Center at the critical level for centralizers in type A
aa r X i v : . [ m a t h . R T ] D ec Center at the critical level for centralizers in type A A. I. Molev
Abstract
We consider the affine vertex algebra at the critical level associated with the centralizerof a nilpotent element in the Lie algebra gl N . Due to a recent result of Arakawa and Premet,the center of this vertex algebra is an algebra of polynomials. We construct a family offree generators of the center in an explicit form. As a corollary, we obtain generators ofthe corresponding quantum shift of argument subalgebras and recover free generators of thecenter of the universal enveloping algebra of the centralizer produced earlier by Brown andBrundan. Let a be a finite-dimensional Lie algebra over C equipped with an invariant symmetric bilinearform. Consider the corresponding affine Kac–Moody algebra b a defined as a central extension ofthe Lie algebra of Laurent polynomials a [ t, t − ] ; see (2.2) below. The vacuum module over b a isa vertex algebra whose center is a commutative associative algebra. In the case of a simple Liealgebra a the structure of the center z ( b a ) at the critical level was described by a celebrated theoremof Feigin and Frenkel [5] (see also [6]) which states that z ( b a ) is an algebra of polynomials ininfinitely many variables. This fact can be regarded as an affine version or chiralization of thewell-known description of the center of the universal enveloping algebra U( a ) as a polynomialalgebra.On the other hand, in the case where a = g e is the centralizer of a nilpotent element e in asimple Lie algebra g , the invariant algebra S( a ) a admits free families of generators under certainadditional conditions, due to the pioneering work of Panyushev, Premet and Yakimova [13]. Inparticular, these conditions hold for all nilpotents in types A and C which thus confirms Premet’sconjecture in those cases. Explicit generators of both the invariant algebra and the center of U( a ) in type A were produced by Brown and Brundan [2] with the use of the shifted Yangians. Itwas shown by Yakimova [16] that the generators of S( a ) a produced in [2] coincide with thosepreviously conjectured in [13].A recent work of Arakawa and Premet [1] provides a chiralization of these results for thecentralizer a which shows that the Feigin–Frenkel theorem extends to the affine vertex algebraat the critical level associated with the Kac–Moody algebra b a . As a consequence, they showedthe existence of the regular quantum shift of argument subalgebras and proved that they are freepolynomial algebras. Moreover, explicit formulas for generators of z ( b a ) were produced in [1] inthe case where a is the centralizer of a minimal nilpotent in gl N .1ur goal in this paper is to give explicit formulas for generators of z ( b a ) for an arbitrarynilpotent element e ∈ gl N (Theorem 2.1). As a corollary, we get generators of the quantumshift of argument subalgebras A χ ∈ U( a ) associated with an element χ ∈ a ∗ thus providingan explicit solution of Vinberg’s quantization problem [15] for centralizers (Corollary 2.6). It isbased on the results of [1] showing that if the element χ is regular, then the classical limit gr A χ coincides with the Mishchenko–Fomenko subalgebra A χ ⊂ S( a ) ; see [9]. We conjectured in [12,Conjecture 5.8] that the subalgebra A χ can also be obtained with the use of the symmetrizationmap. Note that the subalgebra A is the center of U( a ) . In this case our generators essentiallycoincide with those found in [2]; see Corollary 2.7 below.By taking e = 0 in Theorem 2.1 we recover the formulas for the generators of the algebra z ( b gl n ) found in [3] and [4] which we also interpret with the use of an operator ∆ preserving z ( b gl n ) (Corollary 2.4). For more details on this particular case and extensions to the other classical typessee [11]. A different way to produce generators of the Feigin–Frenkel center z ( b g ) was developedin a recent work by Yakimova [17]. Using the notation of [2], suppose that e ∈ g = gl N is a nilpotent matrix with Jordan blocks ofsizes λ , . . . , λ n , where λ · · · λ n and λ + · · · + λ n = N . Consider the corresponding pyramid which is a left-justified array of rows of unit boxes such that the top row contains λ boxes, the next row contains λ boxes, etc. The row-tableau is obtained by writing the numbers , . . . , N into the boxes of the pyramid consecutively by rows from left to right. For instance,the row-tableau corresponds to the Jordan blocks of sizes , , and N = 9 . Let row( a ) and col( a ) denote therow and column number of the box containing the entry a .Let e ab with a, b = 1 , . . . , N be the standard basis elements of g . For any i, j n and λ j − min( λ i , λ j ) r < λ j set E ( r ) ij = X row( a )= i, row( b )= j col( b ) − col( a )= r e ab , summed over a, b ∈ { , . . . , N } . The elements E ( r ) ij form a basis of the Lie algebra a = g e . Thecommutation relations for the basis vectors have the form h E ( r ) ij , E ( s ) kl i = δ kj E ( r + s ) il − δ il E ( r + s ) kj , (2.1)assuming that E ( r ) ij = 0 for r > λ j . 2ote that in the particular case of a rectangular pyramid λ = · · · = λ n = p the Lie algebra a is isomorphic to the truncated polynomial current algebra gl n [ v ] / ( v p = 0) (also known as the Takiff algebra ). The isomorphism is given by E ( r ) ij e ij v r , r = 0 , . . . , p − , i, j n. Equip the Lie algebra a with the invariant symmetric bilinear form h , i defined as the nor-malized Killing form at the critical level on the -th component of a ; see [1, Sec. 5]. Explicitly,it is given by the formulas D E (0) ii , E (0) jj E = min( λ i , λ j ) − δ ij (cid:16) λ + · · · + λ i − + ( n − i + 1) λ i (cid:17) , and if λ i = λ j for some i = j then D E (0) ij , E (0) ji E = − (cid:16) λ + · · · + λ i − + ( n − i + 1) λ i (cid:17) , whereas all remaining values of the form on the basis vectors are zero. Note that the sum in thebrackets equals the number of boxes in the first λ i columns of the pyramid.The corresponding affine Kac–Moody algebra b a is the central extension b a = a [ t, t − ] ⊕ C , (2.2)where a [ t, t − ] is the Lie algebra of Laurent polynomials in t with coefficients in a . For any r ∈ Z and X ∈ g we will write X [ r ] = X t r . The commutation relations of the Lie algebra b a have the form h X [ r ] , Y [ s ] i = [ X, Y ][ r + s ] + r δ r, − s h X, Y i , X, Y ∈ a , and the element is central in b a . The vacuum module at the critical level over b a is the quotient V ( a ) = U( b a ) / I , where I is the left ideal of U( b a ) generated by a [ t ] and the element − . By the Poincaré–Birkhoff–Witt theorem, the vacuum module is isomorphic to the universal enveloping algebra U (cid:16) t − a [ t − ] (cid:17) , as a vector space. This vector space is equipped with a vertex algebra structure;see [7], [8]. Denote by z ( b a ) the center of this vertex algebra which is defined as the subspace z ( b a ) = { v ∈ V ( a ) | a [ t ] v = 0 } . It follows from the axioms of vertex algebra that z ( b a ) is a unital commutative associative algebra.It can be regarded as a commutative subalgebra of U (cid:16) t − a [ t − ] (cid:17) . This subalgebra is invariantwith respect to the translation operator T which is the derivation of the algebra U (cid:16) t − a [ t − ] (cid:17) whose action on the generators is given by T : X [ r ]
7→ − r X [ r − , X ∈ a , r < . z ( b a ) is called a Segal–Sugawara vector . By [1, Thm 1.4], there exists a complete set of Segal–Sugawara vectors S , . . . , S N , which means that all translations T r S l with r > and l = 1 , . . . , N are algebraically independent and any element of z ( b a ) can be written asa polynomial in the shifted vectors; that is, z ( b a ) = C [ T r S l | l = 1 , . . . , N, r > . In the case e = 0 this reduces to the Feigin–Frenkel theorem in type A [5, 6].To construct a complete set of Segal–Sugawara vectors, for all i, j ∈ { , . . . , n } introducepolynomials in a variable u with coefficients in this algebra by E ij ( u ) = E (0) ij [ −
1] + · · · + E ( λ j − ij [ − u λ j − if i > j,E ( λ j − λ i ) ij [ − u λ j − λ i + · · · + E ( λ j − ij [ − u λ j − if i < j. For another variable x calculate the column-determinant cdet x + λ T + E ( u ) E ( u ) . . . E n ( u ) E ( u ) x + λ T + E ( u ) . . . E n ( u ) ... ... . . . ... E n ( u ) E n ( u ) . . . x + λ n T + E nn ( u ) (2.3) = X σ ∈ S n sgn σ · (cid:16) δ σ (1)1 ( x + λ T ) + E σ (1)1 ( u ) (cid:17) . . . (cid:16) δ σ ( n ) n ( x + λ n T ) + E σ ( n ) n ( u ) (cid:17) and write it as a polynomial in x with coefficients in V ( a )[ u ] , x n + φ ( u ) x n − + · · · + φ n ( u ) , φ k ( u ) = X r φ ( r ) k u r . (2.4) Theorem 2.1.
The coefficients φ ( r ) k with k = 1 , . . . , n and λ n − k +2 + · · · + λ n < r + k λ n − k +1 + · · · + λ n (2.5) belong to the center z ( b a ) of the vertex algebra V ( a ) . Moreover, they form a complete set ofSegal–Sugawara vectors for the Lie algebra a .Examples . In the principal nilpotent case with n = 1 and e = e + · · · + e N − N we have φ ( r )1 = E ( r )11 [ −
1] = e r +1 [ −
1] + · · · + e N − r N [ − , r = 0 , . . . , N − . For n = 2 we have cdet x + λ T + E ( u ) E ( u ) E ( u ) x + λ T + E ( u ) = x + φ ( u ) x + φ ( u ) φ ( u ) = E ( u ) + E ( u ) ,φ ( u ) = E ( u ) E ( u ) − E ( u ) E ( u ) + λ E ′ ( u ) , where X ′ ( u ) = T X ( u ) . Therefore, a complete set of Segal–Sugawara vectors for a is given by φ ( r )1 = E ( r )11 [ −
1] + E ( r )22 [ − with r = 0 , , . . . , λ − , and φ ( r )2 = X a + b = r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ( a )11 [ − E ( b )12 [ − E ( a )21 [ − E ( b )22 [ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + λ E ( r )22 [ − with r = λ − , . . . , λ + λ − , where the vertical lines indicate the column-determinant.The minimal nilpotent case e = e n n +1 ∈ gl n +1 corresponds to the pyramid with the n rows , . . . , , . For the coefficients of x n − and x n − in the general expansion (2.4) we have φ ( u ) = E ( u ) + · · · + E n n ( u ) ,φ ( u ) = X i 1] + · · · + E (0) n n [ − 1] = e [ − 1] + · · · + e n +1 n +1 [ − ,φ (1)1 = E (1) nn [ − 1] = e n n +1 [ − , whereas φ (1)2 = n − X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (0) ii [ − E (1) in [ − E (0) ni [ − E (1) nn [ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ( n − E (1) nn [ − , cf. [1, Sec. 5].As with the minimal nilpotent case considered in [1], the proof of Theorem 2.1 follows thesame approach as in the paper [3] which deals with the case e = 0 . We will give some necessaryadditional details in Section 3 which include the use of the derivation ∆ of the algebra U( b a ) determined by its commutation relations with the left multiplication operators h ∆ , X [ r ] i = r X [ r + 1] , h ∆ , i = 0 , (2.6)for X ∈ a . We can regard ∆ as an operator on the vacuum module V ( a ) . It is immediate fromthe commutation relations that ∆ preserves Segal–Sugawara vectors and thus gives rise to theoperator ∆ : z ( b a ) → z ( b a ) . Its use in the proof of Theorem 2.1 will be based on the following property of the coefficients φ ( r ) k ∈ V ( a ) , where we do not assume that they belong to z ( b a ) .5 emma 2.3. Under the action of ∆ we have ∆ : φ ( r ) k (2.7) for all r > λ n − k +2 + · · · + λ n − k + 1 , while for r = λ n − k +2 + · · · + λ n − k + 1 we have ∆ : φ ( r ) k 7→ − ( k − λ + · · · + λ n − k +1 ) φ ( r ) k − . We will prove Lemma 2.3 in Section 3. In particular, it implies that a complete set of Segal–Sugawara vectors in the case λ = · · · = λ n = 1 (that is, for a = gl n ) can be produced from asingle vector. Set φ = cdet T + E [ − E [ − . . . E n [ − E [ − T + E [ − . . . E n [ − ... ... . . . ... E n [ − E n [ − . . . T + E nn [ − , which is the constant term of the polynomial in (2.3); see also [3] and [11, Ch. 7]. Corollary 2.4. The elements ∆ k φ for k = 0 , , . . . , n − form a complete set of Segal–Sugawaravectors for gl n . A slightly less explicit but equivalent expression for the coefficients φ ( r ) k is obtained in thefollowing proposition, where we introduce τ as an element of the extended Lie algebra b a ⊕ C τ which satisfies the commutation relations h τ, X [ r ] i = − r X [ r − , h τ, i = 0 . For all i, j ∈ { , . . . , n } define the elements E ij ( τ ) of the tensor product space V ( a ) ⊗ C [ τ ] by E ij ( τ ) = δ ij τ λ j + E (0) ij [ − τ λ j − + · · · + E ( λ j − ij [ − if i > j,E ( λ j − λ i ) ij [ − τ λ i − + · · · + E ( λ j − ij [ − if i < j. Introduce elements φ ◦ , . . . , φ ◦ N ∈ V ( a ) by using the expansion of the column-determinant of thematrix E ( τ ) = [ E ij ( τ )] , cdet E ( τ ) = τ N + φ ◦ τ N − + · · · + φ ◦ N . We will say that the element E ( b ) ij [ r ] of the Lie algebra t − a [ t − ] has weight b . This defines agrading on the universal enveloping algebra U (cid:16) t − a [ t − ] (cid:17) . Proposition 2.5. For each k = 1 , . . . , n and r satisfying (2.5) , the Segal–Sugawara vector φ ( r ) k coincides with the homogeneous component of maximal weight of the coefficient φ ◦ r + k . roof. Expand the column-determinant cdet E ( τ ) = X σ ∈ S n sgn σ · E σ (1)1 ( τ ) . . . E σ ( n ) n ( τ ) and move all powers of τ to the right by commuting them with the elements E ( b ) ij [ r ] . It is straight-forward to verify that a nonzero contribution to the maximum weight component of φ ◦ r + k can onlycome from commutators involving the leading terms τ λ i of E ii ( τ ) for i = 1 , . . . , n . Moreover,this component has weight equal to r and coincides with φ ( r ) k .By adapting Rybnikov’s construction [14] to the case of the affine Kac–Moody algebra b a asin [1], for any element χ ∈ a ∗ and a variable z consider the homomorphism ̺ χ : U (cid:16) t − a [ t − ] (cid:17) → U( a ) ⊗ C [ z − ] , X [ r ] X z r + δ r, − χ ( X ) , (2.8)for any X ∈ a and r < . If S ∈ z ( b a ) is a homogeneous element of degree d with respect to thegrading defined by deg X [ r ] = − r for r < , define the elements S ( k ) ∈ U( a ) (depending on χ )by the expansion ̺ χ ( S ) = S (0) z − d + · · · + S ( d − z − + S ( d ) . If the variable z takes a particular nonzero value in C , then the formula in (2.8) defines a homo-morphism U (cid:16) t − a [ t − ] (cid:17) → U( a ) . Since z ( b a ) is a commutative subalgebra of U (cid:16) t − a [ t − ] (cid:17) , itsimage under this homomorphism is a commutative subalgebra of U( a ) which we denote by A χ .This subalgebra does not depend on the value of z .It is clear from its definition that the degree of the coefficient φ ( r ) k in (2.4) equals k . Therefore,the respective degrees of the Segal–Sugawara vectors provided by Theorem 2.1 coincide with thedegrees of the basic invariants of the symmetric algebra S( a ) given by , . . . , | {z } λ n , , . . . , | {z } λ n − , . . . , n, . . . , n | {z } λ , as found in [13]; see also [2]. Define elements φ ( a ) k ( m ) ∈ U( a ) by applying the homomorphism ̺ χ to the Segal–Sugawara vectors: ̺ χ : φ ( r ) k φ ( r ) k (0) z − k + · · · + φ ( r ) k ( k − z − + φ ( r ) k ( k ) . More explicitly, set χ ( r ) ij = χ ( E ( r ) ij ) and χ ij ( u ) = χ (0) ij + · · · + χ ( λ j − ij u λ j − if i > j,χ ( λ j − λ i ) ij u λ j − λ i + · · · + χ ( λ j − ij u λ j − if i < j. Furthermore, let E ij ( u ) = E (0) ij + · · · + E ( λ j − ij u λ j − if i > j,E ( λ j − λ i ) ij u λ j − λ i + · · · + E ( λ j − ij u λ j − if i < j. ̺ χ will be written as apolynomial in x , cdet x − λ ∂ z + χ ( u ) + E ( u ) z − . . . χ n ( u ) + E n ( u ) z − ... . . . ... χ n ( u ) + E n ( u ) z − . . . x − λ n ∂ z + χ nn ( u ) + E nn ( u ) z − = x n + φ ( u, z ) x n − + · · · + φ n ( u, z ) , with φ k ( u, z ) = X r,m φ ( r ) k ( m ) u r z − k + m . Recall that the Mishchenko–Fomenko subalgebra A χ of the symmetric algebra S( a ) is aPoisson-commutative subalgebra generated by all χ -shifts of all a -invariants P ∈ S( a ) a , as orig-inally defined in [9]. By using the results of [1] we come to the following. Corollary 2.6. Suppose the element χ ∈ a ∗ is regular. Then the elements φ ( r ) k ( m ) for k = 1 , . . . , n with r satisfying conditions (2.5) and m = 0 , . . . , k − are algebraically independent generatorsof the algebra A χ . Moreover, A χ is a quantization of the subalgebra A χ so that gr A χ = A χ . As another corollary of Theorem 2.1, we recover the algebraically independent generators ofthe center of U( a ) constructed in [2] with the use of the shifted Yangians; see also [10] for theparticular case of rectangular pyramids. The algebra A coincides with the center of U( a ) sothat the formulas for the generators are found by taking χ = 0 in (2.8); cf. [11, Secs 6.5 & 7.2].Write the column-determinant cdet x + ( n − λ + E ( u ) E ( u ) . . . E n ( u ) E ( u ) x + ( n − λ + E ( u ) . . . E n ( u ) ... ... . . . ... E n ( u ) E n ( u ) . . . x + E nn ( u ) as a polynomial in x , x n + Φ ( u ) x n − + · · · + Φ n ( u ) with Φ k ( u ) = X r Φ ( r ) k u r . Corollary 2.7. The coefficients Φ ( r ) k with k = 1 , . . . , n and r satisfying conditions (2.5) arealgebraically independent generators of the center of the algebra U( a ) . The elements Φ ( r ) k are slightly different from the corresponding central elements z r given by[2, Eq. (1.3)]. The mapping E ( p ) ij E ( p ) ij + δ p δ ij cλ i defines an automorphism of the algebra U( a ) for any given constant c . The image of the Casimir element Φ ( r ) k under this automorphismwith c = − n + 1 coincides with z r + k . 8 Proof of Theorem 2.1 To prove the first part of the theorem, we will verify that the coefficients φ ( r ) k ∈ V ( a ) are anni-hilated by a family of elements which generate a [ t ] as a Lie algebra. For such a family we take E (0) i +1 i [0] , E ( λ i +1 − λ i ) i i +1 [0] for i = 1 , . . . , n − , and E ( p ) i i [ s ] for p = 0 , . . . , λ i − all i = 1 , . . . , n and s > .We will apply the generators to the column-determinant cdet E of the matrix E = [ E ij ] , wherewe set E ij = δ ij ( x + λ i T ) + E ij ( u ) . The result of such an application is a polynomial in x of theform ψ ( u ) x n − + · · · + ψ n ( u ) , ψ k ( u ) = X r ψ ( r ) k u r , ψ ( r ) k ∈ V ( a ) . (3.1)It will be sufficient to verify that for k = 1 , . . . , n the degree of the polynomial ψ k ( u ) in u is lessthan λ n − k +2 + · · · + λ n − k + 1 (3.2)(in particular, ψ ( u ) must be equal to zero). We will consider the groups of generators case bycase and use the notation τ i = x + λ i T for brevity. Note the commutation relations h τ i , X [ r ] i = − r λ i X [ r − , X ∈ a . (3.3) Generators E (0) i +1 i [0] . We will rely on some simple properties of column-determinants de-scribed in [3, Lemmas 4.1 & 4.2]. They allow us to write the element E (0) i +1 i [0] cdet E as thedifference of two column-determinants (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E i E i +1 . . . E n . . . . . . . . . . . . . . . . . . e E i +1 1 . . . e E i +1 i e E i +1 i +1 . . . e E i +1 n E i +1 1 . . . E i +1 i E i +1 i +1 . . . E i +1 n . . . . . . . . . . . . . . . . . . E n . . . E n i E n i +1 . . . E n n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E i b E i . . . E n . . . . . . . . . . . . . . . . . . E i . . . E i i b E i i . . . E i n E i +1 1 . . . E i +1 i b E i +1 i . . . E i +1 n . . . . . . . . . . . . . . . . . . E n . . . E n i b E n i . . . E n n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , obtained from cdet E by replacing row i and column i + 1 , respectively, as indicated. Here weset e E i +1 j = E i +1 j for j i and e E i +1 j = δ i +1 j τ i +1 + E ( λ j − λ i ) i +1 j [ − u λ j − λ i + · · · + E ( λ j − i +1 j [ − u λ j − for j > i + 1 , while b E k i = E k i for k > i + 1 and b E k i = δ k i τ i +1 + E ( λ i +1 − λ k ) k i [ − u λ i +1 − λ k + · · · + E ( λ i − k i [ − u λ i − for k i .Start with the first determinant and observe that it stays unchanged if we subtract row i + 1 from row i . The first i entries in the new row i will be equal to zero and it will have the form [0 , . . . , , ˇ E i +1 i +1 , . . . , ˇ E i +1 n ] ˇ E i +1 j = E ( λ j − λ i +1 ) i +1 j [ − u λ j − λ i +1 + · · · + E ( λ j − λ i − i +1 j [ − u λ j − λ i − for j = i + 1 , . . . , n . Write the first determinant as a polynomial in x and consider the coefficientof x n − k . This coefficient is a polynomial in u , and if n − k i − , then its degree does notexceed ( λ n − k +1 − 1) + · · · + ( λ j − λ i − 1) + · · · + ( λ n − . (3.4)This is clear since E il ( u ) is a polynomial in u of degree λ l − , while ˇ E i +1 j is a polynomial ofdegree λ j − λ i − . However, the sum (3.4) is less than (3.2) because λ n − k +1 − λ i − < .Similarly, if n − k > i , then the coefficient of x n − k is a polynomial in u whose degree doesnot exceed the sum ( λ n − k +2 − 1) + · · · + ( λ j − λ i − 1) + · · · + ( λ n − which is less than (3.2) because λ i > . Therefore, the contribution to the polynomial ψ k ( u ) in(3.1) arising from the first determinant does not violate the required degree condition.To reach the same conclusion for the second determinant, use its simultaneous expansionalong columns i and i + 1 . The expansion will involve × column-minors of the form (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ai b E ai E bi b E bi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.5)for a < b . First consider the case with a = i and suppose that λ i = λ i +1 . Then the minor (3.5)equals (cid:16) τ i + E ii ( u ) (cid:17) E bi ( u ) − E bi ( u ) (cid:16) τ i +1 + E ii ( u ) (cid:17) = E bi ( u )( τ i − τ i +1 ) + h τ i + E ii ( u ) , E bi ( u ) i . We have E bi ( u )( τ i − τ i +1 ) = E bi ( u )( λ i − λ i +1 ) T = 0 . Note that since [ E ( r ) ii , E ( s ) bi ] = 0 forthe values with the condition r + s > λ i , the degree of the polynomial h τ i + E ii ( u ) , E bi ( u ) i in u does not exceed λ i − . Furthermore, by (2.1) and (3.3) the coefficient of u λ i − equals λ i E ( λ i − bi [ − − λ i E ( λ i − bi [ − 2] = 0 so that the degree of the polynomial does not exceed λ i − .Now consider the contribution of the × minor under consideration to the coefficient of x n − k in the second determinant. If n − k i − then the contribution is a polynomial in u whosedegree does not exceed ( λ n − k +1 − 1) + · · · + ( λ i − − 1) + ( λ i − 2) + ( λ i +2 − 1) + · · · + ( λ n − which is less than the sum in (3.2) because λ n − k +1 − < λ i +1 . If n − k > i then the contributionto the coefficient of x n − k is a polynomial in u whose degree does not exceed ( λ i − 2) + ( λ n − k +3 − 1) + · · · + ( λ n − which is again less than (3.2). 10ontinuing with the case a = i in (3.5), suppose now that λ i < λ i +1 . Exactly as abovewe find that the minor equals E bi ( u )( τ i − τ i +1 ) plus a polynomial in u of degree λ i +1 − .This implies that the resulting contribution to the coefficient of x n − k is a polynomial in u whosedegree is less than (3.2).The minor (3.5) is equal to zero for i + 1 a < b , while the remaining values of a and b are considered in a way quite similar to the above arguments. This allows us to conclude that thecoefficients φ ( r ) k ∈ V ( a ) satisfying (2.5) are annihilated by all generators of the form E (0) i +1 i [0] . Generators E ( λ i +1 − λ i ) i i +1 [0] . We will argue in the same way as above. It will be convenient tomultiply the expression E ( λ i +1 − λ i ) i i +1 [0] cdet E by u λ i +1 − λ i so that the product will be written as thedifference of two column-determinants (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E i E i +1 . . . E n . . . . . . . . . . . . . . . . . . E i . . . E ii E i i +1 . . . E in e E i . . . e E ii e E i i +1 . . . e E in . . . . . . . . . . . . . . . . . . E n . . . E n i E n i +1 . . . E n n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . b E i +1 E i +1 . . . E n . . . . . . . . . . . . . . . . . . E i . . . b E i i +1 E i i +1 . . . E i n E i +1 1 . . . b E i +1 i +1 E i +1 i +1 . . . E i +1 n . . . . . . . . . . . . . . . . . . E n . . . b E n i +1 E n i +1 . . . E n n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , obtained from cdet E by replacing row i + 1 and column i , respectively, as indicated, where nowwe use the notation e E ij = E ij for j > i + 1 and e E ij = δ ij τ i + E ( λ i +1 − λ i ) ij [ − u λ i +1 − λ i + · · · + E ( λ j − ij [ − u λ j − for j i , while b E k i +1 = E k i +1 for k i and b E k i +1 = δ k i +1 τ i + E ( λ i +1 − λ i ) k i +1 [ − u λ i +1 − λ i + · · · + E ( λ i +1 − k i +1 [ − u λ i +1 − for k > i + 1 . The product u λ i +1 − λ i E ( λ i +1 − λ i ) i i +1 [0] cdet E is a polynomial in x and we need toverify that for k = 1 , . . . , n its coefficient of x n − k is a polynomial in u of degree less than thesum of λ i +1 − λ i and the expression (3.2).The first determinant remains unchanged if we subtract row i from row i + 1 . The last n − i entries in the new row i + 1 will be equal to zero and it will have the form [ ˇ E i , . . . , ˇ E ii , , . . . , with ˇ E ij = E (0) ij [ − 1] + · · · + E ( λ i +1 − λ i − ij [ − u λ i +1 − λ i − for j = 1 , . . . , i . Therefore, by looking at the coefficient of x n − k in the resulting polynomial,we find in the same way as in the case of generators E (0) i +1 i [0] , that the maximum power of u occurring in this coefficient does not exceed λ i +1 − λ i − plus the expression in (3.2). Thismeans the contribution arising from the first determinant satisfies the required degree condition.Now turn to the second determinant and use its simultaneous expansion along columns i and i + 1 . This will involve × column-minors of the form (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b E a i +1 E a i +1 b E b i +1 E b i +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.6)11or a < b . Note that the minor is zero for a < b i . Now suppose that a < b = i + 1 and λ a = λ i +1 . Then λ i = λ i +1 and the minor equals E a i +1 ( u ) (cid:16) τ i +1 + E i +1 i +1 ( u ) (cid:17) − (cid:16) τ i + E i +1 i +1 ( u ) (cid:17) E a i +1 ( u )= E a i +1 ( u )( τ i +1 − τ i ) − h τ i + E i +1 i +1 ( u ) , E a i +1 ( u ) i . Since λ i = λ i +1 the term E a i +1 ( u )( τ i +1 − τ i ) is zero. Furthermore, [ E ( r ) a i +1 , E ( s ) i +1 i +1 ] = 0 for r + s > λ i +1 , and so the degree of the polynomial h τ i + E i +1 i +1 ( u ) , E a i +1 ( u ) i in u does notexceed λ i +1 − . The coefficient of u λ i +1 − is zero by (2.1) and (3.3), hence the degree of thepolynomial does not exceed λ i +1 − .The same calculation as for the minors (3.5) shows that the contribution of the × minor(3.6) under consideration to the second determinant is a polynomial in x such that the coefficientof x n − k is a polynomial in u whose degree does not exceed the sum of λ i +1 − λ i − and theexpression in (3.2).The case λ a < λ i +1 and the remaining values of a and b in (3.6) are considered in a similarway. Generators E (0) ii [1] . Note the commutation relations h E (0) ii [1] , τ j i = λ j E (0) ii [0] implied by (3.3).Hence, we also have the relations h E (0) ii [1] , E ml i = δ ml (cid:16) λ m E (0) ii [0] + κ im (cid:17) + δ mi E il [0] − δ il E mi [0] , where we set κ im = D E (0) ii , E (0) mm E and use the notation E ij [0] = E (0) ij [0] + · · · + E ( λ j − ij [0] u λ j − if i > j,E ( λ j − λ i ) ij [0] u λ j − λ i + · · · + E ( λ j − ij [0] u λ j − if i < j. (3.7)By [3, Lemma 4.1], we get h E (0) ii [1] , cdet E i = n X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . h E (0) ii [1] , E j i . . . E n . . . . . . . . . . . . . . . E n . . . h E (0) ii [1] , E nj i . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.8)which equals the sum n X j =1 κ ij E b b + n X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . λ j E (0) ii [0] . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.9)12lus the difference of two column-determinants (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E n . . . . . . . . . E i [0] . . . E in [0] . . . . . . . . . E n . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E i [0] . . . E n . . . . . . . . . . . . . . . E n . . . E n i [0] . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.10)where E b b denotes the column-determinant of the matrix obtained from E by deleting row andcolumn j . Now we proceed as in [3] relying on Lemma 4.2 therein to evaluate the action ofthe elements of the form E (0) ii [0] , E ij [0] and E mi [0] . For the generator E (0) ii [0] occurring as the ( j, j ) -entry with j = i we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . E (0) ii [0] . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = n X m = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . E im . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − δ j j , is zero for λ i − λ s > λ j . Otherwise, if λ i − λ s < λ j , then the commutator is apolynomial in u whose component of maximal degree equals ( λ i − λ j − λ s ) E ( λ j − s j [ − u λ j − .13imilarly, the maximal degree component of the commutator h E ij [0] , E s i i with s > i > j equals − λ j E ( λ j − s j [ − u λ j − .By looking at the powers of x we find that the contributions of these components to eachpolynomial ψ k ( u ) in (3.1) are polynomials in u whose degrees are less than the expression in(3.2), unless λ j = λ i . Moreover, the same property is shared by the components with powers of u less than λ j − . This allows us to replace all commutators h E ij [0] , E s i i with s = j occurringin the column-determinant, with the polynomial − min( λ j , λ s ) E s j .The same argument implies that we may replace the remaining commutators with i > j bythe rule h E ij [0] , E j i i λ j ( E ii − τ i ) − λ j ( E j j − τ j ) . Bringing the calculations together, we can write the first column-determinant in (3.10) as thesum of two expressions n − X j =1 , j = i ( − i + j min( λ i , λ j ) n X m = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . E im − δ im τ i . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.13)and − i − X j =1 ( − i + j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . λ E j . . . E n . . . . . . . . . . . . . . .. . . . . . λ j ( E j j − τ j ) . . . . . .. . . . . . . . . . . . . . . E n . . . λ j E nj . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.14)Here row i and column j are considered to be deleted in the column-determinants so that therows and columns are labelled, respectively, by the symbols , . . . , b ı, . . . , n and , . . . , b , . . . , n .The only nonzero entry E im − δ im τ i in column m in the column-determinants in (3.13) occurs inrow j , while the displayed middle column in (3.14) occupies column i .Note that permutation of rows in column-determinants results in a changed sign as for usualcommutative determinants. Hence, by moving row j in the expression (3.13) to the position ofthe i -th row we can present this expression in the form − n − X j =1 , j = i min( λ i , λ j ) n X m = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . E im − δ im τ i . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.15)where the entry E im − δ im τ i now occupies the ( i, m ) -position, while the rows and columns ofthe column-determinant are labelled by the symbols , . . . , b , . . . , n .14n the expression (3.14) write each column-determinant as the difference (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . λ E j . . . E n . . . . . . . . . . . . . . .. . . . . . λ j E j j . . . . . .. . . . . . . . . . . . . . . E n . . . λ j E nj . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . λ j τ j . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.16)and move row j in the second determinant to the position of the i -th row so that the correspondingpart of (3.14) will be written as the sum − i − X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . λ j τ j . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where the entry λ j τ j now occupies the ( i, i ) -position.The next step is to show that taking signs into account, we can move column i in the firstcolumn-determinant in (3.16) to the left by permuting it with columns i − , i − , . . . , j + 1 consecutively without changing its contribution to the coefficients of the powers of x . Arguingby induction, suppose that column i was swapped with a few columns on its left. As an inductionstep, we need to show that for any l ∈ { j + 1 , . . . , i − } the sum of two column determinants ofthe form (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E l λ E j . . . E n . . . . . . . . . . . . . . . . . .. . . . . . E j l λ j E j j . . . . . .. . . . . . . . . . . . . . . . . . E n . . . E nl λ j E nj . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . λ E j E l . . . E n . . . . . . . . . . . . . . . . . .. . . . . . λ j E j j E j l . . . . . .. . . . . . . . . . . . . . . . . . E n . . . λ j E nj E nl . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.17)does not contribute to any polynomial ψ k ( u ) in (3.1) beyond powers of u less than (3.2). Usingtheir expansions along the two displayed adjacent columns we note that the sum of any × minors (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E al min( λ a , λ j ) E aj E bl min( λ b , λ j ) E bj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) min( λ a , λ j ) E aj E al min( λ b , λ j ) E bj E bl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.18)with a < b is a polynomial in u whose degree does not exceed λ l − . Therefore, the contributionof the sum (3.17) to ψ k ( u ) is a polynomial in u whose degree is less than the sum in (3.2), exceptfor the case where λ i = 1 . However, in this case we must have λ = · · · = λ i so that thedifference (3.18) is easily checked to be zero (which agrees with the Manin matrix property of153, Lemma 4.3]). Thus, expression (3.14) can be replaced with i − X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . λ E j . . . E n . . . . . . . . . . . . . . .. . . . . . λ j E j j . . . . . .. . . . . . . . . . . . . . . E n . . . λ j E nj . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − i − X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . λ j τ j . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.19)where the middle column in the first sum occupies the position of column j , with the row andcolumn i considered to be deleted, while the middle column occurring in the second sum occupiesthe position of column i with the row and column j considered to be deleted.Observe that the column expansion along column i and the use of relations of the form (3.12)lead to a similar evaluation of the second determinant in (3.10). As in the above arguments wewill only keep the terms which can contribute to the coefficients of the powers x n − k beyondpowers of u less than (3.2). By an appropriate adjustment of the labels of rows and columnsof emerging column-determinants, as compared with the evaluation of the first determinant in(3.10), we can write the second determinant (taking into account the minus sign in front) as thesum of two expressions n X j = i +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . λ E j . . . E n . . . . . . . . . . . . . . .. . . . . . λ i − E i − j . . . . . .. . . . . . λ i E i +1 j . . . . . .. . . . . . . . . . . . . . .. . . . . . λ i ( E j j − τ j ) . . . . . .. . . . . . . . . . . . . . . E n . . . λ i E nj . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.20)and − n X j = i +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . λ E i . . . E n . . . . . . . . . . . . . . .. . . . . . λ i E ii . . . . . .. . . . . . . . . . . . . . . E n . . . λ i E ni . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + n X j = i +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . λ i τ i . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.21)where the rows and columns of the column-determinants in (3.20) and in the second sum in(3.21) are labelled by , . . . , b ı, . . . , n with λ i τ i occurring as the ( j, j ) -entry, while the rows andcolumns in the first sum in (3.21) are labelled by , . . . , b , . . . , n .Bringing together the evaluations of the column-determinants in (3.9) and (3.10) by takinginto account (3.11), (3.15), (3.19), (3.20) and (3.21), we can conclude that, modulo terms not16ontributing to the coefficients of the powers x n − k beyond powers of u less than (3.2), the element E (0) ii [1]cdet E equals the sum of the following expressions: i − X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . ( λ − λ j ) E j . . .. . . . . . . . .. . . ( λ j − − λ j ) E j − j . . .. . . . . .. . . . . . . . .. . . . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + n X j = i +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . ( λ − λ i ) E j . . .. . . . . . . . .. . . ( λ i − − λ i ) E j − j . . .. . . . . .. . . . . . . . .. . . . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.22)where the rows and columns of the column-determinants are labelled by , . . . , b ı, . . . , n and weonly displayed columns labelled by j ; plus n X j = i +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . ( λ i − λ ) E i . . .. . . . . . . . .. . . ( λ i − λ i − ) E i − i . . .. . . . . .. . . . . . . . .. . . . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + n − X j = i +1 ( λ j − λ i ) n X m = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . . . .. . . . . . . . .. . . E im . . .. . . . . . . . .. . . . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.23)where the rows and columns of the column-determinants are labelled by , . . . , b , . . . , n and wedisplayed columns i and m , respectively; together with i − X j =1 λ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . . . .. . . . . . . . .. . . τ i − τ j . . .. . . . . . . . .. . . . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + n X j = i +1 λ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . . . .. . . . . . . . .. . . τ i − τ j . . .. . . . . . . . .. . . . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.24)where the rows and columns of the column-determinants are labelled by , . . . , b , . . . , n in thefirst sum and by , . . . , b ı, . . . , n in the second, and τ i − τ j occurs as the ( i, i ) and ( j, j ) -entry,respectively. It is understood that the remaining non-displayed entries of all column-determinantscoincide with the respective entries of the matrix E .The final part of the arguments is to verify that the contribution of each of the above column-determinants to the coefficient of x n − k in the polynomial E (0) ii [1]cdet E is a polynomial in u whose degree is less than (3.2).Fix a value of j ∈ { , . . . , i − } and consider the corresponding column-determinant in thefirst sum in (3.22). Let the index s < j be such that λ s < λ s +1 = · · · = λ j . We may assume that17 > . Then the determinant takes the form (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . ( λ − λ j ) E j . . .. . . . . . . . .. . . ( λ s − λ j ) E s j . . .. . . . . .. . . . . . . . .. . . . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Consider the coefficient of x n − k . If n − k s − , then its degree as a polynomial in u does notexceed ( λ n − k +1 − 1) + · · · + \ ( λ i − 1) + · · · + ( λ n − . However, this sum is less than (3.2) because λ n − k +1 < λ j λ i . Furthermore, if n − k > s thenthe degree of the polynomial does not exceed ( λ a − 1) + ( λ n − k +2 − 1) + · · · + \ ( λ i − 1) + · · · + ( λ n − for some a ∈ { , . . . , s } because the expansion of the column-determinant as the alternatingsum of the products of its entries has the property that any nonzero product must contain a non-diagonal entry in column a for some a ∈ { , . . . , s } . Again, this degree is less than the sum in(3.2) so that no undesirable contribution to the coefficients of the powers of x comes from thefirst sum in (3.22).The same conclusion for the second sum in (3.22) and the first sum in (3.23) is reached byquite a similar argument.Now fix summation indices m > j > i in the second sum in (3.23) and consider the corre-sponding column-determinant. Similar to the previous case, the resulting coefficient of x n − k is apolynomial in u , and if n − k i − , then its degree does not exceed ( λ n − k +1 − 1) + · · · + \ ( λ j − 1) + · · · + ( λ n − which is less or equal to the expression in (3.2). The equality occurs only for λ n − k +1 = λ j , butthis forces the coefficient λ j − λ i in (3.23) be equal to zero. If n − k > i then the degree of thepolynomial does not exceed ( λ i − 1) + ( λ n − k +2 − 1) + · · · + \ ( λ j − 1) + · · · + ( λ n − because any nonzero product of the entries in the expansion of the column-determinant mustcontain a non-diagonal entry in column i . However, this sum does not exceed the expression in(3.2) with the equality occurring only in the case λ j = λ i . Therefore, no undesirable contributionto the coefficients of the powers of x comes from the second sum in (3.23).Finally, for the column-determinants in (3.24) note that τ i − τ j = ( λ i − λ j ) T so that con-tributions to the coefficients of the powers of x in the resulting polynomial can only occur fromthe action of the derivation T on the polynomials E kl ( u ) . This means that for the evaluation of18he column-determinants we may assume that both the rows and columns labelled by i and j aredeleted. In the same way as above, this implies that the coefficient of x n − k is a polynomial in u whose degree does not exceed the expression in (3.2) with the equality occurring only in the case λ j = λ i = 1 . However, in this case we have τ i − τ j = 0 . Generators E (0) ii [ s ] for s > . Note that due to the fact that the element E (0)11 + · · · + E (0) nn belongs to the kernel of the bilinear form h , i , this step is not necessary for the Takiff case,where λ = · · · = λ n . Indeed, all elements E (0)11 [ s ] + · · · + E (0) nn [ s ] are central in the Lie algebra b a , whereas in a we have E (0) ii − E (0) i +1 i +1 = [ E (0) i i +1 , E (0) i +1 i ] . Therefore, the required vanishingproperties of cdet E for the generators E (0) ii [ s ] for s > follow from the previously consideredcases.Returning to the general values of the λ i , note the relations s E ( p ) ii [ s + 1] = h ∆ , E ( p ) ii [ s ] i (3.25)implied by (2.6). Therefore the required properties E (0) ii [ s ] φ ( r ) k = 0 for s > follow fromLemma 2.3 by a straightforward induction on s and k , where the cases s = 1 and k = 1 are takenas the induction bases. Proof of Lemma . The commutator of two derivations ∆ and T of V ( a ) ∼ = U (cid:16) t − a [ t − ] (cid:17) isanother derivation [∆ , T ] = 2 d, where the action of d is determined by the relations h d, X [ r ] i = r X [ r ] , X ∈ a . We also have the commutation relation [ d, T ] = − T .Applying the operator ∆ to the column-determinant cdet E we get the expansion ∆ cdet E = n X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . −E j [0] . . . E n . . . . . . . . . . . . . . .. . . . . . λ j d − E j j [0] . . . . . .. . . . . . . . . . . . . . . E n . . . −E nj [0] . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.26)where we use notation (3.7). The next step is to evaluate these column-determinants in thevacuum module in the same way as for the determinants (3.10) above, with the use of the relations [ d, E ij ] = −E ◦ ij with E ◦ ij = E ij − δ ij x . The contribution arising from the action of d equals n − X j =1 λ j n X m = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . −E ◦ m . . . E n . . . . . . . . . . . . . . . E n . . . −E ◦ nm . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where rows and columns of the column-determinants are labelled by the numbers , . . . , b , . . . , n ,and we displayed column m in the middle. 19o evaluate the column-determinants in (3.26) containing the terms E ij [0] , apply formulas(3.12). It is clear from these evaluations of the expression in (3.26) that the coefficient of x n − k is a polynomial in u of degree not exceeding the number given in (3.2). This implies relations(2.7) thus proving the first part of the lemma. Furthermore, in proving its the second part, weonly need to keep the leading component in u of degree equal to (3.2). Apply the above formulasobtained for the evaluation of determinants (3.10), including suitable permutations of rows andcolumns of column-determinants, which are valid modulo lower degree terms in u . As a result,we can represent the contribution arising from the terms E ij [0] as the sum of the expressions n − X j =1 n X m = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . λ E ◦ m . . . E n . . . . . . . . . . . . . . .. . . . . . λ j − E ◦ j − m . . . . . .. . . . . . λ j E ◦ j +1 m . . . . . .. . . . . . . . . . . . . . . E n . . . λ j E ◦ nm . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.27)where the rows and columns of the column-determinants are labelled by , . . . , b , . . . , n ; plus thesum n − X j =1 n X i = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . λ E ◦ j . . . E n . . . . . . . . . . . . . . .. . . . . . λ j − E ◦ j − j . . . . . .. . . . . . λ j E ◦ j j . . . . . .. . . . . . . . . . . . . . . E n . . . λ j E ◦ nj . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.28)where the rows and columns of the column-determinants are labelled by , . . . , b ı, . . . , n ; togetherwith n − X j =1 n X i = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . λ j ( τ j − τ i ) . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.29)where the rows and columns of the column-determinants are labelled by , . . . , b , . . . , n and λ j ( τ j − τ i ) occurs as the ( i, i ) -entry.As with the determinants of the form (3.22) and (3.24) considered above, we can see thatno contribution to the leading component in u of degree equal to (3.2) comes from the expres-sion (3.29), while modulo lower degree terms in u , both expressions (3.27) and (3.28) can besimplified by replacing all coefficients λ , . . . , λ j − with λ j .Thus, summarizing the arguments, we may conclude that for r = λ n − k +2 + · · · + λ n − k + 1 the image ∆( φ ( r ) k ) coincides with the coefficient of u r of the polynomial in u which occurs as the20oefficient of x n − k in the expression − n − X j =1 λ j n X m = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E ◦ m . . . E n . . . . . . . . . . . . . . . E n . . . E ◦ nm . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − n − X j =1 λ j n X i = j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E ◦ j . . . E n . . . . . . . . . . . . . . . E n . . . E ◦ nj . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where the rows and columns of the column-determinants are labelled by , . . . , b , . . . , n in thefirst sum and by , . . . , b ı, . . . , n in the second.The coefficient of x n − k in the above expression is a linear combination of the principal minors E a ,...,a k − of the matrix E corresponding to the rows and columns a < · · · < a k − . In particular,observe that the linear combination contains the principal minor E n − k +2 ,...,n with the coefficient − ( k − λ + · · · + λ n − k +1 ) . (3.30)To apply the formula for φ ( r ) k − provided by (2.4) and complete the proof of the lemma we needto verify that all other principal minors containing u r occur in the linear combination with thesame coefficient.Define the parameters s n − k + 2 and n − k + 2 p n by the conditions λ · · · λ s − < λ s = · · · = λ n − k +2 = · · · = λ p < λ p +1 · · · λ n . The principal minors containing u r must have the form E a ,...,a q ,p +1 ,...,n for some s a < · · · < a q p, q = p − n + k − . The coefficient of such a minor in the linear combination equals − q +1 X i =1 ( k − i )( λ a i − +1 + · · · + λ a i − ) − q X i =1 ( p − q − a i + i ) λ a i , where we set a = 0 and a q +1 = p + 1 . This coefficient coincides with (3.30) thus completingthe proof of the lemma.To establish the required vanishing properties of the elements φ ( r ) k with respect to the remain-ing family of generators E ( p ) ii [ s ] with s > , observe that in view of Lemma 2.3 and relations(3.25), it is sufficient to consider the values s = 0 and s = 1 . Generators E ( p ) ii [0] with p = 0 , , . . . , λ i − . For a fixed value or p we have the commutationrelations u p h E ( p ) ii [0] , E ml i = δ mi ( E il − e E il ) − δ il ( E mi − b E mi ) , where we set e E ii = b E ii = 0 , e E il = E ( λ l − λ i ) il [ − u λ l − λ i + · · · + E ( λ l − λ i + p − il [ − u λ l − λ i + p − for i < l,E (0) il [ − 1] + · · · + E ( p − il [ − u p − for i > l, p < λ l ,E (0) il [ − 1] + · · · + E ( λ l − il [ − u λ l − for i > l, p > λ l b E mi = E ( λ i − λ m ) mi [ − u λ i − λ m + · · · + E ( λ i − λ m + p − mi [ − u λ i − λ m + p − for m < i, p < λ m ,E ( λ i − λ m ) mi [ − u λ i − λ m + · · · + E ( λ i − mi [ − u λ i − for m < i, p > λ m ,E (0) mi [ − 1] + · · · + E ( p − mi [ − u p − for m > i. Hence the element E ( p ) ii [0] cdet E equals the difference of two column-determinants (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E n . . . . . . . . . E i − e E i . . . E in − e E in . . . . . . . . . E n . . . E n n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E i − b E i . . . E n . . . . . . . . . . . . . . . E n . . . E n i − b E n i . . . E n n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which simplifies to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . b E i . . . E n . . . . . . . . . . . . . . . E n . . . b E n i . . . E n n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E n . . . . . . . . . e E i . . . e E in . . . . . . . . . E n . . . E n n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.31)It remains to verify that each of these column-determinants is a polynomial in x with the propertythat the coefficient of x n − k is a polynomial in u whose degree is less that p plus the expressionin (3.2).Begin with the first determinant in (3.31) and suppose that n − k i − . Consider theproducts of the entries of the matrix which occur in the expansion of the column-determinant.If such a product contains an entry of the form b E m i with m n − k , then the degree of thecorresponding polynomial in u does not exceed ( λ m − 1) + ( λ n − k +2 − 1) + · · · + ( λ i − λ m + p − 1) + · · · + ( λ n − if p < λ m , and does not exceed ( λ m − 1) + ( λ n − k +2 − 1) + · · · + ( λ i − 1) + · · · + ( λ n − if p > λ m . If the product contains an entry of the form b E m i with n − k + 1 m < i , then thedegree of the corresponding polynomial in u does not exceed ( λ n − k +1 − 1) + · · · + ( λ i − λ m + p − 1) + · · · + ( λ n − or ( λ n − k +1 − 1) + · · · + ( λ i − 1) + · · · + ( λ n − , p < λ m or p > λ m . Similarly, if the product contains an entry of the form b E m i with m > i , then the degree of the corresponding polynomial in u does not exceed ( λ n − k +1 − 1) + · · · + ( p − 1) + · · · + ( λ n − . It is clear that in all cases the degree is less than p plus the expression in (3.2).The verification in the case i > n − k is quite similar. The second determinant in (3.31) isconsidered in the same way. Generators E ( p ) ii [1] with p = 1 , . . . , λ i − . We proceed as in the case p = 0 considered aboveand use similar calculations with some simplifications due to the fact that they do not involvevalues of the bilinear form. We have the relations u p h E ( p ) ii [1] , E ml i = δ ml λ m u p E ( p ) ii [0] + δ mi E il [0] − δ il E mi [0] , where we use the notation E ij [0] = E ( λ j − λ i + p ) ij [0] u λ j − λ i + p + · · · + E ( λ j − ij [0] u λ j − for i < j,E ( p ) ij [0] u p + · · · + E ( λ j − ij [0] u λ j − for i > j, where empty sums are understood as being equal to zero. Relations (3.8) hold in the same formwith E (0) ii [1] replaced by u p E ( p ) ii [1] so that the commutator h u p E ( p ) ii [1] , cdet E i equals the sum n X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . . . . E n . . . . . . . . . . . . . . .. . . . . . λ j u p E ( p ) ii [0] . . . . . .. . . . . . . . . . . . . . . E n . . . . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) plus the difference of two column-determinants (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E n . . . . . . . . . E i [0] . . . E in [0] . . . . . . . . . E n . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E . . . E i [0] . . . E n . . . . . . . . . . . . . . . E n . . . E n i [0] . . . E nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . It turns out that in contrast to the case p = 0 , each of these column-determinants is a polynomialin x with the property that the coefficient of x n − k is a polynomial in u whose degree is less that p plus the expression in (3.2). Indeed, this follows easily from the respective counterparts of thedecompositions (3.11) and (3.12), because the resulting column-determinants retain the propertythat the degree of u of any entry in column l does not exceed λ l − .23e have thus proved that the coefficients φ ( r ) k with k = 1 , . . . , n satisfying conditions (2.5)belong to the center z ( b a ) of the vertex algebra V ( a ) .The proof of the second part of the theorem relies on [1, Thm 3.2]. It reduces the task to theverification that the symbols of the elements φ ( r ) k in the symmetric algebra S (cid:16) t − a [ t − ] (cid:17) coin-cide with the respective images of certain algebraically independent generators of the algebra of a -invariants S( a ) a under the embedding X X [ − for X ∈ a . However, the desired propertyholds for the generators of the algebra S( a ) a produced explicitly in [2, Thm 4.1]. The same gen-erators were produced in an earlier work [13] as a conjecture which was later confirmed in [16].They coincide with the symbols of the respective elements Φ ( r ) k ∈ U( a ) given in Corollary 2.7. References [1] T. Arakawa and A. Premet, Quantizing Mishchenko–Fomenko subalgebras for centralizersvia affine W -algebras , Trans. Mosc. Math. Soc. (2017), 217–234.[2] J. Brown and J. Brundan, Elementary invariants for centralizers of nilpotent matrices , J.Aust. Math. Soc. (2009), 1–15.[3] A. V. Chervov and A. I. Molev, On higher order Sugawara operators , Int. Math. Res. Not.(2009), 1612–1635.[4] A. Chervov and D. Talalaev, Quantum spectral curves, quantum integrable systems and thegeometric Langlands correspondence , arXiv:hep-th/0604128 .[5] B. Feigin and E. Frenkel, Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras , Int. J. Mod. Phys. A , Suppl. 1A (1992), 197–215.[6] E. Frenkel, Langlands correspondence for loop groups , Cambridge Studies in AdvancedMathematics, 103. Cambridge University Press, Cambridge, 2007.[7] E. Frenkel and D. Ben-Zvi, Vertex algebras and algebraic curves , Mathematical Surveysand Monographs, vol. 88, Second ed., American Mathematical Society, Providence, RI,2004.[8] V. Kac, Vertex algebras for beginners , University Lecture Series, 10. American Mathemat-ical Society, Providence, RI, 1997.[9] A. S. Mishchenko and A. T. Fomenko, Euler equation on finite-dimensional Lie groups ,Math. USSR-Izv. (1978), 371–389.[10] A. I. Molev, Casimir elements for certain polynomial current Lie algebras , in “Group 21,Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras," Vol.1, (H.-D. Doebner, W. Scherer, P. Nattermann, Eds). World Scientific, Singapore, 1997,172–176. 2411] A. Molev, Sugawara operators for classical Lie algebras . Mathematical Surveys andMonographs 229. AMS, Providence, RI, 2018.[12] A. Molev and O. Yakimova, Quantisation and nilpotent limits of Mishchenko–Fomenkosubalgebras , Repres. Theory (2019), 350–378.[13] D. Panyushev, A. Premet and O. Yakimova, On symmetric invariants of centralisers inreductive Lie algebras , J. Algebra (2007), 343–391.[14] L. G. Rybnikov,