Generalized Landau-Lifshitz systems and Lie algebras associated with higher genus curves

GENERA LIZED LAND A U-LIFSHITZ SYSTEMS AND L IE ALGEBRAS ASSOCIA TED WITH HIGHER GENUS CUR VES S. IGONIN, J. V AN DE LE UR, G. MANNO, AND V. TRUSH KO V Abstract. The W ahlquist-E stabro o k prolonga tion metho d allows to obtain for some PDEs a Lie algebr a that is resp onsible fo r Lax pa irs and B¨ acklund transformations of certain t ype. W e study the W ahlquist-Esta br o ok alg ebra of the n -dimensional gener al- ization of the Landau-Lifshitz equation and construct an epimorphism from this algebra onto an infinite-dimensio nal q uasigra ded Lie a lgebra L ( n ) of certain matrix -v a lue d func- tions on a n algebr aic curve of genus 1 + ( n − 3 )2 n − 2 . F or n = 3 , 4 , 5 we prov e that the W ahlquist-E stabro o k algebra is isomorphic to the direct sum o f L ( n ) and a 2-dimensio nal ab elian L ie algebra. Using these results, for any n a new family of Miura type transfor - mations (differential substitutions) parametriz e d b y po ints of the above men tioned curve is constructed. As a by-pro duct, we obtain a r epresentation of L ( n ) in terms o f a finite nu mber o f g enerator s and rela tions, which may b e of indep endent interest. 1. Introduction In the last 25 years it has b een w ell understo od ho w to construct integrable PDEs fro m infinite-dimensional Lie algebras (see, e.g., [6, 7, 9, 22 , 25] and references therein). The presen t pap er a ddresses the in v ers e problem: given a system of PDEs, how to determine whether it is related to a n infinite-dimensional Lie algebra and how to reco v er this Lie algebra? A partial answ er to this question is provided b y the so-called W ahlquist-Estabro ok prolongation metho d [4, 26], whic h is an a lg orithmic pro cedure that for a giv en (1 + 1)-dimensional system of PDEs constructs a Lie algebra called the Wahlquist-Estabr o ok algebr a (or the WE algebr a in short). The WE algebra is respo nsible for Lax pairs and B¨ ac klund transformations of certain t ype. The metho d giv es this algebra in terms of generators and relations. F or some PD Es the explicit structure of the WE algebra w as rev ealed, and as a result one obtained in teresting infinite-dimensional Lie algebras (see, e.g., [3, 5, 14, 19] and references therein). In the o r iginal metho d of W a hlquist and Estabro ok the obta ined algebras lac k an y in v arian t co ordinate-free meaning. Recen t ly the WE a lg ebra ha s been included in a sequence of Lie algebras tha t hav e a remark able geometric inte rpretation: they are the analogue of the top o logical fundamen tal group for the category of PDEs [11, 12]. All finite- dimensional quotien ts of these Lie algebras ha v e a co ordinat e- f ree meaning a s symmetry algebras of certain co verings of PDEs [11] (the notio n of co v erings of PDEs b y Kr a silshc hik and Vinogrado v [2, 15] is a far-reac hing geometric g eneralization of W ahlquist-Estabro ok pseudop o ten tials). Moreov er, an effectiv e neces sary condition for existence of a B¨ acklu nd transformation connecting t w o g iv en PDEs ha s b een obtained in terms of these Lie al- gebras [11 ], whic h has allow ed to prov e for the first time ev er tha t some PDEs are not connected by an y B¨ ac klund transformation [1 1]. In o ur opinion, these results strongly suggest to compute and study the WE algebras for more PDEs. 1991 Mathematics Subje ct Classific ation. 37K30 , 37K35. Key wor ds and phr ases. W ahlquist-Estabr o ok pr olonga tion structure s , genera lized Landau-Lifshitz systems, q uasigra ded infinite-dimensional Lie algebr as, Miura type transfor mations. 1 GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 2 In this pap er w e apply the W a hlquist-Estabro ok metho d to the f ollo wing system (1) S t =  S xx + 3 2 h S x , S x i S  x + 3 2 h S, R S i S x , h S, S i = 1 , where S = ( s 1 ( x, t ) , . . . , s n ( x, t )) is a column-v ector of dimension n > 1, h· , · i is the standard scalar pro duct, and R = diag ( r 1 , . . . , r n ) is a constant diagonal matrix with r i 6 = r j for i 6 = j . This system w as in tro duced in [8] a nd p ossesses a La x pair (a zero- curv ature represen tation) para metrized b y p o in ts of the follow ing alg ebraic curv e (2) λ 2 i − λ 2 j = r j − r i , i, j = 1 , . . . , n, in the space C n with co ordinat es λ 1 , . . . , λ n . According to [8], t his curv e is of gen us 1 + ( n − 3)2 n − 2 . System ( 1 ) has also an infinite n um b er of symmetries, conserv ation laws [8], and a B¨ ac klund auto-transformation with a parameter [1]. Soliton-like solutions of (1) can be found in [1]. According to [8], for n = 3 this system coincides with the higher symmetry (the commuting flow) of third order for the we ll-kno wn Landau-Lifshitz equation (see, e.g., [10]). Th us system (1) is an n -dimensional generalization of the Landau-Lifshitz equation. Note that, to our kno wledge, b efore the prese n t pap er the W ahlquist-Estabro ok metho d w as nev er applied to a system of PDEs connected with an algebraic curve of gen us greater than 1, and the explicit structure o f the WE algebra w as not computed for an y system with more than tw o dep enden t v ariables (unkno wn f unctions s i ( x, t )). F or arbitrary n w e construct an epimorphism from the WE algebra of (1) on to the infinite-dimensional Lie a lgebra L ( n ) of certain so n, 1 -v alued functions on the curve (2). Note that L ( n ) is not gra ded, but is quasigraded [16] L ( n ) = ∞ M i =1 ¯ L i , [ ¯ L i , ¯ L j ] ⊂ ¯ L i + j + ¯ L i + j − 2 , dim ¯ L 2 k − 1 = n, dim ¯ L 2 k = n ( n − 1 ) 2 ∀ k ∈ N . F or n = 3 , 4 , 5 w e pro v e that the WE algebra is isomorphic to the direct sum of L ( n ) and a tw o- dimensional ab elian Lie algebra. In particular, fo r n = 3 the WE alg ebra o f (1) is isomorphic to the WE algebra of the anisotropic Landau- Lifshitz equation [19]. F or n = 2 the curv e (2) is rational and system (1) b elongs to t he w ell-studied class of scalar ev olutionary equations [5, 14 ], so we skip the case n = 2 . T o ac hiev e this, we pro v e that t he algebra L ( n ) is isomorphic for any n ≥ 3 to the Lie algebra give n b y generators p 1 , . . . , p n and relatio ns [ p i , [ p i , p k ]] − [ p j , [ p j , p k ]] = ( r j − r i ) p k for i 6 = k , j 6 = k , (3) [ p i , [ p j , p k ]] = 0 for i 6 = k , i 6 = j, j 6 = k . In our opinion, this algebraic result ma y b e of indep enden t interes t. F or n = 3 it w as pro v ed in [19]. It is known that for a given ev olutionary system (4) u i t = F i ( u j , u k x , u l xx , . . . ) , i = 1 , . . . , m, the WE algebra helps also to find a ‘mo dified’ system (5) v i t = H i ( v j , v k x , v l xx , . . . ) , i = 1 , . . . , m, connected with (4) b y a B¨ ac klund transforma t ion of Miura t ype (sometimes also called a differen tial substitution) (6) u i = g i ( v j , v k x , v l xx , . . . ) , i = 1 , . . . , m. GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 3 That is, for a n y solution v 1 , . . . , v m of (5) functions (6) form a solution of (4), and f o r an y solution u 1 , . . . , u m of (4) lo cally there exist functions v 1 , . . . , v m satisfying (5), (6). F or system (1) a mo dified system (5 ) and a transformation (6) w ere not know n. Using a suitable ve ctor field represen tation of the WE algebra, w e find a family of systems (5) and transformations (6) for (1). This fa mily is par a metrized by p o ints of the curv e (2). Ap- paren tly , this is the first example of a family of Miura t yp e t r a nsformations parametrized b y p oints of a curv e of gen us greater than 1. W e w ould like to mak e also the follo wing observ atio n. Clearly , relations (3) lo ok some- what similar to equations (2). And indeed, formula (19) and Theorem 3 b elow explain ho w the generator p i is related to λ i . Note that, as we sho w in Sections 3.1, 3.2, 3.3, at least for n = 3 , 4 , 5 relations (3) arise f rom the internal j et space geometry of system (1) using the W ahlquist-Estabro ok pro cedure. Therefore, t he W ahlquist-Estabro ok metho d and its generalization in [11, 12] may b e useful also for the following problem: g iven a sys- tem of PDEs, whic h is susp ected to b e in tegrable, how to reco v er an algebraic curve that most naturally parametrizes a p ossible Lax pair for this syste m? F or example, applying the W ahlquist-Estabro ok pro cedure to system (1) without an y knowle dge of Lax pairs or algebraic curv es behind (1), one o btains r elat io ns (3) for the WE a lg ebra. Since p i should corresp ond to some matrix-v alued functions on an algebraic curve , lo o king at (3 ) it is not so hard to guess that the curv e should b e of the form (2). It w ould b e v ery in teresting to mak e this observ ation into a rigo rous construction for more PDEs. Sev era l more in tegrable systems asso ciated with the curv e (2) w ere intro duced in [8, 22, 23, 24 ], where the o pp osite approac h is ta k en: they start with an infinite-dimensional Lie algebra very similar to our algebra L ( n ) and construct in tegrable systems from this Lie algebra. Note that a represen tation of the Lie algebra in terms o f a finite n um b er of generators and relations was not o bt a ined in [8, 22, 2 3, 24]. The f unctions S = ( s 1 ( x, t ) , . . . , s n ( x, t )) in (1) and the parameters λ i , r i in (2) ma y tak e v alues in C or R . In this pap er the C -v alued case is studied, but all results a nd pro ofs ar e v alid also in the R - v alued case, if one replaces C b y R in the definitions. The pap er is org a nized as follo ws. In Section 2 w e presen t a rigo rous definition of WE algebras f o r arbitrary ev olutionary systems and, therefore, for a ny systems of PDEs t ha t can b e written in evolutionary form. F or this w e use f o rmal pow er series with co efficien ts in Lie a lgebras. F or example, suc h series o ccur in computations of the WE algebra of the f -Gordon equation u tt − u xx = f ( u ) [21], whic h, as is we ll kno wn, can b e rewritten in ev olutionary form as follo ws (7) u t = q , q t = u xx + f ( u ) . W e discuss also p ossible generalizations of the W ahlquist-Estabro ok ansatz. In Sections 3, 4 the ab o v e men tioned results o n the WE algebra of system (1) are obtained, and in Section 5 the family of Miura t ype transforma t io ns is constructed. The app endix con tains the pro of o f tec hnical Lemma 1. The fo llowing abbreviations are used in the pap er: WE = W ahlquist-Estabro ok, ZCR = zero-curv ature represen tation. 2. The general definition of WE algebras Originally the W ahlquist-Estabro ok prolongatio n metho d w as f orm ulated in terms of differen tial f orms. W e prefer the v ector fields ve rsion of it, which go es as follo ws. F or a giv en m -comp o nen t ev olutionary system of PDEs of or der d ≥ 1 (8) u i t = F i ( u 1 , . . . , u m , u 1 1 , . . . , u m 1 , . . . , u 1 d , . . . , u m d ) , i = 1 , . . . , m, u i k = ∂ k u i ∂ x k , GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 4 consider the infinite-dimensional jet space with the co ordinates (9) x, t, u i k , i = 1 , . . . , m, k = 0 , 1 , 2 , . . . , u i 0 = u i . The to t a l deriv ative op erators D x = ∂ ∂ x + X i, k u i k +1 ∂ ∂ u i k , D t = ∂ ∂ t + X i, k D k +1 x ( F i ) ∂ ∂ u i k are commuting v ector fields on this space. Let α b e an ( m × d ) - matrix with entries α ik ∈ Z + , i = 1 , . . . , m, k = 0 , . . . , d − 1 . Denote the set of suc h matrices b y M . Denote by U α the follow ing pro duct of co ordi- nates (9) U α = Y i =1 ,...,m, k =0 ,...,d − 1  u i k  α ik . Consider tw o formal p o w er series (10) X = X α ∈M A α U α , T = X β ∈M B β U β . in the v ariables (11) u i k , 1 ≤ i ≤ m, 0 ≤ k ≤ d − 1 , where the co efficien ts A α , B β are elemen ts of some (not sp ecified y et) Lie a lg ebra. Th e equation (12) [ D x + X , D t + T ] = D x ( T ) − D t ( X ) + [ X , T ] = 0 in the space of formal p ow er series is equiv alen t to some Lie algebraic relations f o r the elemen ts A α , B β . Here D x ( T ) = X β ∈M B β · D x ( U β ) , D t ( X ) = X α ∈M A α · D t ( U α ) , [ X , T ] = X α,β ∈M [ A α , B β ] · U α · U β . Let F b e the free Lie algebra generated by all the letters A α , B β for α, β ∈ M . The quotien t of F ov er the ab o v e men t io ned relations arising from equation ( 12) is called the Wahlquist-Estabr o o k Lie algebr a of system (8) (or the WE algebr a in brief ). F rom now on A α , B β are elemen ts of the WE algebra. Then (10) is the most general solution of (1 2) pro vided that X, T are p o w er series in v ariables (11). Remark 1. F or man y systems (8) equation (12) implies that X, T are of the form (13) X = k 1 X i =1 C i f i , T = k 2 X j =1 D j g j , k 1 , k 2 ∈ N , where f i , g j are analytic functions of v ariables (11), the functions f 1 , . . . , f k 1 are linearly indep enden t, the f unctions g 1 , . . . , g k 2 are linearly indep enden t, and C i , D j are elemen ts of the WE algebra. Expanding f i , g j as p ow er series in (11), we obtain that A α , B β from (10) are linear com binations of C i , D j and, therefore, in this case C i , D j can b e tak en as ano ther set of generators of the same WE algebra. Ho w eve r, the cases when X , T are formal p o w er series and cannot b e presen ted as finite sums of analytic functions do also o ccur. F o r example, this happ ens f o r t he f - Gordon equation u tt − u xx = f ( u ) [21], which can b e rewritten in ev olutionary form (7). GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 5 Remark 2. A natural question arises, what happ ens if in equation (12) one considers p ow er series X , T in the v ariables u i k , 1 ≤ i ≤ m, 0 ≤ k ≤ s, for ar bit r a ry s ? It turns out t hat if s > d − 1 then before solving (12) one should simplify X , T b y so-called g auge transformations [11, 12], and then one also obtains certain Lie algebras [11, 12], whic h are generally bigger tha n the WE algebra for s = d − 1. F or example, for the K r ichev er-Nov ik o v equation the WE algebra fo r s = d − 1 is tr ivial, but the case s = d do es pro duce an in teresting Lie algebra [14]. Co ordinate-free meaning of t he Lie algebras fo r arbitr a ry s is studied in [11, 1 2 ]. In this w a y one obtains new geometric inv aria nts of PDEs, whic h allow to pro v e, for example, that some PDEs are not connected b y any B¨ ac klund transformations [11, 12]. Let g b e a Lie algebra. Recall [10] that g - v alued functions M , N of a finite n um ber of v ariables (9 ) form a zer o-curvatur e r epr esen tation (Z CR in short) for system (8) if (14) [ D x + M , D t + N ] = D x ( N ) − D t ( M ) + [ M , N ] = 0 . If dim g < ∞ then M , N are supp osed to b e a nalytic or smo oth g -v alued functions, while if dim g = ∞ then M , N are formal p ow er series with co efficien ts in g . F or example, the p ow er series X , T f orm a Z CR with v alues in the WE algebra. It is kno wn that eve ry Lax pair for a (1 + 1 ) -dimensional syste m of PDEs determines a ZCR. In additio n to the W ahlquist-Estabro ok tec hnique, some other metho ds to obtain ZCRs f o r a giv en system of PDEs also exist (see, e.g., [17 , 18, 20] a nd references therein). Since (1 0) is the mo st general solution o f (12) and equation (14) is similar t o (12), w e obtain the follo wing result. Prop osition 1. Supp ose that M , N ar e g -value d functions of variables (11) and form a ZCR. Exp and M , N as p ower se ries in (11) M = X α ∈M M α U α , N = X β ∈M N β U β , M α , N β ∈ g . Then the map A α 7→ M α , B β 7→ N β determines a ho momorphism fr om the WE algebr a to g . If the c o efficients M α , N β , α , β ∈ M , gener ate the whole Lie algebr a g then this homomorphism is surje ctive. 3. Comput a tions for the generalized Landa u- Lifshitz syste m In order to study the WE alg ebra of system (1), we need to resolv e (lo cally) the con- strain t h S, S i = 1 for the vec tor S = ( s 1 ( x, t ) , . . . , s n ( x, t )). F o llowing [8], w e do it as (15) s i = 2 u i 1 + h u, u i , i = 1 , . . . , n − 1 , s n = 1 − h u, u i 1 + h u, u i , where u is an ( n − 1)- dimensional v ector with the comp onen ts u 1 ( x, t ) , . . . , u n − 1 ( x, t ), and h· , · i is the standard scalar pr o duct. Then one can rewrite system (1) as [8] (16) u t = u xxx − 6 h u, u x i ∆ − 1 u xx +  − 6 h u, u xx i ∆ − 1 + 24 h u, u x i 2 ∆ − 2 − 6 h u, u ih u x , u x i ∆ − 2  u x +  6 h u x , u xx i ∆ − 1 − 1 2 h u, u x ih u x , u x i ∆ − 2  u + 3 2  r n + 4∆ − 2 n − 1 X i =1 ( r i − r n )( u i ) 2  u x , where ∆ = 1 + h u, u i and r 1 , . . . , r n are distinct complex num b ers t hat a re the en tries of the matrix R = diag ( r 1 , . . . , r n ) fro m system (1). GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 6 Let E i,j ∈ gl n +1 ( C ) be the matrix with ( i, j )-th en try equal to 1 and all other en tr ies equal to zero. Recall that the Lie subalgebra so n, 1 ⊂ gl n +1 ( C ) has the following basis E i,j − E j,i , i < j ≤ n, E l,n +1 + E n +1 ,l , l = 1 , . . . , n. F rom the results of [8, 25] one can o btain the follow ing so n, 1 -v alued ZCR of system (1) M = n X i =1 s i λ i ( E i,n +1 + E n +1 ,i ) , (17) N = D 2 x ( M ) + [ D x ( M ) , M ] + ( r 1 + λ 2 1 ) M +  1 2 ( S, RS ) + 3 2 ( S x , S x )  M . (18) Here λ 1 , . . . , λ n are complex parameters satisfying equations (2 ). Remark 3. It w a s noticed in [22] that t he form ulas λ = λ 2 i + r i , y = Q n i =1 λ i , pro vide a mapping from the curv e ( 2 ) to the h yperelliptic curv e y 2 = Q n i =1 ( λ − r i ). How ev er, according to [8], the curv e (2) itself is not hy p erelliptic. If S = ( s 1 , . . . , s n ) is giv en b y form ulas (15) then (17), (18) determines a ZCR for system (1 6). In the a lgebra C [ λ 1 , . . . , λ n ] consider the ideal I ⊂ C [ λ 1 , . . . , λ n ] generated b y the p olynomials λ 2 i − λ 2 j + r i − r j , i, j = 1 , . . . , n . Denote b y ¯ λ i the imag e o f λ i in t he quotient algebra Q = C [ λ 1 , . . . , λ n ] /I , whic h is equal to the algebra of p olynomial functions on the curv e (2). Consider the infinite-dimensional L ie algebra o v er C gl n +1 ( C ) ⊗ C Q ∼ = gl n +1 ( Q ) , [ g 1 ⊗ q 1 , g 2 ⊗ q 2 ] = [ g 1 , g 2 ] ⊗ q 1 q 2 , g i ∈ gl n +1 ( C ) , q i ∈ Q, and the elemen ts (19) Q i = ( E i,n +1 + E n +1 ,i ) ⊗ ¯ λ i ∈ so n, 1 ⊗ C Q ⊂ gl n +1 ( C ) ⊗ C Q, i = 1 , . . . , n. Denote by L ( n ) ⊂ so n, 1 ⊗ Q the Lie subalgebra g enerated by Q 1 , . . . , Q n . Ob viously , the elemen t ¯ λ = ¯ λ 2 i + r i ∈ Q do es not dep end on i . F o r i, j = 1 , . . . , n , i 6 = j , and k ∈ N consider the f ollo wing elemen ts of so n, 1 ⊗ Q Q 2 k − 1 i = ( E i,n +1 + E n +1 ,i ) ⊗ ¯ λ k − 1 ¯ λ i , Q 2 k ij = ( E i,j − E j,i ) ⊗ ¯ λ k − 1 ¯ λ i ¯ λ j . F or i, j, a, b = 1 , . . . , n , i 6 = j , a 6 = b , and k 1 , k 2 ∈ N one has (20) [ Q 2 k 1 ij , Q 2 k 2 ab ] = δ aj Q 2( k 1 + k 2 ) ib − δ ib Q 2( k 1 + k 2 ) aj + δ j b Q 2( k 1 + k 2 ) ai − δ ia Q 2( k 1 + k 2 ) j b + r i δ ib Q 2( k 1 + k 2 − 1) aj − r j δ aj Q 2( k 1 + k 2 − 1) ib + r i δ ia Q 2( k 1 + k 2 − 1) j b − r j δ j b Q 2( k 1 + k 2 − 1) ai , (21) [ Q 2 k 1 ij , Q 2 k 2 − 1 a ] = δ aj Q 2 k 1 +2 k 2 − 1 i − δ ia Q 2 k 1 +2 k 2 − 1 j − r j δ aj Q 2 k 1 +2 k 2 − 3 i + r i δ ia Q 2 k 1 +2 k 2 − 3 j , (22) [ Q 2 k 1 − 1 i , Q 2 k 2 − 1 j ] = Q 2( k 1 + k 2 − 1) ij , [ Q 2 k 1 − 1 i , Q 2 k 2 − 1 i ] = 0 . Since Q 1 i = Q i and Q 2 k ij = − Q 2 k j i , fro m (20), (2 1), (22) w e obtain tha t Q 2 k − 1 l , Q 2 k ij , i, j, l = 1 , . . . , n, i < j, k = 1 , 2 , 3 , . . . , span the Lie alg ebra L ( n ). It is easily seen that these elemen ts are linearly independen t o v er C and, therefore, form a basis of L ( n ). F or k ∈ N set ¯ L 2 k − 1 = h Q 2 k − 1 l | l = 1 , . . . , n i , ¯ L 2 k = h Q 2 k ij | i, j = 1 , . . . , n, i < j i . GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 7 Here and b elo w for elemen ts v 1 , . . . , v s of a v ector space the expression h v 1 , . . . , v s i denotes the linear span of v 1 , . . . , v s o v er C . Then from (20), (21), (22) one gets L ( n ) = ∞ M i =1 ¯ L i , [ ¯ L i , ¯ L j ] ⊂ ¯ L i + j + ¯ L i + j − 2 . Therefore, the Lie alg ebra L ( n ) is quasigraded [16]. Clearly , f o rm ulas (17), (18) can b e regarded a s a Z CR with v alues in the algebra L ( n ). In particular, M = P n i =1 s i Q i . Com bining this with Prop osition 1 , we obtain the follo wing. Theorem 1. F or any n ≥ 3 w e have a n epimorphism fr om the WE algebr a of system (1) onto the in finite-dimension al Lie algebr a L ( n ) . Let us giv e a complete description of the WE algebra of (16 ) f o r small n . F or n = 2 system (16) is a scalar equation of the form u t = u xxx + f ( u, u x , u xx ). Since for suc h equations the WE algebras ha v e already b een studied quite extensiv ely (see, e.g, [5, 14] and references therein) and the curv e (2) is rationa l fo r n = 2, we skip the case n = 2. F or n = 3 , 4 , 5 the WE alg ebras are studied b elo w. 3.1. The case n=3. According to Section 2, w e must solv e equation (12) f o r X = X ( u 1 , u 2 , u 1 x , u 2 x , u 1 xx , u 2 xx ) , T = T ( u 1 , u 2 , u 1 x , u 2 x , u 1 xx , u 2 xx ) . If w e differen tiate (12) with resp ect to the v ariables u 1 xxxxx , u 2 xxxxx , u 1 xxxx , u 2 xxxx , we g et that X dep ends only on ( u 1 , u 2 ). Next, differen tiating (12) with resp ect to u 1 xxx , u 2 xxx , u 1 xx , u 2 xx sev eral times, one obta ins X = X ( u 1 , u 2 ) , T = ∂ X ∂ u 1 u 1 xx + ∂ X ∂ u 2 u 2 xx + F 1 ( u 1 , u 2 , u 1 x , u 2 x ) , where F 1 = − 1 2  ∂ 2 X ∂ u 1 ∂ u 1 ∆ + 6 u 1 ∂ X ∂ u 1 − 6 u 2 ∂ X ∂ u 2  ∆ − 1 ( u 1 x ) 2 −  ∂ 2 X ∂ u 1 ∂ u 2 ∆ + 6 u 2 ∂ X ∂ u 1 + 6 u 1 ∂ X ∂ u 2  ∆ − 1 u 1 x u 2 x − 1 2  ∂ 2 X ∂ u 2 ∂ u 2 ∆ + 6 u 2 ∂ X ∂ u 2 − 6 u 1 ∂ X ∂ u 1  ∆ − 1 ( u 2 x ) 2 + [ ∂ X ∂ u 1 , X ] u 1 x + [ ∂ X ∂ u 2 , X ] u 2 x + F 2 ( u 1 , u 2 ) , and ∆ = 1 + ( u 1 ) 2 + ( u 2 ) 2 . Then the expression (23) [ D x + X , D t + T ] = D x ( T ) − D t ( X ) + [ X , T ] b ecomes a third degree p o lynomial in u 1 x , u 2 x with co efficien ts dep ending on u 1 , u 2 . By putting equal to zero the third degree co efficien ts, we get a system of linear PDEs of order 3 for t he function X ( u 1 , u 2 ). The general solution of this system is (24) X =  D 0 + u 1 D 1 + u 2 D 2 +  ( u 1 ) 2 + ( u 2 ) 2  D 3  ∆ − 1 , where D i do not depend on u 1 , u 2 and, therefore, b elong to a set of generators of the WE algebra. T aking in to accoun t, that system (16) a rises from (1 ) by means of (15), it is more con v enien t to rewrite (24) as X =  2 C 1 u 1 + 2 C 2 u 2 + C 3  1 − ( u 1 ) 2 − ( u 2 ) 2   ∆ − 1 + C 0 , GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 8 where C i are also elemen ts of t he WE a lgebra. By putting to zero the co efficien ts of (23) at the monomials ( u 1 x ) 2 and ( u 2 x ) 2 , we obtain (25) [ C 0 , C 1 ] = [ C 0 , C 2 ] = [ C 0 , C 3 ] = 0 . No w, b y putting the co efficien ts of u 1 x and u 2 x equal to zero, one gets a system of t w o first order differen tia l equations for F 2 ( u 1 , u 2 ). Its compatibilit y conditions a re satisfied if and only if the following relations hold [ C 0 , [ C 1 , C 2 ]] = [ C 0 , [ C 1 , C 3 ]] = [ C 0 , [ C 2 , C 3 ]] = 0 , (26) [ C 1 , [ C 2 , C 3 ]] = [ C 2 , [ C 3 , C 1 ]] = [ C 3 , [ C 1 , C 2 ]] = 0 , (27) [ C 1 , [ C 1 , C 3 ]] − [ C 2 , [ C 2 , C 3 ]] = ( r 2 − r 1 ) C 3 , (28) [ C 1 , [ C 1 , C 2 ]] − [ C 3 , [ C 3 , C 2 ]] = ( r 3 − r 1 ) C 2 , (29) [ C 2 , [ C 2 , C 1 ]] − [ C 3 , [ C 3 , C 1 ]] = ( r 3 − r 2 ) C 1 . (30) Note that (26) follows from (2 5) using the Jacobi identit y . Then w e are able to compute F 2 ( u 1 , u 2 ). F 2 =  ( − 2[ C 3 , [ C 1 , C 3 ]] + 3 C 1 r 3 )( u 1 ) 5 + (3 C 2 r 3 − 2 [ C 3 , [ C 2 , C 3 ]])( u 1 ) 4 u 2 + (6 C 1 r 3 − 4 [ C 3 , [ C 1 , C 3 ]])( u 1 ) 3 ( u 2 ) 2 + ( − 4[ C 3 , [ C 2 , C 3 ]] + 6 C 2 r 3 )( u 1 ) 2 ( u 2 ) 3 + ( − 2[ C 3 , [ C 1 , C 3 ]] + 3 C 1 r 3 ) u 1 ( u 2 ) 4 + (3 C 2 r 3 − 2[ C 3 , [ C 2 , C 3 ]])( u 2 ) 5 + (2 C 3 r 2 + 2[ C 2 , [ C 2 , C 3 ]] − 2 C 3 r 1 + 3 C 3 r 3 )( u 1 ) 4 + (4[ C 2 , [ C 2 , C 3 ]] + 2 C 3 r 2 − 2 C 3 r 1 + 6 C 3 r 3 )( u 1 ) 2 ( u 2 ) 2 + (2[ C 2 , [ C 2 , C 3 ]] + 3 C 3 r 3 )( u 2 ) 4 + (4 C 1 r 1 − 4[ C 3 , [ C 1 , C 3 ]] + 2 C 1 r 3 )( u 1 ) 3 + (2 C 2 r 3 − 4 [ C 3 , [ C 2 , C 3 ]] + 4 C 2 r 1 )( u 1 ) 2 u 2 + ( − 4[ C 3 , [ C 1 , C 3 ]] + 4 C 1 r 2 + 2 C 1 r 3 ) u 1 ( u 2 ) 2 + (4 C 2 r 2 + 2 C 2 r 3 − 4 [ C 3 , [ C 2 , C 3 ]])( u 2 ) 3 + (4[ C 2 , [ C 2 , C 3 ]] + 4 C 3 r 2 + 2 C 3 r 1 )( u 1 ) 2 + (4[ C 2 , [ C 2 , C 3 ]] + 6 C 3 r 2 )( u 2 ) 2 + ( − 2[ C 3 , [ C 1 , C 3 ]] + 3 C 1 r 3 ) u 1 + (3 C 2 r 3 − 2 [ C 3 , [ C 2 , C 3 ]]) u 2 + C 3 r 3 + 2 C 3 r 2 + 2[ C 2 , [ C 2 , C 3 ]]  ∆ − 3 + C ′ , where C ′ is another g enerator o f the WE algebra. Then (23) dep ends only on u 1 , u 2 and v anishe s if and o nly if [ C 3 , C ′ ] = 0 , (31) [ C 0 , C ′ ] = 0 , (32) [ C 3 , [ C 2 , [ C 2 , C 3 ]]] = 0 , (33) [ C 3 , [ C 3 , [ C 3 , C 1 ]]] = 3 2 r 3 [ C 1 , C 3 ] + [ C 1 , C ′ ] , (34) [ C 3 , [ C 3 , [ C 3 , C 2 ]]] = 3 2 r 3 [ C 2 , C 3 ] + [ C 2 , C ′ ] , (35) [[ C 2 , C 3 ] , [ C 1 , C 2 ]] = 1 2 r 3 [ C 1 , C 3 ] + [ C 1 , C ′ ] + r 2 [ C 1 , C 3 ] , (36) [ C 2 , [ C 2 , [ C 2 , C 3 ]]] = − 1 2 r 3 [ C 2 , C 3 ] − [ C 2 , C ′ ] − r 2 [ C 2 , C 3 ] . (37) Therefore, the WE algebra of (16) for n = 3 is giv en b y the generators C ′ , C 0 , C 1 , C 2 , C 3 and all relations obtained in t his subsection. Let us simplify the structure of these relations. T aking in to account (25) and (27), relations (31), (3 2 ), (33), (36), (37) imply (38) [ C ′ +  r 2 + 1 2 r 3  C 3 + [ C 2 , [ C 2 , C 3 ]] , C i ] = 0 , i = 0 , 1 , 2 , 3 . GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 9 Then it is easily seen that a ll relatio ns follow from (25), (27), (28), (29), (30), (38) using the Jacobi iden tit y . F or example, let us prov e this for (33) a nd (34), for all o ther relations the statemen t can b e prov ed analogously . Applying ad C 3 to (28), ad C 2 to (29), and ad C 1 to ( 3 0), one obta ins [ C i , [ C j , [ C j , C i ]]] = [ C i , [ C k , [ C k , C i ]]] for { i, j, k } = { 1 , 2 , 3 } . Besides, b y the Jacobi identit y , w e hav e [ C i , [ C j , [ C j , C i ]]] = − [ C j , [ C i , [ C i , C j ]]]. Therefore, [ C 3 , [ C 2 , [ C 2 , C 3 ]]] = − [ C 2 , [ C 3 , [ C 3 , C 2 ]]] = − [ C 2 , [ C 1 , [ C 1 , C 2 ]]] = [ C 1 , [ C 2 , [ C 2 , C 1 ]]] = [ C 1 , [ C 3 , [ C 3 , C 1 ]]] = − [ C 3 , [ C 1 , [ C 1 , C 3 ]]] = − [ C 3 , [ C 2 , [ C 2 , C 3 ]]] , whic h implies (33). Using (30), (27), and the Jacobi iden tit y , one obtains [ C 3 , [ C 3 , [ C 3 , C 1 ]]] = [ C 3 , [ C 2 , [ C 2 , C 1 ]]] + ( r 2 − r 3 )[ C 3 , C 1 ] = [[ C 3 , C 2 ] , [ C 2 , C 1 ]] + ( r 2 − r 3 )[ C 3 , C 1 ] = − [ C 1 , [ C 2 , [ C 2 , C 3 ]]] + ( r 2 − r 3 )[ C 3 , C 1 ] . Com bining this with (38) for i = 1, we get (34) . Th us the WE algebra is isomorphic t o the direct sum g (3) ⊕ A , where g (3) is the Lie algebra giv en by the generators C 1 , C 2 , C 3 and r elat io ns (27 ) , ( 2 8), (29 ) , ( 3 0), and A is the tw o-dimensional ab elian L ie algebra spanned b y C 0 , C ′ +  r 2 + 1 2 r 3  C 3 + [ C 2 , [ C 2 , C 3 ]]. 3.2. The case n=4. According to Section 2, w e must solv e equation (12) f o r X = X ( u 1 , u 2 , u 3 , u 1 x , u 2 x , u 3 x , u 1 xx , u 2 xx , u 3 xx ) , T = T ( u 1 , u 2 , u 3 , u 1 x , u 2 x , u 3 x , u 1 xx , u 2 xx , u 3 xx ) . Similarly to the previous subsection we obtain the following. X =  2 C 1 u 1 + 2 C 2 u 2 + 2 C 3 u 3 + C 4  1 − ( u 1 ) 2 − ( u 2 ) 2 − ( u 3 ) 2   ∆ − 1 + C 0 . T = 3 X i =1 ∂ X ∂ u i u i xx − 1 2 3 X i =1 ∂ 2 X ∂ u i ∂ u i ∆ + 6  ∂ X ∂ u i u i − X j 6 = i ∂ X ∂ u j u j  ! ∆ − 1 ( u i x ) 2 − X 1 ≤ i 1 , L 0 ⊂ L 1 ⊂ L 2 ⊂ · · · L ( n ) = [ i L i . F rom (20), (21), (22) one gets that for all k ∈ N Q 2 s − 1 l , Q 2 s ij , i, j, l = 1 , . . . , n, i < j, s ≤ k , form a basis of L 2 k , Q 2 s − 1 l , Q 2 s − 2 ij , i, j, l = 1 , . . . , n, i < j, s ≤ k , form a basis of L 2 k − 1 . This implies (45) dim( L m /L m − 1 ) =  n, if m is o dd , n ( n − 1) / 2 , if m is ev en . Consider the similar filtratio n on g ( n ) by v ector subspaces g m g 0 = 0 , g 1 = h p 1 , . . . , p n i , g m = g 1 + X i + j ≤ m [ g i , g j ] for m > 1 . Clearly , (46) ϕ ( g m ) = L m . GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 12 Com bining this with (45), w e obtain t ha t it remains to prov e (47) dim( g m / g m − 1 ) ≤  n, if m is o dd , n ( n − 1 ) / 2 , if m is ev en . Indeed, if (47) holds then (45) and (46) imply that ϕ is an isomorphism. F or n = 3 statemen t (47) was pro v ed in [19]. Below w e supp ose n ≥ 4. F or eac h k ∈ N set P 2 k ij = (ad p i ) 2 k − 1 ( p j ) , for i 6 = j, P 2 k − 1 1 = (ad p 2 ) 2 k − 2 ( p 1 ) , P 2 k − 1 i = (ad p 1 ) 2 k − 2 ( p i ) for i = 2 , 3 , . . . , n. W e will use the fo llowing notation for iterated comm utators [ a 1 a 2 . . . a k − 1 a k ] = [ a 1 , [ a 2 , [ . . . , [ a k − 1 , a k ]] . . . ] . Also, for brevit y we replace eac h generator p i b y the corresp onding index i . So, for example, [ ii [ j j k ] l k ] = [ p i , [ p i , [[ p j , [ p j , p k ]] , [ p l , p k ]]]] , P 2 k ij = [ i . . . i | {z } 2 k − 1 j ] , P 2 k − 1 1 = [2 . . . 2 | {z } 2 k − 2 1] , P 2 k − 1 i = [1 . . . 1 | {z } 2 k − 2 i ] for i = 2 , 3 , . . . , n. F or suc h an iterated commutator C of sev eral p i denote by o( C ) the n um b er of p i that app ear in C . F or example, o([ ij ]) = 2 , o([ iij ]) = 3 , o([ ii [ j j k ] l k ]) = 7 , o( P 2 k ij ) = 2 k , o( P 2 k − 1 i ) = 2 k − 1 , o([ P 2 k 1 ij , P 2 k 2 i ′ j ′ ]) = 2( k 1 + k 2 ) . F or tw o suc h iterated comm utators A, B w e write A ≃ B if A − B ∈ g m for m = max(o( A ) , o ( B )) − 1 . In particular, A ≃ 0 means that A ∈ g m for m = o( A ) − 1 . Remark 4. Denote b y ¯ C the ima g e o f C in the quotien t v ector space g o( C ) / g o( C ) − 1 . Since [ g i , g j ] ⊂ g i + j , we can consider the asso ciated graded Lie algebra gr( g ) = M i g i / g i − 1 . So, A ≃ B if and o nly if ¯ A = ¯ B in gr( g ). In Lemma 1 b elow we essen tially obtain some iden tities in the algebra gr( g ). F rom Lemma 1 b y induction on k one obtains that P 2 s − 1 l , P 2 s ij , i, j, l = 1 , . . . , n, i < j, s ≤ k , span g 2 k , P 2 s − 1 l , P 2 s − 2 ij , i, j, l = 1 , . . . , n, i < j, s ≤ k , span g 2 k − 1 , whic h implies (47). This lemma is pro v ed in the App endix. Lemma 1. If i, j, i ′ , j ′ ar e distinct inte gers fr o m { 1 , . . . , n } then for al l k 1 , k 2 ∈ Z + one has [[ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 j ]] ≃ 0 , in p articular , [ P 2 k 1 +1 j , P 2 k 2 +1 j ] ≃ 0 , (48) P 2( k 1 + k 2 +1) ij ≃ − P 2( k 1 + k 2 +1) j i , (49) GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 13 [ P 2 k 1 ij , P 2 k 2 +2 ij ] ≃ 0 for k 1 ≥ 1 , (50) [ P 2 k 1 +1 i , P 2 k 2 +1 j ] ≃ P 2( k 1 + k 2 +1) ij , (51) [ P 2 k 1 +1 i , P 2 k 2 +2 ij ] ≃ P 2( k 1 + k 2 )+3 j , (52) [ P 2 k 1 +1 i , P 2 k 2 +2 i ′ j ′ ] ≃ 0 , (53) [ P 2 k 1 ij , P 2 k 2 +2 i ′ j ′ ] ≃ 0 for k 1 ≥ 1 , (54) [ P 2 k 1 ij , P 2 k 2 +2 ij ′ ] ≃ − P 2( k 1 + k 2 +1) j j ′ for k 1 ≥ 1 . (55)  5. M iura type transforma tions The definition of Miura t ype transformations was giv en in the in tro duction. Consider an ev olutionary system of order d ≥ 1 (56) u i t = F i ( u 1 , . . . , u m , u 1 1 , . . . , u m 1 , . . . , u 1 d , . . . , u m d ) , i = 1 , . . . , m, u i k = ∂ k u i ∂ x k . Supp ose that for t his system w e ha v e a ZCR of the form [ D x + M , D t + N ] = D x ( N ) − D t ( M ) + [ M , N ] = 0 , (57) M = M ( u 1 , . . . , u m ) , N = ( u 1 , . . . , u m , u 1 1 , . . . , u m 1 , . . . , u 1 d − 1 , . . . , u m d − 1 ) with v alues in a Lie algebra g . According to Prop osition 1, suc h ZCR is determined b y a homomorphism from the WE algebra to g . Consider a homomorphism ρ from g to the Lie algebra of v ector fields on an m - dimensional manifo ld W . Let w 1 , . . . , w m b e lo cal co ordinates in W and ρ ( M ) = m X i =1 a i ( w 1 , . . . , w m , u 1 , . . . , u m ) ∂ ∂ w i , (58) ρ ( N ) = m X i =1 b i ( w 1 , . . . , w m , u 1 , . . . , u m , . . . , u 1 d − 1 , . . . , u m d − 1 ) ∂ ∂ w i . (59) Then relatio n (57) implies that t he system ∂ w i ∂ x = a i ( w 1 , . . . , w m , u 1 , . . . , u m ) , (60) ∂ w i ∂ t = b i ( w 1 , . . . , w m , u 1 , . . . , u m , . . . , u 1 d − 1 , . . . , u m d − 1 ) , (61) u i t = F i ( u 1 , . . . , u m , . . . , u 1 d , . . . , u m d ) , i = 1 , . . . , m, is consisten t. Supp ose that from equations (60) one can express lo cally (62) u i = c i  w 1 , . . . , w m , ∂ w 1 ∂ x , . . . , ∂ w m ∂ x  , i = 1 , . . . , m. Substituting this to (61), w e obtain an ev olutionary system for w 1 ( x, t ) , . . . , w m ( x, t ) con- nected b y a Miura type transformation (62) with system (56 ). This is the simplest case of a mor e general metho d t o obtain Miura ty p e transformations fro m ZCRs [1 3 ]. Let us apply this construction t o system (16) and its ZCR (17), (18), where λ 1 , . . . , λ n are again treated as complex parameters satisfying equations (2 ) and S = ( s 1 , . . . , s n ) is giv en b y form ulas ( 1 5). This Z CR tak es v alues in the Lie subalgebra so n, 1 ⊂ gl n +1 ( C ) generated by the elemen ts A i = E i,n +1 + E n +1 ,i , i = 1 , . . . , n . Consider the canonical a ction of the group G L n +1 ( C ) o n the pro jectiv e space C P n . It determines a homomorphism ϕ from gl n +1 ( C ) to the Lie algebra of v ector fields on C P n . GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 14 In the standard co o rdinates w 1 , . . . , w n of an affine c hart C n of C P n this homomorphism reads ϕ : E ij 7→ w i n X l =1 ( δ j,l − δ j,n +1 w l ) ∂ ∂ w l for i, j = 1 , . . . , n + 1 , w n +1 = 1 . In particular, (63) ϕ ( A i ) = ∂ ∂ w i − w i n X l =1 w l ∂ ∂ w l . Clearly , v ector fields (63) are tangen t to t he ( n − 1)-dimensional submanifold W = { ( w 1 , . . . , w n ) | ( w 1 ) 2 + · · · + ( w n ) 2 = 1 } ⊂ C n . On some neigh b orho o d W ′ ⊂ W of the p oint ( w 1 = · · · = w n − 1 = 0 , w n = 1 ) ∈ W w e can take w 1 , . . . , w n − 1 as lo cal co ordinates and (64) w n = p 1 − ( w 1 ) 2 − · · · − ( w n − 1 ) 2 . Th us w e obtain the follo wing homomorphism ρ from so n, 1 to the Lie algebra of v ector fields on W ′ ρ ( A i ) = ϕ ( A i )    W ′ = ∂ ∂ w i − w i n − 1 X l =1 w l ∂ ∂ w l , i = 1 , . . . , n − 1 , ρ ( A n ) = ϕ ( A n )    W ′ = − p 1 − ( w 1 ) 2 − · · · − ( w n − 1 ) 2 n − 1 X l =1 w l ∂ ∂ w l . Applying t his to (17) , one g ets (65) ρ ( M ) = n − 1 X i =1  s i λ i − w i n X j =1 s j λ j w j  ∂ ∂ w i , where one assumes ( 6 4), ( 1 5). Recall that the righ t-hand side of (60) is obtained from (58). Using (65), (1 5), (64), w e get the follow ing for m of equations (60) in our case (66) ∂ w i ∂ x = 1 1 + P n − 1 j =1 ( u j ) 2 2 u i λ i − 2 w i n − 1 X j =1 λ j u j w j − w i λ n p 1 − ( w 1 ) 2 − · · · − ( w n − 1 ) 2  1 − n − 1 X j =1 ( u j ) 2  ! , i = 1 , . . . , n − 1 . Computing ρ ( N ) b y means of (18) and (6 5), from t he co efficien ts of the v ector field (59) w e obtain equations (61) of the follo wing form (67) ∂ w i ∂ t = n − 1 X i =1 s i xx λ i − w i n X j =1 s j xx λ j w j + n X j =1 λ i λ j w j ( s j x s i − s i x s j )+  r 1 + λ 2 1 + 1 2 ( S, RS ) + 3 2 ( S x , S x )  s i λ i − w i n X j =1 s j λ j w j  ! ∂ ∂ w i , i = 1 , . . . , n − 1 , where S = ( s 1 , . . . , s n ) is given b y (15) a nd w n is equal to (64). GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 15 Denote b y a i = a i ( w 1 , . . . , w n − 1 , u 1 , . . . , u n − 1 ) the right-hand side of (6 6). It is easily seen that the ( n − 1) × ( n − 1)-mat r ix || ∂ a i /∂ u j || at the p oint (68) w 1 = · · · = w n − 1 = u 1 = · · · = u n − 1 = 0 is equal to diag (2 λ 1 , . . . , 2 λ n − 1 ). Supp o se that (69) λ i 6 = 0 , i = 1 , . . . , n − 1 , then, b y t he implicit function theorem, on a neigh b orho o d of the p oint (68) from equa- tions (66) w e can express (70) u i = c i  w 1 , . . . , w n − 1 , ∂ w 1 ∂ x , . . . , ∂ w n − 1 ∂ x  , i = 1 , . . . , n − 1 . Substituting (70) t o (67), one gets an ev olutionary system of the form (71) w i t = H i ( w 1 , . . . , w n − 1 , . . . , w 1 xxx , . . . , w n − 1 xxx ) , i = 1 , . . . , n − 1 , connected with system (1 6) b y the Miura t yp e transformation (7 0 ). It r emains t o study the case when (69) do es not hold. That is, λ k = 0 for some 1 ≤ k ≤ n − 1 . Since in equations ( 2) one has r i 6 = r j for i 6 = j , w e obtain that λ i 6 = 0 for i 6 = k . Then at any p oin t of the form (72) u k = 1 , u l = w l = 0 ∀ l 6 = k one has ∂ a i ∂ u j = 0 ∀ i 6 = j, ∂ a i ∂ u i = 2 λ i ∀ i 6 = k , ∂ a k ∂ u k = 2 w k λ n p 1 − ( w k ) 2 . Therefore, at an y po int of the form ( 72) with w k 6 = 0 , ± 1 the matrix || ∂ a i /∂ u j || is no n- singular, and on a neigh b orho o d of suc h p o in t fro m equations (66) w e can again g et expressions of the form (70) and pro ceed as describ ed ab ov e. Appendix: pr oof of Lemma 1 W e prov e this b y induction on k 1 + k 2 . F or k 1 + k 2 = 0 (that is, k 1 = k 2 = 0) the statemen t s follow easily from (42) and (43). Assume that all the statemen ts are v alid for k 1 + k 2 ≤ m for some m ∈ Z + . W e m ust pro v e them f or k 1 + k 2 = m + 1. Belo w l is an arbitrary in t eger suc h tha t 1 ≤ l ≤ n , l 6 = i , l 6 = j . W e use the notation from Section 4. Pro of of (48) . First, note that b y the induction assumption for q ≤ m one ha s [ ll i . . . i | {z } 2 q j ] = [ l l [ P 1 i P 2 q ij ]] ≃ [ l l P 2 q +1 j ] = [ l [ P 1 l P 2 q +1 j ]] ≃ [ l P 2 q +2 lj ] = [ P 1 l P 2 q +2 lj ] ≃ P 2 q +3 j . Similarly , w e obtain (73) [ ll i . . . i | {z } 2 q j ] ≃ [ i . . . i | {z } 2 q +2 j ] ≃ [ l . . . l | {z } 2 q +2 j ] ≃ P 2 q +3 j ∀ q ≤ m. Without loss of generality we can assume k 2 ≥ 1 in ( 48). Using(73) and the Jacobi iden tit y , one gets (74) [[ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 j ]] ≃ [[ i . . . i | {z } 2 k 1 j ][ l l i . . . i | {z } 2 k 2 − 2 j ]] = − [[ l i . . . i | {z } 2 k 1 j ][ l i . . . ij ]] + [ l [ i . . . i | {z } 2 k 1 j ][ l i . . . ij ]] ≃ [[ ll i . . . i | {z } 2 k 1 j ][ i . . . ij ]] − 2[ l [ l i . . . i | {z } 2 k 1 j ][ i . . . ij ]] . GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 16 T o obta in the la st line, w e used the fact tha t [ l [ i . . . i | {z } 2 k 1 j ][ l i . . . i | {z } 2 k 2 − 2 j ]] ≃ − [ l [ l i . . . i | {z } 2 k 1 j ][ i . . . ij ]] , b ecause [[ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 − 2 j ]] ≃ 0 by the induction assumption. Since, b y (73), [ l l i . . . i | {z } 2 k 1 j ] ≃ [ i . . . i | {z } 2 k 1 +2 j ], from (74) one obtains [[ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 j ]] ≃ [[ i . . . i | {z } 2 k 1 +2 j ][ i . . . i | {z } 2 k 2 − 2 j ]] − 2[ l [ l i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 − 2 j ]] . If k 2 ≥ 2, applying the same pro cedure to [[ i . . . i | {z } 2 k 1 +2 j ][ i . . . i | {z } 2 k 2 − 2 j ]] yields [[ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 j ]] ≃ [[ i . . . i | {z } 2 k 1 +4 j ][ i . . . i | {z } 2 k 2 − 4 j ]] − 2[ l [ l i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 − 2 j ]] − 2[ l [ l i . . . i | {z } 2 k 1 +2 j ][ i . . . i | {z } 2 k 2 − 4 j ]] . Th us applying this pro cedure sev eral times to the first summand of the right-hand side, w e obtain (75) [[ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 j ]] ≃ [[ i . . . i | {z } 2( k 1 + k 2 ) j ] j ] − 2 k 2 X s =1 [ l [ l i . . . i | {z } 2( k 1 + s − 1) j ][ i . . . i | {z } 2( k 2 − s ) j ]] . By the induction a ssumption and (73), one has [[ l i . . . i | {z } 2( k 1 + s − 1) j ] i ] ≃ [[ P 1 l P 2( k 1 + s ) − 1 j ] P 1 i ] ≃ [ P 2( k 1 + s ) lj P 1 i ] ≃ 0 . Therefore, (76) [ l [ l i . . . i | {z } 2( k 1 + s − 1) j ][ i . . . i | {z } 2( k 2 − s ) j ]] ≃ [ l i . . . i | {z } 2( k 2 − s ) [ l i . . . i | {z } 2( k 1 + s − 1) j ] j ]] = − [ l i . . . i | {z } 2( k 2 − s ) j l i . . . i | {z } 2( k 1 + s − 1) j ] Since, by the induction assumption and (73), [ l i . . . i | {z } 2( k 2 − s ) j ] ≃ [ P 1 l P 2( k 2 − s )+1 j ] ≃ [ P 2( k 2 − s +1) lj ] ≃ − [ P 1 j P 2( k 2 − s )+1 l ] ≃ − [ j i . . . i | {z } 2( k 2 − s ) l ] , [ ll i . . . i | {z } 2( k 1 + s − 1) j ] ≃ [ i . . . i | {z } 2( k 1 + s ) j ] , equation (76) implies [ l [ l i . . . i | {z } 2( k 1 + s − 1) j ][ i . . . i | {z } 2( k 2 − s ) j ]] ≃ −  l i . . . i | {z } 2( k 2 − s ) j l i . . . i | {z } 2( k 1 + s − 1) j  ≃  j i . . . i | {z } 2( k 2 − s ) ll i . . . i | {z } 2( k 1 + s − 1) j  ≃  j i . . . i | {z } 2( k 1 + k 2 ) j  Com bining this with (75) yields (77)  [ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 j ]  ≃ −  j i . . . i | {z } 2( k 1 + k 2 ) j  − 2 k 2  j i . . . i | {z } 2( k 1 + k 2 ) j  . F or k 1 = 0 this equation implies [ j i . . . i | {z } 2( k 1 + k 2 ) j  ≃ 0. Combin g this with (77), w e obtain (48). Pro of of (49) . By the induction assumption, (48), a nd (42), [ i . . . i | {z } 2 m +1 j ] ≃ − [ j . . . j | {z } 2 m +1 i ] , [ i j . . . j | {z } 2 m i ] ≃ 0 , [[ iij ] j ] ≃ 0 , [ iij ] ≃ [ l l j ] , [ j j i ] ≃ [ l l i ] . Using this and (43) , one gets GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 17 P 2( k 1 + k 2 +1) ij = [ i . . . i | {z } 2 m +3 j ] = [ ii i . . . i | {z } 2 m +1 j ] ≃ − [ ii j . . . j | {z } 2 m +1 i ] ≃ − [[ iij ] j . . . j | {z } 2 m i ] ≃ − [ j . . . j | {z } 2 m [ iij ] i ] = [ j . . . j | {z } 2 m iiij ] ≃ [ j . . . j | {z } 2 m ill j ] = [ j . . . j | {z } 2 m [ il ] l j ] = [ j . . . j | {z } 2 m [[ il ] l ] j ] = − [ j . . . j | {z } 2 m j l l i ] ≃ − [ j . . . j | {z } 2 m j j j i ] = − P 2( k 1 + k 2 +1) j i . Pro of of (50) . By the Jacobi iden tit y , (78) [ P 2 k 1 ij , P 2 k 2 +2 ij ] = [[ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 +1 j ]] = − [[ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 j ]] + [ i [ i . . . i | {z } 2 k 1 − 1 j ][ i . . . i | {z } 2 k 2 j ]] . By the induction a ssumption and (73), [[ i . . . i | {z } 2 k 1 − 1 j ][ i . . . i | {z } 2 k 2 j ]] ≃ [ P 2 k 1 ij P 2 k 2 +1 j ] ≃ P 2( k 1 + k 2 )+1 i . Substituting this to (7 8) and using (48), w e o btain [ P 2 k 1 ij , P 2 k 2 +2 ij ] ≃ − [[ i . . . i | {z } 2 k 1 j ][ i . . . i | {z } 2 k 2 j ]] + [ i, P 2( k 1 + k 2 )+1 i ] ≃ 0 . Pro of of (51) . By (73) and the induction assumption of (53), f o r any q 1 , q 2 ∈ Z + suc h that q 1 + q 2 ≤ 2( k 1 + k 2 ) − 1 and q 1 + q 2 is o dd one has [[ l . . . l | {z } q 1 i ][ l . . . l | {z } q 2 j ]] ≃ 0 . Using this and the Jacobi iden tit y , w e get [[ l . . . l | {z } 2 k 1 i ][ l . . . l | {z } 2 k 2 j ]] ≃ − [[ l . . . l | {z } 2 k 1 +1 i ][ l . . . l | {z } 2 k 2 − 1 j ]] ≃ [[ l . . . l | {z } 2 k 1 +2 i ][ l . . . l | {z } 2 k 2 − 2 j ]] ≃ · · · ≃ [[ l . . . l | {z } 2( k 1 + k 2 ) i ] j ] = − [ j l . . . l | {z } 2( k 1 + k 2 ) i ] Com bining this with (73) and (49) yields [ P 2 k 1 +1 i , P 2 k 2 +1 j ] ≃ [[ l . . . l | {z } 2 k 1 i ][ l . . . l | {z } 2 k 2 j ]] ≃ − [ j l . . . l | {z } 2( k 1 + k 2 ) i ] ≃ − [ j j . . . j | {z } 2( k 1 + k 2 ) i ] = − P 2( k 1 + k 2 +1) j i ≃ P 2( k 1 + k 2 +1) ij . Pro of of (52) . Consider first the case k 1 = 0, then [ P 2 k 1 +1 i , P 2 k 2 +2 ij ] = [ i i . . . i | {z } 2 k 2 +1 j ] . Set c = 1 if j 6 = 1 and c = 2 if j = 1. If i = c then [ i i . . . i | {z } 2 k 2 +1 j ] = P 2( k 1 + k 2 )+3 j and the pro of is complete. Supp ose that i 6 = c , then using (49 ) one gets [ P 2 k 1 +1 i , P 2 k 2 +2 ij ] = [ i i . . . i | {z } 2 k 2 +1 j ] ≃ − [ i j . . . j | {z } 2 k 2 +1 i ] . Since, by (4 8), [ i j . . . j | {z } 2 k 2 i ] ≃ 0, using (73) and [[ ij ] c ] = 0 w e obtain GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 18 [ P 2 k 1 +1 i , P 2 k 2 +2 ij ] ≃ − [ i j . . . j | {z } 2 k 2 +1 i ] ≃ − [[ ij ] j . . . j | {z } 2 k 2 i ] ≃ − [[ ij ] c . . . c | {z } 2 k 2 i ] = − [ c . . . c | {z } 2 k 2 [ ij ] i ] = [ c . . . c | {z } 2 k 2 [ iij ]] ≃ [ c . . . c | {z } 2 k 2 [ ccj ]] = P 2( k 1 + k 2 )+3 j . No w consider the case k 1 ≥ 1. By (73), f o r any l 6 = i , l 6 = j we ha v e [ P 2 k 1 +1 i , P 2 k 2 +2 ij ] = − [ P 2 k 2 +2 ij , P 2 k 1 +1 i ] ≃ − [[ i . . . i | {z } 2 k 2 +1 j ][ l . . . l | {z } 2 k 1 i ] . Since due to the induction a ssumption of (53) o ne has [[ i . . . i | {z } 2 k 2 +1 j ] l ] ≃ 0, using (73) w e get [ P 2 k 1 +1 i , P 2 k 2 +2 ij ] ≃ − [[ i . . . i | {z } 2 k 2 +1 j ][ l . . . l | {z } 2 k 1 i ] ≃ − [ l . . . l | {z } 2 k 1 [ i . . . i | {z } 2 k 2 +1 j ] i ] = [ l . . . l | {z } 2 k 1 i . . . i | {z } 2 k 2 +2 j ] ≃ [ l . . . l | {z } 2( m +2) j ] . In the consideration of the case k 1 = 0 w e show ed tha t for any i 6 = j [ i . . . i | {z } 2( m +2) j ] ≃ P 2 m +5 j . (Recall that m = k 1 + k 2 − 1 ). Therefore, [ l . . . l | {z } 2( m +2) j ] ≃ P 2( k 1 + k 2 )+3 j . Pro of of (53) . Consider first the case k 1 ≥ 1. Since n ≥ 4, there is l ∈ { 1 , . . . , n } suc h that l 6 = i , l 6 = i ′ , l 6 = j ′ . Then, by the induction assumption of (53), [[ i ′ . . . i ′ | {z } 2 k 2 +1 j ′ ] l ] ≃ 0 , [[ i ′ . . . i ′ | {z } 2 k 2 +1 j ′ ] i ] ≃ 0 . Using this and (73) , one gets (79) [ P 2 k 1 +1 i , P 2 k 2 +2 i ′ j ′ ] = − [ P 2 k 2 +2 i ′ j ′ , P 2 k 1 +1 i ] ≃ [[ i ′ . . . i ′ | {z } 2 k 2 +1 j ′ ] l . . . l | {z } 2 k 1 i ] ≃ [ l . . . l [ i ′ . . . i ′ | {z } 2 k 2 +1 j ′ ] i ] ≃ 0 . If we set k 2 = 0 then (79) implies that fo r any distinct inte gers c 1 , c 2 , c 3 , c 4 ∈ { 1 , . . . , n } (80) [[ c 1 c 2 ] c 4 . . . c 4 | {z } 2 m +2 c 3 ] ≃ 0 . Since [ c 4 . . . c 4 | {z } 2 m +2 c 3 ] ≃ [ c 2 . . . c 2 | {z } 2 m +2 c 3 ] due to (73), from (80) we obtain (81) [[ c 1 c 2 ] c 2 . . . c 2 | {z } 2 m +2 c 3 ] ≃ 0 . By the Jacobi iden tit y , ( 81), and (73), (82) [ c 1 c 2 . . . c 2 | {z } 2 m +3 c 3 ] ≃ [ c 2 c 1 c 2 . . . c 2 | {z } 2 m +2 c 3 ] ≃ [ c 2 c 1 . . . c 1 | {z } 2 m +3 c 3 ] . Also, prop ert y (49) implies (83) [ c 1 c 2 . . . c 2 | {z } 2 m +3 c 3 ] ≃ − [ c 1 c 3 . . . c 3 | {z } 2 m +3 c 2 ] . It remains to study the case k 1 = 0. Using (82) and (83), w e get [ P 2 k 1 +1 i , P 2 k 2 +2 i ′ j ′ ] = [ i i ′ . . . i ′ | {z } 2 k 2 +1 j ′ ] ≃ [ i ′ i . . . i | {z } 2 k 2 +1 j ′ ] ≃ − [ i ′ j ′ . . . j ′ i ] ≃ − [ j ′ i ′ . . . i ′ i ] ≃ [ j ′ i . . . ii ′ ] ≃ [ ij ′ . . . j ′ i ′ ] ≃ − [ ii ′ . . . i ′ j ′ ] = − [ P 2 k 1 +1 i , P 2 k 2 +2 i ′ j ′ ] . Therefore, [ P 2 k 1 +1 i , P 2 k 2 +2 i ′ j ′ ] ≃ 0 . GENERALIZED LA NDA U-LIFSHI TZ S YSTEMS A ND LIE ALGEBRAS 19 Pro of of (54) . By (53), [ i, P 2 k 2 +2 i ′ j ′ ] ≃ 0 , [ j, P 2 k 2 +2 i ′ j ′ ] ≃ 0 . Ob viously , this implies (54). Pro of of (55 ) . By pro p ert y (53) , [[ i . . . i | {z } 2 k 1 − 1 j ] j ′ ] ≃ 0 . Using this, equation (73), and (49), one obtains [ P 2 k 1 ij , P 2 k 2 +2 ij ′ ] ≃ − [ P 2 k 1 ij , P 2 k 2 +2 j ′ i ] = − [[ i . . . i | {z } 2 k 1 − 1 j ] j ′ . . . j ′ | {z } 2 k 2 +1 i ] ≃ − [ j ′ . . . j ′ | {z } 2 k 2 +1 [ i . . . i | {z } 2 k 1 − 1 j ] i ] = [ j ′ . . . j ′ | {z } 2 k 2 +1 i . . . i | {z } 2 k 1 j ] ≃ [ j ′ . . . j ′ | {z } 2 k 2 +1 j ′ . . . j ′ | {z } 2 k 1 j ] = P 2( k 1 + k 2 +1) j ′ j ≃ − P 2( k 1 + k 2 +1) j j ′ . A ckno wledgements The authors thank T. Skrypn yk and V. V. Sok olo v for helpful discus sions. W ork of S.I. is supp o rted b y the NW O VENI grant 639 .031.515. JvdL is partially supp orted b y the Europ ean Union through t he FP6 Marie Curie G ran t (ENIGMA) and the Europ ean Science F oundation (MISGAM). S.I. thanks the Max Planck Institute for Mathematics (Bonn, German y) for its hospitalit y and excellen t w orking conditio ns during 02.2006 - 01.2007, when a part of this researc h was done. Reference s [1] M. Ju. Balakhnev. 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Deformations of lo op a lgebras and integrable systems : hierarchies of integrable equa- tions. J. Math. Ph ys. 45 (2004), 457 8–45 9 5. [26] H. D. W ahlquist and F. B. Estabr o ok. Prolo ngation structures of nonlinear ev olution equatio ns . J. Math. Phys. 16 (197 5), 1–7 . Serg ey Igonin, Dep ar tment of Ma thema tics, University o f Utrecht, P.O. Box 8001 0 , 3508 T A Utrecht, The N etherlands E-mail addr ess : igonin @mccm e.ru Johan v an de Leur, Dep ar tment of Ma thema tics, U niversity of U trecht, P. O. Box 80010, 3508 T A Utrecht, The N etherlands E-mail addr ess : J.W.va ndeLe ur@uu .nl Gianni Manno , U n iv ersit ` a degli Studi di Milano-Bicocca, Dip ar timento di Ma tema tica e Applicazioni, Via Cozzi 53, 20125 Milano, It al y E-mail addr ess : gianni .mann o@uni mib.it Vladimir Trushk o v, Un iversity of Peresla vl, So vetska y a 2, 152020 Peresla vl-Zalessky, Y arosla vl region, Russia E-mail addr ess : vladim ir@tr ushko v.pereslavl.ru