Geodesic flow on SO(4), kac-moody Lie algebra and singularities in the complex t-plane

Publicacions Matem`atiques, Vol 43 (1999), 261–279. GEODESIC FLOW ON SO(4), KAC-MOODY LIE ALGEBRA AND SINGULARITIES IN THE COMPLEX t-PLANE A. Lesfari Abstract The article studies geometrically the Euler-Arnold equations associated to geodesic flow on SO(4) for a left invariant diagonal metric. Such metric were first introduced by Manakov [17] and extensively studied by Mishchenko-Fomenko [18] and Dikii [6]. An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke [4] and Haine [8]. In this problem there are four invariants of the motion defining in C4= Lie(SO(4) ⊗C) an affine Abelian surface as complete intersection of four quadrics. The first section is devoted to a Lie algebra theoretical approach, based on the Kostant-Kirillov coadjoint action. This method allows us to linearizes the problem on a two-dimensional Prym variety Prymσ(C) of a genus 3 Riemann surface C. In section 2, the method consists of requiring that the general solutions have the Painlev´e property, i.e., have no movable singularities other than poles. It was first adopted by Kowalewski [10] and has developed and used more systematically [3], [4], [8], [13]. From the asymptotic analysis of the differential equations, we show that the linearization of the EulerArnold equations occurs on a Prym variety Prymσ(Γ)ofananother genus 3 Riemann surface Γ. In the last section the Riemann surfaces are compared explicitly. 1. Lie algebra theoretical method Consider the group SO(4) and its Lie algebra so(4) paired with itself, via the customary inner product X,Y =−1 2tr (X.Y ) 262 A. Lesfari where X=(Xij)1≤i, j≤4= 6  i=1 xiei=   0−x3x2−x4 x30−x1−x5 −x2x10−x6 x4x5x60   ∈so(4). A left invariant metric on SO(4) is defined by a non-singular symmetric linear map Λ:so(4) →so(4),X→ Λ·X and by the following inner product: given two vectors gX and gY in the tangent space SO(4) at the point g∈SO(4) gX,gY =X,Λ−1·Y regardless of g. Then the geodesic flow for this metric takes the following commutator form (Euler-Arnold equations): (1.1) • X=[X,Λ·X],•≡d dt where Λ·X=(λijXij)1≤i, j≤4 = 6  i=1 λixiei=   0−λ3x3λ2x2−λ4x4 λ3x30−λ1x1−λ5x5 −λ2x2λ1x10−λ6x6 λ4x4λ5x5λ6x60   ∈so(4). This flow is Hamiltonian with regard to the usual Kostant-Kirillov symplectic structure induced on the orbit O=Ad∗ g(X)=g−1Xg :g∈SO(4) formed by the coadjoint action Ad∗ g(X) of the group SO(4) on the dual Lie algebra so(4)∗≈so(4). Let z1,z 2∈so(4) and consider ξ1=[X,z1], ξ2=[X,z2] two tangent vectors to the orbit at the point X∈so(4). Then the symplectic structure is defined by ω(X)(ξ1,ξ 2)=X,[z1,z 2]. This orbit is 4-dimensional and is defined by setting two trivial quadratic invariants H1and H2equal to generic constants c1and c2: (1.2) H1=√det X=x1x4+x2x5+x3x6=c1 H2=−1 2tr X2=x2 1+x2 2+···+x2 6=c2. Geodesic flow 263 Fonctions Hdefined on the orbit lead to Hamiltonian vector fields • X=[X,∇H]. In particular (1.3) H=1 2X,λ ·X=1 2λ1x2 1+λ2x2 2+···+λ6x2 6, induces geodesic motion (1.1). The constants of the motion are given by the two quadratic invariants H1,H2(1.2) and the Hamiltonian H(1.3). Since the system is Hamiltonian on a 4-dimensional symplectic manifold {H1=c1}∩{H2=c2} to make it completely integrable, one needs one independent invariant. Under Manakov’s conditions [17]: (1.4)                                                  λ1=β2−β3 α2−α3 λ2=β1−β3 α1−α3 λ3=β1−β2 α1−α2 λ4=β1−β4 α1−α4 λ5=β2−β43 α2−α4 λ6=β3−β4 α3−α4 ,α i,β iarbitrary,  i<j (αi−βj)=0 the Lax flow (1.1) can be transformed into the following Lax-type equation (with an indeterminate h): (1.5) • (X+αh)=[X+αh, ΛX+βh] α= diag (α1,... ,α 4) β= diag (β1,... ,β 4)      • X=[X,Λ·X]⇔(1.1) [X,β]+[α, Λ·X]=0⇔(1.4) [α, β] = 0 trivially satisfied for diagonal matrices. 264 A. Lesfari Consider the Kac-Moody extension (n=4) L= gl(n)=N  −∞ Aihi:Ai∈gl(n),Narbitrary of gl(n) with the bracket [A(h),B(h)] = Aihi,Bjhj= k  i+j=k [Ai,B j] hk and the nondegenerate, invariant inner product A(h),B(h)=Aihi,Bjhj= i+j=−1 tr(AiBj). This Lie algebra has a natural decomposition L=L−∞,−1+L0,∞,Lij =   i≥0 Akhk  . Observe that L⊥ −∞,−1=L−∞,−1and L⊥ 0,∞=L0,∞where ⊥is taken with respect the form above. The infinite-dimensional Lie group underlying L−∞,−1acts coadjointly on the dual Kac-Moody Lie algebra L∗ −∞,−1≈ L⊥ 0,∞=L0,∞, according to the rule of customary conjugation followed by registering the non-negative powers of honly. The orbits described in this way come equipped with a symplectic structure with Poisson bracket {H1,H 2}(α)=α, ∇L∗ −∞,−1H1,∇L∗ −∞,−1H2 where α∈L ∗ −∞,−1and ∇L∗ −∞,−1H∈L −∞,−1. The functions defined on this orbit are all in involution and the flow (1.5) evolves on the coadjoint orbit through the point X+ah ∈L 0,∞,X∈so(4). By the AdlerKostant-Symes theorem [1], [9], [24], the coefficients of zihiappearing in the Riemann surface: (1.6) C:(z,h)∈C2: det (X+ah −zI)=0  associated to the equation (1.5), are invariant of the system in involution for the symplectic structure of this orbit. Also the flows generated Geodesic flow 265 by these invariants can be realized as straight lines on the Abelian variety defined by the periods of the Riemann surface C. Explicitly, equation (1.6) looks as follows (1.7) C:H2 1(X)+H4(X)h2−H3(X)zh+H2(X)z2+ 4  i=1 (αih−z)=0 with H1(X)=c1,H2(X)=c2defined by (1.2), H3(X)=2H=c3by (1.3) and a 4th quadratic invariant of the form (1.8) H4(X)=µ1x2 1+µ2x2 2+···+µ6x2 6=c4 where µ1=γ2−γ3 α2−α3 ,µ 4=γ1−γ4 α1−α4 µ2=γ1−γ3 α1−α3 ,µ 5=γ2−γ4 α2−α4 µ3=γ1−γ2 α1−α2 ,µ 6=γ3−γ4 α3−α4 . For generic choice of the ci,Cis a Riemann surface of genus 3 and it has a natural involution σ:C→C,(z,h)→ (−z,−h) due to the skew-symmetry of the matrix X. Therefore the Jacobian variety Jac(C)ofC(cf. [7] for definitions) splits up into an even and old part: the even part is an elliptic curve C0=C/σ and the odd part is a 2-dimensional Abelian surface Prymσ(C) called the Prym variety: Jac(C)=C0+ Prymσ(C). The van Moerbeke-Mumford linearization method [19] provides then an algebraic map from the complex affine variety 4 ! i=1 {Hi(X)=ci}⊂C6 to the Jacobi variety Jac(C). By the antisymmetry of C, this map sends this variety to the Prym variety Prymσ(C): 4 ! i=1 {Hi(X)=ci}→Prymσ(C),p→ 3  k=1 sk and the complex flows generated by the constants of the motion are straight lines on Prymσ(C). Finally, we have the Theorem 1. Let Prymσ(C)be the Prym variety of the Riemann surface C(1.7). Under conditions (1.4), the Euler-Arnold equations (1.1) can be linearized on Prymα(C). 266 A. Lesfari 2. Structure of the singularities in the complex t-plane and the integrability of the Euler-Arnold equations First we recall several basics concepts. Consider a completely integrable Hamiltonian system (2.1) XH:• x=JH(x),x∈R2n+k,•≡d dt,≡∂ ∂x where His the Hamiltonian and J=J(x) is a skew-symmetric matrix with polynomial entries in x, for which the corresponding Poisson bracket {Hi,H j}="∂Hi ∂x ,J∂Hj ∂x # satisfies the Jacobi identity. Let gtbe the corresponding phase flow. The system possesses n+kindependent polynomial invariants H1,... ,H n+k (Casimir functions) of which klead to zero vector fields JH n+i(x)=0, 1≤i≤k, the nremaining ones are in involution (i.e., {Hi,H j}= 0). For most values of ci∈R, the invariant manifolds n+k $ i=1 Hi=ci,x∈R2n+k are compact, connected and by a theorem of Arnold-Liouville [5], are diffeomorphic to real tori Rn/Lattice on which the flows gt i(x) defined by the vector fields XHi,1≤i≤n, are straight lines motions. Let now x∈C2n+k,t∈Cand Z⊂C2n+ka non-empty Zariski open set. Note that the map Ψ:(H1,... ,H n+k):C2n+k→Cn+k is submersive on Z, i.e., dH1(x),... ,dH n+k(x) are linearly independent on Z. Let I=ΨC2n+k\Z =c=(ci)∈Cn+k:∃x∈Ψ−1(c) with dH1(x)∧···∧dHn+k(x)=0 be the set of critical values of Ψ and let Ibe the Zariski closure of Iin Cn+k. Recall [4], [22] that the system (2.1) is algebraically completely integrable if, for every c∈Cn+k\I, the fibre A=Ψ −1(c) is the affine part of an Abelian variety % A∼ =Cn/Lattice, the flows gt i(x), x∈A, t∈C, defined by the vector fields XHi,1≤i≤nare straight lines motions on Cn/Lattice and the coordinates xi=xi(t1,... ,t n) are meromorphic in (t1,... ,t n). Geodesic flow 267 Adler and van Moerbeke [3], [4] have developed and used the following necessary algebraic complete integrability criterion, inspired by the work of S. Kowalewski [10]: if the Hamiltonian system (2.1) is algebraically completely integrable, then each xiblows up for some value of t∈Cand whenever it blows up, the solution x(t) behaves as a Laurent series xi=t−ki&x(0) i+x(1) it+x(2) it2+···',k i∈Z,some ki>0 which admits dim(phase space) −1=m−1 free parameters. To explain the criterion, if the Hamiltonian flow (2.1) is algebraically completely integrable, it means that the variables xiare meromorphic on the torus Cn/Lattice and by compactness they must blow up along a codimension one subvariety (a divisor) S⊂Cn/Lattice. By the algebraic complete integrability definition, the flow (2.1) is a straight line motion on Cn/Lattice and thus it must hit the divisor Sin at least one place. Moreover through every point of S, there is a straight line motion and therefore a laurent expansion around that point of intersection. Hence the differential equation must admit Laurent expansions which depend on the n−1 parameters defining Sand the n+k constants cidefining the torus Cn/Lattice, the total count is therefore m−1 = dim(phase space) −1 parameters. The system (1.1) can be written in the form (2.1), with m=6,His given by (1.3), J=       0−x3x20−x6x5 x30−x1x60−x4 −x2x10−x5x40 0−x6x50−x3x2 x60−x4x30−x1 −x5x40−x2x10       ∈so(6) and is explicitely given by • x1=(λ3−λ2)x2x3+(λ6−λ5)x5x6 • x2=(λ1−λ3)x1x3+(λ4−λ4)x4x6 • x3=(λ2−λ1)x1x2+(λ5−λ4)x4x5 • x4=(λ3−λ5)x3x5+(λ6−λ2)x2x6 • x5=(λ4−λ3)x3x4+(λ1−λ6)x1x6 • x6=(λ2−λ4)x2x4+(λ5−λ1)x1x5. 268 A. Lesfari Here we have n=2,k=2,Z=x∈C6:Ψ(x)∈C4\Iis a nonempty Zariski open set in C6and (2.2) A=Ψ −1(c)= 4 ! i=1 Hi(x)=ci,x∈C6 where Hi(x) are given in (1.2), (1.3), (1.8). The invariant variety A(2.2) is the fibre of a morphism from C6to C4,thusAis a smooth affine surface for generic c=(c1,... ,c 4)∈C4 and the main problem will be to complete Ainto an Abelian surface. So, the question I address is how does one find the compactification of Ainto an Abelian surface? This compactification is not trivial and the simplest one obtained as a closure: A= 4 ! i=1 Hi(x)=cix2 0⊂P6 i.e., x1x4+x2x5+x3x6=c1x2 0 x2 1+x2 2+···+x2 6=c2x2 0 λ1x2 1+λ2x2 2+···+λ6x2 6=c3x2 0 µ1x2 1+µ2x2 2+···+µ6x2 6=c4x2 0 where [x0,x 1,... ,x 6] are homogeneous coordinates on P6, does not lead to this result. (In the following we will not distinguish between x1as a homogeneous coordinates [x0,x 1] and as an affine coordinate x1/x0.) Indeed, an Abelian surface is not simply-connected and therefore cannot be projective complete intersection. In other words, if Ais to be the affine part of an Abelian surface, Amust have a singularity somewhere along the locus at infinity I=A!{x0=0}. A direct calculation shows that Iis an ordinary double curve of Aexcept at 16 ordinary pinch points of A; the variety Ahas a local analytic equation x2=yz2. The reduced curve Iris a smooth elliptic curve. Now, it’s only after blowing up Aalong the curve Irthat one gets the desired Abelian surface. Geodesic flow 269 Theorem 2. The divisor of poles of the functions x1,x 2,... ,x 6is a Riemann surface Sof genus 9. For generic constants, the surface A (2.2) is the affine part of an Abelian surface % Aobtained by gluing to A the divisor S. Proof: Consider points at infinity which are limit points of trajectories of the flow. There is a Laurent decomposition of such asymptotic solutions, (2.3) X(t)=t−1&X(0) +X(1)t+X(2)t2+···' which depend on dim(phase space) −1 = 5 free parameters. Putting (2.1) into (1.1), solving inductively for the X(k), one finds at the 0th step a non-linear equation, (2.4) X(0) +X(0),Λ·X(0)=0 and at the kth step, a linear system of equations (L −kI) X(k)=(0 for k=1 quadratic polynomial in X(1),... ,X(k−1) for k>1 where L denotes the linear map L(Y)=Y,Λ·X(0)+X(0),Λ·Y+Y= Jacobian map of (2.4). One parameter appear at the 0th step, i.e., in the resolution of (2.4) and the 4 remaining ones at the kth step, k=1,... ,4. Taking into account only solutions trajectories lying on the surface A, we obtain one-parameter families which are parameterized by a Riemann surface. To be precise we search for the set Sof Laurent solutions (2.3) restricted to the affine invariant surface A, i.e., (2.5) S= closure of the continuous components of {Laurent solutions X(t) such that Hi(X(t)) = ci,1≤i≤4} = 4 ! i=1 t0−coefficient of Hi(X(t)) = ci = a Riemann surface whose affine equation is      w2+c1&x(0) 5x(0) 6'2+c2&x(0) 4x(0) 6'2+c3&x(0) 4x(0) 5'2+c4x(0) 4x(0) 5x(0) 6 ≡w2+F&x(0) 4,x (0) 5,x (0) 6'=0 276 A. Lesfari 3. Main observation We know from section 1, that the linearization of the Euler-Arnold equations (1.1) takes place on the Prym variety Prymσ(C) of the genus 3 Riemann surface C(1.7); the latter is a double ramified cover of an elliptic curve C0. Also, from the asymptotic analysis (section 2) of the equations (1.1), the intersection A(2.2) of the four invariants (1.2), (1.3), (1.8) completes into an Abelian surface % Aupon adding a Riemann surface S(2.5) of genus 9, which is a 4-fold unramified cover of a Riemann surface Γ(2.11) of genus 3; the latter is a double ramified cover of an elliptic curve Γ0. The Abelian surface % Acan also be identified as the Prym variety Prymσ(Γ) and the problem linearizes on Prymσ(Γ). From the fondamental exponential sequence 0→Z→O% A exp →O ∗ % A→0 we get the map ···→H1&% A,O∗ % A'→H2&% A,Z'→··· i.e., the first Chern class of a line bundle on % A. Therefore the group Pico&% A'of holomorphic line bundles on % Awith Chern class zero (any line bundle with Chern class zero can be realized by constant multipliers) is given by Pico&% A'=H1&% A,O% A'/H1&% A,Z' and is naturally isomorphic to the dual Abelian surface % Aof % A (means the dual Abelian surface). The relationship between % Aand % Ais symmetric like the relationship between two vectors spaces set up a bilinear pairing. It is interesting to observe that the Abelian surfaces % A= Prymσ(Γ) obtained from the asymptotic analysis of the differential equations and Prymσ(C) obtained from the orbits in the Kac-Moody Lie algebra are not identical but only isogeneous, i.e., one can be obtained from the other by doubling some periods and leaving other unchanged. The precise relation between these two Abelian surfaces is % A= (Prymσ(C)) i.e., they are dual of each other. In fact, the functions x1,... ,x 6are themselves meromorphic on % A, while only their squares are on Prymσ(C). Geodesic flow 277 The final point we want to make is that the relationship between the Riemann surfaces Γand Cis quite intricate. As usual we let Θ the theta divisor on Jac(Γ), we have Prymσ(C)\Π=Θ∩Prymσ(C)=Γ with Π a Zariski open set of Prymσ(C). 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