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CRM-1854 (1993) MOMENT MAPS TO LOOP ALGEBRAS CLASSICAL R–MATRIX AND INTEGRABLE SYSTEMS† J. Harnad1 Department of Mathematics and Statistics, Concordia University 3 7141 Sherbrooke W., Montréal, Canada H4B 1R6, and 9 9 Centre de recherches mathématiques, Université de Montréal 1 C. P. 6128-A, Montréal, Canada H3C 3J7 n and a J M. A. Wisse2 8 2 Département de mathématiques et de statistique, Université de Montréal C.P. 6128-A, Montréal, P.Q., Canada H3C 3J7 2 v 4 0 1 Abstract 1 0 AclassofPoissonembeddingsofreduced,finitedimensionalsymplecticvectorspaces 3 into the dual space g∗ of a loop algebra, with Lie Poisson structure determined by the 9 R / classical split R–matrix R=P+−P− is introduced. These may be viewed as equivariant h moment maps inducieng natural Hamiltonian actions of the “dual” group GR =G+×G− t - of a loop group Gon the symplectic space. The R–matrix version of the Adler-Kostant- p Symes theorem is used to induce commuting flows determinedby isospecteral equeationseof e h Lax type. The ceompatibility conditions determine finite dimensional classes of solutions : to integrable systems of PDE’s, which can be integrated using the standard Liouville- v i Arnold approach. This involves an appropriately chosen “spectral Darboux” (canonical) X coordinate system in which there is a complete separation of variables. As an example, r the method is applied to the determination of finite dimensional quasi-periodic solutions a of the sine-Gordon equation. 0. Introduction. In [AHP, AHH1-AHH3] a unified framework was developed, describing a wide class of integrable finite dimensional Hamiltonian systems, as well as finite dimensional solutions † Talk given by J. H. at the NSERC-CAP Summer Institute in Theoretical Physics, Workshop on Quantum Groups, Integrable Models and Statistical Systems. Kingston, Canada, July 13 - 17th 1992. Research sup- ported inpart by the Natural Sciencesand EngineeringResearch Council of Canada and the Fonds FCAR du Québec. 1e-mail address: [email protected] or [email protected] 2e-mail address: [email protected] 1 2 J. HARNAD AND M.-A. WISSE to integrable systems of PDE’s. The basic idea is to represent all these systems in terms of commuting isospectral flows determined by Lax equations within finite dimensional Poisson subspaces of loop algebras consisting of orbits that are rational in the loop parameter. The commuting flows are generated by Hamiltonians that are spectral invariants, or equivalently Ad∗–invariants on the space g+∗ dual to the half of the loop algebra admitting holomorphic extensions to the interior disk of a circle within the complex plane of the loop parameter. According to the Adler-Kostaent-Symes (AKS) theorem, such invariants generate commuting flows with respect to the Lie Poisson structure on g+∗ and are determined by equations of Lax type. e Finite dimensional, generically quasi-periodic solutions to integrable systems of PDE’s arise within this framework as the compatibility conditions satisfied by the matrix elements, within a given representation. The procedure developed in [AHP, AHH1-AHH3] consists of three steps; first, a suitably defined moment mapping is used to embed symplectic vector spaces quotiented by Hamiltonian group actions as Poisson subspaces of the space g+∗. The image consists of rational coadjoint orbits with fixed poles. The AKS theorem is then used to define the Hamiltonians and determine the dynamical equations, with the ring of ienvariants generated by the invariant spectral curve given by the characteristic equation of the loop algebra element. The generic systems so obtained may be subjected to further reductions under both discrete and continuous Hamiltonian symmetry groups. Finally, a suitably defined Dar- boux (canonical) coordinate system is introduced, associated to the divisor of zeroes of the matrix representating the loop algebra element on the spectral curve. Within this “spectral Darboux” coordinate system, there is a complete separation of variables which, through application of the Liouville-Arnold integration method on the invariant Lagrangian manifolds, reduces the problem of integration to quadratures given by Abelian integrals. The resulting flows are thus seen to be linearized by the Abel map in the natural linear coordinate system on the Jacobian variety of the spectral curve, or some quotient thereof. Through the Jacobi inversion method, the matrix elements may be expressed in terms of theta functions, thereby producing, through classical Hamiltonian methods, the types of solutions typically obtained through more sophisticated algebro-geometric means [AvM, KN, D, AHH1] . This approach has been applied to a wide variety of integrable systems, both finite and infinite dimensional, such as: the cubically nonlinear Schrödinger equation and various multi- component generalizations thereof [AHP, AHH3, HW1, W1], the sine-Gordon equation [HW2, TW], the massive Thirring model [W2] the Neumann and Rosochatius systems [AHP, AHH1, AHH3] and various generalized tops [AHH2, HH]. In the subsequent section, an extended form of the moment map embedding method of MOMENT MAPS TO LOOP ALGEBRAS 3 [AHP, AHH2] will be given, based on the classical R–matrix technique, in which the two halves of the loop algebra g+, g− are treated on an equal footing. In section 3, the process of discrete reduction, together with the Liouville-Arnold linearization method, will be illustrated for the case of the sine-Goerdoneequation. Full details for the latter may be found in [HW2]. 1. Moment Maps to Loop Algebras and the Classical R–Matrix. We first establish notations for loop groups and algebras. Let (M,ω) denote the symplectic vector space whose elements are pairs (F,G) of N ×r matrices M = {(F,G) ∈ MN,r ×MN,r}, (1.1) with symplectic form: ω = tr (dFT ∧dG). (1.2) Let G denote the group of smooth loops in Gl(r) (real or complex), or some subgroup thereof, viewed as invertible matrix-valued smooth functions g(λ) defined on a smooth, simple closed curvee Γ enclosing the origin of the complex λ plane. Let G+, G− be the subgroups of loops admitting holomorphic extensions, respectively, to the interior and exterior regions Γ+, Γ− (including ∞), such that for g ∈ G−, g(∞) = I. The Lieealgeberas corresponding to G, G±, are denoted g and g± respectively. Their elements are smooth gl(r) valued functions on Γ, with elements X ∈ g+ admittinge holomorphic extensions to Γ+ and X ∈ g− to Γe−,ethe + − latter satisfyieng X(e∞) = 0. e e We identify g as a dense subspace of the dual space g∗ through the pairing: 1 dλ e < X,Y > := tr (X(λ)Ye(λ)) , 2πi λ IΓ X ∈ g∗ , Y ∈ g. (1.3) Using the vector space decompositeion e g = g− ⊕g+, (1.4) this allows us to identify the dual spaceseas e e g+∗ ∼ g , g−∗ ∼ g , (1.5) − + where g , g denote, respectiveley, theesubspacees of geconsisting of elements admitting a − + holomorphic extension to Γ− or Γ+, with X ∈ g satisfying X (0) = 0. + + + e e e e 4 J. HARNAD AND M.-A. WISSE Let P : g −→ g± ± (1.6) P : X 7−→ X ± ± e e be the projections to the subspaces g± relative to the decomposition (1.4) and define the endomorphism R : g → g as the difference: e e e R := P+ −P−. (1.7) Then R is a classical R-matrix [S], in the sense that the bracket [ , ] defined on g by: R 1 1 [X,Y] := [RX,Y]+ [X,RY] e (1.8) R 2 2 is skew symmetric and satisfies the Jacobi identity, determining a new Lie algebra structure on the same space as g, which we denote g . The corresponding group is the “dual group” R G = G− × G+ associated to G. The adjoint and coadjoint actions of G on g ∼ g∗ are R R R R given by: e e e e e e e e e Ad (g) : (X +X ) 7−→ g X g−1 +g X g−1 (1.9a) R + − + + + − − − Ad∗(g) : (Y +Y ) 7−→ (g X g−1) +(g X g−1) , (1.9b) R + − − + − + + − + − X ∈ g±, Y ∈ g , g ∈ G±. (1.9) ± ± ± ± The Lie Poisson bracket on g∗R ∼ g∗edual to the Leie bracket [ , ]eR is: e {ef,g}|X =< [df,dg]R,X > (1.10) for smooth functions f,g on g∗ (with the usual identifications, df| , dg| ∈ g∗ ∼ g). R X X Let A ∈ gl(N) be a fixed N × N matrix having no eigenvalues on Γ, and define the e e e following action of G on M R e G : M −→ M R g(λ) : (F,G) −→ (F ,G ) (1.11) g g e where (F ,G ) are defined by: g g 1 F := F − (A−λI)−1F g−1(λ)−g−1(λ) dλ (1.12a) g 2πi + − IΓ 1 (cid:0) (cid:1) G := G− (AT −λI)−1G gT(λ)−gT(λ) dλ. (1.12b) g 2πi + − IΓ (cid:0) (cid:1) MOMENT MAPS TO LOOP ALGEBRAS 5 It is straightforward to verify, using the identity (A−λI)−1 −(A−σI)−1 (A−λI)−1(A−σI)−1 = (1.13) λ−σ and residue calculus, that the G composition rule is satisfied, so (1.11), (1.12a,b) does, R indeed define a G –action on M. A similar calculation shows that this action preserves the R e symplectic form (1.2) and is, in fact, generated as a Hamiltonian action by the equivariant e moment map: J : M −→ g∗ A,Y R J (F,G) = λY +λGT(A−λI)−1F, (1.14) A,Y e e where Y ∈ gl(r) is any r×r ematrix. The constant term λY may be included without affecting the equivariance of the moment map (1.14) since λY is an infinitesimal character for the Lie algebra g : R < λY,[X,Y] >= 0, ∀X,Y ∈ g . (1.15) R R e The splitting J (F,G) = J (F,G) + J (F,G), J (F,G) ∈ g is determined by the pole A,Y + − ± ± e structure, withtheλY termplusthepolesateigenvaluesofAinΓ− includedinthe(g−)∗ ∼ g + part and the peoles at eigenvaelues in Γ+ ine the (g+)∗e∼ g part.e − e e The fibres of the Poisson map J are the orbits of the stability subgroup G := A,Y A e e Stab(A) ⊂ Gl(N) under the Hamiltonian Gl(N) action defined by e g : M −→ M g : (F,G) 7−→ (gF,(gT)−1G). (1.16) On a suitably defined open, dense set, this action is free, allowing us to define the quotient Poisson space M/G . Since the map J : M → g∗ passes to the quotient, defining a A A,Y R 1–1 Poisson map J : M/G → g∗ , we may identify M/G with the image Poisson space A,Y A R A gY ⊂ g∗ consisting of elements N(λ) of tehe form e A R e e n ni N e i,l N(λ) = λY +λ i , (1.17) (λ−αi)li Xi=1lXi=1 where {α } are the eigenvalues of A, {l } are the dimensions of the corresponding i i=1,...n i i=1,...n Jordan blocks of generalized eigenspaces, and the ranks and Jordan structures of the r × r matrices N are determined by the multiplicities of the eigenvalues of A and its Jordan i,l i structure. 6 J. HARNAD AND M.-A. WISSE The Hamiltonianequations on the phase space gY ∼ M/G are generated by elements of A A the ring IY consisting of Ad∗–invariant polynomials on g∗, restricted to the Poisson subspace A gY ⊂ g∗. According to the Adler-Kostant-Symes (AKS) theorem, in its R–matrix form [S], A R the elements of this ring Poisson commute, generatingecommuting Hamiltonian flows, and Hamileton’s equations for φ ∈ IY have the Lax form: A dN(λ) = [(dφ) ,N] = −[(dφ) ,N], + − dt (dφ) ∈ g±, dφ = (dφ) +(dφ) . (1.18) ± + − This implies that the spectral curvee S, with affine part defined by the characteristic equation det(L(λ)−zI) := P(λ,z) = 0, (1.19) where n a(λ) L(λ) := N(λ), a(λ) := (λ−α )ni (1.20) i λ i=1 Y is invariant under these flows, and its coefficients generate the ring IY. A Integrable systems of PDE’s then arise as the equations satisfied by the matrix elements of N implied by the compatibility conditions d(dφ) d(dψ) + + − +[d(φ) ,d(ψ) ] = 0, (1.21) + + dx dt φ, ψ ∈ IY, A where t, x are the respective flow parameters for the Hamitonians φ and ψ. The flows may be integrated through a standard algebro-geometric construction that leads to a linearizing map to the Jacobi variety of the spectral curve S ([KN, D, AHH1]). To obtain interesting examples, the generic systems so obtained must usually be further reduced by certain continuous or discrete symmetry groups. The easiest way to arrive at the linearization involves the introduction of a suitably defined “spectral Darboux” (canonical) coordinate system associated to the divisor of zeroes of the eigenvectors of L(λ) on the spectral curve. The general construction is detailed in [AHH3]. Its application to the particular problem of determining finite dimensional quasi-periodic solutionsto the sine-Gordon equationis described in the next section. Full details for this case may be found in [HW2]. MOMENT MAPS TO LOOP ALGEBRAS 7 2. Quasiperiodic Solutions of the Sine-Gordon Equation. To obtain the sine-Gordon equation ∂2u ∂2u − = sin(u) (2.1) ∂x2 ∂t2 (1) we shall need commuting isospectral flows in the twisted loop algebra su(2) := su (2). This is the subalgebra of gl(2) consisting of elements X(λ) that are invariant under the three involutive endomorphisms: b e e σ : X(λ) 7−→ JXT(λ)J (2.2a) 1 σ : X(λ) 7−→ −X†(λ¯) (2.2b) 2 σ : X(λ) 7−→ τX(−λ)τ (2.2c) 3 where 1 0 0 −1 τ := , J := . (2.3) 0 −1 1 0 (cid:18) (cid:19) (cid:18) (cid:19) The first of these implies that X is traceless, the second that it is in su(2) and the third that it is in the “twisted” subalgebra su(2). e We choose the matrix Y as b 0 −1 Y = J = , (2.4) 1 0 (cid:18) (cid:19) Γ as a circle centred at the origin of the λ–plane, and A as a diagonal 2N ×2N dimensional matrix, having distinct eigenvalues, all inside Γ+, of the form A = diag (α ,α¯ ,...,α ,α¯ ,−α ,...,−α¯ ,iβ ,−iβ ,...,iβ ,−iβ ), (2.5) 1 1 p p 1 p 2p+1 2p+1 N N where {α } have nonvanishing real and imaginary parts and {β } are real. j j=1,...,p j j=2p+1,...,N The space M = {(F,G) ∈ M2N,2 × M2N,2} must correspondingly be reduced by the endomorphisms Σ : (F,G) 7−→ (GJ,−FJ) (2.6a) 1 Σ : (F,G) 7−→ (−JG¯,JF¯) (2.6b) 2 Σ : (F,G) 7−→ (−iKFτ,iKGτ) (2.6c) 3 where J = diag (J,J,...,J) (2.7) 8 J. HARNAD AND M.-A. WISSE is the block diagonal 2N ×2N matrix with 2×2 blocks equal to J and 0 I 2p −I 0 2p   K = J (2.8) .  ..     J      is the 2N × 2N matrix consisting of a 4p × 4p block formed from I , the 2p × 2p identity 2p matrix and its negative, and N −2p diagonal 2×2 blocks J. The moment map JY of eq. (1.14) then intertwines the automorphism groups generated A F G by Σ ,Σ ,Σ and by σ ,σ ,σ . Denoting by 2j−1 , 2j−1 the consecutive pairs 1 2 3 1e 2 3 F G 2j 2j (cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19) of 2×2 blocks in (F,G), the fixed point set M ⊂ M under Σ ,Σ ,Σ consists of (F,G) with Σ 1 2 3 2×2 blocks of the form F ϕ γ¯ G −γ¯ ϕ 2j−1 = j j , 2j−1 = j j , j = 1,...,p (2.9a) F γ −ϕ¯ G ϕ¯ γ 2j j j 2j j j (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) F iϕ −iγ¯ G iγ¯ iϕ 2j+2p−1 = j j , 2j+2p−1 = j j , j = 1,...,p F iγ iϕ¯ G −iϕ¯ iγ (cid:18) 2j+2p (cid:19) (cid:18) j j (cid:19) (cid:18) 2j+2p (cid:19) (cid:18) j j (cid:19) (2.9b) F iγ γ¯ G −γ¯ iγ 2j−1 = j j , 2j−1 = j j , j = 2p+1,...,N. F γ iγ¯ G −iγ¯ γ (cid:18) 2j (cid:19) (cid:18) j j (cid:19) (cid:18) 2j (cid:19) (cid:18) j j (cid:19) (2.9c) We have Σ∗ω = ω, Σ∗ω = ω¯, Σ∗ω = ω (2.10) 1 2 3 so M is a real symplectic space, with symplectic form: Σ p N ω = 4 (dγ ∧dϕ¯ +dγ¯ ∧dϕ )+4i dγ¯ ∧dγ . (2.11) j j j j j j j=1 j=2p+1 X X b The restriction of JY to M , which we denote J, is given by A σ 0 −1 b(λ) −c(λ) J(Fe,G) := N(λ) = λ b+2λ ∈ su(2). (2.12) ¯ 1 0 c¯(λ) −b(λ) (cid:18) (cid:19) (cid:18) (cid:19) where b b p N 2 −ϕ γ¯ ϕ¯ γ |γ | j j j j j b(λ) = λ + +iλ α2 −λ2 α¯2 −λ2 β2 +λ2 j=1 j j ! j=2p+1 j X X (2.13) c(λ) = p αjγ¯i2 + α¯jϕ¯2j −i N βjγ¯j2 . α2 −λ2 α¯2 −λ2 β2 +λ2 j=1 j j ! j=2p+1 j X X MOMENT MAPS TO LOOP ALGEBRAS 9 Now, choose Hamiltonians H ,H ∈ IY as: ξ η A 1 a(λ) H (X) = − tr (X(λ))2 dλ (2.14a) ξ 4πi λ2 IΓ (cid:18) (cid:19) 1 a(λ) H (X) = tr (X(λ))2 dλ, (2.14b) η 4πi λ2N IΓ (cid:18) (cid:19) where p N a(λ) = [(λ2 −α2)(λ2 −α¯2)] (λ2 +β2). (2.15) j j k j=1 k=2p+1 Y Y Then the Lax form of Hamilton’s equations may be written as dN = −[dH (N) ,N] (2.16a) ξ − dξ dN = [dH (N) ,N], (2.16b) η + dη where, setting a(λ) L(λ) = N(λ) = λ2N−1L +λ2N−2L +···+L +a(λ)Y, (2.17) 0 1 2N−1 λ we have 1 dH (N) = − (L +a(0)Y) (2.18a) ξ − 2N−1 λ dH (N) = L +λY. (2.18b) η + 0 The spectral curve is of the form det(L(λ)−zI) = P(λ,z) = z2 +a(λ)P(λ) = 0 (2.19a) P(λ) = P +λ2P +···+λ2N−2P +λ2N, (2.19b) 0 1 N−1 with all coefficients P in IY and, in particular i A H = P , H = −P . (2.20) ξ 0 η N−1 Choosing the invariant level set 1 a(0)P = det(L +a(0)Y) = , (2.21) 0 2N−1 16 10 J. HARNAD AND M.-A. WISSE implies that 1 0 eiu L +a(0)Y = , (2.22) 2N−1 4 −e−iu 0 (cid:18) (cid:19) where u is real. Setting ξ = x+t, η = x−t, (2.23) the compatibility equations for (2.16a,b) then imply that u satisfies the sine-Gordon equation (2.1). Since the spectral curve C determined by (2.19a,b) is invariant under the involution (z,λ) 7→ (z,−λ), it is a two-sheeted covering of the hyperelliptic curve with genus N − 1 whose affine part given is by z2 +a(E)P(E) = 0, (2.24) where e e P(λ2) := P(λ), a(λ2) := a(λ), λ2 = E. (2.25) We also define functions b, c by e e b(λ2) = b(λ), c(λ2) = c(λ). (2.26) e e Setting z := zλ, we obtain the genus N hyperelliptic curve C with affine part given by e e z2 +Ea(E)P(E) = 0, (2.27) e e which has two additional branch points at E = 0,∞. e e e Following the general method of [AHH3], we define on C the divisor of degree N with coordinates (E ,ζ ) determined by solving the equation µ µ µ=1,...,N e 2c(E )+1 = 0 (2.28) µ for {E } and substituting in µ µ=1,...,N e P(E ) 2b(E ) µ µ ζ = − = . (2.29) µ s Eµa(Eµ) Eµ e e This defines the zeroes of the eigenvectors of L(λ) on tphe spectral curve C. In terms of these, e we have: N e u = −iln 2 E +ǫπ (2.30) µ ! µ=1 Y where ǫ = 1,0 for N even or odd, respectively. The {E ,ζ } may be viewed as µ µ µ=1,...,N ∗ coordinate functions on the coadjoint orbit through N = J ∈ su (2), and the following result R is key to the linearization of the flows (cf. [HW2] for details): b b