TODA-LLY COOL STUFF BARBARA SHIPMAN

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1 TODA-LLY COOL STUFF BARBARA SHIPMAN. Introduction to the Toda Lattice Finite non-periodic Toda lattice Toda, 967: Dynamical system: a system of n collinear particles with positions q,..., q n Repelling force between adjacent particles: F r = e r Potential Energy: F r = V r, so V r = e r Total energy: H p, q = n n p 2 i + e q i+ q i 2 Hamilton s equations: The resulting equations are: q j = p j i= i= ṗ j = H q j, q j = H p j ṗ = e q 2 q ṗ j = e q j+ q j + e q j q j ṗ n = e qn q n It is not clear how to solve this system. Change of variables and inverse scattering solution Flaschka, 972. Recall any equation like L = [L, B], then the spectrum of L is preserved under this flow. The reason is that tr L k = tr L k = tr [ L k, B ] prove by induction, which is zero. L, B is called a Lax pair. Let Let a j = e q j+ q j > 0, j =...n b j = p j, j =...n; b j = c b. a X J = X =... Jacobi matrix,.. Π N X = 0 0 a n a a n 0 b n

2 2 BARBARA Then the Jacobi matrix has distinct, real eigenvalues, and the dynamical system now is Ẋ t = [ X t, Π N X t ], H t = 2 tr X 2 Constants of motion: H k = k tr X k, k = 2,..., n, also tr X. So it is completely integrable. Note each H k has its own flow [X Ẋ t = t, Π N X t ] k. The flows give coordinates on n-dimensional isospectral set within the phase space that is 2n-dimensional. There are many generalizations of this obtained by extending or changing the phase space on which the Toda lattice is defined. There are many differences in structure of the isospectral sets, some of which are subtle, some of which are qualitatively very different. Many have used these flows to study the topology of the isospectral sets in different versions of the Toda lattice. A standard inverse scattering scheme can be used to solve these equations. That is, suppose in general that we have the matrix equation L = [L, B]. This is equivalent to the following system of equations: Lφ = φλ φ = Bφ where φ is the matrix of eigenvectors of L, and Λ is the diagonal matrix of corresponding eigenvalues which are constant under the flow, since the flow is isospectral. Note that from the first equation and the fact that we have Lφ + L φ = φλ, or Lφ LBφ = BLφ, so L = [L, B]. Note that all steps are reversible, so that these systems are equivalent. Now, suppose we are given the initial matrix L 0, such that L 0 φ 0 = φ 0 Λ. Since L = φλφ, we can express Bφ entirely in terms of φ and Λ. Then you get an IVP for a system of ODEs for φ t. You can solve that, and then you can of course obtain L t. Symmetric version of Toda lattice: b α b. α X s = αn = D a..... D, α n b n a n b n where a i = α 2 i > 0, D = diag, α,..., α n. New equations: Ẋ s = [X s, Skew X s ] H s = 2 tr X 2 s,

3 where Skew X s = TODA LATTICE 3 0 α. α..... α n α n 0 There is a - correspondence between the symmetric X s and the Jacobian X J from before. There are several Factorization Solutions : For the Jacobian version of the problem: Ẋ = [ X, Π N X ] if we write then the solution is X 0 = X 0 ; e tx 0 = n t b + t 0 0 = 0 0, 0 0 X t = n t X 0 n t. The flow exists for all time t R, as long as a i > 0 for all i. If not all of the a i are > 0, then the factorization may fail. But you can still do an inverse scattering solution Kodama, McLaughlin, Ye, 996. The symmetric version: if we write then the solution is Ẋ = [X, Skew X] X 0 = X 0 ; e tx 0 = k t b + t = SO n matrix 0, 0 0 X t = k t X 0 k t. The flow exists for all time, and factorization always works, for *any* α i R Symes, 979. Work that has been done on Toda lattice: Finite nonperiodic Toda lattice Finite periodic Toda lattice where you add force between first and last particle semi-infinite Toda lattices full-symmetric real X s : Deift, Li, Nanda, Tomei tridiagonal, real X s : Flaschka, Tomei, Kodama, Symes, McLaughlin, Moser

4 4 BARBARA 0 complex: full Hessenberg Jacobi form X J = : Ercolani, Flaschka, Shipman, Singer complex tridiagonal Jacobi form: Ercolani, Flaschka, Haine real tridiagonal Jacobi form: Kodama, Ye, Flaschke, Gekhtman Nonabelian Toda lattices: Bloch, Gekhtman, Koelling complex semisimple Lie algebras: Gekhtman, Shapiro 2. Geometry of the Toda flows 2.. Example where factorization fails. Let X 0 = e tx 0 + t t t 0 = = t t = n t b + t. t +t + t t 0 +t Then Then X t = n t X 0 n t = e X 0 = If t, let B = upper triangular matrices +t +t 2 +t 2 0, t, t X t n t modb in SL 2 B flag mfld = n t b + t modb = e tx 0 modb The latter is defined for all t R. In the flag manifold, when t =, we have e X 0 0 modb = 0 modb Note that SL 2 B = N B B N 0 0 = big cell pt = circle B B So we have completed the flow.

5 TODA LATTICE Generalizations and geometry. Jacobi complex version: Let J be the set of matrices b. a X = such that b i = 0, a i, b i C The complex dimension is 2n 2. The Hamiltonians are H k = k + tr X k+ for k =,..., n. From these you get flows Ẋ t k = [ X t k, Π X k t k ], t k C. These n Hamiltonians cut out an isospectral subset T Λ. If a j = 0, this is preserved under the flow. Let e t kx k = n t k b t k = lower triangular, s on diag upper triangular B. Then the solution is X t k = n t k X o n t k. We use the flows to study geometry of T Λ. The fixed points of the flows are where all a j = 0. Take X T Λ, and say the characteristic poly is a n λ n s 2 λ n 2 s 3 λ n 3... s n. The companion matrix is the rational canonical form C Λ = s n... s 3 s Theorem 2.. Kostant There exists a unique L = 0 such that X = LC Λ L. The -dimensional eigenspace of each λ is spanned by λ v λ = λ 2.. λ n Let J Λ be the Jordan canonical form in T Λ with fixed ordering of eigenvalues. Let W be the matrix of generalized eigenvectors so that C Λ = W J Λ W. b n

6 6 BARBARA Take X 0 T Λ. Then we write X 0 = LC Λ L = LW J Λ W L We will map X 0 to the flag manifold SL n C mod B. The map is Consider X 0 W L modb. X t k = n t k X o n t k = n LW J Λ W L n, so L n is the unique L from the Kostant Theorem. Thus, it maps to W L n t k modb = W L e t kx k 0 modb The right hand side is a complete flow completion of the flow X t k.we see W L n t k modb = W L e t klw J Λ W L k = e t kj Λ k W L modb This the k th flow in SL n C mod B. So the flow acts on the initial condition. These flows generate an n -dimensional group action. If the eigenvalues are distinct, this gives the action of the diagonal subgroup on the flag manifold. This group A Λ is the stabilizer of J Λ in SL n C under conjugation. If the eigenvalues are not distinct, it is more complicated than the diagonal matrices. α α 2... α n. A Λ = blocks of... α2. α The moment map will map into a polytope in the weight lattice corresponding to the group. 3. The moment map Let G be a complex semisimple Lie group. Let H be a Cartan subgroup maximal torus maximal abelian diagonalizable eg diagonal subgroup in SL n, C. Let g and h be the corresponding Lie algebras. Let h R = Hom R h, R. Let σ : g gl V be a representation of g G. Note that h acts on V with simultaneous eigenspaces. So V = α A V α, where A is the set of weights of V. Note that if g = sl n, C, h h =... : h i = 0 For every H h, if v α is an eigenvector of h then h n H v = α H v α

7 TODA LATTICE 7 So α h R, ie α : h R is called the weight, and v α is called a weight vector. And V α is the span of eigenvectors corresponding to weight α. For example, if g = sl 3, C. Then h = h h 2 : h i = 0 h. 3 Let if v α = e, so α h = h, so V α = span {e }. Let V be an irreducible representation of g. Then consider the projectivization P V. Then g acts on P V. There is a highest weight vector v α. Then you obtain an orbit G [v α ] = G P P V. Consider the adjoint representation of sl 3, C on V = sl 3, C. Then P sl 3, C = CP 7. Then choose a highest weight vector: v α = = e e e e e 2. Then g SL 3, C acts on this weight vector by g e e e 2 = ge ge ge 2. = first column span of first two columns So the orbit is a flag manifold, and the stabilizer is B = Next we want to use the moment map to understand the Toda flows. The moment map is a map µ : G P h R. For X G [v α ], we write it as a linear combination of unit weight vectors: X = α A Π α X v α Then we let µ [X] = α A Π α X 2 α α A Π α X 2 Then the image µ G P is the weight polytope of V. Then H acts on G P by conjugation. The fixed points of the H-action are the vertices of weight polytope. For our example, they are the six points e i e i e j, and the weight polytope is a hexagon. Theorem 3.. Convexity theorem - Atiyah Given [X] G [v α ]. Then the closure of the torus orbits H [X] = orbit H [X] with lower-dimensional orbits of H in the boundary.

8 8 BARBARA Department of Mathematics, University of Texas at Arlington, Arlington, Texas, USA address: