From pencils of Novikov algebras of Stäckel type to soliton hierarchies

Fr om pencils of No vik o v algebras of St ä ckel type to soliton hierar chies Maciej Błaszak † , Krzysztof Marciniak ∗ ‡ , and Błażej M. Szablik o wski † † ISQI, F acult y of Physics and Astronom y , Adam Mic kiewicz Universit y Uniw ersytetu Poznańskiego 2, 61-614 P oznań, P oland blaszakm@amu.edu.pl, bszablik@amu.edu.pl ORCID: 0000-0002-3951-2850 , 0000-0002-4001-6328 ‡ Departmen t of Science and T echnology Link öping Universit y , Campus Norrköping 601 74 Norrk öping, Sweden krzma@itn.liu.se ORCID: 0000-0003-3280-0160 Marc h 27, 2026 Abstract In this article w e construct ev olutionary soliton hierarchies from p encils of Novik ov al- gebras of Stäck el type. W e start by defining a sp ecial class of asso ciativ e Novik o v algebras, whic h we call Novik ov algebras of Stäck el type, as they are asso ciated with classical Stäck el metrics in Viète co ordinates. W e obtain sufficient conditions for p encils of these algebras so that the corresp onding Dubro vin-No vik ov Hamiltonian operators can b e cen trally extended, pro ducing sets of pairwise compatible Poisson op erators. These op erators lead to coupled K orteweg-de V ries (cKdV) and coupled Harry Dym (cHD) hierarchies, as well as to a trian- gular cKdV hierarc hy and a triangular cHD hierarch y . Keyw ords: No vik o v algebras, central extensions, soliton hierarchies, multi-Hamiltonian structures MSC: 37K10, 35Q51, 17D25 1 In tro duction In this article w e construct soliton hierarc hies of evolutionary type from p encils of asso ciativ e No viko v algebras of Stäck el t yp e. Using this approac h, w e reconstruct the coupled Kortew eg- de V ries (cKdV) and coupled Harry Dym (cHD) hierarchies [ 1 – 3 ] as well as the triangular cKdV and triangular cHD hierarc hies. There are v arious wa ys of constructing soliton hierarchies from appropriate algebraic struc- tures. F or example, in [ 11 ] the authors used lo op algebras and r -matrix theory to produce compatible Poisson brack ets leading to cKdV and cHD hierarchies. In [ 7 ] F rob enius algebras w ere applied to multi-component third-order lo cal Poisson structures. In the article [ 14 ], the ∗ Corresp onding author. 1 authors p erformed the construction of (1 + 1) -dimensional integrable bi-Hamiltonian systems asso ciated with Novik o v algebras. The obtained systems were multi-component generalizations of the Camassa-Holm equation [ 9 ] that can b e interpreted as Euler equations on the resp ectiv e cen trally extended Lie algebras. The homogeneous first-order Hamiltonian op erators [ 4 , 12 ], which are a sp ecial case of the Dubro vin-Novik o v op erators of h ydro dynamic t yp e [ 10 ], hav e a v ery natural underlying algebraic structure. The conditions for a homogeneous op erator Π ij = 1 2 ( b ij k + b j i k ) u k d dx + 1 2 b ij k u k x , to b e Hamiltonian are such that the b ij k are the structure constants of a Novik o v algebra [ 4 ]. Moreo ver, these op erators can b e defined through Lie-Poisson structures asso ciated with the so-called translationally inv arian t Lie algebras, which are in one-to-one correspondence with No viko v algebras. F or more information ab out this and directly related topics, see [ 14 ] and the recen t works [ 15 – 17 ]. The asso ciated translationally inv ariant Lie algebra can b e centrally extended. The con- dition for the existence of co cycles (either first-order or third-order Gelfand-F uks co cycles) is equiv alent to the existence of symmetric bilinear forms on the Novik o v algebra satisfying cer- tain compatibility conditions (quasi-F rob enius and F rob enius). Second-order co cycles result in an tisymmetric bilinear forms, which again satisfy certain algebraic relations [ 4 , 14 ]. In this article w e in tro duce the concept of p encils of c ommutative Novikov algebr as of Stäckel typ e in order to construct centrally extended Poisson p encils of Dubrovin-No vik o v type, which lead to soliton hierarc hies of evolutionary type. The article has the following structure. In Section 2 w e review some known facts ab out No viko v algebras and central extensions of related Poisson op erators. In Section 3 we consider particular asso ciativ e Novik o v algebras that w e call of Stäck el type, as their first-order central extensions contain flat Stäck el metrics. In Section 4 w e com bine these single algebras into p encils of algebras of Stäc k el type, whic h yield cen tral extensions of Poisson p encils of Dubro vin-Novik o v t yp e, containing terms of first and third order. In the main theorem of this pap er (Theorem 2 ) w e establish sufficient conditions for the construction of such cen tral extensions. These p encils in turn lead to soliton hierarc hies of evolutionary type. In Section 5 we apply our theory to construct (i) the cKdV hierarch y , (ii) the cHD hierarch y (b oth in the con ven tion used in [ 3 ]), (iii) the triangular cKdV hierarc hy , and (iv) the triangular cHD hierarch y . 2 No vik o v algebras, the asso ciated P oisson op erators and their cen tral extensions Definition 1. A finite-dimensional algebr a A over R is c al le d a Novik o v algebra if it is right- c ommutative: ( a ◦ b ) ◦ c = ( a ◦ c ) ◦ b, (1) and left-symmetric (quasi-asso ciative): ( a ◦ b ) ◦ c − a ◦ ( b ◦ c ) = ( b ◦ a ) ◦ c − b ◦ ( a ◦ c ) . (2) Her e a, b, c ∈ A and ◦ denotes the multiplic ation in A . The quasi-asso ciativit y condition implies that an y non-commutativ e No viko v algebra A is 2 Lie-admissible, that is, the commutator [ a, b ] = a ◦ b − b ◦ a defines the structure of a Lie algebra on the underlying v ector space A . Assume dim A = n and let us c ho ose a basis e 1 , . . . , e n in A . Let us denote the corresp onding structure constan ts of the algebra A by b ij k . Thus 1 ( a ◦ b ) k = b ij k a i b j or e i ◦ e j = b ij k e k , where a, b ∈ A . Then, the corresp onding n × n matrix A with co efficients A ij = b ij s e s (3) is the (m ultiplication) characteristic matrix of the algebra A . Remark 1. If the algebr a A is c ommutative the structur e c onstants of A ar e symmetric, that is b ij k = b j i k , while the c onditions ( 1 ) and ( 2 ) r e duc e to the asso ciativity c ondition ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) . F or an y Novik o v algebra A we can consider the algebra L A of all smo oth A -v alued functions on x ∈ S 1 . This algebra is equipp ed with the Lie brack et [ [ a, b ] ] = a x ◦ b − b x ◦ a, (4) where now a, b ∈ L A so that a and b dep end on x ∈ S 1 . Throughout the article we will use the letters a, b, c, . . . to denote elements of A as well as elements of L A , which will b e clear from the con text. In fact, the brack et ( 4 ) is a Lie brac k et if and only if A is a Novik o v algebra [ 4 ]. Consider no w the following first-order op erator Π ij = 1 2  b ij k + b j i k  q k d dx + 1 2 b ij k q k x , i, j = 1 , . . . , n, (5) where x ∈ S 1 and q = ( q 1 , . . . , q n ) , with q i = q i ( x ) . The op erator ( 5 ) acts on L A . It is Poisson if and only if b ij k are the structure constan ts of a Novi ko v algebra [ 4 , 12 ]. The asso ciated P oisson brack et is of Lie-Poisson type and is defined, for any pair of func- tionals H , F on L ∗ A , b y {H , F } [ q ] := Z S 1 δ H δ q i Π ij δ F δ q j dx ≡ ⟨ q , [ [ δ q H , δ q F ] ] ⟩ , q ∈ L ∗ A , δ q H , δ q F ∈ L A , (6) with δ q H = δ H δ q i e i , δ q F = δ F δ q i e i . The pairing b et ween L ∗ A and L A is giv en by ⟨ q , a ⟩ = Z S 1 ( q , a ) dx, q ∈ L ∗ A , a ∈ L A , where ( · , · ) denotes the dual pairing b etw een A ∗ and A . Let us define a 2 -co cycle on L A as a bilinear form ω : L A × L A → R such that ω is skew- 1 Throughout the pap er we use the Einstein summation conv ention, except in some cases where the summation sym b ol is used explicitly . 3 symmetric: ω ( a, b ) = − ω ( b, a ) (7) and satisfies the cyclic condition ω ([ [ a, b ] ] , c ) + ω ([ [ b, c ] ] , a ) + ω ([ [ c, a ] ] , b ) = 0 . (8) With eac h such 2 -co cycle one can asso ciate the following central extension of the P oisson brac k et ( 6 ): {H , F } ω [ q ] = ⟨ q , [ [ δ q H , δ q F ] ] ⟩ + ω ( δ q H , δ q F ) . There are three t yp es of differential 2 -co cycles on L A [ 4 ]. A symmetric bilinear form Z on A generates a 2 -co cycle of or der 1 on L A giv en by ω ( a, b ) = Z S 1 Z ( a x , b ) dx if and only if the quasi-F rob enius condition Z ( a ◦ b, c ) = Z ( a, c ◦ b ) (9) is satisfied for an y a, b, c ∈ A . Suc h a co cycle yields the following extended Poisson op erator: P ij = Π ij + Z ij d dx , Z ij = Z j i . F urther, an anti-symmetric bilinear form Z on A generates a 2 -co cycle of or der 2 on L A giv en by ω ( a, b ) = Z S 1 Z ( a xx , b ) dx if and only if Z satisfies the quasi-F rob enius condition ( 9 ) and, additionally , the cyclic condition Z ( a ◦ b, c ) + Z ( b ◦ c, a ) + Z ( c ◦ a, b ) = 0 (10) for all a, b, c ∈ A . Notice that in the commutativ e case, for an an ti-symmetric Z , the quasi-F rob enius condition ( 9 ) together with the cyclic condition ( 10 ) reduce to the single condition of the form Z ( a ◦ b, c ) = 0 , (11) where a, b, c ∈ A are arbitrary . This is due to the fact that in this situation ( 10 ) reads 0 = Z ( a ◦ b, c ) − Z ( a ◦ b, c ) − Z ( a ◦ b, c ) = − Z ( a ◦ b, c ) so ( 11 ) follo ws. This co cycle yields the follo wing extended Poisson op erator: P ij = Π ij + Z ij d 2 dx 2 , Z ij = − Z j i . Finally , a symmetric bilinear form Z on A generates a 2 -co cycle of order 3 on L A giv en by ω ( a, b ) = Z S 1 Z ( a xxx , b ) dx 4 if and only if Z satisfies the quasi-F rob enius condition ( 9 ) and, additionally , the condition Z ( a, b ◦ c ) = Z ( a, c ◦ b ) . (12) This co cycle yields the follo wing extended Poisson op erator: P ij = Π ij + Z ij d 3 dx 3 , Z ij = Z j i . Note that in the comm utative case the condition ( 12 ) is alwa ys satisfied. Therefore, in this case the conditions for co cycles of order 1 and order 3 coincide and are given b y the same quasi-F rob enius condition ( 9 ), whic h can b e written as the (standard) F rob enius condition Z ( a ◦ b, c ) = Z ( a, b ◦ c ) , a, b, c ∈ A , (13) or, equiv alently , as Z ( e i ◦ e j , e k ) = Z ( e i , e j ◦ e k ) , i, j, k = 1 , . . . , n. (14) Let Z ij := Z ( e i , e j ) . Then the F rob enius condition ( 13 ) reduces in co ordinates to the following homogeneous system of linear equations for the symmetric form Z ij = Z j i : b ij s Z sk − Z is b j k s = 0 , i, j, k = 1 , . . . , n. (15) Moreo ver, since the conditions ( 7 ) and ( 8 ) are linear in ω , an arbitrary linear combination of the ab o ve co cycles leads to a corresp onding centrally extended Poisson op erator P ij as w ell. 3 No vik o v algebras of Stäc k el t yp e Consider a family of n -dimensional algebras A m = ( R m , ◦ m ) , defined for each m ∈ { 0 , . . . , n } , with the m ultiplication e i ◦ m e j =          e i + j + m − n − 1 , for i, j ∈ { 1 , . . . , n − m } ≡ I m 1 , − e i + j + m − n − 1 , for i, j ∈ { n − m + 1 , . . . , n } ≡ I m 2 , 0 , otherwise . (16) Th us the structure constants of A m are giv en by e i ◦ m e j = ( b m ) ij s e s , ( b m ) ij s =          δ i + j + m − n − 1 s , i, j ∈ I m 1 , − δ i + j + m − n − 1 s , i, j ∈ I m 2 , 0 , otherwise . (17) As ( b m ) ij s = ( b m ) j i s in ( 17 ), it follows that every A m is comm utative. In fact, each A m is a No viko v algebra, since the following assertion holds (cf. Remark 1 ). Lemma 1. A l l A m ar e asso ciative. This lemma is a straightforw ard consequence of Remark 2 b elow. The algebra A m will henceforth b e called the m -th Novik o v algebra of Stäc kel type, due to considerations b elow. Moreo ver, A n is the only one among the A m that has a unit y element, namely − e 1 . 5 Example 1. F or n = 4 , the multiplic ation matric es A m define d in ( 3 ) by ( A m ) ij = ( b m ) ij s e s , with the structur e c onstants ( 17 ) , ar e A 0 =       0 0 0 0 0 0 0 e 1 0 0 e 1 e 2 0 e 1 e 2 e 3       , A 1 =       0 0 0 0 0 0 e 1 0 0 e 1 e 2 0 0 0 0 − e 4       , A 2 =       0 0 0 0 0 e 1 0 0 0 0 − e 3 − e 4 0 0 − e 4 0       , A 3 =       0 0 0 0 0 − e 2 − e 3 − e 4 0 − e 3 − e 4 0 0 − e 4 0 0       , A 4 =       − e 1 − e 2 − e 3 − e 4 − e 2 − e 3 − e 4 0 − e 3 − e 4 0 0 − e 4 0 0 0       . Remark 2. Note that the algebr a A n c an b e r epr esente d as n -dimensional algebr a of trunc ate d p olynomials I n ∼ = R [ x ] / ⟨ x n ⟩ , e i ◦ e j = − e i + j − 1 , (18) wher e e i = − x i − 1 , for i = 1 , . . . , n , ar e b asis ve ctors. Similarly, the algebr a A 0 c an b e r epr esente d as the sub algebr a of trunc ate d p olynomials without fr e e (c onstant) terms ˜ I n ∼ = x R [ x ] / ⟨ x n +1 ⟩ , e i ◦ e j = e i + j − n − 1 , wher e e i = x n +1 − i , for i = 1 , . . . , n , ar e b asis ve ctors. The ab ove algebr as ar e obviously c om- mutative and asso ciative. Note that ˜ I 1 is a trivial algebr a. Then, al l the n -dimensional algebr as fr om the family A m have the fol lowing structur e: A 0 ≡ ˜ I n , A m ∼ = ˜ I n − m ⊕ I m for m ∈ { 1 , . . . , n − 1 } , A n ≡ I n . The minus sign in the definitions ( 16 ) and ( 18 ) is not mer ely a matter of c onvention, but it is r e quir e d by c omp atibility c onditions, se e Se ction 4 . In the case of our Novik o v algebras of Stäc kel type it is p ossible to find a general solution Z of the F rob enius condition ( 15 ). Theorem 1. The gener al n -p ar ameter solution of the F r ob enius c ondition ( 15 ) for the symmetric biline ar form Z m on the m -th algebr a A m , define d by ( 16 ) , is given by ( Z m ) ij =          φ i + j + m − n − 1 m , i, j ∈ I m 1 , − φ i + j + m − n − 1 m , i, j ∈ I m 2 , 0 , otherwise , (19) wher e ( Z m ) ij = Z m ( e i , e j ) and φ s m ar e arbitr ary r e al c onstants. Here and in what follows w e use the notation φ i m = 0 for i < 0 and for i > n . The pro of is giv en in the App endix. Thus, each form Z m ≡ Z m ( φ ) dep ends on n parameters φ 0 m , . . . , φ n − m − 1 m , 6 φ n − m +1 m , . . . , φ n m and is explicitly giv en by Z m ( φ ) =             φ 0 m . . . . . . φ 0 m · · · φ n − m − 1 m 0 ( n − m ) × m 0 m × ( n − m ) − φ n − m +1 m · · · − φ n m . . . . . . − φ n m             , m = 0 , . . . , n. (20) Lemma 2. No A m has a 2 -c o cycle of or der 2 . The pro of of this lemma is in the App endix. Consequen tly , the P oisson op erator corresp onding to each algebra A m Π ij m = ( b m ) ij k q k d dx + 1 2 ( b m ) ij k q k x (21) (cf. ( 5 )) can b e cen trally extended to the 2 n -parameter Poisson op erator P ij m = Π ij m + ( Z m ) ij ( φ ) d dx + ( Z m ) ij ( ψ ) d 3 dx 3 , whic h in the matrix form can b e presented as P m = G m ( q , φ ) d dx + 1 2 [ G m ( q , φ )] x + Z m ( ψ ) d 3 dx 3 , (22) where Z m ( ψ ) is defined by ( 20 ) but with a new set of n parameters ψ i m , while G ij m ( q , φ ) := ( b m ) ij k q k + ( Z m ) ij ( φ ) , so that G m ( q , φ ) =               φ 0 m . . . q 1 + φ 1 m . . . . . . . . . φ 0 m q 1 + φ 1 m · · · q n − m − 1 + φ n − m − 1 m 0 ( n − m ) × m 0 m × ( n − m ) − q n − m +1 − φ n − m +1 m · · · − q n − φ n m . . . . . . − q n − φ n m               . F rom no w on, G m ≡ G m ( q , φ ) will b e considered as a flat con trav arian t metric on a pseudo- Euclidean space with co ordinates ( q 1 , . . . , q n ) . Note that ( 22 ) is the most general differential central extension of the Poisson brack et ( 21 ). Also, for a fixed m , the shift q i + φ i m 7→ q i , i = 1 , . . . , n, (23) 7 transforms G m to the form G m ( q , φ ) =               φ m . . . q 1 . . . . . . . . . φ m q 1 · · · q n − m − 1 0 ( n − m ) × m 0 m × ( n − m ) − q n − m +1 · · · − q n . . . . . . − q n               . Here we denote the only remaining parameter φ 0 m b y φ m . This means that the first-order cen tral extension of the P oisson op erator ( 21 ) is 1 -parameter for m = 0 , . . . , n − 1 and trivial for m = n . Example 2. F or n = 4 the matric es G m in ( 3 ) take the form G 0 =       0 0 0 φ 0 0 0 φ 0 q 1 0 φ 0 q 1 q 2 φ 0 q 1 q 2 q 3       , G 1 =       0 0 φ 1 0 0 φ 1 q 1 0 φ 1 q 1 q 2 0 0 0 0 − q 4       , G 2 =       0 φ 2 0 0 φ 2 q 1 0 0 0 0 − q 3 − q 4 0 0 − q 4 0       , G 3 =       φ 3 0 0 0 0 − q 2 − q 3 − q 4 0 − q 3 − q 4 0 0 − q 4 0 0       , G 4 =       − q 1 − q 2 − q 3 − q 4 − q 2 − q 3 − q 4 0 − q 3 − q 4 0 0 − q 4 0 0 0       . Eac h of the metrics G m in ( 3 ) attains, after the rescaling q ′ i = φ − 1 m q i , and assuming that φ m  = 0 (this rescaling is not necessary for G n ), the form of a Stäck el metric in Viète coordinates [ 5 ]. F urther transformation from Viète coordinates to separation co ordinates ( λ 1 , . . . , λ n ) tak es the form q ′ i = ( − 1) i s i ( λ 1 , . . . , λ n ) , i = 1 , . . . , n, where s i ( λ 1 , . . . , λ n ) are the elementary symmetric p olynomials. In the separation co ordinates ( λ 1 , . . . , λ n ) the metric G m attains the diagonal form G m = φ − 1 m     λ m 1 ∆ 1 0 . . . 0 λ m n ∆ n     , ∆ i = Y j  = i ( λ i − λ j ) . 4 No vik o v p encils of Stäc k el t yp e Let us no w define the algebra A with the multiplication ◦ giv en b y arbitrary linear com binations of the m ultiplications from the algebras A m : e i ◦ e j := n X m =0 α m e i ◦ m e j , (24) 8 where the parameters α m ∈ R . The asso ciated structure constants are b ij s = n X m =0 α m ( b m ) ij s , e i ◦ e j = b ij s e s . (25) The algebra A is commutativ e and b elow w e show that it is still asso ciative. Hence it is also a No viko v algebra, dep ending no w on n + 1 arbitrary parameters α m , m = 0 , . . . , n . W e will call the algebra A a No viko v p encil of Stäck el t yp e. Lemma 3. The algebr a A is asso ciative for arbitr ary choic e of the p ar ameters α m . Equivalently, the multiplic ations fr om the asso ciative algebr as A m ar e mutual ly c omp atible, i.e. ( a ◦ m b ) ◦ p c + ( a ◦ p b ) ◦ m c = a ◦ m ( b ◦ p c ) + a ◦ p ( b ◦ m c ) , (26) for al l a, b, c ∈ A and m, p = 0 , . . . , n . The pro of is in the App endix. Conclusion 1. The op er ator Π = n X m =0 α m Π m , wher e Π m ar e n + 1 Poisson op er ators of the form ( 21 ) , is Poisson for al l values of α m , so that al l Π m ar e p airwise c omp atible. Due to the comm utativity of A , the F rob enius condition for the Novik o v p encil A has also the form ( 14 ) with the m ultiplication ◦ defined by ( 24 ). The theorem b elow shows that in this situation there exists a particular solution Z of ( 14 ) that has the form of a p encil of all Z m with exactly the same co efficien ts α m as in the No viko v p encil ( 25 ). Theorem 2. The p encil Z = n X m =0 α m Z m , (27) wher e the biline ar forms Z m ar e given by ( 19 ) , satisfies the F r ob enius c ondition ( 13 ) on the algebr a A define d by ( 24 ) , for any choic e of the p ar ameters α m , pr ovide d that φ s m = φ s , s = 0 , . . . , n. (28) The condition ( 28 ) means that all the bilinear forms Z m in ( 27 ) share the same set of parameters. Explicitly , the bilinear forms Z m in the p encil ( 27 ) ha ve the form ( Z m ) ij =          φ i + j + m − n − 1 , i, j ∈ I m 1 , − φ i + j + m − n − 1 , i, j ∈ I m 2 , 0 , otherwise , (29) so that Z dep ends on n + 1 parameters φ 0 , . . . , φ n , and each Z m dep ends on the same parameters except for φ n − m . The pro of is in the App endix. Remark 3. The F r ob enius c ondition ( 14 ) is e quivalent to demanding Z m ( e i ◦ p e j , e k ) + Z p ( e i ◦ m e j , e k ) = Z m ( e i , e j ◦ p e k ) + Z p ( e i , e j ◦ m e k ) , i, j, k = 1 , . . . , n, 9 for al l p airs m, p = 0 , . . . , n . Corollary 1. The op er ator P = n P m =0 α m P m is Poisson, with P m = Z m ( ψ 0 , . . . , ψ n ) d 3 dx 3 + Π m + Z m ( φ 0 , . . . , φ n ) d dx , m = 0 , . . . , n. (30) Her e ψ 0 , . . . , ψ n and φ 0 , . . . , φ n ar e two indep endent sets of p ar ameters use d to c onstruct the first- and thir d-or der c entr al extensions by ( 29 ) , and thus al l P m ar e p airwise c omp atible. Moreo ver, since φ s m do not dep end on m (cf. ( 28 )), there exists a common shift (cf. shifts ( 23 )) q i + φ i → q i , i = 1 , . . . , n, that turns all P m in ( 30 ) in to the following ( n + 1) -parameter form: P m = Z m ( ψ 0 , . . . , ψ n ) d 3 dx 3 + Π m + Z m ( φ ) d dx , m = 0 , . . . , n, (31) where Z m ( φ ) =          φ . . . φ 0 ( n − m ) × m 0 m × ( n − m ) 0 m × m          , m = 0 , . . . , n, (with φ 0 = φ , note that Z n = 0 ) and Z m ( ψ ) =               ψ 0 . . . . . . ψ 0 · · · ψ n − m − 1 0 ( n − m ) × m 0 m × ( n − m ) − ψ n − m +1 · · · − ψ n . . . . . . − ψ n               , m = 0 , . . . , n. Th us, ( n + 1) compatible Poisson op erators of third order ( 31 ) can b e written in a compact form P m =               j 0 . . . . . . j 0 · · · j n − m − 1 0 ( n − m ) × m 0 m × ( n − m ) − j n − m +1 · · · − j n . . . . . . − j n               , m = 0 , . . . , n, where j 0 = φ d dx + ψ 1 d 3 dx 3 , j k = 1 2  q k d dx + d dx q k  + ψ k d 3 dx 3 , k = 1 , . . . , n. 10 Note that the op erator P can b e written in the form (cf. ( 22 )) P = G ( q , φ ) d dx + 1 2 [ G ( q , φ )] x + Z ( ψ ) d 3 dx 3 , where G ij ( q , φ ) := b ij k q k + ( Z ) ij ( φ ) = n X m =0 α m G ij m ( q , φ ) is the most general flat Stäc kel metric [ 5 , 8 ]. 5 Multi-Hamiltonian hierarc hies of ev olutionary t yp e The set of compatible Hamiltonian op erators P m in ( 4 ) leads to v arious m ulti-Hamiltonian hierarc hies. 5.1 Coupled Harry Dym hierarch y First, w e show that the set ( 4 ) contains kno wn p ositive and negative coupled Harry Dym (cHD) hierarc hies [ 2 , 3 , 6 ]. In order to fit the notation to that known from the literature, let n = N , P m = B N − m , q i = u i , ψ 1 = ψ , ψ i = 1 4 ε i , i = 1 , . . . , N − 1 and ψ N = 0 . Th us B m =              J 0 . . . . . . J 0 · · · J m − 1 0 m × ( N − m ) 0 ( N − m ) × m − J m +1 · · · − J N . . . . . . − J N              , m = 0 , . . . , N , (32) where J 0 = φ∂ + ψ ∂ 3 , J k = 1 2 ( u k ∂ + ∂ u k ) + 1 4 ε k ∂ 3 , k = 1 , . . . , N − 1 , J N = 1 2 ( u N ∂ + ∂ u N ) = u 1 2 N ∂ u 1 2 N , and ∂ ≡ ∂ ∂ x . First, notice that the Casimir of B 0 is C 0 =  0 , . . . , 0 , au − 1 2 N  T and the Casimir of B N , whic h is x -indep endent, takes the form C N = ( c, 0 , . . . , 0) T , where a and c are arbitrary constants. Besides, the op erators B m satisfy the infinite recursion B k +1 = R B k , k ⩾ 0 , where all B k with k > N are non-lo cal and where the recursion op erator R and its inv erse ha ve the form R =          0 · · · 0 − J 0 J − 1 N 1 . . . 1 − J 1 J − 1 N . . . − J N − 1 J − 1 N          , R − 1 =          − J 1 J − 1 0 . . . − J N − 1 J − 1 0 1 . . . 1 J N J − 1 0 0 · · · 0          . (33) 11 Then, the p ositiv e (lo cal) cHD hierarch y has the form u t r = K r = R r − 1 K 1 , r = 1 , 2 , . . . , (34) where u = ( u 1 , . . . , u N ) T and for a = 1 K 1 = B N C 0 =             u − 1 2 N  xxx + φ  u − 1 2 N  x 1 4 ε 1  u − 1 2 N  xxx + u 1  u − 1 2 N  x + 1 2 u − 1 2 N ( u 1 ) x . . . 1 4 ε N − 1  u − 1 2 N  xxx + u N − 1  u − 1 2 N  x + 1 2 u − 1 2 N ( u N − 1 ) x            . The negative (non-lo cal) cHD hierarc h y exists when φ = 0 (so in ( 31 ) there is no central extension of the first-order in this case) and has the form u t − r = K − r = R − r K 0 , r = 0 , 1 , . . . , where for c = − 2 K 0 = B 0 C N =       ( u 1 ) x ( u 2 ) x . . . ( u N ) x       and where the first non trivial vector field is K − 1 =       ( u 2 ) x − 1 4 ε 1 ( u 1 ) x − u 1 ∂ − 1 u 1 − 1 2 ( u 1 ) x ∂ − 2 u 1 . . . ( u N ) x − 1 4 ε N − 1 ( u 1 ) x − u N − 1 ∂ − 1 u 1 − 1 2 ( u N − 1 ) x ∂ − 2 u 1 − u N ∂ − 1 u 1 − 1 2 ( u N ) x ∂ − 2 u 1       . 5.2 Coupled K ortew eg-de V ries hierarch y Next, w e show that the set ( 4 ) also contains known p ositive and negativ e coupled Kortew eg-de V ries (cKdV) hierarchies [ 1 , 3 , 6 ]. Again, in order to fit the notation to that known from the literature, let n = N , P m = B m , q i = − u N − i , ψ i = − 1 4 ε n − i , i = 1 , . . . , N , ψ 0 = 0 . Then, Poisson tensors ( 4 ) tak e again the form ( 32 ) and the recursion op erator attains again the same form ( 33 ), but no w J k = 1 2 ( u k ∂ + ∂ u k ) + 1 4 ε k ∂ 3 , k = 0 , . . . , N − 1 , J N = − φ∂ . 12 Notice that a Casimir of B 0 is now C 0 = (0 , . . . , 0 , c ) T . The p ositive (lo cal) cKdV hierarch y has the form ( 34 ), where u = ( u 0 , . . . , u N − 1 ) T , c = − 2 , φ = 1 , K 1 = B N C 0 =       ( u 0 ) x ( u 1 ) x . . . ( u N − 1 ) x       and the first non trivial vector field is K 2 = RK 1 =       J 0 u N − 1 ( u 0 ) x + J 1 u N − 1 . . . ( u N − 2 ) x + J N − 1 u N − 1       =       1 4 ε 0 ( u N − 1 ) xxx + u 0 ( u N − 1 ) x + 1 2 u N − 1 ( u 0 ) x ( u 0 ) x + 1 4 ε 1 ( u N − 1 ) xxx + u 1 ( u N − 1 ) x + 1 2 u N − 1 ( u 1 ) x . . . ( u N − 2 ) x + 1 4 ε N − 1 ( u N − 1 ) xxx + u N − 1 ( u N − 1 ) x + 1 2 u N − 1 ( u N − 1 ) x       . The inv erse (non-lo cal) cKdV hierarc hy exists when ε 0 = 0 . Then, the Casimir of B N is C N = ( au − 1 2 0 , 0 , . . . , 0) T , and for a = − 1 the in v erse hierarch y starts from K − 1 = B 0 C N =          J 1 u − 1 2 0 . . . J N − 1 u − 1 2 0 J N u − 1 2 0          =            1 4 ε 1  u − 1 2 0  xxx + u 1  u − 1 2 0  x + 1 2 u − 1 2 0 ( u 1 ) x . . . 1 4 ε N − 1  u − 1 2 0  xxx + u N − 1  u − 1 2 0  x + 1 2 u − 1 2 0 ( u N − 1 ) x −  u − 1 2 0  x            . 5.3 T riangular coupled Harry Dym hierarch y A third p ossibility occurs when we consider the op erator P n in ( 4 ) in the following w a y . In order to see the resem blance b et ween the hierarc hy obtained b elow, whic h we will call the triangular cHD hierarc hy , and the cHD hierarc h y ab ov e, let us use the following notation: n = N , q i = − u N − i +1 , i = 1 , . . . , N . and c ho ose ψ i = − 1 4 , i = 0 , . . . , N (w e recall that Z n = 0 ). In the v ariables u = ( u 1 , . . . , u N ) T the op erator P n b ecomes then P N = 1 4     1 . . . . . . 1 · · · 1     ∂ 3 +     b 1 . . . . . . b 1 · · · b N     , b i = 1 2 ( u i ∂ + ∂ u i ) = u 1 2 i ∂ u 1 2 i . (35) 13 It is a sum of t wo compatible (due to the theory in the previous section) P oisson op erators π 0 =     b 1 . . . . . . b 1 · · · b N     , π 1 = 1 4     1 . . . . . . 1 · · · 1     ∂ 3 , with the Casimir C 0 = ( u − 1 2 1 , 0 , . . . , 0) T and the x -indep endent Casimir C 1 = ( c 1 , . . . , c N ) T , resp ectiv ely . F rom these op erators we can construct the following lo cal hierarc hy u t r = K r = R r − 1 K 1 , r = 1 , 2 , . . . where R = π 1 π − 1 0 , K 1 = π 1 C 0 =        0 . . . 0 1 4 ( u − 1 2 1 ) xxx        , and the follo wing non-lo cal hierarch y u t − r = K − r = R 1 − r K − 1 , r = 1 , 2 , . . . , where R − 1 = π 0 π − 1 1 , K − 1 = π 0 C 1 =       ( u 1 ) x ( u 1 ) x + ( u 2 ) x . . . P N i =1 ( u i ) x       . F or N = 1 it yields standard lo cal and non-lo cal Harry Dym (HD) hierarchies with u 1 = u , π 0 = u 1 2 ∂ u 1 2 and π 1 = 1 4 ∂ 3 , with the first flo ws given by K 1 = π 1 C 0 = 1 4 ( u − 1 2 ) xxx , K 2 = RK 1 = − 1 16  u − 1 2 ( u − 1 2 ) 2 x  xxx = − 1 64 ( u − 7 2 u 2 x ) xxx , . . . and b y ( c 1 = 2 ) K − 1 = u x , K − 2 = R − 1 K − 1 = 2 u x ∂ − 2 u + 4 u∂ − 1 u, . . . resp ectiv ely . F or N > 1 the lo cal hierarc hy is degenerate, as the matrix of R has zeros ov er the diagonal and the recursion op erator of HD on the diagonal so that the lo cal hierarch y of v ector fields tak es the form K r = (0 , . . . , 0 , K r [ u 1 ]) T . On the other hand, the non-lo cal hierarc hy has 14 a triangular non-lo cal coupled Harry Dym form, as ( R ij ) − 1 =          R − 1 1 , i = j, R − 1 i − j +1 − R − 1 i − j , i > j, 0 , i < j, where R − 1 k = 4 u 1 2 k ∂ u 1 2 k ∂ − 3 = 4 u k ∂ − 2 + 2( u k ) x ∂ − 3 . Then, the comp onents of the first tw o flows are giv en by ( K − 1 ) i = i X j =1 ( u j ) x , ( K − 2 ) i = i X j =1 R − 1 j ( u i − j +1 ) x , i = 1 , . . . , N , so that K − 1 =          ( u 1 ) x ( u 1 + u 2 ) x ( u 1 + u 2 + u 3 ) x . . . ( u 1 + u 2 + . . . + u N ) x          , and K − 2 =          2( u 1 ) x ∂ − 2 u 1 + 4 u 1 ∂ − 1 u 1 2  ( u 1 ) x ∂ − 2 u 2 + ( u 2 ) x ∂ − 2 u 1  + 4  u 1 ∂ − 1 u 2 + u 2 ∂ − 1 u 1  2  ( u 1 ) x ∂ − 2 u 3 + ( u 2 ) x ∂ − 2 u 2 + ( u 3 ) x ∂ − 2 u 1  + 4  u 1 ∂ − 1 u 3 + u 2 ∂ − 1 u 2 + u 3 ∂ − 1 u 1  . . . ( K − 2 ) N          . 5.4 T riangular coupled K ortew eg-de V ries hierarch y The last p ossibilit y o ccurs if we take again the Poisson tensor P N in ( 35 ) P N = π 1 = 1 4     1 . . . . . . 1 · · · 1     ∂ 3 +     b 1 . . . . . . b 1 · · · b N     and the first order P oisson tensor π 0 =     1 . . . . . . 1 · · · 1     ∂ , compatible with π 1 since π 0 is the first-order central extension of the op erator Π n . Only π 0 has a lo cal Casimir of the form ( c 1 , . . . , c N ) T . It generates a lo cal hierarch y , the triangular coupled KdV hierarc hy of the form u t r = K r = R r − 1 K 1 , r = 1 , 2 , . . . , 15 where u = ( u 1 , . . . , u N ) T , R = π 1 π − 1 0 , K 1 = π 1 C 0 =       ( u 1 ) x ( u 1 ) x + ( u 2 ) x . . . P N i =1 ( u i ) x       , with the c hoice c i = 2 , i = 1 , . . . , N and R ij =          R 1 , i = j, R i − j +1 − R i − j , i > j, 0 , i < j, R k = 1 4 ∂ 2 + u k + 1 2 ( u k ) x ∂ − 1 . Then, the comp onen ts of the first tw o flows of this hierarch y are ( K 1 ) i = i X j =1 ( u j ) x , ( K 2 ) i = i X j =1 R j ( u i − j +1 ) x , . . . , i = 1 , . . . , N , so that K 1 =          ( u 1 ) x ( u 1 + u 2 ) x ( u 1 + u 2 + u 3 ) x . . . ( u 1 + u 2 + . . . + u N ) x          , K 2 =          1 4 ( u 1 ) xxx + 3 2 u 1 ( u 1 ) x 1 4 ( u 1 + u 2 ) xxx + 3 2 ( u 1 u 2 ) x 1 4 ( u 1 + u 2 + u 3 ) xxx + 3 2  u 1 u 3 + 1 2 u 2 2  x . . . ( K 2 ) N          , . . . The triangular hierarchies constructed in Subsections 3 and 4 are generated by the Novik o v algebra A n , where the m ultiplication matrix A n con tains all basic elemen ts e i . F or the remaining No viko v algebras A m , m = 0 , . . . , n − 1 , the multiplication matrices A m do not contain all basic elemen ts e i and, consequen tly , the constructed triangular systems are degenerate and thus not in teresting. 6 App endix A Pro of of Theorem 1 Assume first that all the indices i, j, k ∈ I m 1 . Then, given ( 17 ), the left-hand side of ( 15 ) reads n − m − 1 X s =0  δ i + j + m − n − 1 s ( Z m ) sk − δ j + k + m − n − 1 s ( Z m ) is  = ( Z m ) i + j + m − n − 1 ,k − ( Z m ) i,j + k + m − n − 1 ( 19 ) = φ i + j + k +2 m − 2 n − 2 m − φ i + j + k +2 m − 2 n − 2 m = 0 , due to the fact that in this case i + j + m − n − 1 ⩽ n − m − 1 . Similar calculations sho w that ( 15 ) is satisfied in the case i, j, k ∈ I m 2 : n X s = n − m +1  δ i + j + m − n − 1 s ( Z m ) sk − δ j + k + m − n − 1 s ( Z m ) is  = ( Z m ) i + j + m − n − 1 ,k − ( Z m ) i,j + k + m − n − 1 ( 19 ) = φ i + j + k +2 m − 2 n − 2 m − φ i + j + k +2 m − 2 n − 2 m = 0 , 16 due to the fact that in this case n − m + 1 ⩽ i + j + m − n − 1 . Finally , let us assume that one of the indices, sa y k , b elongs to the other index set. Assume thus that i, j ∈ I m 1 while k ∈ I m 2 (all other such situations are prov ed analogously). Then, giv en ( 17 ), the first sum on the left-hand side of ( 15 ) is ( b m ) ij s ( Z m ) sk = n X s = n − m +1 δ i + j + m − n − 1 s ( Z m ) sk = 0 as the index i + j + m − n − 1 ⩽ n − m − 1 for i, j ∈ I m 1 . The second sum on the left-hand side of ( 15 ) is n − m − 1 X s =0 ( b m ) j k s ( Z m ) is and it is also 0 since j ∈ I m 1 and k ∈ I m 2 . Finally , a direct computation shows that under the symmetry assumption Z pq = Z q p the matrix of the system ( 15 ) has co-rank n and thus ( 19 ) is the ( n -parameter) general solution of ( 15 ). Pro of of Lemma 2 The condition ( 11 ) reads (again, no summation o v er m ) ( b m ) ij k ( Z m ) ks = 0 for i, j, s = 1 , . . . , n. (A.1) W e will now show that it necessarily implies that Z m = 0 . F or a fixed m and an y s , supp ose that i, j ∈ I m 1 . Then, ( A.1 ) reads 0 = ( b m ) ij k ( Z m ) ks = n X k =1 δ i + j + m − n − 1 k ( Z m ) ks = ( Z m ) i + j + m − n − 1 ,s , whic h implies ( Z m ) αs = 0 for 1 ⩽ α ⩽ n − m − 1 (and all s ) since in this case 1 + m − n ⩽ i + j + m − n − 1 ⩽ n − m − 1 . F urther, for i, j ∈ I m 2 , ( A.1 ) reads 0 = ( b m ) ij k ( Z m ) ks = − n X k =1 δ i + j + m − n − 1 k ( Z m ) ks = − ( Z m ) i + j + m − n − 1 ,s , whic h implies that ( Z m ) αs = 0 for n − m + 1 ⩽ α ⩽ n + m − 1 (and all s ), since in this case n − m + 1 ⩽ i + j + m − n − 1 ⩽ n + m − 1 . Th us, in ( Z m ) αs all the ro ws except the row n − m (and, in the case m = 0 , the ro w n ) v anish. Since Z m is an tisymmetric, it follows that Z m = 0 . So, no non trivial Z m exist that yield an order 2 co cycle. Pro of of Lemma 3 The asso ciativit y condition ( 26 ) is equiv alen t to ( e i ◦ m e j ) ◦ p e k + ( e i ◦ p e j ) ◦ m e k = e i ◦ m ( e j ◦ p e k ) + e i ◦ p ( e j ◦ m e k ) , (A.2) for all i, j, k = 1 , . . . , n and for all m, p = 0 , . . . , n . Explicitly , the condition ( A.2 ) reads ( b m ) ij s ( b p ) sk r + ( b p ) ij s ( b m ) sk r − ( b m ) is r ( b p ) j k s − ( b p ) is r ( b m ) j k s = 0 , (A.3) for all i, j, k , r = 1 , . . . , n and for all m, p = 0 , . . . , n . F or m = p this lemma reduces to Lemma 1 . Assume thus that m > p (so that n − m < n − p ). There are, up to p erm utations and 17 analogous situations, only four different cases: 1. i, j, k ∈ I m 1 = { 1 , . . . , n − m } ⊂ I p 1 , 2. i, j ∈ I m 1 = { 1 , . . . , n − m } , k ∈ I p 1 ∩ I m 2 = { n − m + 1 , . . . , n − p } , 3. i ∈ I m 1 , j ∈ I p 1 ∩ I m 2 , k ∈ I p 2 ⊂ I m 2 , and 4. i ∈ I m 1 , j, k ∈ I p 1 ∩ I m 2 . In the case 1 all terms in ( A.3 ) are equal so it is satisfied. F or example, the first term b ecomes ( b m ) ij s ( b p ) sk r = n P s =1 δ i + j + m − n − 1 s δ s + k + p − n − 1 r = δ i + j + k + m + p − 2 n − 2 r . All the other terms in ( A.3 ) yield the same expression as all the indices i, j, k ∈ I m 1 ∩ I p 1 . In the case 2, the first term in ( A.3 ) reads ( b m ) ij s ( b p ) sk r = n − p P s =1 δ i + j + m − n − 1 s δ s + k + p − n − 1 r = δ i + j + k + m + p − 2 n − 2 r and is equal exactly to the third term, as the third term b ecomes ( b m ) is r ( b p ) j k s = n − m P s =1 δ i + s + m − n − 1 r δ j + k + p − n − 1 s = δ i + j + k + m + p − 2 n − 2 r . The second term ( b p ) ij s ( b m ) sk r is equal to zero, as ( b p ) ij s = δ i + j + p − n − 1 s is non-zero only for s = i + j + p − n − 1 ∈ I m 1 and then ( b m ) sk r b ecomes zero as k ∈ I m 2 . F or analogous reasons, the fourth term is also zero. Thus, also in this case ( A.3 ) is satisfied. In the case 3 all four terms are zero. The reason is as follows. The first term ( b m ) ij s ( b p ) sk r is zero since ( b m ) ij s = 0 as i ∈ I m 1 while j ∈ I m 2 . The second term ( b p ) ij s ( b m ) sk r is zero as ( b p ) ij s = δ i + j + p − n − 1 s is nonzero only if s = i + j + p − n − 1 < n − m , i.e. only if s ∈ I m 1 , and then ( b m ) sk r = 0 since k ∈ I p 2 ⊂ I m 2 . The third term ( b m ) is r ( b p ) j k s is zero since ( b p ) j k s = 0 as j ∈ I p 1 while k ∈ I p 2 . The fourth term ( b p ) is r ( b m ) j k s is zero as ( b p ) is r ( b m ) j k s = − n − m P s =1 δ i + s + p − n − 1 r δ j + k + m − n − 1 s = 0 , since j + k + m − n − 1 ⩾ n − p + 1 > n − m + 1 so the sum disapp ears. So, also in the case 3 the formula ( A.3 ) is satisfied. Finally , let us analyze the case 4. The first term ( b m ) ij s ( b p ) sk r is zero since i ∈ I m 1 while j ∈ I m 2 so ( b m ) ij s = 0 . The second term ( b p ) ij s ( b m ) sk r is zero, as ( b p ) ij s ( b m ) sk r = − n P s = n − m +1 δ i + j + p − n − 1 s δ s + k + m − n − 1 r = 0 due to the fact that i + j + p − n − 1 < n − m + 1 so that the sum disapp ears. Finally , the last t wo terms in ( A.3 ) cancel each other. The third term is ( b m ) is r ( b p ) j k s = n − m P s =1 δ i + s + m − n − 1 r δ j + k + p − n − 1 s = δ i + j + k + m + p − 2 n − 2 r , while the fourth term is ( b p ) is r ( b m ) j k s = − n − p P s =1 δ i + s + p − n − 1 r δ j + k + m − n − 1 s = − δ i + j + k + m + p − 2 n − 2 r , so they cancel eac h other. Thus, the lemma is prov ed. 18 Pro of of Theorem 2 Giv en ( 27 ), ( 14 ) can b e rewritten as 0 = Z  α p ( b p ) ij s e s , e k  − Z  e i , α p ( b p ) j k s e s  = n X p =0 α p h ( b p ) ij s Z ( e s , e k ) − ( b p ) j k s Z ( e i , e s ) i = n X m =0 n X p =0 α p α m h ( b p ) ij s ( Z m ) sk − ( b p ) j k s ( Z m ) is i = n X m =0 α 2 m cancels by Theorem 1 z }| { h ( b m ) ij s ( Z m ) sk − ( b m ) j k s ( Z m ) is i + n X m =0 X m>p α m α p h ( b p ) ij s ( Z m ) sk − ( b p ) j k s ( Z m ) is + ( b m ) ij s ( Z p ) sk − ( b m ) j k s ( Z p ) is i . Due to ( 17 ) it is equiv alent to the condition ( b p ) ij s ( Z m ) sk − ( b p ) j k s ( Z m ) is + ( b m ) ij s ( Z p ) sk − ( b m ) j k s ( Z p ) is = 0 . (A.4) for all m, p = 0 , . . . , n and all i, j, k = 1 , . . . , n (the index s v aries from 1 to n ). Note that the condition in Remark 3 yields directly the formula ( A.4 ). F or m = p this formula reduces to ( 15 ) in the con text of Theorem 1 and has b een pro v ed ab o ve in this app endix. Let us thus assume that m > p . There are, up to p ermutations and analogous situations, only three different cases: 1. i, j, k ∈ I m 1 = { 1 , . . . , n − m } , 2. i, j ∈ I m 1 = { 1 , . . . , n − m } , k ∈ I p 1 ∩ I m 2 = { n − m + 1 , . . . , n − p } , and 3. i ∈ I m 1 , j ∈ I p 1 ∩ I m 2 , k ∈ I p 2 . Consider the case 1: i, j, k ∈ I m 1 = { 1 , . . . , n − m } . Let us calculate separately all the terms in ( A.4 ): ( b p ) ij s ( Z m ) sk = n − m X s =1 δ i + j + p − n − 1 s ( Z m ) sk = ( Z m ) i + j + p − n − 1 ,k = φ i + j + k + m + p − 2 n − 2 , since k ∈ I m 1 and ( i + j + p − n − 1) ∈ I m 1 as i + j + p − n − 1 ⩽ n − 2 m + p − 1 < n − m − 1 , ( b p ) j k s ( Z m ) is = n − m X s =1 δ j + k + p − n − 1 s ( Z m ) is = ( Z m ) i,j + k + p − n − 1 = φ i + j + k + p + m − 2 n − 2 , since i ∈ I m 1 and ( j + k + p − n − 1) ∈ I m 1 as j + k + p − n − 1 ⩽ n − 2 m + p − 1 < n − m − 1 , ( b m ) ij s ( Z p ) sk = n − m X s =1 δ i + j + m − n − 1 s ( Z p ) sk = ( Z p ) i + j + m − n − 1 ,k = φ i + j + k + m + p − 2 n − 2 , since k ∈ I m 1 ⊂ I p 1 and ( i + j + m − n − 1) ∈ I p 1 as i + j + m − n − 1 ⩽ n − m − 1 ⩽ n − p , and finally ( b m ) j k s ( Z p ) is = n − m X s =1 δ j + k + m − n − 1 s ( Z p ) is = ( Z p ) i,j + k + m − n − 1 = φ i + j + k + m + p − 2 n − 2 , since i ∈ I m 1 ⊂ I p 1 and j + k + m − n − 1 ∈ I p 1 as j + k + m − n − 1 ⩽ n − m − 1 ⩽ n − p . In consequence, all terms in ( A.4 ) are equal (the same is true when i, j, k ∈ I p 2 = { n − p + 1 , . . . , n } ) and th us ( A.4 ) is satisfied. 19 Consider no w the case 2: i, j ∈ I m 1 = { 1 , . . . , n − m } , k ∈ I p 1 ∩ I m 2 = { n − m + 1 , . . . , n − p } . Again, let us calculate eac h term in ( A.4 ): ( b p ) ij s ( Z m ) sk I m 1 ⊂ I p 1 = n X s = n − m +1 δ i + j + p − n − 1 s ( Z m ) sk = 0 since k ∈ I m 2 and i + j + p − n − 1 ∈ I m 1 as i + j + p − n − 1 ⩽ n − 2 m + p − 1 < n − m − 1 , ( b p ) j k s ( Z m ) is I m 1 ⊂ I p 1 = n − m X s =1 δ j + k + p − n − 1 s ( Z m ) is = ( Z m ) i,j + k + p − n − 1 = φ i + j + k + p + m − 2 n − 2 , since i ∈ I m 1 and ( j + k + p − n − 1) ∈ I m 1 as j + k + p − n − 1 ⩽ n − m − 1 , ( b m ) ij s ( Z p ) sk s,k ∈ I p 1 = n − p X s =1 δ i + j + m − n − 1 s ( Z p ) sk = ( Z p ) i + j + m − n − 1 ,k = φ i + j + k + p + m − 2 n − 2 , since k ∈ I p 1 and ( i + j + m − n − 1) ∈ I p 1 as i + j + m − n − 1 ⩽ n − m − 1 , and finally ( b m ) j k s ( Z p ) is = 0 since j ∈ I m 1 and k ∈ I m 2 . Th us, the first and the fourth terms in ( A.4 ) are equal to zero while the middle terms cancel eac h other. Hence, ( A.4 ) is satisfied also in case 2. Finally , consider the case 3: i ∈ I m 1 = { 1 , . . . , n − m } , j ∈ I p 1 ∩ I m 2 = { n − m + 1 , . . . , n − p } , k ∈ I p 2 = { n − p + 1 , . . . , n } . Then, ( b p ) ij s ( Z m ) sk k ∈ I p 2 ⊂ I m 2 = n X s = n − m +1 δ i + j + p − n − 1 s ( Z m ) sk = 0 since k ∈ I p 2 and ( i + j + p − n − 1) ∈ I m 1 ⊂ I p 1 as i + j + p − n − 1 ⩽ n − m − 1 < n − p, ( b p ) j k s ( Z m ) is = 0 since j ∈ I p 1 and k ∈ I p 2 , ( b m ) ij s ( Z p ) sk = 0 since i ∈ I m 1 and j ∈ I m 2 , and, finally ( b m ) j k s ( Z p ) is i ∈ I m 1 ⊂ I p 1 = − n − m X s =1 δ j + k + m − n − 1 s ( Z p ) is = 0 , since i ∈ I m 1 and ( j + k + m − n − 1) ∈ I m 2 as j + k + m − n − 1 ⩾ n − p + 1 > n − m + 1 . Th us, in this case all the terms in ( A.4 ) v anish and therefore ( A.4 ) is satisfied. This means that the p encil ( 27 ) with Z m giv en by ( 29 ) is a particular solution of the F rob enius condition ( 15 ). The theorem is pro ved. 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