 1 ≤ i, j ≤ m + n and signs σ i ∈ {±} , we firstly set T (0) i,j ; σ ,...,σ n +1 := δ i,j σ i and then for r ≥ T ( r ) i,j ; σ ,...,σ m + n := r X s =1 X i ,...,i s j ,...,j s σ row( j ) · · · σ row( j s − ) ( − pa( i )+ ··· +pa( i s ) ˜ e i ,j · · · ˜ e i s ,j s (9.5)where the second sum is taken over all i , . . . , i s , j , . . . , j s ∈ I such that INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 39 (1) deg( e i ,j ) + · · · + deg( e i s ,j s ) = r ;(2) col( i t ) ≤ col( j t ) for each t = 1 , . . . , s ;(3) if σ row( j t ) = +, then col( j t ) < col( i t +1 ) for each t = 1 , . . . , s − σ row( j t ) = − , then col( j t ) ≥ col( i t +1 ) for each t = 1 , . . . , s − i ) = i , row( j s ) = j ;(6) row( j t ) = row( i t +1 ) for each t = 1 , . . . , s − T ( r ) i,j ; σ ,...,σ m + n belongs to F r U ( p ).For an integer 0 ≤ x ≤ m + n , we set the shorthand notation T ( r ) i,j ; x := T ( r ) i,j ; σ ,...,σ m + n where σ i = − if i ≤ x ,+ if x < i .We further define the following series for all 1 ≤ i, j ≤ m + n : T i,j ; x ( u ) := X r ≥ T ( r ) i,j ; x u − r ∈ U ( p )[[ u − ]] . (9.6)The following lemma can be established by exactly the same approach as [BK2, Lemma 9.2],where the use of super column height perfectly solves the subtle sign issue. We omit the detailsince the argument there is quite formal and does not depend on the underlying associativesuperalgebra in which the calculations are performed. Lemma 9.1. [BK2, Lemma 9.2] Let ≤ i, j, x, y ≤ m + n be integers with x < y . (1) If x < i ≤ y < j ≤ m + n then T i,j ; x ( u ) = y X k = x +1 T i,k ; x ( u ) T k,j ; y ( u ) . (2) If x < j ≤ y < i ≤ m + n then T i,j ; x ( u ) = y X k = x +1 T i,k ; y ( u ) T k,j ; x ( u ) . (3) If x < y < i ≤ m + n and y < j ≤ m + n , then T i,j ; x ( u ) = T i,j ; y ( u ) + y X k,ℓ = x +1 T i,k ; y ( u ) T k,ℓ ; x ( u ) T ℓ,j ; y ( u ) . (4) If x < i ≤ y ≤ m + n and x < j ≤ y , then y X k = x +1 T i,k ; x ( u ) T k,j ; y ( u ) = − δ i,j . Define an invertible ( m + n ) × ( m + n ) matrix with entries in U ( p )[[ u − ]] by T ( u ) := (cid:0) T i,j ;0 ( u ) (cid:1) ≤ i,j ≤ m + n Fix a composition µ = ( µ , µ , . . . , µ z ) of m + n . Applying the Gauss decomposition in § T ( u ) = F ( u ) D ( u ) E ( u )where D ( u ) is a diagonal block matrix, E ( u ) is an upper unitriangular block matrix, and F ( u ) is a lower unitriangular block matrix, with respect to µ .The diagonal blocks of D ( u ) define matrices D ( u ) , . . . , D z ( u ), the upper diagonal blocks of E ( u ) define matrices E , ( u ) , . . . , E z − ,z ( u ), and the lower diagonal matrices of F ( u ) definematrices F , ( u ) , . . . , F z,z − ( u ), respectively. Set E b ( u ) = E b,b +1 ( u ), F b ( u ) = F b +1 ,b ( u ) for1 ≤ b ≤ z − D ′ a ( u ) := D a ( u ) − for all 1 ≤ a ≤ z . The entries of these matrices in turndefine the following series: D a ; i,j ( u ) = X r ≥ D ( r ) a ; i,j u − r , D ′ a ; i,j ( u ) = X r ≥ D ′ ( r ) a ; i,j u − r ,E b ; h,k ( u ) = X r ≥ E ( r ) b ; h,k u − r , F b ; k,h ( u ) = X r ≥ F ( r ) b ; k,h u − r , for all 1 ≤ a ≤ z , 1 ≤ b ≤ z − 1, 1 ≤ i, j ≤ µ a , 1 ≤ h ≤ µ b , 1 ≤ k ≤ µ b +1 .Nevertheless, all of these elements, depending on the fixed choice of µ , are parallel to theelements in Y m | n with the same notations given in § 3, except that the elements defined herebelong to U ( p ). Theorem 9.2. With µ = ( µ , . . . , µ z ) be fixed as above. For any admissible indices a, b, i, j, h, k ,we have D a ; i,j ( u ) = T µ + ··· + µ a − + i,µ + ··· + µ a − + j ; µ + ··· + µ a − ( u ) ,D ′ a ; i,j ( u ) = − T µ + ··· + µ a − + i,µ + ··· + µ a − + j ; µ + ··· + µ a ( u ) ,E b ; h,k ( u ) = T µ + ··· + µ b − + h,µ + ··· + µ b + k ; µ + ··· + µ b ( u ) ,F b ; k,h ( u ) = T µ + ··· + µ b + k,µ + ··· + µ b − + h ; µ + ··· + µ b ( u ) . Proof. Note that it suffices to show the identities for D, E and F , since the one for D ′ followsfrom the one for D and Lemma 9.1(4). We prove our statement by induction on the lengthof µ . The initial case is µ = ( m + n ), which is trivial since T ( u ) = D ( u ).Now let µ = ( µ , . . . , µ z ) be a composition of length z ≥ 2. Define a new composition ν = ( ν , . . . , ν z − ) of length z − ν i = µ i for all 1 ≤ i ≤ z − ν z − = µ z − + µ z ; INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 41 that is, merge the last two parts of µ . By the induction hypothesis, we have ν D a ( u ) = (cid:0) T ν + ··· + ν a − + i,ν + ··· + ν a − + j ; ν + ··· + ν a − ( u ) (cid:1) ≤ i,j ≤ ν a , ∀ ≤ a ≤ z − , ν E b ( u ) = (cid:0) T ν + ··· + ν b − + h,ν + ··· + ν b + k ; ν + ··· + ν b ( u ) (cid:1) ≤ h ≤ ν b , ≤ k ≤ ν b +1 , ∀ ≤ b ≤ z − , ν F b ( u ) = (cid:0) T ν + ··· + ν b + k,ν + ··· + ν b − + h ; ν + ··· + ν b ( u ) (cid:1) ≤ k ≤ ν b +1 , ≤ h ≤ ν b , ∀ ≤ b ≤ z − , where we add a superscript ν to emphasize that these elements are defined with respect to ν . Note that ν D a ( u ) = µ D a ( u ) for all 1 ≤ a ≤ z − ν E b ( u ) = µ E b ( u ), ν F b ( u ) = µ F b ( u )for all 1 ≤ b ≤ z − µ E z − ( u ) equals to the submatrix consisting of the first µ z − columns of ν E z − ( u ), while µ F z − ( u ) equals to the submatrix consisting of the top µ z − rowsof ν F z − ( u ). Both of them are of the form described in the theorem. It remains to check theidentities for µ D z − ( u ), µ D z ( u ), µ E z − ( u ) and µ F z − ( u ).Define matrices P, Q, R and S by P = (cid:0) T µ + ··· + µ z − + i,µ + ··· + µ z − + j ; µ + ··· + µ z − ( u ) (cid:1) ≤ i,j ≤ µ z − ,Q = (cid:0) T µ + ··· + µ z − + i,µ + ··· + µ z − + µ z − + j ; µ + ··· + µ z − + µ z − ( u ) (cid:1) ≤ i ≤ µ z − , ≤ j ≤ µ z ,R = (cid:0) T µ + ··· + µ z − + µ z − + i,µ + ··· + µ z − + j ; µ + ··· + µ z − + µ z − ( u ) (cid:1) ≤ i ≤ µ z , ≤ j ≤ µ z − ,S = (cid:0) T µ + ··· + µ z − + µ z − + i,µ + ··· + µ z − + µ z − + j ; µ + ··· + µ z − + µ z − ( u ) (cid:1) ≤ i,j ≤ µ z . By Lemma 9.1 with x = µ + . . . + µ z − and y = µ + . . . + µ z − , we have ν D z − ( u ) = I µ z − R I µ z ! P S ! I µ z − Q I µ z ! = P P QRP S + RP Q ! . Now the explicit descriptions of the matrices µ D z − ( u ), µ D z ( u ), µ E z − ( u ) and µ F z − ( u )follows from Lemma 5.11, which completes the induction argument. (cid:3) In the extreme case that µ = (1 m + n ), we write simply D ( r ) i , D ′ ( r ) i , E ( r ) j and F ( r ) j for theelements D ( r ) i ;1 , , D ′ ( r ) i ;1 , , E ( r ) j ;1 , and F ( r ) j ;1 , of U ( p ) for all 1 ≤ i ≤ m + n , 1 ≤ j ≤ m + n − r ≥ 1, respectively. Corollary 9.3. D ( r ) i = T ( r ) i,i ; i − , E ( r ) j = T ( r ) j,j +1; j , F ( r ) j = T ( r ) j +1 ,j ; j and D ′ ( r ) i = − T ( r ) i,i ; i . Main theorem Let π be a pyramid associated with a 0 m n -sequence Υ which corresponds to a good pairin gl M | N and let ( σ, ℓ, Υ) be the triple associated to π given by Proposition 2.8. Let Y ℓm | n ( σ )denote the shifted super Yangian of level ℓ associated to π equipped with the canonicalfiltration and let W π denote the finite W -superalgebra associated to π equipped with theKazhdan filtration . Suppose also that µ = ( µ , . . . , µ z ) is an admissible shape for σ , and recall the shorthandnotations s µa,b and p µa from (5.1) and (8.2). We have the elements D ( r ) a ; i,j , D ′ ( r ) a ; i,j , E ( r ) b ; h,k and F ( r ) b ; k,h of U ( p ) defined by Theorem 9.2 according to this fixed shape µ . On the other hand,we also have the parabolic generators D ( r ) a ; i,j , D ′ ( r ) a ; i,j , E ( r ) b ; h,k and F ( r ) b ; k,h in Y ℓµ ( σ ) as defined in § 8. We are ready to present the main result of this article. Theorem 10.1. Let π be a pyramid and let ( σ, ℓ, Υ) be the corresponding triple given byProposition . For any shape µ = ( µ , . . . , µ z ) admissible to σ , there exists is a uniqueisomorphism Y ℓm | n ( σ ) ∼ → W π of filtered superalgebras such that the generators { D ( r ) a ; i,j | ≤ a ≤ z, ≤ i, j ≤ µ a , r > } , { E ( r ) b ; h,k | ≤ b < z, ≤ h ≤ µ b , ≤ k ≤ µ b +1 , r > s µb,b +1 } , { F ( r ) b ; k,h | ≤ a < z, ≤ k ≤ µ b +1 , ≤ h ≤ µ b , r > s µb +1 ,b } of Y ℓµ ( σ ) are mapped to corresponding elements of U ( p ) denoted by the same symbols. Inparticular, these elements of U ( p ) are m -invariants and they form a generating set for W π . Similar to the argument in [BK2], the proof of Theorem 10.1 is processed by inductionon the number ℓ − t , where ℓ is the length of the bottom row and t is the length of the toprow of π . Our initial case is ℓ = t . In this case, the pyramid is of rectangular shape so theassociated shift matrix is the zero matrix. Hence the shifted super Yangian is the whole Y m | n itself, and its quotient is exactly the truncated super Yangian Y ℓm | n . As mentioned in § π is not of rectangular shape so that ℓ ≥ ℓ − t > 0. Byinduction on the length of the shape and Lemma 5.11, it suffices to prove the special casewhen µ is the minimal admissible shape for σ .Let H denote the absolute height of the shortest column of π . Since π is a pyramid, either H = | q | or H = | q ℓ | . There are two cases: • Case R: H = | q ℓ | ≤ | q | . • Case L: H = | q | < | q ℓ | .We will explain the proof of Case R in detail and only sketch the proof of Case L, which canbe obtained by a very similar argument with mild modifications.From now on we assume that Case R holds. Recall that we numbered the boxes of π usingthe index set I := { < · · · < M < < . . . < N } in the standard way: down columns from left to right, where i (respectively, i ) stands forthe boxes labeled with + (respectively, − ). Suppose that there are p (respectively, q ) boxes INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 43 labeled with + (respectively, − ) in the right-most column of π . Since µ is minimal admissible,we have H = p + q = µ z .Let ˙ π be the pyramid obtained by removing the right-most column of π . We know thatthe removed boxes of π are numbered with M − p + 1 , M − p + 2 , . . . , M, N − q + 1 , N − q + 2 , . . . , N , and their order in the right-most column is determined by Υ z , the last H digits of the0 m n -sequence Υ.By our assumption, the bottom H rows of π forms a rectangle, call it π H . A key observa-tion [Pe2, Remark 3.5] is that permuting the rows of the rectangle π H will not change thecorresponding even good pair ( e π , h π ); see also Remark 2.5. Although our argument in factworks in general, for convenience, we assume that the last H digits of Υ is the standard one:Υ z = p z }| { · · · q z }| { · · · . As a result, the right-most two columns of π is of the form... M − p + 1 M − p + 1 M − p + 2 M − p + 2... ... M − p MN − q + 1 N − q + 1 N − q + 2 N − q + 2... ... N − q N Let ˙ σ = ( ˙ s i,j ) ≤ i,j ≤ m + n be the shift matrix defined by (6 . 1) where its associated pyramidis ˙ π . Define ˙ p , ˙ m and ˙ e in ˙ g = gl M − p | N − q according to (2.1) and (2.4) and let ˙ χ : ˙ m → C bethe character x ( x, ˙ e ).Let ˙ D ( r ) a ; i,j , ˙ D ′ ( r ) a ; i,j , ˙ E ( r ) b ; h,k and ˙ F ( r ) b ; k,h denote the elements of U ( ˙ p ) as defined in § µ , which is admissible for both of σ and ˙ σ . By the induction hypothesis,Theorem 10.1 holds for ˙ π , so the following elements of U ( ˙ p ) are invariant under the ˙ χ -twisted action of ˙ m ; in other words, they belong to the finite W -superalgebra W ˙ π : { ˙ D ( r ) a ; i,j , ˙ D ′ ( r ) a ; i,j } for 1 ≤ a ≤ z, ≤ i, j ≤ µ a and r > { ˙ E ( r ) b ; h,k } for 1 ≤ b ≤ z − , ≤ h ≤ µ a , ≤ k ≤ µ a +1 and r > s µb,b +1 − δ b +1 ,z ; { ˙ F ( r ) b ; k,h } for 1 ≤ b ≤ z − , ≤ k ≤ µ a +1 , ≤ h ≤ µ a and r > s µb +1 ,b . We embed U ( ˙ g ) into U ( g ) in the following manner: for all i, j in the index set˙ I := { , . . . , M − p, , . . . , N − q } , the generators ˜ e ij of U ( ˙ g ) defined by (9.2) with respect to the pyramid ˙ π are assigned to thegenerators ˜ e ij of U ( g ) defined with respect to π .This embedding in turns embeds U ( ˙ p ) into U ( p ) and ˙ m into m , respectively. Moreover,the character ˙ χ of ˙ m is precisely the restriction of the character χ of m . As a consequence,the ˙ χ -twisted action of ˙ m on U ( ˙ p ) equals to the restriction of the χ -twist action of m on U ( p ).For convenience, define the index sets J = { M − p + i | ≤ i ≤ p } ∪ { N − q + j | ≤ j ≤ q } , J = { M − p + i | ≤ i ≤ p } ∪ { N − q + j | ≤ j ≤ q } . Note that they are the numbers appearing in the right-most and the second right-mostcolumns of the rectangle π H , respectively.Define the bijection R : { , , . . . , p + q } → J by setting R ( f ) to be the number assignedto the f -th box in the right-most column of the rectangle π H . Similarly, define the bijection R : { , , . . . , p + q } → J which assigns R ( f ) to be the number appearing in the left of R ( f ). For example, R (1) = M − p +1, R ( p + q ) = N and R ( p + q ) = N − q . In particular,define η : J → { , , . . . , p + q } (10.1)to be the inverse map of R .The relations between the elements D ( r ) a ; i,j , E ( r ) b ; h,k , F ( r ) b ; k,h of U ( p ) given by π and the elements˙ D ( r ) a ; i,j , ˙ E ( r ) b ; h,k , ˙ F ( r ) b ; k,h of U ( ˙ p ) given by ˙ π are described in the following lemma, which is probablythe most crucial step in the proof of our main theorem. INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 45 Lemma 10.2. The following equations hold for all ≤ a ≤ z , ≤ b ≤ z − , ≤ i, j ≤ µ a , ≤ h ≤ µ b , ≤ k ≤ µ b +1 , all r > that makes sense, and any fixed ≤ g ≤ H : D ( r ) a ; i,j = ˙ D ( r ) a ; i,j + δ a,z H X f =1 ( − | f | z ˙ D ( r − a ; i,f ˜ e R ( f ) ,R ( j ) + [ ˙ D ( r − a ; i,g , ˜ e R ( g ) ,R ( j ) ] ! , (10.2) E ( r ) b ; h,k = ˙ E ( r ) b ; h,k + δ b +1 ,z H X f =1 ( − | f | z ˙ E ( r − b ; h,f ˜ e R ( f ) ,R ( k ) + [ ˙ E ( r − b ; h,g , ˜ e R ( g ) ,R ( k ) ] ! , (10.3) F ( r ) b ; k,h = ˙ F ( r ) b ; k,h , (10.4) where for (10.3) we are assuming that r > s µz − ,z if b + 1 = z .Proof. It can be observed from the explicit description of the elements T ( r ) i,j ; x in (9.5) with thehelp from Theorem 9.2 together with our assumption on the right-most two columns of therectangle π H . (cid:3) The inductive descriptions provided in Lemma 10.2, together with the induction hypothe-sis, allow us to deduce the following several lemmas and eventually to show that the elements D ( r ) a ; i,j , E ( r ) b ; h,k and F ( r ) b ; k,h of U ( p ) are m -invariants when the indices are appropriate. Lemma 10.3. The following elements of U ( p ) are m -invariant: (i) D ( r ) a ; i,j and D ′ ( r ) a ; i,j for ≤ a ≤ z − , ≤ i, j ≤ µ a and r > ; (ii) E ( r ) b ; h,k for ≤ a ≤ z − , ≤ h ≤ µ b , ≤ k ≤ µ b +1 and r > s µb,b +1 ; (iii) F ( r ) b ; k,h for ≤ a ≤ z − , ≤ k ≤ µ b +1 , ≤ h ≤ µ b and r > s µb +1 ,b .Proof. All of these elements in U ( p ) coincide with the elements with the same name in U ( ˙ p )by Lemma 10.2. Hence they are ˙ m -invariant by the induction hypothesis. Define˙ m c := m \ ˙ m . It remains to show that these elements are invariant under the χ -twisted action for all e e f,g in ˙ m c only. Note that e e f,g ∈ ˙ m c if and only if g ∈ ˙ I and f ∈ J .By Theorem 9.2 and (9.5) again, all elements in the description of the lemma are linearcombinations of supermonomials of the form ˜ e i ,j · · · ˜ e i r ,j r in U ( ˙ p ) with i s ∈ ˙ I and j s ∈ ˙ I \ J for all 1 ≤ s ≤ r .By (9.4), χ (˜ e f,g ) = 0 for all g ∈ ˙ I \ J and f ∈ J . This implies that all such supermonomialsare invariant under the χ -twisted action of all e e f,g ∈ ˙ m c and our lemma follows. (cid:3) Lemma 10.4. The following elements of U ( p ) are ˙ m -invariant: (1) D ( r ) z ; i,j for ≤ i, j ≤ µ z and r > . (2) E ( r ) z − i,j for ≤ i ≤ µ z − , ≤ j ≤ µ z and r > s µz − ,z . Proof. (1) By (10.2), we obtain D ( r ) z ; i,j = ˙ D ( r ) z ; i,j + H X f =1 ( − | f | z ˙ D ( r − z ; i,f ˜ e R ( f ) ,R ( j ) + [ ˙ D ( r − z ; i,g , ˜ e R ( g ) ,R ( j ) ] . For any x ∈ ˙ m , we have [ x, e e R ( f ) ,R ( j ) ] = 0 = [ x, e e R ( g ) ,R ( j ) ]. Using this result together withthe induction hypothesis, one deduces that pr χ ([ x, D ( r ) z ; i,j ]) = 0. The proof of (2) is similarby starting with (10.3). (cid:3) Lemma 10.5. (1) D (1) z ; i,j is ˙ m c -invariant for all ≤ i, j ≤ µ z . (2) Suppose s µz − ,z = 1 . Then D (2) z ; i,j is ˙ m c -invariant for all ≤ i, j ≤ µ z . (3) Suppose s µz − ,z = 1 . Then E (2) z − h,k is ˙ m c -invariant for all ≤ h ≤ µ z − and ≤ k ≤ µ z .Proof. We only give the detail of the proof of (1) here, where (2) and (3) can be deduced ina similar fashion.By Theorem 9.2, (9.5) and (10.2), we have D (1) z ; i,j = ˙ D (1) z ; i,j + ( − | i | z ˜ e R ( i ) ,R ( j ) = X ≤ k ≤ ℓ − (cid:0) X p k ,q k ( − | i | z ˜ e p k ,q k (cid:1) + ( − | i | z ˜ e R ( i ) ,R ( j ) , where the second sum is taken over all p k , q k ∈ ˙ I satisfying the following conditions(i) col( p k ) = col( q k ) = k ,(ii) row( p k ) = µ + · · · + µ z − + i ,(iii) row( q k ) = µ + · · · + µ z − + j .Let ˜ e f,g ∈ ˙ m c be arbitrary given so that we have g ∈ ˙ I and f ∈ J .Suppose first that row( g ) = µ + . . . + µ z − + i . Then we have [˜ e f,g , ˜ e p k ,q k ] = 0 for any p k , q k appearing in the sum. Moreover, [˜ e f,g , ˜ e R ( i ) ,R ( j ) ] = ± δ f,R ( j ) ˜ e R ( i ) ,g , which belongs to thekernel of χ by (9.4). It follows that pr χ ([˜ e f,g , D (1) z ; i,j ]) = 0.Assume now that row( g ) = µ + . . . + µ z − + i . Then g equals exactly one p k appearing inthe sum and hence (cid:2) ˜ e f,g , X ≤ k ≤ ℓ − (cid:0) X p k ,q k ∈ ˙ I ( − | i | z ˜ e p k ,q k (cid:1)(cid:3) = ( − | i | z ˜ e f,q k for a certain 1 ≤ k ≤ ℓ − q k ) = ℓ − 1. Then ˜ e f,q k belongs to ker χ by (9.4). Also, since g = p k and col( q k ) = col( p k ) = ℓ − 1, the term[˜ e f,g , ˜ e R ( i ) ,R ( j ) ] = ± δ f,R ( j ) ˜ e R ( i ) ,g belongs to the ker χ . Then we have pr χ [˜ e f,g , D (1) z ; i,j ] = 0. INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 47 Finally, assume that row( g ) = µ + . . . + µ z − + i and col( q k ) = ℓ − 1. It implies that g = p k = R ( i ). By definition, we have[˜ e f,R ( i ) , D (1) z ; i,j ] = ( − | i | z ˜ e f,R ( j ) + δ f,R ( j ) ( − | j | z ˜ e R ( i ) ,R ( i ) , which belongs to the kernel of χ by (9.4). This completes the proof of (1). (cid:3) Lemma 10.6. Suppose that s µz − ,z = 1 . Then the following identities hold in U ( p ) for r > : (1) E ( r +1) z − h,k = ( − | g | z − [ D (2) z − h,g , E ( r ) z − g,k ] − µ z − X f =1 D (1) z − h,f E ( r ) z − f,k , (2) D ( r +1) z ; i,j = ( − | g | z − [ F (2) z − i,g , E ( r ) z − g,j ] − r +1 X t =1 D ( r +1 − t ) z ; i,j D ′ ( r ) z − g,g . Proof. By the induction hypothesis and (5.6), for any r > ≤ g ≤ µ z − , we have[ ˙ D (2) z − h,g , ˙ E ( r ) z − g,k ] = ( − | g | z − ˙ E ( r +1) z − h,k + ( − | g | z − µ z − X p =1 ˙ D (1) z ; h,p ˙ E ( r ) z − p,k . (10.5)Also, (10.3) implies that E ( r ) z − g,k = ˙ E ( r ) z − g,k + H X f =1 ( − | f | z ˙ E ( r − z − g,f ˜ e R ( f ) ,R ( k ) + [ ˙ E ( r − z − g,j , ˜ e R ( j ) ,R ( k ) ] (10.6)It is clear that [ ˙ D (2) z − h,g , ˜ e R ( f ) ,R ( k ) ] = 0. Also, due to (9.5) and Theorem 9.2, the expansionof ˙ D (2) z − h,g into supermonomials will never involve any matrix unit of the form ˜ e ? ,R ( j ) andit follows that [ ˙ D (2) z − h,g , ˜ e R ( j ) ,R ( k ) ] = 0. Computing the supercommutator of (10.6) with D (2) z − h,g = ˙ D (2) z − h,g and using (10.5), we have[ D (2) z − h,g , E ( r ) z − g,k ] = [ ˙ D (2) z − h,g , ˙ E ( r ) z − g,k ] + H X f =1 ( − | f | z [ ˙ D (2) z − h,g , ˙ E ( r − z − g,f ]˜ e R ( f ) ,R ( k ) + (cid:2) [ D (2) z − h,g , E ( r − z − g,j ] , ˜ e R ( j ) ,R ( k ) (cid:3) = ( − | g | z − ˙ E ( r +1) z ; h,k + ( − | g | z − µ z − X p =1 ˙ D (1) z − h,p ˙ E ( r ) z − p,k + H X f =1 ( − | f | z (cid:16) ( − | g | z − ˙ E ( r +1) z ; h,f + ( − | g | z − µ z − X p =1 ˙ D (1) z − h,p ˙ E ( r ) z − p,f (cid:17) ˜ e R ( f ) ,R ( k ) + h ( − | g | z − ˙ E ( r +1) z ; h,j + ( − | g | z − µ z − X p =1 ˙ D (1) z − h,p ˙ E ( r ) z − p,j , ˜ e R ( j ) ,R ( k ) i . Using (10.3) a few times, one shows that the above equals to( − | g | z − (cid:0) E ( r +1) z − h,k + µ z − X p =1 D (1) z − h,p E ( r ) z − p,k (cid:1) and the equality (1) is established.Now we deal with (2). By the induction hypothesis and (5.8), we have[ ˙ F (2) z − i,g , ˙ E ( r ) z − g,j ] = ( − | g | z − ( r +1 X t =0 ˙ D ( r +1 − t ) z ; i,j ˙ D ′ ( t ) z − g,g )= ( − | g | z − ˙ D ( r +1) z ; i,j + ( − | g | z − r +1 X t =1 ˙ D ( r +1 − t ) z ; i,j ˙ D ′ ( t ) z − g,g . (10.7)Changing the indices in equation (10.6), we have E ( r ) z − g,j = ˙ E ( r ) z − g,j + H X f =1 ( − | f | z ˙ E ( r − z − g,f ˜ e R ( f ) ,R ( j ) + [ ˙ E ( r − z − g,h , ˜ e R ( h ) ,R ( j ) ] (10.8)Note that the expansion of ˙ F (2) z − i,g into supermonomials will never involve any matrix unitof the forms ˜ e ? ,R ( h ) , ˜ e R ( h ) , ? or ˜ e R ( h ) , ? , and hence [ ˙ F (2) z − i,g , ˜ e R ( f ) ,R ( j ) ] = [ ˙ F (2) z − i,g , ˜ e R ( h ) ,R ( j ) ] =0. As a consequence, we perform the following calculation using the fact that F (2) z − i,g =˙ F (2) z − i,g together with (10.8):[ F (2) z − i,g ,E ( r ) z − g,j ] = [ ˙ F (2) z − i,g , ˙ E ( r ) z − g,j ] + H X f =1 ( − | f | z [ ˙ F (2) z − i,g , ˙ E ( r − z − g,f ]˜ e R ( f ) ,R ( j ) + (cid:2) [ ˙ F (2) z − i,g , ˙ E ( r − z − g,h ] , ˜ e R ( h ) ,R ( j ) (cid:3) = ( − | g | z − ˙ D ( r +1) z ; i,j + ( − | g | z − r +1 X t =1 ˙ D ( r +1 − t ) z ; i,j ˙ D ′ ( t ) z − g,g + H X f =1 ( − | f | z (cid:16) ( − | g | z − ˙ D ( r ) z ; i,f + ( − | g | z − r X t =1 ˙ D ( r − t ) z ; i,f ˙ D ′ ( t ) z − g,g (cid:17) ˜ e R ( f ) ,R ( j ) + h ( − | g | z − ˙ D ( r ) z ; i,h + ( − | g | z − r X t =1 ˙ D ( r − t ) z ; i,h ˙ D ′ ( t ) z − g,g , ˜ e R ( h ) ,R ( j ) i Using (10.2) a few times, the above can be rewritten as( − | g | z − D ( r +1) z ; i,j + ( − | g | z − r +1 X t =1 D ( r +1 − t ) z ; i,j ˙ D ′ ( t ) z − g,g and our assertion (2) follows. (cid:3) Lemma 10.7. Suppose s µz − ,z = 1 . Then INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 49 (1) D ( r ) z ; i,j are m -invariant for all r ≥ and ≤ i, j ≤ µ z . (2) E ( r ) z − h,k are m -invariant for all r > and ≤ h ≤ µ z − , ≤ k ≤ µ z .Proof. By Lemma 10.4, these elements are ˙ m -invariant. It remains to check that they are˙ m c -invariant, but that follows from Lemma 10.5, Lemma 10.6 and induction on r . (cid:3) Lemma 10.8. Suppose that s µz − ,z > . Then the following elements are invariant under the χ -twisted action of ˜ e R ( x ) ,R ( y ) for all ≤ x, y ≤ H . (1) D ( r ) z ; i,j for all r ≥ and ≤ i, j ≤ µ z . (2) E ( r ) z − h,k for all r > s µz − ,z and ≤ h ≤ µ z − , ≤ k ≤ µ z .Proof. Let ¨ π be the pyramid obtained by deleting the right-most two columns of π . Define¨ p , ¨ m and ¨ e ∈ gl M − p | N − q as before, and embed U (¨ g ) into U ( ˙ g ) as how we embed U ( ˙ g ) into U ( g ). The induction hypothesis applies to the pyramid ¨ π hence we know that the elements¨ D ( r ) z ; i,j in W ¨ π are ¨ m -invariant under the ˙ χ -twisted action.Applying Lemma 10.2 to π and ˙ π , we have D ( r ) z ; i,j = ˙ D ( r ) z ; i,j + H X f =1 ( − | f | z ˙ D ( r − z ; i,f ˜ e R ( f ) ,R ( j ) + [ ˙ D ( r − z ; i,g , ˜ e R ( g ) ,R ( j ) ] (10.9)and ˙ D ( r ) z ; i,j = ¨ D ( r ) z ; i,j + H X f =1 ( − | f | z ¨ D ( r − z ; i,f ˜ e R ( f ) ,R ( j ) + [ ¨ D ( r − z ; i,g , ˜ e R ( g ) ,R ( j ) ] (10.10)where R ( g ) is defined to be the number assigned to g -th box in the third right-most columnof the rectangle π H .Substituting (10.10) into (10.9) and simplifying the result by (9.3), one deduces that forall r ≥ D ( r ) z ; i,j = A + B + C + D + E + F + G + H , where A = ¨ D ( r ) z ; i,j , B = H X k =1 ( − | k | z ¨ D ( r − z ; i,k ˜ e R ( k ) ,R ( j ) ,C = [ ¨ D ( r − z ; i,g , ˜ e R ( g ) ,R ( j ) ] , D = H X k =1 ( − | k | z ¨ D ( r − z ; i,k ˜ e R ( k ) ,R ( j ) E = H X h,k =1 ( − | h | z + | k | z ¨ D ( r − z ; i,h ˜ e R ( h ) ,R ( k ) ˜ e R ( k ) ,R ( j ) , F = H X k =1 ( − | k | z ¨ D ( r − z ; i,k ˜ e R ( k ) ,R ( j ) ,G = H X k =1 [ ¨ D ( r − z ; i,g , ˜ e R ( g ) ,R ( k ) ]˜ e R ( k ) ,R ( j ) , H = [ ¨ D ( r − z ; i,g , ˜ e R ( g ) ,R ( j ) ] . Let X = ˜ e R ( x ) ,R ( y ) for some 1 ≤ x, y ≤ H . Note that X commutes with all elementsin U (¨ p ). Using (9.1), (9.3) and (9.4), we can explicitly compute their images under the composition pr χ ◦ ad X as follows:pr χ ([ X, A ]) = 0 , pr χ ([ X, B ]) = δ xj ( − | x | z + | y | z ¨ D ( r − z ; i,y , pr χ ([ X, C ]) = 0 , pr χ ([ X, D ]) = δ xj ( − | x | z + | y | z ¨ D ( r − z ; i,y , pr χ ([ X, E ]) = ( − | x | z + | y | z δ xj ( p − q ) ¨ D ( r − z ; i,y + ( − | y | z +1 ¨ D ( r − z ; i,y ˜ e R ( x ) ,R ( j ) + δ xj H X k =1 ( − ( | x | z + | y | z )( | k | z + | j | z )+ | k | z ¨ D ( r − z ; i,k ˜ e R ( k ) ,R ( y ) , pr χ ([ X, F ]) = − ( − | x | z + | y | z δ xj ( p − q ) ¨ D ( r − z ; i,y + ( − | y | z ¨ D ( r − z ; i,y ˜ e R ( x ) ,R ( j ) − δ xj H X k =1 ( − ( | x | z + | y | z )( | k | z + | j | z )+ | k | z ¨ D ( r − z ; i,k ˜ e R ( k ) ,R ( y ) , pr χ ([ X, G ]) = ( − | y | z + | j | z δ xj [ ¨ D ( r − z ; i,g , ˜ e R ( g ) ,R ( f ) ] , pr χ ([ X, H ]) = − ( − | y | z + | j | z δ xj [ ¨ D ( r − z ; i,g , ˜ e R ( g ) ,R ( f ) ] . As a consequence, pr χ ([ X, D ( r ) z ; i,j ]) = 0. The proof of (2) is similar. (cid:3) Proposition 10.9. The following elements of U ( p ) are m -invariant with respect to the χ -twisted action: { D ( r ) a ; i,j } ≤ a ≤ z, ≤ i,j ≤ µ a ,r> , { E ( r ) b ; h,k } ≤ b If follows from the induction hypothesis and Lemma 10.3–Lemma 10.8. (cid:3) A consequence of Proposition 10.9 is that the elements in the description of Theorem 10.1are actually elements of W π . Moreover, by the induction hypothesis, we may identify Y ℓ − µ ( ˙ σ ) = Y ℓ − m | n ( ˙ σ ) with W ˙ π ⊆ U ( ˙ p ) and the generators ˙ D ( r ) a : i,j , ˙ E ( r ) b ; h,k and ˙ F ( r ) b ; k,h in Y ℓ − µ ( ˙ σ )coincide with the elements of W ˙ π denoted by the same notations. Now we are going to makeuse of the useful monomorphism ∆ R : Y ℓm | n ( σ ) → U ( ˙ p ) ⊗ U ( gl p | q ) obtained in Theorem 8.2.By Corollary 8.4, for each d ≥ 0, we havedim ∆ R ( F d Y ℓm | n ( σ )) = dim F d Y ℓm | n ( σ ) = dim F d S ( g e ) , (10.11)where F d S ( g e ) is the sum of all graded elements in S ( g e ) of degree ≤ d with respect to theKazhdan grading.Define the general parabolic generators E ( r ) a,b ; i,j and F ( r ) b,a ; j,i in F r U ( p ) by equations (5.19)and (5.20) recursively, where the index k could be chosen arbitrarily there. Let X d denote INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 51 the subspace of U ( p ) spanned by all supermonomials in the elements { D ( r ) a ; i,j } ≤ a ≤ z, ≤ i,j ≤ µ a , ≤ r ≤ s µa,a , { E ( r ) a,b ; h,k } ≤ a
0. In particular, ψ R : W π → U ( ˙ p ) ⊗ gl p | q is an injective homomorphism. Comparing ψ R with the map ∆ R defined in Theorem 6.1(1), we see that ψ R ( D ( r ) a ; i,j ) = ∆ R ( D ( r ) a ; i,j ), where the elements D ( r ) a ; i,j onthe left-hand side are the elements of W π and the elements D ( r ) a ; i,j on the right-hand side arethe generators of Y ℓm | n ( σ ). Similarly, ψ R ( E ( r ) b ; h,k ) = ∆ R ( E ( r ) b ; h,k ) and ψ R ( F ( r ) b ; k,h ) = ∆ R ( F ( r ) b ; k,h ) forall admissible indices b, h, k, r .Finally, the composition ψ − R ◦ ∆ R : Y ℓm | n ( σ ) → W π is exactly the filtered superalgebraisomorphism described in Theorem 10.1 and the elements listed in Theorem 10.1 indeedgenerate W π . This completes the induction step of our main theorem under the assumptionof Case R.Next we sketch how to complete the induction step under the assumption of Case L. Inthis case, we enumerate the bricks of π down columns from right to left . Note that different ways of enumerating are just choosing different bases to describe gl M | N ∼ = End( C M | N ) so wemay choose the way most suitable for our purpose.Let ˙ π denote the pyramid obtained from π by deleting the left-most column of π . Let I ,˙ I , J and J be the same index sets as defined in Case R. It is clear that the deleted bricksare still numbered with elements in J . Moreover, we may again assume that the left-mosttwo columns of π is of the form ... M − p + 1 M − p + 1 M − p + 2 M − p + 2... ... M M − pN − q + 1 N − q + 1 N − q + 2 N − q + 2... ... N N − q Similarly we define the bijection L : { , , . . . , p + q } → J by setting L ( f ) to be thenumber assigned to the f -th box in the left-most column of the rectangle π H , and define thebijection L : { , , . . . , p + q } → J by assigning L ( f ) to be the number appearing in the right of L ( f ). In particular, denote by ξ : J → { , , . . . , p + q } (10.12)the inverse map of L .Let ˙ σ be the shift matrix obtained from (6.2), where the corresponding pyramid is exactly˙ π , and define ˙ p , ˙ m , ˙ e ∈ ˙ g := gl M − p | N − q via (2.1) and (2.4) with respect to ˙ π . Note that inCase L we embed U ( ˙ g ) into U ( g ) by the natural embedding , which already sends the elements˜ e ij of U ( ˙ g ) to the elements ˜ e ij of U ( g ) for all i, j ∈ ˙ I .Under the natural embedding, the superalgebra W ˙ π = U ( ˙ p ) ˙ m is a subalgebra of U ( ˙ p ) ⊂ U ( p ) and the ˙ χ -twisted action of ˙ m on U ( ˙ p ) is exactly the same with the restriction of the χ -twisted action of m on U ( p ). Let ˙ D ( r ) a ; i,j , ˙ D ′ ( r ) a ; i,j , ˙ E ( r ) b ; h,k and ˙ F ( r ) b ; k,h denote the elements of U ( ˙ p )as defined in § µ which is the minimal admissible shape of σ butalso admissible for ˙ σ . By the induction hypothesis, all of these elements are ˙ m -invariant.From now we follow exactly the same idea in Case R to complete the proof. By thefollowing crucial lemma, which is the analogue of Lemma 10.2, we may express the elements D ( r ) a ; i,j , D ′ ( r ) a ; i,j , E ( r ) b ; h,k and F ( r ) b ; k,h in U ( p ) in terms of ˙ D ( r ) a ; i,j , ˙ D ′ ( r ) a ; i,j , ˙ E ( r ) b ; h,k and ˙ F ( r ) b ; k,h . Then bysimilar case-by-case discussions and computations as before, we can prove that all of the INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 53 elements D ( r ) a ; i,j , D ′ ( r ) a ; i,j , E ( r ) b ; h,k and F ( r ) b ; k,h are indeed m -invariant under our current setting inCase L. We provide only the most crucial lemma below since its proof and other argumentsare almost identical as in the earlier case. Lemma 10.10. The following equalities hold for all all admissible a, b, i, j, h, k, r and anyfixed ≤ g ≤ H : D ( r ) a ; i,j = ˙ D ( r ) a ; i,j + δ a,z ( − | i | z H X f =1 ˜ e L ( i ) ,L ( f ) ˙ D ( r − z ; f,j + [˜ e L ( i ) ,L ( g ) , ˙ D ( r − z ; g,j ] ! , (10.13) E ( r ) b ; h,k = ˙ E ( r ) b ; h,k , (10.14) F ( r ) b ; k,h = ˙ F ( r ) b ; k,h + δ b,z − ( − | i | z H X f =1 ˜ e L ( k ) ,L ( f ) ˙ F ( r − z − f,h + [˜ e L ( k ) ,L ( g ) , ˙ F ( r − z − g,h ] ! , (10.15) where for (10.15) we are assuming that r > s µz,z − if b = z − . With the help of Lemma 10.10, one can deduce that the statement of Proposition 10.9 stillholds in Case L. Finally, define a superalgebra homomorphism ψ L : U ( p ) → U ( gl p | q ) ⊗ U ( ˙ p )by ψ L (˜ e i,j ) := ˜ e ξ ( i ) ,ξ ( j ) ⊗ i ) = col( j ) = 1,0 if col( i ) = 1 , col( j ) ≥ ⊗ ˜ e i,j if 2 ≤ col( i ) ≤ col( j ),where the function ξ is defined by (10.12). Using Lemma 10.10 again, we have that ψ L ( D ( r ) a ; i,j ) = 1 ⊗ ˙ D ( r ) a ; i,j + δ a,z H X f =1 ( − | f | z ˜ e i,f ⊗ ˙ D ( r − a ; f,j ψ L ( E ( r ) b ; h,k ) = 1 ⊗ ˙ E ( r ) b ; h,k ,ψ L ( F ( r ) b,k,h ) = 1 ⊗ ˙ F ( r ) b ; k,h + δ b +1 ,z H X f =1 ( − | f | z ˜ e k,f ⊗ ˙ F ( r − b ; f,h . Using exactly the same argument as in Case R, one shows that the map ψ L is injectiveand the composition ψ − L ◦ ∆ L : Y ℓm | n ( σ ) → W π gives the required isomorphism of filteredsuperalgebras. This completes the proof of Theorem 10.1. Corollary 10.11. Let π be a pyramid corresponding to an even good pair and ~π be a pyramidobtained by horizontally shifting rows of π . Let W π and W ~π denote the associated finite W -superalgebras, respectively. Then there exists a superalgebra isomorphism ι : W π → W ~π defined on parabolic generators with respect to an admissible shape µ by (5.23) . In otherwords, the definition of a finite W -superalgebra associated to an even good pair depends onlyon e up to isomorphism. Proof. This is an immediate consequence of (8.4) and the isomorphism in Theorem 10.1. (cid:3) Remark 10.12. A more general result of Corollary was obtained in [Zh] by a verydifferent approach. It is proved that the definition of type A finite W -superalgebra is inde-pendent of the choices of the good Z -grading (which may not be even) up to isomorphism,generalizing the results of [BG, GG] . References [Ar] T. Arakawa, Introduction to W-algebras and their representation theory, Perspectives in Lie Theory ,Springer INdAM Series (2017), 179–250.[BFN] A. Braverman, M. Finkelberg and H. Nakajima, Coulomb branches of 3 d N = 4 quiver gauge theoriesand slices in the affine Grassmannian (with appendices by A. Braverman, M. Finkelberg, J. Kamnitzer,R. Kodera, H. Nakajima, B. Webster, A. Weekes), Adv. Theor. Math. Phys. (2019), no. 1, 75–166.[BR] C. Briot and E. Ragoucy, W -superalgebras as truncations of super-Yangians, J. Phys. A (2003),1057-1081.[BBG] J. Brown, J. Brundan and S. Goodwin, Principal W -algebras for GL( m | n ), Algebra and NumberTheory. (2013), 1849-1882.[BG] J. Brundan and S. Goodwin, Good grading polytopes, Proc. Lond. Math. Soc. (3) , No. 1, (2007),155-180.[BGK] J. Brundan, S. Goodwin and A. Kleshchev, Highest weight theory for finite W-algebras, Int. Math.Res. Not. IMRN (2008), No. 15, Art. ID rnn051.[BK1] J. Brundan and A. Kleshchev, Parabolic Presentations of the Yangian Y ( gl n ), Comm. Math. Phys. (2005), 191-220.[BK2] J. Brundan and A. Kleshchev, Shifted Yangians and finite W -algebras, Adv. Math. (2006), 136-195.[BK3] J. Brundan and A. Kleshchev, Representations of Shifted Yangians and Finite W-algebras, Mem.Amer. Math. Soc. (2008).[BK4] J. Brundan and A. Kleshchev, Schur-Weyl duality for higher levels, Sel. Math. (2008), 1-57.[CW] S.-J. Cheng and W. Wang, Dualities and Representations of Lie Superalgebras , Graduate Studies inMathematics , AMS, 2013.[C1] I. Cherednik, A new interpretation of Gelfand-Tzetlin bases, Duke Math. J. (1987), 563-577.[C2] I. Cherednik, Quantum groups as hidden symmetries of classic representation theory, DifferentialGeometric Methods in Theoretical Physics (Chester, 1988), World Sci. Publishing, Teaneck, NJ, 1989,pp. 47-54.[Dr1] V. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. (1985),254-258.[Dr2] V. Drinfeld, A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. (1988),212-216.[EK] A. Elashvili and V. Kac, Classification of good gradings of simple Lie algebras, Lie groups and invarianttheory (E. B. Vinberg ed.), Amer. Math. Socl. Transl. (2005), 85–104.[FKPRW] M. Finkelberg, J. Kamnitzer, K. Pham, L. Rybnikov and A. Weekes, Comultiplication for shiftedYangians and quantum open Toda lattice, Adv. Math. (2018), 349–389. INITE W -SUPERALGEBRAS VIA SUPER YANGIANS 55 [FPT] R. Frassek, V. Pestun and A. Tsymbaliuk, Lax matrices from antidominantly shifted Yangians andquantum affine algebras, preprint, arXiv:2001.04929[FSS] L. Frappat, A. Sciarrino and P. Sorba, Structure of basic Lie superalgebras and of their affine exten-sions. Comm. Math. Phys. (1989), 457–500.[GG] W. Gan and V. Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. IMRN (2002),243–255.[GR] I. Gelfand and V. Retakh, Quasideterminants, I, Selecta Math. (1997), 517–546.[Go] L. Gow, Gauss Decomposition of the Yangian Y ( gl m | n ), Comm. Math. Phys. (2007), 799-825.[Ho] C. Hoyt, Good gradings of basic Lie superalgebras, Israel J. Math. (2012), 251-280.[KWWY] J. Kamnitzer, B. Webster, A. Weekes and O. Yacobi, Yangians and quantizations of slices in theaffine Grassmannian, Algebra Number Theory (2014), no. 4, 857–893.[Ko] B. Kostant, On Whittaker vectors and representation theory, Invent. Math. (1978), 101-184.[Lo] I. Losev, Finite W-algebras, Proceedings of the International Congress of Mathematicians , Vol. III, pp.1281-1307, Hindustan Book Agency, New Delhi, 2010.[Mo] A. Molev, Yangians and classical Lie algebras , Mathematical Surveys and Monographs, AmericanMathematical Society, Providence, RI, 2007.[Na1] M. Nazarov, Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. (1991),123-131.[Na2] M. Nazarov, Yangian of the Queer Lie Superalgebra, Comm. Math. Phys. (1999), 195-223.[Pe1] Y. Peng, Parabolic presentations of the super Yangian Y ( gl M | N ), Comm. Math. Phys. (2011),229-259.[Pe2] Y. Peng, Finite W -superalgebras and truncated super Yangians, Lett. Math. Phys. (2014), 89-102.[Pe3] Y. Peng, On shifted super Yangians and a class of finite W -superalgebras, J. Algebra. (2015),520-562.[Pe4] Y. Peng, Parabolic presentations of the super Yangian Y ( gl M | N ) associated with arbitrary 01-sequences, Comm. Math. Phys. (2016), 313-347.[PS1] E. Poletaeva and V. Serganova, On Kostant’s theorem for the Lie superalgebra Q(n), Adv. Math. (2016), 320-359.[PS2] E. Poletaeva and V. Serganova, On the finite W-algebra for the Lie superalgebra Q(n) in the non-regular case, J. Math. Phys. (2017), no. 11, 111701.[Pr1] A. Premet, Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeilerconjecture, Invent. Math. (1995), 79-117.[Pr2] A. Premet, Special transverse slices and their enveloping algebras, Adv. Math. (2002), 1-55.[RS] E. Ragoucy and P. Sorba, Yangian realisations from finite W -algebras, Comm. Math. Phys. (1999),551-572.[St] V. Stukopin, Yangians of Lie superalgebras of type A(m, n), (Russian) Funktsional. Anal. i Prilozhen. (1994), no. 3, 85-88; translation in Funct. Anal. Appl. (1994), no. 3, 217-219.[Ts] A. Tsymbaliuk, Shuffle algebra realizations of type A super Yangians and quantum affine superalgebrasfor all Cartan data, preprint, arXiv:1909.13732[Wa] W. Wang, Nilpotent orbits and finite W -algebras, Fields Institute Communications Series (2011),71-105.[WZ1] W. Wang and L. Zhao, Representations of Lie superalgebras in prime characteristic I, Proc. LondonMath. Soc. (2009), 145–167. [WZ2] W. Wang and L. Zhao, Representations of Lie superalgebras in prime characteristic II: the queerseries, J. Pure Appl. Algebra. (2011), 2515-2532.[WK] B. Weisfeiler and V. Kac, On irreducible representations of Lie p-algebras, Func. Anal. Appl. (1971),111–117.[ZS1] Y. Zeng and B. Shu, Finite W-superalgebras for basic Lie superalgebras, J. Algebra. (2015),188-234.[ZS2] Y. Zeng and B. Shu, On Kac-Weisfeiler modules for general and special linear Lie superalgebras, IsraelJ. Math. (2016), 471-490.[Zh] L. Zhao, Finite W-superalgebras for queer Lie superalgebras, J. Pure Appl. Algebra. (2014),1184-1194. Department of Mathematics, National Central University, Chung-Li, Taiwan, 32054 E-mail address ::