aa r X i v : . [ m a t h . QA ] A ug FINITE W -SUPERALGEBRAS AND TRUNCATED SUPERYANGIANS YUNG-NING PENG
Abstract.
Let Y ℓm | n be the super Yangian of general linear Lie superalgebrafor gl m | n . Let e ∈ gl mℓ | nℓ be a “rectangular” nilpotent element and W e be thefinite W -superalgebra associated to e . We show that Y ℓm | n is isomorphic to W e . Introduction
The connection between Yangians and finite W -algebras of type A was first no-ticed by mathematical physicists Briot, Ragoucy and Sorba in [17, 4] under somerestrictions, and then constructed in general cases by Brundan and Kleshchev ex-plicitly in [3]. In this way, a realization of finite W -algebra in terms of truncatedYangian is obtained, and this provides a useful tool for the study of the represen-tation theory of finite W -algebras. In this note, we establish such a connectionbetween finite W -superalgebras and super Yangians of type A where the nilpotentelement e is rectangular (cf. § Y m | n be the super Yangian for gl m | n and Y ℓm | n be the truncated super Yan-gian of level ℓ for some non-negative integer ℓ . Our main result is that thereexists an isomorphism of filtered superalgebras between Y ℓm | n and W e , the finite W -superalgebra associated to a rectangular e ∈ gl mℓ | nℓ . Such a connection wasfirstly obtained in [6]. In this article, we provide a new proof in a different ap-proach, which is similar to [3].In a recent paper [2], such a connection between the super Yangian Y | and thefinite W -superalgebra associated to a principal nilpotent element e is obtained. Inparticular, the specialization of our result when m = n = 1 coincides with thespecial case in [2] when e is both principal and rectangular.2. Super Yangian Y m | n and its truncation Y ℓm | n Let m and n be non-negative integers. Let b be an ǫ - δ sequence of gl m | n intro-duced in [8, 13, 7]. To be precise, b is a sequence consisting of exactly m δ ’s and n ǫ ’s, both indistinguishable. For example, δǫδǫǫ is such a sequence of gl | .Note that the set of such sequences is in one-to-one correspondence with theclasses of the simple systems of gl m | n modulo the Weyl group action, and each ǫ - δ sequence b gives rise to a distinguished Borel subalgebra, which will be also Mathematics Subject Classification.
Key words and phrases.
Yangian, finite W -algebra. denoted by b . In particular, the sequence b st := m z }| { δ . . . δ n z }| { ǫ . . . ǫ corresponds to thestandard Borel subalgebra consisting of upper triangular matrices. Remark 2.1.
Contrary to the gl n case, there are many inequivalent simple rootsystems and hence many inequivalent Dynkin diagrams for gl m | n . It is also truethat each ǫ - δ sequence corresponds to exactly one of the Dynkin diagrams. As anexample, the following are two inequivalent Dynkin diagrams for gl | (equivalently, sl | ), where their corresponding simple systems and ǫ - δ sequences are listed: (cid:13) N (cid:13) (cid:13) N N N (cid:13) δ − δ δ − ǫ ǫ − ǫ ǫ − ǫ δ − ǫ ǫ − δ δ − ǫ ǫ − ǫ b = b st = δδǫǫǫ b = δǫδǫǫ Here (cid:13) denotes an even simple root, N denotes an odd simple root, δ i and ǫ j areelements of h ∗ sending a matrix to its i -th and (2+ j )-th diagonal entry, respectively.Such a phenomenon can also be observed in other types of Lie superalgebras. Werefer the reader to [9, 7] for the detail.For each 1 ≤ i ≤ m + n , define a number | i | ∈ Z , called the parity of i , asfollows: | i | := (cid:26) i -th position of b is δ ,1 if the i -th position of b is ǫ . (2.1)For a given b , the super Yangian Y m | n (cf. [14]) is the associative Z -gradedalgebra (i.e., superalgebra) over C with generators n t ( r ) ij | ≤ i, j ≤ m + n ; r ≥ o , (2.2)subject to certain relations.To write down the relations, we firstly define the parity of t ( r ) ij to be | i | + | j | , and t ( r ) ij is called an even (odd, respectively) element if its parity is 0 (1, respectively).The defining relations are[ t ( r ) ij , t ( s ) hk ] = ( − | i | | j | + | i | | h | + | j | | h | min( r,s ) − X t =0 (cid:16) t ( t ) hj t ( r + s − − t ) ik − t ( r + s − − t ) hj t ( t ) ik (cid:17) , (2.3)where the bracket is understood as the supercommutator. By convention, we set t (0) ij := δ ij .In the case when m = 0 or n = 0, it reduces to the usual Yangian. The generatorsin (2.2) are called the RTT generators while the relations (2.3) are called the RTTrelations. As in the case of gl m | n , Y m | n are isomorphic for all b . Note that theoriginal definition in [14] corresponds to the case when b = b st .For all 1 ≤ i, j ≤ m + n , we define the formal power series t ij ( u ) := X r ≥ t ( r ) ij u − r . INITE W -SUPERALGEBRAS AND TRUNCATED SUPER YANGIANS 3 It is well-known (cf. [14]) that Y m | n is a Hopf-algebra, where the comultiplication∆ : Y m | n → Y m | n ⊗ Y m | n is defined by∆( t ( r ) ij ) = r X s =0 m + n X k =1 t ( r − s ) ik ⊗ t ( s ) kj , (2.4)and one has the evaluation homomorphism ev : Y m | n → U ( gl m | n ) defined byev (cid:0) t ij ( u ) (cid:1) := δ ij + ( − | i | e i,j , (2.5)where e i,j denotes the elementary matrix in gl m | n . Definition 2.2.
Let ℓ be a non-negative integer and I ℓ be the 2-sided ideal of Y m | n generated by the elements { t ( r ) ij | ≤ i, j ≤ m + n, r > ℓ } . The truncated superYangian of level ℓ , denoted by Y ℓm | n , is defined to be the quotient Y m | n /I ℓ .Note that Y m | n , I ℓ and the quotient Y ℓm | n are all independent of the choice of b . The image of t ( r ) ij in the quotient Y ℓm | n will be denoted by t ( r ) ij ; b , since it will beidentified with certain element depending on b later; see (4.5).Define κ = ev, and for any integer ℓ ≥
2, we define the following homomorphism κ ℓ := ( ℓ − copies z }| { ev ⊗ · · · ⊗ ev) ◦ ∆ ℓ − : Y m | n → U ( gl m | n ) ⊗ ℓ , (2.6)then we have κ ℓ ( t ( r ) ij ) = X ≤ s < ···
Let κ ℓ , I ℓ be as above. (1) The kernel of κ ℓ is exactly I ℓ . (2) The set of all supermonomials in the elements of Y ℓm | n n t ( r ) ij ; b | ≤ i, j ≤ m + n, ≤ r ≤ ℓ o taken in some fixed order forms a basis for Y ℓm | n . (3) The set of all supermonomials in the elements of Y m | n n t ( r ) ij | ≤ i, j ≤ m + n, r ≥ o taken in some fixed order forms a basis for Y m | n . As a consequence, we may identify Y ℓm | n with the image of Y m | n under the map κ ℓ . In particular, the induced homomorphism κ ℓ : Y ℓm | n → U ( gl m | n ) ⊗ ℓ (2.8)is injective. YUNG-NING PENG
Define the canonical filtration of Y m | n F Y m | n ⊆ F Y m | n ⊂ F Y m | n ⊆ · · · by deg t ( r ) ij := r , i.e., F d Y m | n is the span of all supermonomials in the generators t ( r ) ij of total degree ≤ d. It is clear from (2.3) that the associated graded algebra gr Y m | n is supercommutative. We also obtain the canonical filtration on Y ℓm | n induced fromthe natural quotient map Y m | n ։ Y ℓm | n .3. Finite W -superalgebras Let M and N be non-negative integers and let g = gl M | N . Every elements of g inthe context are considered Z -homogeneous unless mentioned specifically. Recallthat g acts on C M | N via the standard representation, which we will denote by ψ : g → End( C M | N ). Let ( · , · ) denote the non-degenerate even supersymmetricinvariant bilinear form on g defined by( x, y ) := str (cid:0) ψ ( x ) ◦ ψ ( y ) (cid:1) , ∀ x, y ∈ g , (3.1)where str means the super trace. See [7, 9, 12] for more comprehensive studiesabout Lie superalgebras. Remark 3.1.
There exists another well-known even supersymmetric invariant bi-linear form on g , called the Killing form , defined by < x, y > := str (cid:0) Ψ( x ) ◦ Ψ( y ) (cid:1) , ∀ x, y ∈ g , where Ψ : g → End g denotes the adjoint representation of g . The Killing form isnon-degenerate for all M = N and it plays an important rule in the classificationof simple Lie superalgebras [12]; however, it is degenerate if M = N and so it isnot suitable for our purpose. Note that the form (3.1) is indeed non-degenerateeven when M = N .Let e be an even nilpotent element in g . It can be shown (cf. [18, 11]) that thereexists (not uniquely) a semisimple element h ∈ g such that ad h : g → g gives a good Z -grading of g for e , which means the following conditions are 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 ≥ − even ; that is, wealways assume that g ( i ) = 0 for all i / ∈ Z .Define the nilpotent subalgebras of g as follows: p := M j ≥ g ( j ) , m := M j< g ( j ) . (3.2)Let χ ∈ g ∗ be defined by χ ( y ) := ( y, e ), for all y ∈ g . Then the restrictionof χ on m gives a one dimensional U ( m )-module. Let I χ denote the left ideal INITE W -SUPERALGEBRAS AND TRUNCATED SUPER YANGIANS 5 of U ( g ) generated by { a − χ ( a ) | a ∈ m } . By PBW theorem of U ( g ), we have U ( g ) = U ( p ) ⊕ I χ . Let pr χ : U ( g ) → U ( p ) be the natural projection and we canidentify U ( g ) /I χ ∼ = U ( p ). Furthermore we define the χ -twisted action of m on U ( p )by a · y := pr χ ([ a, y ]) for all a ∈ m and y ∈ U ( p ) . An element y ∈ U ( p ) is said to be m -invariant if a · y = 0 for all a ∈ m .The finite W -superalgebra (which we will omit the term “finite” from now on)is defined to be the subspace of m -invariants of U ( p ) under the χ -twisted action;that is, 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 } . At this point, the definition of W -superalgebra depends on the nilpotent element e and a semisimple element h which gives a good Z -grading for e . Example 3.2.
If we take the nilpotent element e = 0, then χ = 0, g = g (0) = p , m = 0, W e,h = U ( p ) = U ( g ).Now we introduce certain combinatorial objects called ( m, n ) -colored rectan-gles [11, 2] (which are in fact special cases of the so called pyramids ). Theseobjects provide a diagrammatic way to record the information needed to define W -superalgebras.Let π be a rectangular Young diagram with m + n boxes as its height and ℓ boxesas its base. We choose arbitrary m rows and color the boxes in these rows by +,while we color the other n rows by − . Such a diagram is called an ( m, n )- coloredrectangle , or a rectangle for short. For example, π = − − − −− − − − + + + + − − − − + + + +Set M = mℓ and N = nℓ . We enumerate the + boxes by the numbers { , . . . , M } down columns from left to right, and enumerate the − boxes bythe numbers { , . . . , N } in the same fashion. In fact, we may enumerate the boxesby an arbitrary order as long as the parities are preserved so we just choose theeasiest way according to our purpose. Moreover, we image that each box of π isof size 2 × π is built on the x -axis where the center of π is exactly on the YUNG-NING PENG origin. For example, π = 3 6 9 122 5 8 112 4 6 81 4 7 101 3 5 7 • − − e ( π ) and a semisimple h ( π )in g = gl M | N which gives a good Z -grading of g for e from a given rectangle π . Let J = { < . . . < M < < . . . < N } be an ordered index set and let { e i | i ∈ J } bea basis of C M | N . We identify gl M | N ∼ = End ( C M | N ) with the ( M + N ) × ( M + N )matrices over C by this basis of C M | N with respect to the order e i < e j if i < j in J . Define the element e ( π ) := X i,j ∈ J e i,j ∈ gl M | N , (3.4)summing over all adjacent pairs i j of boxes in π. It is clear that such an element e ( π ) is even nilpotent.Let f col( i ) denote the x -coordinate of the box numbered with i ∈ J . For instance,in our example (3.3), f col(1) = − f col(8) = 1. Then we define the followingdiagonal matrix h ( π ) := − diag (cid:0) f col(1) , . . . , f col( M ) , f col(1) , . . . , f col( N ) (cid:1) ∈ g . (3.5)One may check directly that ad h ( π ) gives a good Z -grading of g for e ( π ). Remark 3.3.
In general, there are other even good Z -gradings for e . However, ifour e = e ( π ) is obtained by a rectangle π according to (3.4), then such a gradingis unique (cf. [11, Theorem 7.2]).Now we characterize those e obtained from (3.4) for some rectangle π . Consider e = e M ⊕ e N ∈ End C M | N , (3.6)where e M and e N are the restrictions of e to C M | and C | N , respectively. Let µ = ( µ , µ , . . . ) and ν = ( ν , ν , . . . ) be the partitions representing the Jordan typeof e M and e N , respectively. Definition 3.4.
An element e is called rectangular if it is even nilpotent and theJordan blocks of e M and e N are all of the same size ℓ , i.e., µ = ( m − copies z }| { ℓ, . . . , ℓ ) and ν = ( n − copies z }| { ℓ, . . . , ℓ ) for some for some non-negative integers ℓ, m, n . INITE W -SUPERALGEBRAS AND TRUNCATED SUPER YANGIANS 7 Clearly, e is of the form (3.4) for some rectangle π if and only if e is rectangular.Assume now e is rectangular. We define a new partition λ by collecting all partsof µ and ν together and reorder them by an arbitrary order. Since all the partsare the same number ℓ , we use ℓ to denote the parts obtained from µ .For example, consider gl | , µ = (4 ,
4) and ν = (4 , , λ is λ = (4 , , , , ǫ - δ sequence b from λ : if the i -th position of λ is ℓ (re-spectively, ℓ ), then the i -th position of b is ǫ (respectively, δ ). For example, the λ above corresponds to the ǫ - δ sequence b = δǫδǫǫ .Then we color the rectangle of height m + n base ℓ with respect to b : we colorthe i -th row of π by + (respectively, − ) if the i -th position of b is δ (respectively, ǫ ) where the rows are counted from top to bottom. After coloring the rows, weenumerate the boxes of π by exactly the same fashion explained in the paragraphbefore (3.3).Therefore, we have a bijection between the set of ( m, n )-colored rectangles ofbase ℓ and the set of pairs ( e, b ) where e is a rectangular element in gl mℓ | nℓ and b is an ǫ - δ sequence containing exactly m δ ’s and n ǫ ’s.Let π be a fixed ( m, n )-colored rectangle and e ( π ) denote the rectangular elementassociates to π . We will denote by W π := W e ( π ) the W -superalgebra associatedto e ( π ). Note that we may omit h ( π ) in our notation with no ambiguity due toRemark 3.3. Remark 3.5.
An interesting observation is that W π is independent of the choicesof the sequence b because any other sequence b ′ yields to the same e and hencethe same W -superalgebra. This is parallel to the fact that Y ℓm | n is independent ofthe choice of b .Now we label the columns of π from left to right by 1 , . . . , ℓ , and for any i ∈ J we let col( i ) denote the column where i appears. Define the Kazhdan filtration of U ( gl M | N ) · · · ⊆ F d U ( gl M | N ) ⊆ F d +1 U ( gl M | N ) ⊆ · · · by declaring deg( e i,j ) := col( j ) − col( i ) + 1 (3.7)for each i, j ∈ J and F d U ( gl M | N ) is 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 ( gl M | N ) denote the associated gradedsuperalgebra. The natural projection gl M | N ։ p induces a grading on W π .On the other hand, let g e denote the centralizer of e in g = gl M | N and let S ( g e )denote the associated supersymmetric algebra. We define the Kazhdan filtrationon S ( g e ) by the same setting (3.7). The following proposition was observed in [19],where the mild assumption on e there is satisfied when e is rectangular. Proposition 3.6. [19, Remark 3.9] gr W π ∼ = S ( g e ) as Kazhdan graded superalge-bras. The following proposition is a well-known result about g e . As remarked in [2],the result is similar to the Lie algebra case gl M + N since e is even. YUNG-NING PENG
Proposition 3.7.
Let π be an (m,n)-colored rectangle and e = e ( π ) be the associ-ated rectangular nilpotent element. For all ≤ i, j ≤ m + n and r > , define c ( r ) i,j := X ≤ h,k ≤ m + n row ( h )= i, row ( k )= j col ( k ) − col ( h )= r − ( − | i | e h,k ∈ g = gl M | N . Then the set of vectors { c ( r ) i,j | ≤ i, j ≤ m + n, ≤ r ≤ ℓ } forms a basis for g e . Corollary 3.8.
Consider Y ℓm | n with the canonical filtration and S ( g e ) with theKazhdan filtration. Let F d Y ℓm | n and F d S ( g e ) denote the associated filtered algebras,respectively. Then for each d ≥ , we have dim F d Y ℓm | n = dim F d S ( g e ) .Proof. Follows from Proposition 2.3, Proposition 3.7 and induction on d . (cid:3) Isomorphism between Y ℓm | n and W π Let π be a given ( m, n )-colored rectangle with base ℓ and b be the ǫ - δ sequencedetermined by the colors of rows of π . We now define some elements in U ( p ). Itturns out later that they are m -invariant, i.e., belong to W π .For each 1 ≤ r ≤ ℓ , set ρ r := − ( ℓ − r )( m − n ) . (4.1)For 1 ≤ i, j ≤ m + n , define˜ e i,j := ( − col( j ) − col( i ) ( e i,j + δ ij ( − | i | ρ col( i ) ) , (4.2)where | i | is determined by b as in (2.1).One may check that[˜ e i,j , ˜ e h,k ] = (˜ e i,k − δ ik ( − | i | ρ col( i ) ) δ hj − ( − ( | i | + | j | )( | h | + | k | ) δ i,k (˜ e h,j − δ hj ( − | j | ρ col( j ) ) . (4.3)Also, for any 1 ≤ i, j ≤ m + n , we have χ (˜ e i,j ) = (cid:26) ( − | i | +1 if row( i ) = row( j ) and col( i ) = col( j ) + 1;0 otherwise. (4.4)For 1 ≤ i, j ≤ m + n , we let t (0) ij ; b := δ i,j and for 1 ≤ r ≤ ℓ , define t ( r ) ij ; b := r X s =1 X i ,...,i s j ,...,j s ( − | i | + ··· + | i s | ˜ e i ,j · · · ˜ e i s ,j s (4.5)where the second sum is over all i , . . . , i s , j , . . . , j s ∈ J such that(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) col( j t ) < col( i t +1 ) for each t = 1 , . . . , s − i ) = i , row( j s ) = j ;(5) row( j t ) = row( i t +1 ) for each t = 1 , . . . , s − INITE W -SUPERALGEBRAS AND TRUNCATED SUPER YANGIANS 9 First note that these elements depends on the choice of b . Also, the restrictions (1)and (2) imply that t ( r ) ij ; b belongs to F r U ( p ) with respect to the Kazhdan grading.Define the following series for all 1 ≤ i, j ≤ m + n : t ij ; b ( u ) := ℓ X r =0 t ( r ) ij ; b u − r ∈ U ( p )[[ u − ]] . (4.6)Now we prove that these distinguished elements in U ( p ) are in fact m -invariantunder the χ -twisted action. Let T ( M at ℓ ) be the tensor algebra of the ℓ × ℓ matricesspace over C and g = gl M | N where M = mℓ , N = nℓ . For all 1 ≤ i, j ≤ m + n ,define a C -linear map t ij ; b : T ( M at ℓ ) → U ( g ) inductively by t ij ; b (1) := δ i,j , t ij ; b ( e a,b ) := ( − | i | e i⋆a,j⋆b ,t ij ; b ( x ⊗ x ⊗ . . . ⊗ x r ) := X ≤ i ,i ,...,i r − ≤ m + n t ii ; b ( x ) t i i ; b ( x ) · · · t i r − j ; b ( x r ) , (4.7)for 1 ≤ a, b ≤ ℓ , r ≥ x , . . . , x r ∈ M at ℓ , where i ⋆ a is defined to be thenumber in the ( i, a )-th position of π , where we label the rows and columns fromtop to bottom and from left to right. For example, let π be the rectangle in (3.3),then t b ( e , ) = ( − | | e ⋆ , ⋆ = − e , . For an indeterminate u , we extend thescalars from C to C [ u ] in the obvious way. Lemma 4.1.
For each ≤ i, j, h, k ≤ m + n and x, y , . . . , y r ∈ M at ℓ , we have [ t ij ; b ( x ) , t hk ; b ( y ⊗ · · · ⊗ y r )] =( − | i | | j | + | i | | h | + | j | | h | (cid:0) r X s =1 t hj ; b ( y ⊗ · · · ⊗ y s − ) t ik ; b ( xy s ⊗ · · · ⊗ y r ) − t hj ; b ( y ⊗ · · · ⊗ y s x ) t ik ; b ( y s +1 ⊗ · · · ⊗ y r ) (cid:1) , (4.8) where the products xy s and y s x on the right are the matrix products in M at ℓ .Proof. By (4.7) and induction on r . (cid:3) Introducing the following matrix A ( u ) with entries in the algebra T ( M at ℓ )[ u ]: A ( u ) = u + e , + ρ e , e , · · · e ,ℓ u + e , + ρ e , e ,ℓ u + e ℓ − ,ℓ − + ρ ℓ − e ℓ − ,ℓ · · · u + e ℓ,ℓ + ρ ℓ . A key observation is that for all 1 ≤ i, j ≤ m + n and 0 ≤ r ≤ ℓ , the element t ( r ) ij ; b of U ( p ) defined by (4.5) equals to the coefficient of the term u ℓ − r in t ij ; b (cid:0) rdet A ( u ) (cid:1) ,where rdet A := X τ ∈ S l sgn( τ ) a ,τ (1) a ,τ (2) · · · a l,τ ( l ) , for a matrix A = ( a i,j ) ≤ i,j ≤ ℓ . We also let A p,q ( u ) stand for the submatrix of A ( u )consisting only of rows and columns numbered by p, . . . , q . Proposition 4.2.
For all ≤ i, j ≤ m + n , ≤ r ≤ ℓ and a fixed b , the elements t ( r ) ij ; b of U ( p ) are m -invariant under the χ -twisted action.Proof. Firstly we observe that the statement is trivial when ℓ ≤
1, hence we mayassume ℓ ≥
2. Note that m is generated by elements of the form t ij ; b ( e c +1 ,c ), henceit suffices to show thatpr χ (cid:16)(cid:2) t ij ; b ( e c +1 ,c ) , t hk ; b (cid:0) rdet A ( u ) (cid:1)(cid:3)(cid:17) = 0for all 1 ≤ i, j, h, k ≤ m + n and 1 ≤ c ≤ ℓ −
1. In this proof, we omit the fixed b in our notation which shall cause no confusion.By (4.8), up to an irrelevant sign, we have h t ij (cid:0) e c +1 ,c (cid:1) , t hk (cid:0) rdet A ( u ) (cid:1)i = t hj (cid:0) rdet A ,c − ( u ) (cid:1) t ik (rdet e c +1 ,c e c +1 ,c +1 · · · e c +1 ,ℓ u + e c +1 ,c +1 + ρ c +1 · · · e c +1 ,ℓ ... . . . ...0 · · · u + e ℓ,ℓ + ρ ℓ ) − t hj (rdet u + e , + ρ · · · e ,c e ,c u + e c,c + ρ c e c,c · · · e c +1 ,c ) t ik (cid:0) rdet A c +2 ,ℓ ( u ) (cid:1) . A crucial observation is that for 1 ≤ i, j ≤ m + n and 1 ≤ c ≤ ℓ −
1, we have t ij (cid:0) e c +1 ,c ( u + e c +1 ,c +1 + ρ c +1 ) (cid:1) − t ij (cid:0) u + e c +1 ,c +1 + ρ c (cid:1) ≡ I χ ) . (4.9)Indeed, it is enough to check (4.9) when ℓ = 2. The trick here is to rewrite thequadratic terms in U ( g ) by supercommtator relations. Then the term ρ c givesexactly the required constant so that the left-hand side of (4.9) can be expressedby a linear combination of elements in I χ .Therefore, h t ij (cid:0) e c +1 ,c (cid:1) , t hk (cid:0) rdet A ( u ) (cid:1)i ≡ t hj (cid:0) rdet A ,c − ( u ) (cid:1) t ik (rdet e c +1 ,c +1 · · · e c +1 ,ℓ u + e c +1 ,c +1 + ρ c · · · e c +1 ,ℓ ... . . . ...0 · · · u + e ℓ,ℓ + ρ ℓ ) − t hj (rdet u + e , + ρ · · · e ,c e ,c u + e c,c + ρ c e c,c · · · ) t ik (cid:0) rdet A c +2 ,ℓ ( u ) (cid:1) INITE W -SUPERALGEBRAS AND TRUNCATED SUPER YANGIANS 11 modulo I χ . Making the obvious row and column operations gives thatrdet e c +1 ,c +1 · · · e c +1 ,ℓ u + e c +1 ,c +1 + ρ c · · · e c +1 ,ℓ ... . . . ...0 · · · u + e ℓ,ℓ + ρ ℓ = ( u + ρ c ) rdet A c +2 ,ℓ ( u ) , rdet u + e , + ρ · · · e ,c e ,c u + e c,c + ρ c e c,c · · · = ( u + ρ c ) rdet A ,c − ( u ) . Substituting these back shows that pr χ (cid:16)(cid:2) t ij ( e c +1 ,c ) , t hk (cid:0) rdet A ( u ) (cid:1)(cid:3)(cid:17) = 0. (cid:3) Set a new notation t ij ; b ( e r,r ) = ( − | i | e [ r ] i,j ∈ U ( h ), where h ∼ = ( gl m | n ) ⊕ ℓ . Clearly, h has a basis { e [ r ] i,j | ≤ r ≤ ℓ, ≤ i, j ≤ m + n } . Define η : U ( h ) → U ( h ) , e [ r ] i,j e [ r ] i,j − δ ij ρ r . Let ξ : U ( p ) → U ( h ) be the algebra homomorphism induced from the naturalprojection p ։ h . Define the map µ := η ◦ ξ : U ( p ) → U ( h ) ∼ = U ( gl m | n ) ⊗ ℓ . Lemma 4.3.
For ≤ i, j ≤ m + n and r > , µ ( t ( r ) ij ; b ) = X ≤ s < ···
Applying the map e r,s δ r,s ( e r,r − ρ r ) to the matrix A ( u ) gives a diagonalmatrix with determinant ( u + e , )( u + e , ) · · · ( u + e ℓ,ℓ ), where its u ℓ − r -coefficientequals to P ≤ s < ···
Theorem 4.4.
Let π be an ( m, n ) -colored rectangle and b be the ǫ - δ sequencedetermined by the rows of π . Then there exists an isomorphism Y ℓm | n ∼ = W π offiltered superalgebras such that the generators { t ( r ) ij ; b | ≤ i, j ≤ m + n, ≤ r ≤ ℓ } of Y ℓm | n are sent to the elements of W π with the same names defined by (4.5).Proof. Again, the result is trivial when ℓ ≤ ℓ ≥
2. ByProposition 2.3, the set of all supermonomials in the elements { t ( r ) ij ; b | ≤ i, j ≤ m + n, ≤ r ≤ ℓ } of Y ℓm | n taken in some fixed order and of total degree ≤ d forms a basis for F d Y ℓm | n (with respect to the canonical filtration). Since κ ℓ is injective, by Corollary 3.8 wehave dim κ ℓ ( F d Y ℓm | n ) = dim F d Y ℓm | n = dim F d S ( g e ) , where S ( g e ) is equipped with the Kazhdan grading.Let X d denote the subspace of U ( p ) spanned by all supermonomials in the el-ements { t ( r ) ij ; b | ≤ i, j ≤ m + n, ≤ r ≤ ℓ } defined by (4.5) taken in some fixedorder and of total degree ≤ d . By Lemma 4.3 and the discussion above, we have µ ( X d ) = κ ℓ ( F d Y ℓm | n ).Proposition 4.2 assures that X d ⊆ F d W π . Together with Proposition 3.6, wehave dim F d S ( g e ) = dim µ ( X d ) ≤ dim X d ≤ dim F d W π ≤ dim F d S ( g e ) . Thus equality holds everywhere and hence X d = F d W π . Moreover, µ is injectiveand µ ( t ( r ) ij ; b ) = κ ℓ ( t ( r ) ij ; b ) for all 1 ≤ i, j ≤ m + n and 0 ≤ r ≤ ℓ . The composition µ − ◦ κ ℓ : Y ℓm | n → W π gives the required isomorphism. (cid:3) Remark 4.5.
The connection between W -algebras of so ( n ) and sp ( n ) and theircorresponding (twisted) Yangians has not been studied in full generality. Somepartial results can be found in [5, 16]; see also [1] for an approach similar to thisarticle. Acknowledgements.
The author would like to thank the anonymous reviewersfor their valuable comments and suggestions to improve the quality of this ar-ticle. The author is supported by post-doctoral fellowship of the Institution ofMathematics, Academia Sinica, Taipei, Taiwan.
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