Particle-like, dyx-coaxial and trix-coaxial Lie algebra structures for a multi-dimensional continuous Toda type system

We prove that with a $(2+1)$-dimensional Toda type system are associated algebraic skeletons which are (compatible assemblings) of particle-like Lie algebras of dyons and triadons type. We obtain trix-coaxial and dyx-coaxial Lie algebra structures for the system from algebraic skeletons of some particular choice for compatible associated absolute parallelisms. In particular, by a first choice of the absolute parallelism, we associate with the $(2+1)$-dimensional Toda type system a trix-coaxial Lie algebra structure made of two (compatible) base triadons constituting a $2$-catena. Furthermore, by a second choice of the absolute parallelism, we associate a dyx-coaxial Lie algebra structure made of two (compatible) base dyons, as well as particle-like Lie algebra structures made of single $3$-dyons.


Introduction
Toda type systems are nonlinear models which play a role in a variety of physical and, more in general, natural phenomena.
The problem of integrability of nonlinear models has been recognized to be related to their algebraic properties in discrete and continuous, as well as, classical and quantum formulations. Algebraic properties can be interpreted as the counterpart of the concept of integrability given as of having 'enough' conservation laws to exhaustively describe the underling field or associated dynamics. Indeed, from an historical point of view, algebraic-geometric approaches are based on the requirement for the existence of conservation laws which emerge from internal symmetries (given in terms of algebraic structures).
In the Seventies, in fact, Wahlquist and Estabrook [25,5] proposed a technique for systematically deriving, from an integrable system, what they called a 'prolongation structure' in terms of a set of 'pseudopotentials' related to the existence of an infinite set of associated conservation laws. They also conjectured that, as a characterizing feature of the integrability property, the structure was 'open' i.e. not a set of structure relations of a finite-dimensional Lie group. Since then, 'open' Lie algebras have been extensively studied in order to distinguish them from freely generated infinite-dimensional Lie algebras.
Their interest in the study of integrability is in the fact that Lax pairs of the inverse spectral transform containing an isospectral parameter can be obtained by an homomorphism of the infinite-dimensional open Lie algebra in a finite-dimensional 'closed' Lie algebra. In their approach, conservation laws are written in terms of 'prolongation' forms and integrability is intended as a Frobenius integrability condition for a 'prolonged' ideal of differential forms describing intrinsically the given nonlinear model in the sense ofÉ. Cartan.
Attempting a description of symmetries in terms of Lie algebras implies the appearance of an homogeneous space and thus the interpretation of prolongation forms as Cartan-Ehresmann connections. It is clear that here the unknowns are both conservation laws and symmetries, and it is clear that the main point in this is how to realize the form of the conservation laws and thus the explicit expression of the prolongation forms. Different prolongation ideals give rise to both different algebraic structures (symmetries) and corresponding conservation laws. By an inverse procedure based on the intrinsic duality between Lie algebras and differential systems [4], open Lie algebraic structures can 'generate' whole families of different nonlinear systems bound by the same internal symmetry structure.
In a series of papers [11,12,13,15,16,17], we explicated an algebraic-geometric interpretation of the above mentioned 'prolongation' procedure in terms of towers with infinitesimal algebraic skeletons (in the sense of [10]) and we will refer to that framework in this paper. It is noteworthy that slight modification of the internal symmetry properties generates new models which can contain possible integrable subcases. For example, activatorsubstrate systems have been obtained by performing a slight modification of the internal symmetry algebra of twisted reaction-diffusion equations [13]. The structure itself with which prolongation forms are postulated can produce open algebraic structures or just Lie algebras. Our aim in this work is to investigate some common features of them and to show the emerging of particle-like Lie algebras structure as symmetry structures of integrable systems (associated with Poisson structures which can be compatible or not). Indeed, in general, infinite dimensional open Lie algebras are the main object of the search in view of the application of the inverse spectral transform to obtain soliton solutions, Bäcklund transformations and so on. Integrable systems, admitting infinite-dimensional prolongation Lie algebras can also admit finite-dimensional Lie algebras, which still can be related to some kind of internal symmetries of the systems themselves and to associated conservation laws. Prolongation forms bringing to finite dimensional Lie algebras (without a spectral parameter) are generally discarded when searching for a Lax pair to be used within the inverse spectral transform.
In this note we show that such (otherwise discarded) symmetries deserve a more careful study. We take a (2 + 1)-dimensional Toda type system as a study case and show that it posses algebraic properties related to the recently introduced concept of particle-like Lie algebra structures [23,24].
Vinogradov developed a completely abstract theory of compatibility of Lie algebra structures starting from the corresponding compatibility theory of Poisson structures. Although the mathematical aspects of the theory are quite involved the nice point is that simple criteria of compatibility or non compatibility have been obtained which somehow have a certain grade of automatism.
Furthermore, as for the physical side, Vinogradov speculated that this particle-like structures could be related with the ultimate particle structure of the matter: he noted that since 'the symmetry algebra u(2) = so(3) of a nucleon can be assembled in one step from three triadons [...] one might think that this structure of the symmetry reflects the fact that a nucleon is made from three "quarks" '. This is of course only a speculation, but it also suggests a quite fascinating new perspective on internal symmetries of integrable systems.
2 Internal symmetries of Toda type systems in (2 + 1) dimensions Consider the (2 + 1)-dimensional system, a continuous (or long-wave) approximation of a spatially two-dimensional Toda lattice [22]: where u = u(x, y, z) is a real field, x, y, z are real local coordinates (if we want, z playing the rôle of a 'time') and the subscripts mean partial derivatives. It can be seen as the limit for γ → ∞ of the more general model u xx + u yy + (1 + u/γ) γ−1 zz = 0 covering (for γ = 0, 1) various continuous approximations of lattice models, among them the Fermi-Pasta-Ulam (γ = 3) [1]. This model is almost ubiquitus, it appears in differential geometry; in mathematical and theoretical physics (Newman and Penrose); in the theory of Hamiltonian systems; in general relativity; in the large n limit of the sl(n) Toda lattice; in extended conformal symmetries, and theory of gravitational instantons; in strings theory and statistical mechanics etc. (see e.g. [3,8,18,20]).
It can be seen as the particular case with d = 1 of so-called 2d-dimensional Toda-type systems [21] from a 'continuum Lie algebra' by means of a zero curvature representation u ww = K(e u ), (in our particular case w = x+iy and K is the differential operator given by K = ∂ 2 ∂z 2 ). In particular, it has been studied in the context of symmetry reductions [2,6] and a (1+1)-dimensional version in the context of prolongation structures [1]. The (2 + 1)-dimensional system has been associated with a Kač-Moody Lie algebra and related to Saveliev's continuum Lie algebras of particular kind [16].

Skeletons for the (2+1) Toda system
Let us first recall a few mathematical tools constituting the background for a detailed treatment of which we refer to [15,16,17] and [10,19].
From one side global properties of partial differential equations such as internal symmetries and invariance properties having an issue in dynamics can be described by mathematical tools which enable us to deal with global properties at large scales, connecting local data to global ones. On the other side transformations of configurations of a system can be globally studied by means of the theory of the action of Lie groups on manifolds. The differential content carried by a Lie group (and its Lie algebra) and by its structure equations provides differential equations.
We observe that two ingredients constitute the nonlinear phenomena: symmetries on the one side (algebraic content) and changes in time and space on the other side (differential content). In particular, to keep account of the 'interaction' of both aspects, we recognize a refined structure of open Lie algebraic structures associated with them: we introduce a notion which generalizes the concept of a homogeneous space, i.e. that of an algebraic skeleton E = g ⊕ V on a finite-dimensional vector space V , with g a possibly infinite dimensional Lie algebra. The further step is introducing a tower with such 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 is a (possibly infinite-dimensional) vector space not necessarily equipped with a Lie algebra structure, g is the Lie algebra of G, and ρ is a representation of 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 algebraic structure is fundamental in order to be able to generate whole families of nonlinear differential systems, starting from it.
We now consider a suitably constructed differentiable structure which is somewhat modelled on the skeleton above. Let us introduce a differentiable manifold 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 conditions to find a homomorphism between the open infinite dimensional Lie algebra, constructed by ρ, and a quotient Lie algebra. Let us call k the (possibly infinite dimensional) Lie algebra obtained as the direct sum of such a quotient Lie algebra with g. From the differentiable side, a tower P (Z, G) on Z with skeleton (E, G, ρ) is an absolute parallelism ω on P valued in E, invariant with respect to ρ and reproducing elements of g from the fundamental vector fields induced on P , i.e. R * g ω = ρ(g) −1 ω, for g ∈ G; ω(Ã) = A, for A ∈ g; here R g denotes the right translation andÃ the fundamental vector field induced on P from A. In general, the absolute parallelism does not define a Lie algebra homomorphism.
Let then k be a Lie algebra and g a Lie subalgebra of k. Let G be a Lie group with Lie algebra g and P (Z, G) be a principal fiber bundle with structure group G over a manifold Z as above. A Cartan connection in P of type (k, G) is a 1-form ω on P with values in k such that ω| TpP : T p P → k is an isomorphism ∀p ∈ P , R * g ω = Ad(g) −1 ω for g ∈ G and reproducing elements of g from the fundamental vector fields induced on P . It is clear that a Cartan connection (P , Z, G, ω) of type (k, G) is a special case of a tower on Z.
Following [16], we recall how to get both some skeletons and towers over them associated with the system (1).
On a manifold with local coordinates (x, y, z, u, p, q, r), we introduce the closed differential ideal defined by the set of 3-forms: It is easy 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 equivalent to the Toda system under study.
By an ansatz introduced in [15], we look for suitable 2-forms (generating associated conservation laws) , and the absolute parallelism forms are given by N (N arbitrary), and H k , F k and G k are, respectively, the pseudopotentials and functions to be determined, while A k m and B k m denote the elements of two N × N constant regular matrices 1 .
The integrability condition for the ideal generated by forms θ j and Ω k finally yields It turns out that N k (ξ m ) can be written in terms of the others [14]. As a consequence, the desired representation for the skeleton is provided by the following equations (we omit the indices for simplicity) [14,16].
Note that here L depends only on ξ m , while P and M still have a dependence on u determined by the first two differential equations. A tower with P and M given in terms of L has been obtained by suitable operator Bessel coefficients [14]; however, it is a non trivial task to characterize explicitly its algebraic skeleton by means of the representation provided by the relations [M, P ] = 0. Particular choices for the absolute parallelism can provide us explicit representations of the prolongation skeleton; in particular a Kač-Moody Lie algebra has been obtained [16]. In the following we will concentrate on those choices that generate particle-like Lie algebra structures.

Particle-like Lie algebra structure
Recently, Vinogradov proved that any Lie algebra over an algebraically closed field or over IR can be assembled in a number of steps from two elementary constituents, that he called dyons and triadons [23]. He considered the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons [24]. Here for the convenience of the reader we recall some basic facts of the theory.  An n-dyon is the direct sum of a dyon with an n − 2-dimensional abelian Lie algebra, n ≥ 2, (i.e. there is only one non vanishing bracket and it is a dyon). Analogously an n-triadon is the direct sum of a triadon with an n−3dimensional abelian Lie algebra, n ≥ 3 (i.e. there is only one non vanishing bracket 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 coaxial Lie algebra structure. A Lie algebra structure will be called trix-coaxial (respectively, dyx-coaxial) if it consists only of base triadons (respectively, base dyons). A coaxial Lie algebra g may be presented as a linear combination, We now recall Proposition 3.1 of [24] stating some necessary and sufficient conditions for the compatibility or incompatibility of particle-like Lie algebra structures: • Two base triadons are non-trivially compatible if and only if they have a common center, a common end, or both. if its origin coincides with one of the ends of the triadon. For further notation and vocabulary we refer the reader to Vinogrados's papers.

Trix-coaxial, dyx-coaxial and particle-like Lie algebra structures for the Toda system
We prove that with a (2 + 1)-dimensional Toda type system are associated algebraic skeletons which are compatible assemblings of particle-like Lie algebras of dyons and triadons type. We obtain trix-coaxial and dyx-coaxial Lie algebra structures for the system from skeletons of some particular choice for compatible associated absolute parallelisms. In particular, we find a trix-coaxial Lie algebra structure made of two (compatible) base triadons constituting a 2-catena (see Proposition 3.1, pag 5 [24]). Let us indeed now look for special skeletons. Let us then put L = X 1 ,M = X 2 ,P = X 3 , [X 1 , X 3 ] = X 4 . From the above we have the following prolongation closed Lie algebra

Trix-coaxial Lie algebra structures
The above is a trix-coaxial Lie algebra structure made of two compatible 4-triadons.

Dyx-coaxial and particle-like Lie algebra structures
In the following we analyze with more detail the case of choice P (u, ξ) = ln uP (ξ), M(u, ξ) = M(e u , ξ) studied in [16] also leading to an infinite di-mensional skeleton homomorphic to a Kač-Moody Lie algebra. We carefully distinguish the various cases.
This choice of the absolute parallelism associates with the Toda system (1) dyx-coaxial and particle-like Lie algebra structures.
Here the dots means that we can continue this structure by introducing new generators still obtaining the peculiar relations of the type (4) which distinguish this algebraic structure from a freely generated Lie algebra (see the discussion in [13]).

Proposition 3.5
The homomorphism X 4 = λX 2 and X 5 = µX 3 associates with the Toda system (1) dyx-coaxial and particle-like Lie algebra structures as well as an infinite-dimensional Lie algebra homomorphic with a Kač-Moody Lie algebra.
Proof. We essentially distinguish the two cases X 8 = 0 and X 8 = 0, together with various different subcases.
Remark 3. 6 We can try to check the compatibility of the above particlelike Lie algebra structures. The latter Lie algebra (7) is constituted of two mutually compatible dyons (dyx-coaxial Lie algebra), whose the first is given by (6), while the Lie algebras (5) and (6) are made of a single dyon and they are not compatible. Indeed we note that the first dyon of (7) is not compatible with (5), while the second dyon of (7) is.
The question now is can we still construct a different dyx-coaxial Lie algebra from the original skeleton? For example the following would be a dyx-coaxial Lie algebra of compatible dyons We ask if we can get it from the prolongation skeleton by a suitable quotienting, i.e. if it is somehow compatible with (or derivable from) the skeleton structure. However, we note that if we put X 4 = 0 from the beginning (which is the case when we assume that [X 1 , X 2 ] = 0), and if X 8 = νX 3 , this would imply also [X 1 , X 3 ] = 0, then we would get (5) back. Thus it appears that the case X 8 = νX 3 corresponds or to a particle-like Lie algebra structure or to a Kač-Moody type Lie algebra (it is noteworthy that the latter is anyway an infinite-dimensional loop Lie algebra of a dyxcoaxial 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 4 = 0 from the beginning, and X 8 = 0, and look for a quotient lie algebra given by X 8 = −γX 2 , , X 5 = µX 3 , and we obtain the Lie algebra structure depending on two parameters By appling the Jacobi identity we get µγX 2 = 0 which, if we require X 2 = 0, is verified either for µ = 0 and γ = 0 (see below (10)) or for µ = 0 and γ = 0 (see below (11)), or for µ = 0 and γ = 0 (trivial case of an abelian Lie algebra).

Proposition 3.7
The case X 4 = 0 from the beginning, and with X 8 = 0, provides us with two base 3-dyons. Proof.
We get the 3-dyon Remark 3.8 It appears that the dyx-coaxial Lie algebra (7) can be assembled by one step from the case (6) and the latter one, (11), by putting µ = λ,.
We note in particular that (9) can not be seen as a dyx-family of dyons since the two dyons [X 1 , X 3 ] = µX 3 , [X 3 , X 2 ] = −γX 2 are incompatible and indeed if we apply the Jacobi identity we get particle-like structures made of single base dyons. It seems therefore that the prolongation skeleton is homomorphic with quotient finite dimensional Lie algebras which have always the structure of a family of compatible dyons or single base 3-dyons. We note that the first dyon of (7) is compatible with (10), while the second dyon of (7) is not.
Summing up we were able to associate with the infinitesimal skeleton (3) a dyx-coaxial Lie algebra structure (7) and particle-like Lie algebra structures made of three base 3-dyons which are only partially compatible among them, i.e.
• the first dyon of (7) is compatible with (10), while the second dyon of (7) is not. • the first dyon of (7) is not compatible with (5), while the second dyon of (7) is. Note that (5), (6), (10) and (11) are not all compatible among them, even they are not compatible in triples, but they are only compatible when took in couples.

Concluding remarks
The structure of trix-coaxial and dyx-coaxial Lie algebras assembled in one step from couples of particle-like Lie algebra structures appears as an intrinsic feature of the Toda system (1), at least associated with the chosen absolute parallelisms. Indeed the similitude transformations seems to be the fundamental internal symmetries of the system (see e.g. [2]).
However, it is important to realize that they could not be obtained from the skeleton (3) by the choice of an homomorphism, and therefore they are not identified as internal symmetries of the Toda system by the choice of the absolute parallelism given by Lemma 3.4. The question if the choice of other forms of the absolute parallelism could identify them is open and will be the object of future investigations.