Particle-like, dyx-coaxial and trix-coaxial Lie algebra structures for a multi-dimensional continuous Toda type system
aa r X i v : . [ n li n . S I] A ug Particle-like, dyx-coaxial and trix-coaxial Lie algebra structuresfor a multi-dimensional continuous Toda type system ∗ Marcella Palese † and Ekkehart Winterroth ‡ Department of Mathematics, University of Torinovia C. Alberto 10, 10123 Torino, Italye–mail: [email protected], [email protected]
Abstract
We prove that with a (2 + 1)-dimensional Toda type system areassociated algebraic skeletons which are (compatible assemblings) ofparticle-like Lie algebras of dyons and triadons type. We obtain trix-coaxial and dyx-coaxial Lie algebra structures for the system from al-gebraic skeletons of some particular choice for compatible associatedabsolute parallelisms. In particular, by a first choice of the absoluteparallelism, we associate with the (2 + 1)-dimensional Toda type sys-tem a trix-coaxial Lie algebra structure made of two (compatible) basetriadons constituting a 2-catena. Furthermore, by a second choice ofthe absolute parallelism, we associate a dyx-coaxial Lie algebra struc-ture made of two (compatible) base dyons, as well as particle-like Liealgebra structures made of single 3-dyons. Some explicit examples ofapplications such as conservation laws related to special solutions, andan inverse spectral problem are worked out.
Key words : Particle-like Lie algebra structure, infinitesimal skeleton, tower,Toda system. ∗ Dedicated to Hartwig in occasion of his eighteenth birthday. † Corresponding Author ‡ Lepage Research Institute, 17. novembra 1, 08116 Preˇsov, Slovak Republic . Palese and E. Winterroth Toda type systems are nonlinear models which play a role in a variety ofphysical and, more in general, natural phenomena.The problem of integrability of nonlinear models has been recognized tobe related to their algebraic properties in discrete and continuous, as well as,classical and quantum formulations. Algebraic properties can be interpretedas the counterpart of the concept of integrability given as of having ‘enough’conservation laws to exhaustively describe the underling field or associateddynamics. Indeed, from an historical point of view, algebraic-geometric ap-proaches are based on the requirement for the existence of conservation lawswhich emerge from internal symmetries (given in terms of algebraic struc-tures).In the Seventies, in fact, Wahlquist and Estabrook [35, 5] proposed atechnique for systematically deriving, from an integrable system, what theycalled a ‘prolongation structure’ in terms of a set of ‘pseudopotentials’ relatedto the existence of an infinite set of associated conservation laws. They alsoconjectured that, as a characterizing feature of the integrability property, thestructure was ‘open’ i.e . not a set of structure relations of a finite–dimensionalLie group. Since then, ‘open’ Lie algebras have been extensively studiedin order to distinguish them from freely generated infinite-dimensional Liealgebras.Their interest in the study of integrability is in the fact that Lax pairsof the inverse spectral transform containing an isospectral parameter can beobtained by an homomorphism of the infinite-dimensional open Lie algebrain a finite-dimensional ‘closed’ Lie algebra. In their approach, conservationlaws are written in terms of ‘prolongation’ forms and integrability is intendedas a Frobenius integrability condition for a ‘prolonged’ ideal of differentialforms describing intrinsically the given nonlinear model in the sense of ´E.Cartan.Attempting a description of symmetries in terms of Lie algebras impliesthe appearance of an homogeneous space and thus the interpretation of pro-longation forms as Cartan–Ehresmann connections . It is clear that here theunknowns are both conservation laws and symmetries, and the main point inthis is how to realize the form of the conservation laws and thus the explicitexpression of the prolongation forms. Different prolongation ideals give riseto both different algebraic structures (symmetries) and corresponding conser-vation laws. By an inverse procedure based on the intrinsic duality between . Palese and E. Winterroth towers with infinitesimal algebraic skeletons (in the sense of [16])and we will refer to that framework in this paper. It is noteworthy thatslight modification of the internal symmetry properties generates new mod-els which can contain possible integrable subcases. For example, activator-substrate systems have been obtained by performing a slight modification ofthe internal symmetry algebra of twisted reaction-diffusion equations [22].The structure itself with which the tower forms are postulated can pro-duce open algebraic structures or just Lie algebras. Our aim in this work isto investigate some common features of them and to show the emerging ofparticle-like Lie algebras structure as symmetry structures of integrable sys-tems (associated with Poisson structures the compatibility of which is worthyof study [6, 12]). Indeed, in general, infinite dimensional open Lie algebras arethe main object of the search in view of the application of the inverse spectraltransform to obtain soliton solutions, B¨acklund transformations and so on;recent examples of applications can be found e.g . in [9, 10, 18, 36, 37, 38].Although these features are not our prominent task in this paper, an inverseproblem will be obtained in Section 3.3.1.Prolongation forms bringing to finite dimensional Lie algebras (withouta spectral parameter) are generally discarded when searching for a Lax pairto be used within the inverse spectral transform.However, integrable systems, admitting infinite-dimensional prolongationLie algebras can also admit finite-dimensional Lie algebras, which still canbe related to some kind of internal symmetries of the systems themselvesand to associated conservation laws, or even to B¨acklund transformations.We refer, in particular, to the paper “More prolongation structures” by C.Hoenselaers [8], which pointed out two important features of the algebraicstructures obtained by the method of Wahlquist and Estabrook.First feature: it can be that the prolongation forms can not always besolved in such a way that one obtains commutators among vector fields de-pending only on the ‘pseudopotentials’ coordinates. A typical example is, infact, the most general prolongation problem associated by such a procedureto equation (1): in [23] the prolongation problem was formally solved by in-troducing suitable operators of Bessel type, however a prolongation algebra . Palese and E. Winterroth equations, linear problems, B¨acklund transformations, arelocal coordinates expressions of common intrinsic structures ; this is of helpalso in practical questions: solutions of systems can be obtained by simplersystems having in common (part of) skeletons. In Section 3.2.1 we show thateach one of the two compatible 4-triadon constituting the skeleton given bya trix-coaxial Lie algebra structure generates the same conservation law andrelated special solutions. This justifies the possible choice of a more restricted(instead of the most general one) form of the tower (then of the algebraicskeleton) still getting ‘solutions’ (with this term meaning analytical solutionsas well as particular conservation laws) of the original equation.Second feature: even if the prolongation algebra is a finite dimensional(even abelian in his example) Lie algebra, nevertheless there can exist B¨acklundtransformations. It is shown that the exterior differential associated to theprolongation structure of the NLS equation being of genus 3, and it is stressedthat we can choose dependent and independent variables in an arbitray way.We can also lower the genus (in our skeleton formulation this means thechoice of different representations ρ or even different vector spaces V ) so ob-taining a reduced ideal where one of the independent coordinates is turnedinto a dependent coordinate. This turning a global symmetry in a local oneprovides Miura type transformations between the modified NLS and anothersystem, the prolongation structure of which is finite dimensional and there isonly one nontrivial potential entering a B¨acklund transformation acting onthe modified NLS equation.In few words the Wahlquist-Estabrook method not always produces infi-nite dimensional open Lie algebras, but it could be that by that ‘procedure’we get only a part of an algebraic skeleton. Therefore we can not automat-ically infer that, being the prolongation structure finite dimensional, thenthe system is not integrable. The results in [8] are a counterexample, which . Palese and E. Winterroth particle-likeLie algebra structures [33, 34].Vinogradov developed a completely abstract theory of compatibility ofLie algebra structures starting from the corresponding compatibility theoryof Poisson structures. Although the mathematical aspects of the theory arequite involved the nice point is that simple criteria of compatibility or noncompatibility have been obtained which somehow have a certain grade ofautomatism.Furthermore, as for the physical side, Vinogradov speculated that thisparticle-like structures could be related to the ultimate particle structure ofthe matter: he noted that since‘the symmetry algebra u (2) = so (3) of a nucleon can be assembledin one step from three triadons [...] one might think that thisstructure of the symmetry reflects the fact that a nucleon is madefrom three “quarks” ’.This is of course only a speculation, but it also suggests a quite fascinatingnew perspective on internal symmetries of integrable systems. (2 + 1) dimensions Consider the (2 + 1)-dimensional system, a continuous (or long-wave) ap-proximation of a spatially two-dimensional Toda lattice [31]: u xx + u yy + ( e u ) zz = 0 , (1)where u = u ( x, y, z ) is a real field, x, y, z are real local coordinates (if we want, z playing the rˆole of a ‘time’) and the subscripts mean partial derivatives. It . Palese and E. Winterroth γ → ∞ of the more general model u xx + u yy + (cid:2) (1 + u/γ ) γ − (cid:3) zz = 0covering (for γ = 0 ,
1) various continuous approximations of lattice models,among them the Fermi-Pasta-Ulam ( γ = 3) [1]. This model is almost ubiq-uitus , it appears in differential geometry; in mathematical and theoreticalphysics (Newman and Penrose); in the theory of Hamiltonian systems; ingeneral relativity; in the large n limit of the sl ( n ) Toda lattice; in extendedconformal symmetries, and theory of gravitational instantons; in strings the-ory and statistical mechanics etc. (see e.g . [3, 13, 27, 29]).It can be seen as the particular case with d = 1 of so-called 2 d -dimensionalToda-type systems [30] obtained from a ‘continuum Lie algebra’ by means of azero curvature representation u w ¯ w = K ( e u ), (in our particular case w = x + iy and K is the differential operator given by K = ∂ ∂z ). In particular, it hasbeen studied in the context of symmetry reductions [2, 7] and a (1 + 1)-dimensional version in the context of prolongation structures [1]. The (2+1)-dimensional system has been associated with a Kaˇc–Moody Lie algebra andrelated to Saveliev’s continuum Lie algebras of particular kind [25].The Toda system (1) can be put in the complex form ∂ ζ ∂ ¯ ζ u = − / ∂ z e u , by the transformations ζ = g ( η ), ¯ ζ = ¯ g (¯ η ), u = ˜ u − ln( g ′ ¯ g ′ ), where ζ = x + iy ,¯ ζ = x − iy , ∂ ζ = ( ∂ x − i∂ y ), ∂ ¯ ζ = ( ∂ x + i∂ y ), g ′ = g η ( η ), ¯ g ′ = g ¯ η (¯ η ) and g ( η ) is an arbitrary holomorphic function of η = x ′ + iy ′ . A Lax pair for thiscomplex form of the 2 D Toda equation has been found; see e.g . Manakovand Santini [15] and references therein; original references are [30], as well as[39].
Let us first recall a few mathematical tools constituting the background fora detailed treatment of which we refer to [24, 25, 26] and [16, 28].From one side global properties of partial differential equations such asinternal symmetries and invariance properties having an issue in dynamicscan be described by mathematical tools which enable us to deal with globalproperties at large scales, connecting local data to global ones. On the otherside transformations of configurations of a system can be globally studied by . Palese and E. Winterroth i.e . that of an algebraicskeleton E = g ⊕ V on a finite-dimensional vector space V , with g a possiblyinfinite dimensional Lie algebra. The further step is introducing a tower withsuch a skeleton.An algebraic skeleton on a finite-dimensional vector space V is a triple( E , G , ρ ), with G a (possibly infinite-dimensional) Lie group, E = g ⊕ V isa (possibly infinite-dimensional) vector space not necessarily equipped witha Lie algebra structure , g is the Lie algebra of G , and ρ is a representationof g on E such that it reduces to the adjoint representation of g on itself.The fact that E is not a direct sum of Lie algebras, but an open algebraicstructure is fundamental in order to be able to generate whole families ofnonlinear differential systems, starting from it.We now consider a suitably constructed differentiable structure which issomewhat modelled on the skeleton above. Let us introduce a differentiablemanifold P on which a Lie group G , with Lie algebra g , acts on the right; P is a principal bundle P → Z ≃ P / G . By construction, we have that Z is a manifold of type V , i.e . ∀ z ∈ Z , T z Z ≃ V .Suppose we have a way to define a representation ρ of the Lie algebra g on T z Z ≃ V , in such a way that it could be possible under certain conditionsto find a homomorphism between the open infinite dimensional Lie algebra,constructed by ρ , and a quotient Lie algebra. Let us call k the (possiblyinfinite dimensional) Lie algebra obtained as the direct sum of such a quotientLie algebra with g . From the differentiable side, a tower P ( Z , G ) on Z withskeleton ( E , G , ρ ) is an absolute parallelism ω on P valued in E , invariantwith respect to ρ and reproducing elements of g from the fundamental vectorfields induced on P , i.e . R ∗ g ω = ρ ( g ) − ω , for g ∈ G ; ω ( ˜ A ) = A , for A ∈ g ;here R g denotes the right translation and ˜ A the fundamental vector fieldinduced on P from A . In general, the absolute parallelism does not define aLie algebra homomorphism. . Palese and E. Winterroth k be a Lie algebra and g a Lie subalgebra of k . Let G be aLie group with Lie algebra g and P ( Z , G ) be a principal fiber bundle withstructure group G over a manifold Z as above. A Cartan connection in P oftype ( k , G ) is a 1–form ω on P with values in k such that ω | T p P : T p P → k is an isomorphism ∀ p ∈ P , R ∗ g ω = Ad ( g ) − ω for g ∈ G and reproducingelements of g from the fundamental vector fields induced on P . It is clearthat a Cartan connection ( P , Z , G , ω ) of type ( k , G ) is a special case of atower on Z .The vector space V is finite dimensional and generated by some of thevector fields in the prolongation structure. It has the property that eachbracket of some of remaining vector fields of the prolongation structure (freelygenerating an infinite dimensional Lie algebra g ) with its generators is againin V . In particular unknown commutators in the freely generated Lie algebraare related in such a way that their assigned relations are elements of V .As an example of application of such an abstract formulation to the realworld we refer e.g . to [22], whereby activator-substrate systems have beenobtained by performing a slight modification of the internal symmetry alge-bra of twisted reaction-diffusion equations: the necessary condition for thegeneration of stable patterns (related to general integrability properties inthe limit of a null normalized diffusion constant) are formulated in terms of‘closeness’ properties within the symmetry algebra vector space.Following [25], we recall how to get both some skeletons and towers overthem associated with the system (1).On a manifold with local coordinates ( x, y, z, u, p, q, r ), we introduce theclosed differential ideal defined by the set of 3–forms: θ = du ∧ dx ∧ dy − rdx ∧ dy ∧ dz , θ = du ∧ dy ∧ dz − pdx ∧ dy ∧ dz , θ = du ∧ dx ∧ dz + qdx ∧ dy ∧ dz , θ = dp ∧ dy ∧ dz − dq ∧ dx ∧ dz + e u dr ∧ dx ∧ dy + e u r dx ∧ dy ∧ dz . It iseasy to verify that on every integral submanifold defined by u = u ( x, y, z ), p = u x , q = u y , r = u z , with dx ∧ dy ∧ dz = 0, the above ideal is equivalentto the Toda system under study.By an ansatz first introduced in [24], we look for suitable 2–forms (gen-erating associated conservation laws)Ω k = θ km ∧ ω m where θ km = − ˆ A km dx − ˆ B km dy − ˆ C km dz , with ˆ A km , ˆ B km , ˆ C km elements of N × N constant regular matrices, and the absolute parallelism forms are given by ω m = d ˆ ξ m + ˆ F m dx + ˆ G m dy + ˆ H m dz , (2) . Palese and E. Winterroth i.e . Ω k = H k ( u, u x , u y , u z ; ξ m ) dx ∧ dy + F k ( u, u x , u y , u z ; ξ m ) dx ∧ dz + (3)+ G k ( u, u x , u y , u z ; ξ m ) dy ∧ dz + A km dξ m ∧ dx + B km dξ m ∧ dz + dξ k ∧ dy , where ξ = { ξ m } , k, m = 1 , , . . . , N (N arbitrary), and H k , F k and G k are,respectively, the pseudopotentials and functions to be determined, while A km and B km denote the elements of two N × N constant regular matrices relatedto the previous ones and we have rescaled the coordinates ξ k . In particularnote that (see also [24, 20]) F k = ˆ C km ˆ F m − ˆ A km ˆ H m , (4) G k = ˆ C km ˆ G m − ˆ B km ˆ H m , (5) H k = ˆ B km ˆ F m − ˆ A km ˆ G m , (6) ξ k = ˆ C km ˆ ξ m . (7)The integrability condition for the ideal generated by forms θ j and Ω k finally yields H k = e u u z L k ( ξ m ) + P k ( u, ξ m ) , (8) F k = − u y L k ( ξ m ) + Q k ( u, ξ m ) , (9) G k = u x L k ( ξ m ) + M k ( u, ξ m ) , (10)where L k , P k , Q k , M k are functions of integration.It turns out that Q k ( u, ξ m ) can be written in terms of the others. In-deed we have (see e.g . [17, 32]) H k = A km G m − B km F m so that e u u z L k ( ξ l ) + P k ( u, ξ l ) = A km ( u x L m ( ξ l ) + M m ( u, ξ l )) − B km ( − u y L m ( ξ l ) + Q m ( u, ξ l )), i.e .B km Q mu ( u, ξ l ) = e u u z L k ( ξ l ) + P ku ( u, ξ l ) + A km M mu ( u, ξ l ), which can be inte-grated once the dependence on of P k ( u, ξ l ) and M k ( u, ξ l ) on u is given. As aconsequence, the desired representation ρ for the skeleton is provided by thefollowing equations (we omit the indices for simplicity) [23, 25]. P u = e u [ L, M ] , M u = − [ L, P ] , [ M, P ] = 0 . (11)Note that here L depends only on ξ m , while P and M still have a dependenceon u determined by the first two differential equations. A tower with P and M . Palese and E. Winterroth L has been obtained by suitable operator Bessel coefficients [23].Note that formally this tower shall provide the Lax pair of an inversespectral problem; however, it is a non trivial task to characterize explicitlyits algebraic skeleton by means of the representation provided by the relations[ M, P ] = 0, i.e . to obtain a spectral problem in a manageable form.Particular choices for the absolute parallelism can provide us explicitrepresentations of the prolongation skeleton; in particular a Kaˇc–Moody Liealgebra has been obtained [25] (see Proposition 3.5, case 1 . ( b ) below). In thefollowing we will concentrate on those choices that generate particle-like Liealgebra structures. We shall see that it is yet possible to obtain a spectralproblem with a particular choice of the tower. Recently, Vinogradov proved that any Lie algebra over an algebraically closedfield or over IR can be assembled in a number of steps from two elementaryconstituents, that he called dyons and triadons [33]. He considered the prob-lems of the construction and classification of those Lie algebras which can beassembled in one step from base dyons and triadons, called coaxial Lie alge-bras. The base dyons and triadons are Lie algebra structures that have onlyone non-trivial structure constant in a given basis, while coaxial Lie algebrasare linear combinations of pairwise compatible base dyons and triadons [34].Here for the convenience of the reader we recall some basic facts of the theoryin the original Vnogradov’s notation. Definition 3.1
Lie algebra structures g and g on a vector space V arecalled compatible if [ , ] g + [ , ] g is also a Lie algebra product.A Lie algebra g is called simply assembled from Lie algebra structures g , . . . , g m on | g | = V if the Lie algebras g i ’s are pairwise compatible and[ , ] g = [ , ] g + . . . [ , ] g m . Note that if the Lie algebras g i ’s are compatible,then any linear combination of compatible Lie algebras commutators is a Liealgebra commutator (or product) . Definition 3.2
Fix a basis B = e , . . . e n in the representation vector spaceof a given Lie algebra. Let i, j, and k be integers, 1 ≤ i, j, k ≤ n , no twoof them equal, and denote by { i, j | k } (respectively, { i | j } ) the n -triadon(respectively, the n -dyon) such that [ e i , e j ] = − [ e j , e i ] = e k (respectively, . Palese and E. Winterroth e i , e j ] = − [ e j , e i ] = e j ) are the only non-trivial Lie commutators of basisvectors. Vinogradov called them ‘base triadon’ and ‘base dyon’, respectivelyor by the unifying term ‘base lieon’.An n -dyon is the direct sum of a dyon with an n − n ≥
2, ( i.e . there is only one non vanishing bracket and it is adyon). Analogously an n -triadon is the direct sum of a triadon with an n − n ≥ i.e . there is only one non vanishingbracket and it is a triadon). They can also be referred generically as n -lieons.A linear combination of pairwise compatible base lieons is called a coaxialLie algebra structure . A Lie algebra structure will be called trix-coaxial (re-spectively, dyx-coaxial ) if it consists only of base triadons (respectively, basedyons). A coaxial Lie algebra g may be presented as a linear combination, g = X α ( i,j | k ) { i, j | k } + X β ( m | n ) { m | n } of pairwise compatible base lieons.The vectors e i , e j , and e k (respectively, e i , e j ) are called the vertices of thetriadon { i, j | k } (respectively, of the dyon { i | j } ). The vectors e i and e j arecalled the ends of the triadon { i, j | k } , while e k is the center of the triadon .The origin and the end of the dyon { i | j } are e i and e j , respectively. Thebase triadons { i, j | k } and { j, i | k } = - { i, j | k } are not distinguished since theyhave identical compatibility properties.We now recall Proposition 3 . • Two base triadons are non-trivially compatible if and only if they havea common center, a common end, or both. • Two base dyons are incompatible if and only if the origin of one is theend of the other and they have no other common vertices. • A base dyon is non-trivially compatible with a base triadon if and onlyif its origin coincides with one of the ends of the triadon.For further notation and vocabulary we refer the reader to Vinogrados’spapers. . Palese and E. Winterroth We prove that with a (2 + 1)-dimensional Toda type system are associatedalgebraic skeletons which are compatible assemblings of particle-like Lie alge-bras of dyons and triadons type. We obtain trix-coaxial and dyx-coaxial Liealgebra structures for the system from skeletons of some particular choicefor compatible associated absolute parallelisms. In particular, we find atrix-coaxial Lie algebra structure made of two (compatible) base triadonsconstituting a 2-catena (see Proposition 3.1, pag 5 [34]).Let us indeed now look for special skeletons.
Proposition 3.3
Associate with the Toda type system (1) is a trix-coaxialLie algebra structure made of two (compatible) base triadons constituting a -catena. Proof.
If we look for operators P ( u, ξ ) = e u ¯ P ( ξ ), M ( u, ξ ) = M ( e u , ξ ),we get M ( e u ; ξ ) = − e u [ L ( ξ ) , ¯ P ( ξ )]+ ¯ M ( ξ ) and thus ¯ P ( ξ ) = − e u [ L ( ξ ) , [ L ( ξ ) , ¯ P ( ξ )]]+[ L ( ξ ) , ¯ M ( ξ )]. There are additional relations determined by the third prolon-gation equation [ − e u [ L ( ξ ) , ¯ P ( ξ )] + ¯ M ( ξ ) , e u ¯ P ( ξ )] = 0.Let us then put L = X , ¯ M = X , ¯ P = X , [ X , X ] = X . From theabove we have the following prolongation closed Lie algebra[ X , X ] = X , [ X , X ] = X , [ X , X ] = [ X , X ] = [ X , X ] = [ X , X ] = 0 . The above is a trix-coaxial Lie algebra structure made of two compatible4-triadons.Indeed, by taking X = 0, we get [ X , X ] = X , [ X , X ] = [ X , X ] = 0and [ X , X ] = [ X , X ] = [ X , X ] = 0 trivially.On the other hand by taking X = 0, we get [ X , X ] = X , [ X , X ] =[ X , X ] = 0 and [ X , X ] = [ X , X ] = [ X , X ] = 0 trivially.According with [34] the two 4-triadons above are non trivially compatiblehaving a common end X , and they constitute a 2-catena. -catena Let us now explicate the tower corresponding to such 4-triadons. . Palese and E. Winterroth A km = B km = δ km , were δ km is theKronecker symbol. By substituting the above commutators into equations(8) and (10) (the expression of (9) being constrained in this case by therelation F = G + H ), we get H = e u u z X + e u X , (12) G = u x X − e u [ X , X ] + X , (13)Now from equation (3), by sectioning we obtain H k − ξ ky = − ξ kx , (14) G k + ξ ky = ξ kz , (15)(together with F k + ξ kx = ξ kz which depends on the two others).We note that each one of 4-triadons above can be represented in a spaceof local coordinates ξ k providing conservation laws related to two compatiblePoisson structures.Indeed let us consider the 4-triadon given by X = 0. A representationin the coordinates { ξ , ξ , ξ } is given by X = ξ ∂/∂ξ , X = ξ ∂/∂ξ and X = − ξ ∂/∂ξ .The tower corresponding to this case gives e u u z ξ ∂/∂ξ + e u ξ ∂/∂ξ − ξ ky ∂/∂ξ k = − ξ kx ∂/∂ξ k ,u x ξ ∂/∂ξ + e u ξ ∂/∂ξ + ξ ky ∂/∂ξ k = ξ kz ∂/∂ξ k , which gives us the system ξ x = ξ y + M ξ (16) ξ z = ξ y + N ξ (17)where ξ = ( ξ , ξ , ξ ) T , M and N are 3 × M = − e u u z , M = − e u , N = u x , N = e u , and all the other entries are zeros.In view of (4)-(6) and (2) this system can be interpreted as a conservation . Palese and E. Winterroth ξ yy − ξ xx = ( e u u z ξ ) x + ( e u u z ξ ) y , (18) ξ yy − ξ xx = ( e u ξ ) x + ( e u ξ ) y , (19) ξ zz − ξ yy = ( e u ξ + u x ξ ) z + ( e u ξ + u x ξ ) y , (20)where e u u z ξ , e u ξ and e u ξ + u x ξ can be recognized as charge/currentdensities. On the other hand, the system can be simplified since ξ y = ξ z ; wehave that (19) can also be written as ξ zz − ξ xx = ( e u ξ ) x + ( e u ξ ) y , (21)from which we obtain the Maxwell-type equation ξ yy = ξ zz . (22)Further manipulations can be made by using ξ y = ξ y = ξ z .We remark that the same conservation law and related outcomes areobtained by the 4-triadon given by X = 0. Therefore existence of thattower with a finite dimensional skeleton which is a 2-catena says us thatthe two Poisson structures corresponding to each 4-triadon are compatiblealso in a sense which is interpretable from a physical point of view: theyare structures associated with the same Toda system, and more preciselywith the same conservation law and related special solutions. Compatibilityof Poisson structures is beyond the scope of this paper, however this resultsuggest interesting links between special solutions and compatible Poissonstructures, which will be the object of further investigations (in particularfor meron-like configurations or gravitational instantons). In the following we analyze with more detail the case of choice P ( u, ξ ) =ln u ¯ P ( ξ ), M ( u, ξ ) = M ( e u , ξ ) studied in [25] also leading to an infinite di-mensional skeleton homomorphic to a Kaˇc-Moody Lie algebra. We carefullydistinguish the various cases. . Palese and E. Winterroth Lemma 3.4
Let P ( u, ξ ) = ln u ¯ P ( ξ ) , M ( u, ξ ) = M ( e u , ξ ) . We get the follow-ing infinitesimal algebraic skeleton with the structure of an open Lie algebra : [ X , X ] = X , [ X , X ] = X , [ X , X ] = [ X , X ] , [ X , X ] = [ X , X ] , (23)[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = [ X , X ] = [ X , X ] = [ X , X ] = 0 , . . . Proof.
Put L = X ( ξ ).By derivation we get M ( e u , ξ ) = − (ln u − u [ X ( ξ ) , X ( ξ )] + X ( ξ ), and P ( u, ξ ) = ue u ln u [ X ( ξ ) , M ].For u = 0 , P, M ] = 0 we get [[ X , M ] , M ] = 0 ;from which we get[[ X , X ] , X ] = 0 , [ X , [ X , X ]] , X ] + [[ X , X ] , [ X , X ]] = 0 , [[ X , [ X , X ]] , [ X , X ]] = 0 . By putting [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X ,[ X , X ] = X , we obtain an infinite dimensional skeleton as follows[ X , X ] = [ X , X ] = [ X , X ] = [ X , X ] = 0 , [ X , X ] = [ X , X ] , [ X , X ] = [ X , X ] , (24) . . . Here the dots means that we can continue this structure by introducingnew generators still obtaining the peculiar relations of the type (24) whichdistinguish this algebraic structure from a freely generated Lie algebra (seethe discussion in [22]). . Palese and E. Winterroth Proposition 3.5
The homomorphism X = λX and X = µX associateswith the Toda system (1) dyx-coaxial and particle-like Lie algebra structuresas well as an infinite-dimensional Lie algebra homomorphic with a Kaˇc-Moody Lie algebra. Proof.
We essentially distinguish the two cases X = 0 and X = 0,together with various different subcases.1. if [ X , X ] = X = 0, then µ = − λ must old; we can distinguish differentsubcases(a) in general the case X = 0 and µ = − λ = 0 can provide infinite-dimensional Lie algebras homomorphic with Kaˇc-Moody type Liealgebras.(b) the particular case X = νX and µ = − λ = 1 ( i.e . X = X and X = − X ) giving an infinite-dimensional Lie algebra homomorphicwith a Kaˇc-Moody Lie algebra was obtained in [25].(c) the particular case X = νX and µ = − λ = 0 ( i.e . X = X = 0;see [25]) gives a particle-like Lie algebra as a base 3-dyon:[ X , X ] = 0 , [ X , X ] = 0 , [ X , X ] = νX . (25)2. if [ X , X ] = X = 0, then X = X , X = X , and we distinguish thefollowing different subcases (the case µ = λ = 0 giving an abelian Liealgebra):(a) the case µ = 0 and λ = 0 provides us with a particle-like Lie algebraas a base 3-dyon:[ X , X ] = λX , [ X , X ] = 0 , [ X , X ] = 0 . (26)(b) the case λ = µ = 0 provides a dyx-coaxial Lie algebra structure asan assembling of two compatible base 3-dyons[ X , X ] = λX , [ X , X ] = λX , [ X , X ] = 0 . (27)(c) the particular case λ = µ = 1 ( i.e . X = X and X = X ) gives[ X , X ] = X , [ X , X ] = X , [ X , X ] = 0 , and it was obtained in [25]. . Palese and E. Winterroth Remark 3.6
We can try to check the compatibility of the above particle-like Lie algebra structures. The latter Lie algebra (27) is constituted of twomutually compatible dyons (dyx-coaxial Lie algebra), whose the first is givenby (26), while the Lie algebras (25) and (26) are made of a single dyon andthey are not compatible. Indeed we note that the first dyon of (27) is notcompatible with (25), while the second dyon of (27) is.The question now is can we still construct a different dyx-coaxial Liealgebra from the original skeleton? For example the following would be adyx-coaxial Lie algebra of compatible dyons[ X , X ] = λX , [ X , X ] = νX , [ X , X ] = 0 (28)We ask if we can get it from the prolongation skeleton by a suitable quoti-enting, i.e . if it is somehow compatible with (or derivable from) the skeletonstructure. However, we note that if we put X = 0 from the beginning (whichis the case when we assume that [ X , X ] = 0), and if X = νX , this wouldimply also [ X , X ] = 0, then we would get (25) back.Thus it appears that the case X = νX corresponds or to a particle-likeLie algebra structure or to a Kaˇc-Moody type Lie algebra (it is noteworthythat the latter is anyway an infinite-dimensional loop Lie algebra of a dyx-coaxial Lie algebra) and these two cases appear to be non compatible.Let us then investigate from a more general point of view this feature.We ask whether we can look for different quotient homomorphisms.Let now consider the case X = 0 from the beginning, and X = 0, andlook for a quotient lie algebra given by X = − γX , , X = µX , and weobtain the Lie algebra structure depending on two parameters[ X , X ] = 0 , [ X , X ] = µX , [ X , X ] = − γX . (29)By appling the Jacobi identity we get µγX = 0 which, if we require X = 0,is verified either for µ = 0 and γ = 0 (see below (30)) or for µ = 0 and γ = 0 (see below (31)), or for µ = 0 and γ = 0 (trivial case of an abelian Liealgebra). Proposition 3.7
The case X = 0 from the beginning, and with X = 0 ,provides us with two base -dyons. Proof. . Palese and E. Winterroth µ = 0 and γ = 0.By putting X = − γX , and X = X = X = X = 0 we get the 3-dyon[ X , X ] = 0 , [ X , X ] = 0 , [ X , X ] = − γX . (30)The above dyon is incompatible with (25) while it is compatible with(26).2. the case µ = 0 and γ = 0.We get the 3-dyon[ X , X ] = 0 , [ X , X ] = µX , [ X , X ] = 0 . (31) Remark 3.8
It appears that the dyx-coaxial Lie algebra (27) can be as-sembled by one step from the case (26) and the latter one, (31), by putting µ = λ ,.We note in particular that (29) can not be seen as a dyx-family of dyonssince the two dyons [ X , X ] = µX , [ X , X ] = − γX are incompatible andindeed if we apply the Jacobi identity we get particle-like structures made ofsingle base dyons.It seems therefore that the prolongation skeleton is homomorphic withquotient finite dimensional Lie algebras which have always the structure ofa family of compatible dyons or single base 3-dyons. We note that the firstdyon of (27) is compatible with (30), while the second dyon of (27) is not.Summing up we were able to associate with the infinitesimal skeleton(23) a dyx-coaxial Lie algebra structure (27) and particle-like Lie algebrastructures made of three base 3-dyons which are only partially compatibleamong them, i.e . • the first dyon of (27) is compatible with (30), while the second dyon of(27) is not. • the first dyon of (27) is not compatible with (25), while the second dyonof (27) is.Note that (25), (26), (30) and (31) are not all compatible among them,even they are not compatible in triples, but they are only compatible whentook in couples. . Palese and E. Winterroth -dyons Following a procedure similar to that of Section 3.2.1 we shall now derive aLax pair related to the Lie algebra (27) to which the tower skeleton (24) ishomomorphic.We refer again to equations (8) and (10). The tower associated with case2 . ( b ) in Proposition 3.5 becomes in this specific case H = e u u z X − u e u ln u (ln u − X , [ X , X ]] + ue u ln u [ X , X ] , (32) G = u x X − u (ln u − X , X ] + X , (33)By taking into account the representation ρ given by relations (27) we thenget H = e u u z X − λ u e u ln u (ln u − X + λue u ln uX , (34) G = u x X − λu (ln u − X + X , (35)Now, let us represent the dyx-coaxial Lie algebra above in a space of‘ pseudopotentials ’ ξ k by X = − ξ ∂/∂ξ + ξ ∂/∂ξ , X = − λξ ∂/∂ξ , X = λξ ∂/∂ξ .Again by sectioning the tower, equations (14) and (15) provide the fol-lowing e u ( u z ξ + λ u ln uξ ) ∂/∂ξ − e u u z ξ ∂/∂ξ ++ λ u e u ln u (ln u − ξ ∂/∂ξ + ξ ky ∂/∂ξ k = ξ kx ∂/∂ξ k , − u x ( ξ + λξ ) ∂/∂ξ + u x ξ ∂/∂ξ − λ u (ln u − ξ ∂/∂ξ ++ ξ ky ∂/∂ξ k = ξ kz ∂/∂ξ k , which gives us the inverse spectral problem ξ x = ξ y + ˆ M ξ (36) ξ z = ξ y + ˆ N ξ (37) . Palese and E. Winterroth ξ = ( ξ , ξ , ξ ) T , ˆ M and ˆ N are 3 × M = e u u z ,ˆ M = e u λ u ln u , ˆ M = − e u u z , ˆ M = λ u e u ln u (ln u −
1) ˆ N = − u x ,ˆ N = − λu x , ˆ N = u x , ˆ N = − λ u (ln u −
1) and all the other entries arezeros. Here λ plays the role of a spectral parameter and, in view of (4)-(6) and(2), ˆ M and ˆ N can be considered a Lax pair related to the multidimensionalToda system (1) (for spectral problems related to multidimensional nonlinearsystem see, e.g . [17, 32]). This Lax pair should be compared with [15].Compatibility of Lie algebraic structures being expressions of compatibil-ity of the corresponding Poisson structures, we note here that Fernandes [6]studied the relationship between the master symmetries and bi-Hamiltonianstructure of the Toda lattice. We stress that dyons provide indeed particularexamples of master symmetries of related ordinary differential equations. The structure of trix-coaxial and dyx-coaxial Lie algebras assembled in onestep from couples of particle-like Lie algebra structures appears as an in-trinsic feature of the Toda system (1), at least associated with the chosenabsolute parallelisms. Indeed the similitude transformations seems to be thefundamental internal symmetries of the system (see e.g . [2]).As final remark, since (30) is compatible with (26), and since (25) is com-patible with (31), we could construct the following dyx-coaxial Lie algebras:[ X , X ] = λX , [ X , X ] = 0 , [ X , X ] = − γX , (38)and [ X , X ] = 0 , [ X , X ] = µX , [ X , X ] = νX . (39)However, it is important to realize that they could not be obtained from theskeleton (23) by the choice of an homomorphism, and therefore they are notidentified as internal symmetries of the Toda system by the choice of theabsolute parallelism given by Lemma 3.4. The question if the choice of otherforms of the absolute parallelism could identify them is open and will be theobject of future investigations. Acknowledgements
Research partially supported by Department of Mathematics - Universityof Torino through the projects PALM RILO 16 01 and FERM RILO 17 01 . Palese and E. Winterroth
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