wykresx.eps Acta Polytechnica Vol. 51 No. 1/2011 Infinitesimal Algebraic Skeletons for a (2 + 1)-dimensional Toda Type System M. Palese, E. Winterroth Abstract A tower for a (2+ 1)-dimensional Toda type system is constructed in terms of a series expansion of operators which can be interpreted as generalized Bessel coefficients; the result is formulated as an analog of the Baker-Campbell-Hausdorff formula. We tackle the problem of the construction of infinitesimal algebraic skeletons for such a tower and discuss some open problems arising along our approach. Keywords: Toda type system, integrability, infinitesimal skeleton, tower, Cartan connection. 1 Introduction Nonlinear models, and in particular Toda type sys- tems, play a role in a variety of physical phenomena. As is well known, the problem of their integrability is far from being trivial. It is nowadays well recognized that the algebraic properties of nonlinear systems are relevant from the point of view of integrability. A huge scientific production within this topic has de- veloped in both discrete and continuous, as well as, classical and quantistic models. It is nevertheless im- portant not to forget the origin of this interest: for a nonlinear system, it lies in the concept of integra- bility as of having ‘enough’ conservation laws to ex- austively describe the dynamics (an idea which origi- nates in the inverse of the Noether Theorem II in the calculus of variations). Historically, the algebraic- geometric approach is based on the requirement for the existence of conservation laws which leads to the existence of symmetries (in terms of algebraic struc- tures). In this light, Wahlquist and Estabrook [15, 5] pro- posed a technique for systematically deriving what they called a ‘prolongation structure’ in terms of a set of ‘pseudopotentials’ related with the existence of an infinite set of associated conservation laws, and they also conjectured that 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 ex- tensively studied in order to distinguish them from freely generated infinite-dimensional Lie algebras. In their approach, conservation laws are writ- ten in terms of ‘prolongation’ forms and integrabil- ity is intended as an integrability condition for a ‘prolonged’ differential ideal. Attempting a descrip- tion 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 should be stressed that the unknowns are both conservation laws and sym- metries, 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 formulations of the prolongation ideal bring to both different algebraic structures (symmetries) and corresponding conservation laws: of course, the structure with which prolongation forms are postu- lated can produce Lie algebras or more general alge- braic structures. We use the algebraic properties of Toda type systems as a ‘laboratory’ to explicate an algebraic-geometric interpretation of the above men- tioned ‘prolongation’ procedure in terms of towers with infinitesimal algebraic skeletons [9]. Consider the (2 + 1)-dimensional system, a con- tinuous (or long-wave) approximation of a spatially two-dimensional Toda lattice [14]: uxx + uyy + (e u)zz = 0 , (1) where u = u(x, y, z) is a real field, x, y, z are real lo- cal 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 uxx+uyy+ [ (1 + u/γ)γ−1 ] zz = 0, cov- ering (for γ different from 0, 1) various continuous approximations of lattice models, among them the Fermi-Pasta-Ulam (γ = 3). It appears in differential geometry: Kaehler metrics [8]; in mathematical and theoretical physics (see, e.g. Newman and Penrose as well as [12]); in the theory of Hamiltonian systems, in general relativity: heavenly spaces (real, self-dual, euclidean Einstein spaces with one rotational Killing symmetry, [12, 4]); in the large n limit of the sl(n) Toda lattice [11] (from the constrainedWess-Zumino- Supported by the University of Torino through 2009 research project ‘Conservation laws in classical and quantum gravity’. 54 Acta Polytechnica Vol. 51 No. 1/2011 Novikov-Witten model): extended conformal symme- tries (2D CFT ) and reductions of four dimensional theory of gravitational instantons; in strings theory and statistical mechanics. It can be seen as the par- ticular case with d = 1 of so-called 2d-dimensional Toda-type systems [13] from a ‘continuum Lie al- gebra’ by means of a zero curvature representation uww̄ = K(eu), (in our particular case w = x+ iy and K is the differential operator given by K = ∂2 ∂z2 ). In particular, it has been studied in the context of sym- metry reductions [1, 6] and a (1+1)-dimensional ver- sion in the context of prolongation structures which only partially lead to results [2]. 2 Towers with skeletons for Toda type systems The notion of an (infinitesimal) algebraic skeleton is an abstraction of some algebraic aspects of homoge- neous spaces. Let then V denote a finite-dimensional vector space. An algebraic skeleton on V is a triple (E, G, ρ), with G a (possibly infinite-dimensional) Lie group, E = V ⊕ g, g the Lie algebra of G, and ρ a repre- sentation of G on E (infinitesimally of g on E) such that ρ(g)x = Ad(g)x, for g ∈ G, x ∈ g. Let Z be a manifold of type V (i.e. ∀z ∈ Z, TzZ � V ). We say that a principal fibre bundle P (Z, G) provided with an absolute parallelism ω on P is a tower on Z with skeleton (E, G, ρ) if ω takes values in E and satisfies: R∗ gω = ρ(g)−1ω, for g ∈ G; ω(Ã) = A, for A ∈ g; here Rg denotes the right trans- lation and à the fundamental vector field induced on P from A. In general, the absolute parallelism does not define a Lie algebra homomorphism. Let g be a Lie algebra and k be a Lie subalgebra of g. Let K be a Lie group with Lie algebra k and let P (Z, K) be a principal fibre bundle with struc- ture groupK over a manifold Z, as above. A Cartan connection in P of type (g, K) is a 1-form ω on P with values in g satisfying the following conditions: – ω|TuP : TuP → g is an isomorphism ∀u ∈ P ; – R∗ gω = Ad(g)−1ω for g ∈ K; – ω(Ã) = A for A ∈ k. A Cartan connection (P , Z, K, ω) of type (g, K) is a tower on Z. Remark that since, a priori, the prolongation alge- bra does not close into a Lie algebra the starting point for the prolongation procedure is only a tower with an absolute parallelism, and not a Cartan connec- tion. Thus, in principle, Estabrook-Wahlquist pro- longation forms are absolute parallelism forms. The corresponding open Lie algebra structure can be pro- vided with the structure of an infinitesimal algebraic skeleton on a suitable space. First we have to prove that a finite dimensional space V and a Lie algebra g exist satisfying the definition of a skeleton, i.e. in particular that a suitable representation ρ can be de- fined. The representation is obtained by means of an integrability condition for the absolute parallelism of a tower on a manifold Z (of type V ), with skeleton (E, V , g). Note that if E has in addition the structure of a Lie algebra this is exactly a Cartan connection of type (E, g); in fact, the spectral linear problem is nothing but the construction of a Cartan connection from this absolute parallelism. As an example, let us now introduce on a manifold with local coordinates (x, y, z, u, p, q, r) the closed dif- ferential ideal defined by the set of 3-forms: θ1 = du∧ dx∧dy−rdx∧dy∧dz, θ2 = du∧dy∧dz−pdx∧dy∧dz, θ3 = du∧dx∧dz+ qdx∧dy∧dz, θ4 = dp∧dy∧dz− dq ∧ dx∧ dz + eudr ∧ dx ∧ dy+ eur2dx∧ dy ∧ dz. It is easy to verify that on every integral submanifold defined by u = u(x, y, z), p = ux, q = uy, r = uz, with dx ∧ dy ∧ dz �= 0, the above ideal is equivalent to the Toda system under study. In terms of absolute parallelism forms, 2-forms generating associated conservation laws can be de- fined as follows: Ωk = Hk(u, ux, uy, uz; ξ m)dx ∧ dy + F k(u, ux, uy, uz; ξm)dx ∧ dz + Gk(u, ux, uy, uz; ξ m)dy ∧ dz + Ak mdξm ∧ dx+Bk mdξm ∧ dz + dξk ∧ dy , where ξ = {ξm}, k, m = 1, 2, . . . ,N (N arbitrary), and Hk, F k and Gk are, respectively, the pseu- dopotential (coordinates in the space V ) and func- tions to be determined, while Ak m and Bk m denote the elements of two N × N constant regular matri- ces. In fact, we remark that Ωk = θk m ∧ ωm, where θk m = −Āk mdx−B̄k mdy− C̄k mdz, and the absolute par- allelism forms are given by1 ωm = dξ̄m + F̄mdx+ Ḡmdy + H̄mdz . The integrability condition for the ideal generated by forms θj and Ω k finally yields Hk = euuzL k(ξm)+ P k(u, ξm), F k = − uyL k(ξm) + Nk(ξm), Gk = uxLk(ξm) +Mk(u, ξm), where Lk, P k, Nk, Mk are functions of integration. As a consequence, the de- sired representation for the skeleton is provided by the following equations (we omit the indices for sim- plicity). Pu = eu[L, M ] , Mu = −[L, P ] , [M, P ] = 0 .(2) We will consider L, P , M as regular operators so that Lie brackets can be interpreted as commu- tators. We can now look for an exact solution in 1F k = C̄k mF̄ m − Āk mH̄m, Gk = C̄k mḠm − B̄k mH̄m, Hk = B̄k mF̄ m − Āk mḠm, ξk = C̄k mξ̄m 55 Acta Polytechnica Vol. 51 No. 1/2011 order to give the representation explicitly. For any operator D = Dj ∂ ∂ξj , by introducing L[D] = [L, D], we define the n-th power of the operator L by set- ting Ln[D] = [L, [L, . . . , [L, D] . . .], where L appears n-times, and L0[D] = D. Put t = 2e u 2 . A solution of the prolongation equa- tions regular at t = 0 (i.e. at u → −∞) is then given by P = t 2 J1(tL[P0]) , M = J0(tL[M0]) , (3) where J0(·) and J1(·) are formal operator expansions given by J0(tL[M0]) = ∞∑ m=0 (−1)m (m!)2 ( t 2 )2m L2m[M0], J1(tL[P0]) = ∞∑ m=0 (−1)m m!(m+ 1)! ( t 2 )1+2m L1+2m[P0] and M0 ≡ M0(ξ) = M(t; ξ) |t=0 and P0 ≡ P0(ξ) is such that [L, P0] = [L, M0] [10]. By defining operator Bessel coefficients Jm(tX), as the coefficients of the formal expansion e t 2X(z−1/z) = ∞∑ m=−∞ zmJm(tX) (for Bessel functions a standard reference is [16]), we can prove recurrence and derivation formulae by means of which we pro- vide an equivalent solution to our prolongation equa- tions in terms of L: P = t 2 ∞∑ k=−∞ Jk+1(tL)P0Jk(tL) , M = ∞∑ k=−∞ Jk(tL)M0Jk(tL) , based on the formulae J1(tL[P0]) = ∞∑ k=−∞ Jk+1(tL)P0Jk(tL), J0(tL[M0]) = ∞∑ k=−∞ Jk(tL)M0Jk(tL), which are in fact analogous to the Baker-Campbell- Hausdorff expansion [10]. These expansions together with [M, P ] = 0 provide the desired representation and at the same time define a tower with absolute parallelism. The main problem with this tower (which is some- how the most general one) is that it is a non trivial task to characterize explicitly its algebraic skeleton by means of the representation provided by the re- lations [M, P ] = 0. On the other hand, it is well known that the Toda equation can be solved by the inverse scattering transform [7]. However, the as- sociated linear spectral problem was never derived from an infinitesimal algebraic skeleton and in par- ticular as the construction of a Cartan connection from a tower with algebraic skeleton; thus it would be important to derive both the Toda system and re- lated spectral problem(s) (i.e. conservation laws and symmetries) starting from a tower with an algebraic skeleton. In this perspective, particular solutions of the corresponding Estabrook-Wahlquist prolongation problem can assume a relevant role: they correspond to particular choices for the absolute parallelism and can provide us explicit representations of the prolon- gation skeleton. 2.1 Skeletons If we look for operators P (u, ξ) and M(u, ξ) depend- ing on u only through the exponential function, i.e. P (u, ξ) = euP̄ (ξ), M(u, ξ) = M(eu, ξ), the pro- longation equations can now be written as: Pu = eu[L, M ] = ∂P ∂eu eu, Mu = −[L, P ] = ∂M ∂eu eu; on the other hand, we have ∂P ∂eu = P̄ (ξ) = [L(ξ), M(eu; ξ)], ∂M ∂eu = −[L(ξ), P̄ (ξ)]. From the second equation, we get M(eu; ξ) = −eu[L(ξ), P̄ (ξ)] + M̄(ξ) and thus P̄ (ξ) = −eu[L(ξ), [L(ξ), P̄ (ξ)]] + [L(ξ), M̄(ξ)]. We see then that we are able to obtain commutation relations: P̄ (ξ) = [L(ξ), M̄(ξ)], [L(ξ), [L(ξ), P̄ (ξ)]] = 0. There are additional relations determined by the third prolongation equation [−eu[L(ξ), P̄ (ξ)] + M̄(ξ), euP̄ (ξ)] = 0, so that we have [[L, P̄ ], P̄ ] = 0, [M̄, P̄ ] = 0. For the sake of convenience we put L = X1, M̄ = X2, P̄ = X3, [X1, X3] = X4 and we then have the following prolongation closed Lie alge- bra: [X1, X2] = X3 , [X1, X3] = X4 [X1, X4] = [X2, X3] = [X2, X4] = [X3, X4] = 0 , Note that if X4 = μX2 we obtain a quotient Lie al- gebra corresponding to the Euclidean group in the plane and we get a Cartan connection. Suppose now that P (u, ξ) = lnuP̄ (ξ), M(u, ξ) = M(eu, ξ). We derive then Pu = eu[L, M ] = d(ln uP̄ (ξ)) deu eu = 1 u P̄ (ξ), Mu = ∂M ∂eu eu = −[L, P ] = −[L, lnuP̄ (ξ)]; so that ∂M ∂eu = − lnu eu [L, P̄ (ξ)], from which we get M(eu, ξ) = −(lnu − 1)u[L(ξ), P̄ (ξ)] + M̄(ξ), and P (u, ξ) = ueu lnu[L, M ]. From [P, M ] = 0 we get, for u �= 0, 1 (which are trivial solutions of the Toda system), [[L, M ], M ] = 0 ; on the other hand substituting the above expression for M we get [[L, M̄ ], M̄ ] = 0 , 56 Acta Polytechnica Vol. 51 No. 1/2011 [[L, [L, P̄ ]], M̄ ] + [[L, M̄ ], [L, P̄ ]] = 0 , [[L, [L, P̄ ]], [L, P̄ ]] = 0 . by putting again for the sake of convenience L = X1, M̄ = X2, P̄ = X3, then we get the following in- finitesimal algebraic skeleton with the structure of an open Lie algebra: [X1, X2] = X4 , [X1, X3] = X5 , [X4, X5] = [X2, X7] , [X3, X4] = [X2, X5] , [X1, X4] = X6 , [X1, X5] = X7 , [X2, X3] = X8 , [X1, X8] = [X2, X4] = [X2, X6] = [X3, X7] = 0 , . . . We observe that by the homomorphismX4 = X5 = 0 and X8 = νX3 we get a closed Lie algebra (which is different from the Lie algebra corresponding to the Euclidean group in the plane obtained above): [X1, X2] = 0 , [X1, X3] = 0 , [X2, X3] = νX3 . which, by means of a suitable realization, can also provide us with a different Cartan connection (thus a different spectral problem and different conserva- tion laws); on the other hand, we can find a closed Lie algebra by means of the following homomorphism X4 = X2 and X5 = X3, and then we have [X1, X2] = X2 , [X1, X3] = X3 , [X4, X5] = [X2, X3] = X8 = 0 , [X3, X4] = [X3, X2] = −[X2, X3] = [X2, X5] = [X2, X3] , and we also deduce that X6 = X4 = X2, X7 = X3 and that [X1, X8] = [X2, X4] = [X2, X6] = [X3, X7] = 0 are all identically satisfied. It is easy to see that the two different cases above are both given by the homomorphism given by re- quiring X4 = λX2 and X5 = μX3. It is easy to realize that μ = −λ must old and there are the two cases λ = 0 with X8 = νX3 giving the first case, and X8 = 0 with λ = 1 giving the second case, respec- tively. For any λ �= 0 we have a closed Lie algebra de- pending on the parameter λ: [X1, X2] = λX2 , [X1, X3] = −λX3 , [X2, X3] = 0 . Furthermore, by putting in the prolongation skeleton X4 = X2 and X5 = −X3 it is possible to realize the prolongation skeleton as a Kač-Moody Lie algebra of the type [hi, hj ] = 0 , [hi, X+j ] = κijX+j , [hi, X−j ] = −κijX−j , [X+i, X−j] = δijhi , where we put [X2, X3] = X8, [X8, X2] = X9, [X8, X3] = X10, [X8, X9] = X11 [X8, X10] = X12 and {X1, X13, . . .} = hi, {X8, . . .} = hj , {X2, X9, X11, . . .} = X+i, {X3, X10, X12, . . .} = X−j. We also put [X8, X11] = X11, [X8, X12] = −X12, and so on. We then also have [X8, X13] = 0 and it is easy to realize that [X1, X9] = X9, [X1, X10] = −X10, [X1, X11] = X11, [X1, X12] = −X12, and so on; thus characterizing the Cartan matrix κij . It would be of interest to study the relation of skeletons with generalization of continuum Lie al- gebras to the case when the local algebra does not generate g(E;K, S) as a whole, where g(E;K, S) are Saveliev’s continuum Lie algebras and they are defined as follows. Let E be a vector space parametrizing Lie algebras gi, i = 0,+1,−1, ĝ ≡ g−1 ⊕ g0 ⊕ g+1, such that [X0(φ), X0(ψ)] = 0, [X+1(φ), X−1(ψ)] = X0(S(φ, ψ)), [X0(φ), X+1(ψ)] = X+1(K(φ, ψ)), [X0(φ), X−1(ψ)] = −X−1(K(φ, ψ)), with K, S bilinear maps E × E → E satisfying conditions equivalent to the Jacobi identity. Take g′(E;K, S) as the Lie algebra freely generated by a local part ĝ and then the quotient g(E;K, S) = g′(E;K, S)/J , J the largest homogeneous ideal hav- ing a trivial intersection with g0. In fact, such an algebra becomes the Kač-Moody algebra above when E = C n, K = Cartan matrix k, S = I. The relation with the Virasoro algebra without a central charge could be also considered in this light. 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Alberto 10, 10123 Torino, Italy Ekkehart Winterroth E-mail: ekkehart.winterroth@unito.it Department of Mathematics University of Torino via C. Alberto 10, 10123 Torino, Italy 58