HAMILTONIAN STRUCTURE OF

NONABELIAN TODA LATTICE

M. Gekhtman

1. Introduction

In this paper we answer affirmatively the question posed in [RS] on the Liouville complete integrability of the finite non-periodic nonabelian Toda lattice. The nonabelian Toda lattice

\begin{matrix}Ȧ_{k} = B_{k+1}A_{k} − A_{k}B_{k} \\ Ḃ_{k} = A_{k} − A_{k−1} \end{matrix}  (1.1)

where A_{k}, B_{k} are n × n matrices and A_{k} are invertible, was introduced by Polyakov as a discretization of the principal chiral field equation. In the double-infinite case for a suitable class of initial data it was solved via the inverse scattering method in [Br MLR]. Krichever [Kr] found the solution in theta-functions for the periodic nonabelian Toda lattice. In this case the complete integrability in the Liouville sense was also established (see [RS]). The direct and inverse spectral problems developed in [B1] for second-order self-adjoint difference expressions with operator coefficients were used in [BGSh], [BG] to extend the method proposed by Berezansky in [B2] for solving semi-infinite scalar Toda lattice to the case of nonabelian equations similar to (1.1) associated with semi-infinite self-adjoint Jacobi matrices with operator coefficients. A generalization of the inverse spectral problem to the case of nonsymmetric difference operators with operator coefficients was applied in [G1-G3] to integrate the nonabelian Toda lattice and some of its reductions in the semiinfinite and finite non-periodic cases. Another approach, based on a factorization of ordinary differential operators with noncommuting coefficients, was applied in [EGR] to integrate both the finite nonabelian Toda lattice and its two-dimensional generalization.

Let 1 be the n × n identity matrix. The treatment of (1.1) is based on its Lax representation

L̇ = [L, L_{−}],  (1.2)

where in the finite non-periodic case we shall be concerned with, L is a block tridiagonal matrix

L = ( \begin{matrix}B_{0} & 1 & & 0 \\ A_{0} & ... & ... \\ & ... & ... & 1 \\ 0 & & A_{N−1} & B_{N} \end{matrix} )  (1.3)

Typeset by AMS-TEX

and L_{−} is its block lower triangular part. In (1.1) this corresponds to letting k run from 0 to N and setting boundary conditions A_{−1} = A_{N} = 0. In sections 2 and 3 we review, following [G1, G2], essential steps of the inverse problem method for solving (1.2). In particular, we define the n × n matrix rational function

M(λ) = P^{−1}(λ)Q(λ) = Q̃(λ)P̃^{−1}(λ)  (1.4)

that determines L uniquely and allows one to linearize (1.2). Here P (λ) = λ^{N+1}1+ …, Q(λ) = λ^{N}1 + …, P̃(λ) = λ^{N+1}1 + …, Q̃(λ) = λ^{N}1 + … are n × n matrix polynomials.

The Hamiltonian description of (1.2) was first given by Reiman in [R] in terms of the Hamiltonian reduction. In the beginning of section 4 we derive this description following the Adler-Kostant-Symes formalism and describe generic symplectic leaves of the affine space of matrices of the form (1.3) w.r.t. the corresponding Poisson structure. Then we compute the induced Poisson structure on rational matrix functions (1.4) and use it to construct a maximal family of Poisson-commuting independent integrals of the nonabelian Toda lattice.

Notations. In what follows a block matrix means a matrix whose entries are n × n blocks. C_{jk} denotes a (j, k) block entry of a block matrix C, while e_{αβ} denotes the n × n matrix (δ_{μα}δ_{νβ})_{μ,ν=1}^{n}.

The gradient ∇f of a function on matrices is always defined via the traceform. For a rational matrix function W (λ), (W (λ))_{+} is its polynomial part and (W (λ))_{−} = W (λ) − (W (λ))_{+}. For a function W (λ, μ) , (W (λ, μ))_{+,λ} denotes its polynomial part in λ. We denote by ϵ the block strictly upper triangular part of L:

ϵ = ( \begin{matrix}0 & 1 & & 0 \\ & ... & ... \\ & & ... & 1 \\ 0 & & & 0 \end{matrix} )  (1.5)

2. Inverse problem data

With each block-tridiagonal matrix (1.3) associated is the so-called Weyl function (cf. [B1], chap. VII)

M(λ) = Δ_{0}^{T} R_{λ}(L)Δ_{0},  (2.1)

where R_{λ}(L) = (L−λ1)^{−1} and Δ_{0} is the n(N +1)×n matrix Δ_{0} = col[1, 0, …, 0]. In other words, M(λ) is the upper left block entry of the resolvent R_{λ}(L). Clearly, M(λ) is a rational matrix-valued function whose Laurent expansion has a form

M(λ) = − ∑_{k=0}^{∞} \frac{1}{λ^{k+1}} h_{k}, h_{0} = 1.  (2.2)

Denote by H_{k} a block-Hankel matrix (h_{i+j})_{i,j=0}^{k}. The following theorem along with its semi-infinite version can be found in [G1, G2, Sh].

Theorem 2.1. There is a 1-to-1 correspondence between matrices (1.3) with invertible A_{k}, k = 0, …, N − 1 and n × n matrix-valued functions (2.2) satisfying conditions

(i)

det H_{k} ≠ 0, k = 0, …, N  (2.3)

(ii) there exist F_{0}, …, F_{N} such that

h_{k+N+1} = ∑_{j=0}^{n} F_{j}h_{k+j}, k = 0, 1, …  (2.4)

Effective procedures for solving the inverse problem of restoring L from its Weyl function M(λ) or, equivalently from the moment sequence (h_{j})_{j=0}^{∞} were proposed in [G1, G2, Sh].

Note that (2.4) is equivalent to the statement that Q(λ) = P (λ)M(λ), where P (λ) = λ^{N+1}1 − ∑_{j=0}^{N} F; λ^{j}, is a polynomial in λ of degree N. Given L with all A_{k} invertible, polynomials P (λ), Q(λ) can be constructed as follows. Consider a truncated eigenvalue problem

(L − λ1) col[P_{0}(λ), … P_{N}(λ)] = col[0, …, 0, x],  (2.5)

where P_{0}(λ) = 1. This problem has a solution if and only if x = −P (λ). It follows that

det(L − λ1) = det(P (λ)).  (2.6)

Coefficients of P (λ) can be found from (2.5) recursively. In particular, F_{N} = B_{0} + … + B_{N}.

Q(λ) can be constructed in the same way as P (λ) from the matrix L, obtained by deleting the first row and the first column from L. Thus, one has a factorization

M(λ) = P^{−1}(λ)Q(λ)  (2.7)

Similarly, M(λ) can be factorized into

M(λ) = Q̃(λ)P̃^{−1}(λ),  (2.8)

where P̃, Q̃ are monic matrix polynomials of degree N + 1 and N resp. To obtain (2.8), one can consider L̃ = D^{−1}LD, where

D = diag(1, A_{0}, A_{1}A_{0}, …, A_{n−1} ⋯ A_{0}) .

Then L̃ is a block tri-diagonal matrix with unit blocks under the diagonal and Δ_{0}^{T} R_{λ}(L̃)Δ_{0} = Δ_{0}^{T} R_{λ}(λ)Δ_{0} = M(λ). Similar to (2.5), P̃(λ) can be found from

[P̃_{0}(λ), …, P̃_{N}(λ)](L̃ − λ1) = [0, …, 0, −P̃(λ)],  (2.9)

where P̃_{0}(λ) = 1. It is important to note, that P̃_{0}(λ) = λ^{N+1}1+(B_{0} + A_{0}^{−1}B_{1}A_{0} + … + (A_{0} ⋯ A_{N−1})^{−1}B_{N}A_{0} ⋯ A_{N−1}) λ^{N} +…

(2.10)

Polynomials P (λ), Q(λ); P̃(λ), Q̃(λ) in (2.7), (2.8) are defined uniquely and thus, any pair (P (λ), Q(λ)), (resp. (P̃(λ), Q̃(λ)) such that M(λ) defined by (2.7) (resp. (2.8)) satisfies the nondegeneracy condition (2.3), determines unambiguously blocktridiagonal matrix L.

To conclude this section we formulate the following corollary of conditions (2.30), (2.40) that will be useful later.

Lemma 2.2. Let S(λ) be a polynomial of degree m such that (S(λ)M(λ))_{−} = 0. Then there exists a polynomial S_{1}(λ). such that S(λ) = S_{1}(λ)P (λ). In particular, if m < N + 1 then S(λ) = 0.

Proof. Since P (λ) is monic, we can write S(λ) = r(λ) + S_{1}(λ)P (λ), where r(λ) is a polynomial of degree less than N + 1. Then (r(λ)M(λ))_{−} = 0, which implies [r_{0}, …, r_{k}]H_{k} = 0, where we assumed r(λ) = r_{k}λ^{k} + … + r_{0}. Since k < N + 1, this contradicts (2.3). ■

3. Evolution equations

As it was mentioned in the Introduction the nonabelian Toda lattice has a Lax representation (1.2). Consider a general Lax equation

L̇ = [L, C],  (3.1)

that preserves the tridiagonal form (1.3) of L. This condition imposes strong restrictions on the auxiliary matrix A. Namely, it was shown in [G1], that C is completely determined by its block entries C_{0i}, i = 0, …, N and C_{j0}, j = 1, …, N. More precisely, we have the following reccurence relations for C_{jk}:

\begin{aligned}C_{j+1,k+j+1} = C_{j,k+j} & + C_{j,k+1+j}B_{j+k+1} − B_{j}C_{j,k+1+j} \\ & + C_{j,k+2+j}A_{j+k+1} − A_{j−1}C_{j−1,k+1+j} \end{aligned}  (3.2)

and

\begin{aligned}C_{j,j−k} = (A_{j−1}C_{j−1,j−k−1} & + B_{j}C_{j,j−k−1} + C_{j+1,j−1−k} − C_{j,j−k−2} \\ & − C_{j,j−k−1}B_{j−k−1})A_{j−k−1}^{−1}. \end{aligned}  (3.3)

In (3.2) k runs from N to 0 and j from 0 to n − k, while in (3.3) k runs from N to 1 and j from k to N.

In many instances equation (3.1) can be explicitly integrated via the inverse problem outlined in the previous section. Indeed, (2.1) and (2.2) imply that

h_{k} = (L^{k})_{00} ,

which justifies the term moments for h_{k}. Then, one can show ([G1-G3]) that the Lax equation (3.1) forces the following evolution of h_{k}.

Proposition 3.1. Let

\begin{matrix}ψ_{+}(λ) = ∑_{j=0}^{N} C_{0,j}P_{j}(λ) := ∑_{j=0}^{N} ψ_{j}λ^{j}, \\ ψ_{−}(λ) = ∑_{j=0}^{N} P̃_{k}(λ)(A_{k−1} … A_{0})^{−1}C_{k,0} := ∑_{j=0}^{N} ψ_{−j}λ^{j}, \end{matrix}  (3.4)

where P_{j}(λ), P̃_{k}(λ) are defined by (2.5), (2.9). If L obeys (3.1) then the evolution of h_{k} is given by

ḣ_{k} = ∑_{j=0}^{N}(h_{k+j}ψ_{−j} − ψ_{j}h_{k+j}).  (3.5)

In the particular case of the nonabelian Toda lattice the only non-zero element among A_{0j}, A_{k0} is A_{10} = A_{0}. Then ψ_{+}(λ) = 0, ψ_{−}(λ) = P̃_{1}(λ) = λ1 − B_{0} and we have

ḣ_{k} = h_{k+1} − h_{k}h_{1}  (3.6)

Here we used the equality B_{0} = (L)_{00} = h_{1}.

Proposition 3.2 ([G2, G3]). Let h_{k} involves according to (3.60) and let F_{0}, … F_{N} be defined as in (2.4). Then

(i) Ḟ_{k} = 0, k = 0, …, N (ii) solution to (3.6) is given by h_{k} = h̃_{k}h̃_{0}^{−1}, where the matrix H̃_{N} = (h̃_{i+j})_{i,j=0}^{N}

solves the linear equation with constant coefficients

˙H̃_{N} = ( \begin{matrix}0 1 & & 0 \\ ⋮ & ... \\ 0 & & 1 \\ F_{0} & … & F_{N} \end{matrix} ) H̃_{N}.

Proposition 3.2 shows how to linearize the nonabelian Toda lattice. But what is more important in the context of the present paper, it provides us with a large number of first integrals given by matrix elements of the polynomial P (λ) defined in (2.7). However, as it will be seen below, these integrals are not sufficient to prove the complete integrability of the nonabelian Toda lattice in the Liouville sense.

Proposition 3.3. Coefficients of the polynomial det(P̃(λ)+z1) are constant along trajectories of the nonabelian Toda flow.

Proof. Let P̃(λ) = λ^{N+1} − ∑_{j=0}^{N} F̃_{j}λ^{j}. Then, similar to (2.4) we have

h_{k+N+1} = ∑_{j=0}^{N} h_{k+j}F̃_{j}, k = 0, 1, …

Then (3.6) implies

\begin{aligned}∑_{j=0}^{N} h_{k+j} ˙F̃_{j} & = h_{k+N+2} − h_{k+N+1}h_{1} − ∑_{j=0}^{N}(h_{k+j+1} − h_{k+j}h_{1})F̃_{j} \\ & = ∑_{j=0}^{N} h_{k+j}[h_{1}, F̃_{j}]. \end{aligned}

Considering k = 0, …, N and using the invertibility of H_{N} = (h_{i+j})_{i,j=0}^{N}, we obtain

˙F̃_{j} = [h_{1}, F̃_{j}] or, equivalently, ˙ P̃(λ) = [h_{1}, P̃(λ)] and the statement follows. ■

4. Hamiltonian structure and compete integrability of the nonabelian Toda Lattice

Proposition 3.1 allows in many cases to linearize the Lax equation (3.1). However, (3.1) with an arbitrary matrix C satisfying (3.2), (3.3) is too general if we want to study the integrability of (1.2) in the Liouville sense. In particular, (3.1) does not even have to be Hamiltonian. In this section we shall derive the Poisson structure induced on the inverse problem data {M(λ); P (λ), Q(λ); P̃(λ), Q̃(λ)} by that of the nonabelian Toda lattice and use it to find Hamiltonians generating a maximal commutative family of isospectral flows that commute with (1.2). This flows will necessarily have the Lax form.

In what follows it will be convenient to use the tensor notations for Poisson brackets of matrix elements of matrix-valued functions (see [FT, ch. 3, § 1]): for n × n matrices A and B , {A ,^{⊗} B} is defined by

{A ,^{⊗} B}_{ij,i′j′} = {A_{ii′}, B_{jj′}}.  (4.1)

Following [FT], we list properties of (4.1) that will be used later:

{A ,^{⊗} B} = −P{B ,^{⊗} A}P,  (4.2)
{A ,^{⊗} BC} = {A ,^{⊗} B}(1 ⊗ C) + (1 ⊗ B){A ,^{⊗} C},  (4.3)
{A^{−1} ,^{⊗} B} = −(A^{−1} ⊗ 1){A ,^{⊗} B}(A^{−1} ⊗ 1).  (4.4)

Here P is the permutation operator in ℂ^{n} ⊗ ℂ^{n} : P(x ⊗ y) = y ⊗ x. Note also that P satisfies the identity

P(A ⊗ B) = (B ⊗ A)P  (4.5)

Proposition 4.1. The nonabelian Toda lattice (1.1) is generated by the Hamiltonian

H = \frac{1}{2}T r(L^{2}) = T r( ∑_{j=0}^{N−1} A_{j} + \frac{1}{2} ∑_{j=0}^{N} B_{j}^{2})

in the Poisson structure given by

\begin{matrix}\{A_{j} ,^{⊗} A_{k}\} = 0, \{B_{j} ,^{⊗} B_{k}\} = δ_{jk}[1 ⊗ B_{j}, P], \\ \{A_{j} ,^{⊗} B_{k}\} = P(δ_{j+1,k}(A_{j} ⊗ 1) − δ_{jk}(1 ⊗ A_{j})) \end{matrix}  (4.6)

Proof. The Lax form (1.2) of the nonabelian Toda lattice suggests that it fits into the Adler-Kostant-Symes scheme (see [K, Sy]). Indeed, if we consider the decomposition

sℓ(n(N + 1)) = g_{+} + g_{−},  (4.7)

where g_{+} is a subalgebra of block-upper triangular matrices and g_{−} is a subalgebra of block-strictly lower triangular matrices with n × n blocks, then the identification

ϵ + g_{−}^{⊥} ≅ g_{+}^{∗},  (4.8)

where ϵ is as in (1.5) and an orthogonal complement is taken w.r.t. the trace-from, equips the affine space ϵ + g_{−}^{⊥} of block-lower Hessenberg matrices with a Poisson bracket obtained as a pull-back of the Kirillov-Kostant one on g_{+}^{∗}:

{f_{1}, f_{2}}(X) = T r(X, [π_{+}∇f_{1}(X), π_{+}∇f_{2}(X)]),  (4.9)

where X ∈ ϵ + g_{−}^{⊥} and π_{+} is a projection onto g_{+} parallel to g_{−}.

The Poisson brackets (4.6) follow from (4.9) and the fact that set of blocktridiagonal matrices is a Poisson submanifold in ϵ + g_{−}^{⊥}. Then the statement in Proposition 1 is a direct consequence of the AKS theorem ([K,Sy]). ■

Remark. When j = k, the second equation in (4.6) coincides with the definition of the Lie-Poisson bracket on gℓ(n).

Denote by J the set of block-tridiagonal elements in ϵ+g_{−}^{⊥}. As it was mentioned above, J is a Poisson submanifold of ϵ + g_{−}^{⊥} equipped with the bracket (4.9). However, it is not symplectic: the Poisson structure (4.6) is degenerate on J. Symplectic leaves of (4.6) are orbits in J of the co-adjoint action of the group G_{+} of invertible block-upper triangular matrices in SL(n(N + 1)). We shall consider orbits through elements L in J satisfying the following two genericity conditions:

1.

A_{j}, j = 0, …, N − 1 are invertible.  (4.10)

2. Matrix

Λ_{L} = B_{0} + A_{0}^{−1}B_{1}A_{0} + … + A_{0}^{−1} ⋯ A_{N−1}^{−1}B_{N}A_{N−1} ⋯ A_{0}  (4.11)

has distinct eigenvalues.

Proposition 4.2. The orbit O_{L} of the co-adjoint action of G_{+} through the element L ∈ J satisfying conditions 1 and 2 has a dimension

dim O_{L} = 2Nn^{2} + n(n − 1)  (4.12)

and is completely determined by eigenvalues of Λ_{L}.

Proof. Under the identification (4.8) the co-adjoint action of G_{+} on ϵ + g_{−}^{⊥} takes the form

Ad_{β}^{∗}X = π_{g_{−}^{⊥}}(Ad_{β}^{−1}X) + ϵ.  (4.13)

Let L ∈ J satisfies condition 1. We define a sequence of elements β_{0}, …, β_{N−1} ∈ G_{+} as follows: β_{k} = 1 + E_{k,k+1} ⊗ (γ_{k}A_{k}^{−1}), k = 0, …, N − 1, where γ_{k} ∈ gℓ_{n} are given recursively by

γ_{0} = B_{0}, γ_{k+1} = (B_{k+1} + A_{k}γ_{k}A_{k}^{−1}), k = 0, …, N − 2.

Then one computes

Ad_{β_{0}...β_{N−1}}^{∗}L = ( \begin{matrix}0 & 1 & & 0 \\ A_{0} & 0 & ... \\ & ... & ... & 1 \\ 0 & & A_{N−1} & γ_{N} \end{matrix} )  (4.14)

where

γ_{N} = B_{N} + A_{N−1}γ_{N−1}A_{N−1}^{−1}
= A_{N−1} … A_{0}B_{0}A_{0}^{−1} ⋯ A_{N−1}^{−1} + … + A_{N−1}B_{N−1}A_{N−1}^{−1} + B_{N}
= (A_{N−1} ⋯ A_{0})Λ_{L}(A_{N−1} ⋯ A_{0})^{−1} .

Acting further by

D = diag(1, A_{0}, A_{1}A_{0}, …, A_{N1} ⋯ A_{0}) ,

we see that

L_{0} := Ad_{β_{0}...β_{N−1}D}^{∗}L = ( \begin{matrix}0 & 1 & & 0 \\ 1 & 0 & ... \\ & ... & ... & 1 \\ 0 & & 1 & Λ_{L} \end{matrix} )  (4.15)

A stabilizer G_{L_{0}} of the element L_{0} is a subgroup of block-upper triangular matrices with zero second diagonal and with a diagonal part of the form diag (Y, Y, …, Y), where Y belongs to the subgroup G_{Λ_{L}} ⊂ GL(n) stabilizing Λ_{L} under the adjoint action of SL(n).

Furthermore, since the property (4.10) is preserved under the co-adjoint action of G_{+}, every element of O_{L} can be brought to the form (4.15). This, along with the fact that

dim G_{L_{0}} = n^{2} · \frac{N(N − 1)}{2} + dim G_{Λ_{L}},

implies

dim O_{L} = 2N n^{2} + dim O_{Λ_{L}},

where O_{Λ_{L}} is the orbit through Λ_{L} of the adjoint action of SL(n). Then (4.12) follows from the condition 2. ■

Remark. Though under certain conditions solutions of the nonabelian Toda lattice can be conjugated to a symmetric form [BG], we should emphasize the difference in geometric features of the nonabelian Toda lattice and symmetric Toda flows on block tri-diagonal matrices considered as a particular case of full symmetric Toda flows [DLNT]. In the latter case, symplectic leaves are orbits of the co-adjoint action of the upper Borel subgroup, condition (4.11) is not preserved by this action and the set of block-tridiagonal symmetric matrices satisfying (4.10) is stratified into many symplectic leaves whose type and dimension depend on types of Bruhat cells containing A_{k}, k = 0, …, N −1. On the other hand, adjoint invariants of the Borel subgroup are not necessarily constant along the flows of (1.2), so the parabolic chain construction of commuting integrals proposed in [DLNT] does not work for the nonabelian Toda lattice.

Moreover, Propositions 3.2, 3.3 suggest that one can use the inverse problem data defined by (2.1), (2.7), (2.8) to construct a maximal Poisson-commutative family of integrals.

Proposition 4.3. The Poisson structure (4.6) induces the following Poisson brackets for matrix elements of M(λ):

\begin{aligned}\{M(λ) ,^{⊗} M(μ)\} & = \frac{1}{λ − μ} (M(λ) − M(μ)) ⊗ (M(λ) − M(μ)) P \\ & + (M(λ)M(μ) ⊗ M(μ) − M(λ) ⊗ M(μ)M(λ)) P \end{aligned}  (4.16)

Proof. By (2.1),

δM(λ) = −Δ_{0}^{T} R_{λ}(L)δLR_{λ}(L)Δ_{0}.

Therefore,

∇M_{jk}(λ) = −R_{λ}(L)Δ_{0}e_{kj}Δ_{0}^{T} R_{λ}(L).

Recall that (4.6) is equivalent to (4.9), which we rewrite in the form

{f_{1}, f_{2}}(L) = T r(L([∇f_{1}(L), ∇f_{2}(L)] − [∇F_{1}(L), π_{−}∇f_{2}(L)]
−[π_{−}∇f_{2}(L), ∇f_{1}(L)])).  (4.17)

Firstly, we compute

T r(L[∇M_{jk}(λ), ∇M_{αβ}(μ)]) = T r([L − λ1, ∇M_{jk}(λ)]∇M_{αβ}(μ))
= −T r ((Δ_{0}e_{kj}Δ_{0}^{T} R_{λ}(L) − R_{λ}(L)Δ_{0}e_{kj}Δ_{0}^{T} ) ∇M_{αβ}(M))

By virtue of the identity R_{λ}(L)R_{μ}(L) = \frac{1}{λ−μ}(R_{λ}(L) − R_{μ}(L)), the last expression is equal to

\frac{1}{λ − μ} (M(λ)_{jβ}M(μ)_{αk} − M(λ)_{αk}M(μ)_{jβ})
= [\frac{P}{λ − μ}, M(λ) ⊗ M(μ)]_{jk,αβ} .  (4.18)

Secondly,

T r(L[∇M_{jk}(λ), π_{−}∇M_{αβ}(M)]) =
T r ((Δ_{0}e_{kj}Δ_{0}^{T} R_{λ}(L) − R_{λ}(L)Δ_{0}e_{kj}Δ_{0}^{T})π_{−}(R_{μ}^{(L)}Δ_{0}e_{βα}Δ_{0}^{T} R_{μ}(L)))

= T r (e_{kj} ∑_{ν=1}^{N}(R_{λ}(L))_{0ν}(R_{μ}(L))_{ν0}e_{βα}M(μ))

= T r (e_{kj}(Δ_{0}^{T} R_{λ}(L)R_{μ}(L)Δ_{0} − M(λ)M(μ))e_{βα}M(μ))

= T r ((\frac{1}{λ − μ}(M(λ) − M(μ)) − M(λ)M(μ))e_{βα}M(μ)e_{kj})

Similarly,

T r ([L, ∇M_{αβ}(L)], π_{−}∇M_{jk}(λ)) =

T r ((\frac{1}{λ − μ}(M(λ) − M(μ)) − M(μ)M(λ)) e_{kj}M(λ)e_{βα}) .

Substituting back into (4.17) we obtain the desired formula (4.16). ■

Now we can use the uniqueness of the factorization M(λ) = P^{−1}(λ)Q(λ) = Q̃(λ)P̃^{−1}(λ) in order to derive the following

Proposition 4.4. The Poisson brackets for matrix elements of P (λ), Q(λ), Q̃(λ), P̃(λ) are given by equations

\{P (λ) ,^{⊗} P (μ)\} = [\frac{1}{λ − μ}P, P (λ) ⊗ P (μ)]  (4.19)
\{Q(λ) ,^{⊗} Q(μ)\} = [\frac{1}{λ − μ}P, Q(λ) ⊗ Q(μ)]  (4.20)
\{P (λ) ,^{⊗} Q(μ)\} = \frac{1}{λ − μ}(Q(μ) ⊗ P (λ) − Q(λ) ⊗ P (μ))P − Q(λ) ⊗ Q(μ)P(4.21)

\{P̃(λ) ,^{⊗} P̃(μ)\} = [P̃(λ) ⊗ P̃(μ), \frac{1}{λ − μ}P] + [P̃(λ) ⊗ 1, (Q̃(μ) ⊗ 1)P]
− [1 ⊗ P̃(μ), (1 ⊗ Q̃(λ))P]  (4.22)
\{Q̃(λ) ,^{⊗} Q̃(μ)\} = [Q̃(λ) ⊗ Q̃(μ), \frac{P}{λ − μ}]  (4.23)
\{P̃(λ) ,^{⊗} Q̃(μ)\} = \frac{1}{λ − μ}(P̃(λ) ⊗ Q̃(μ) − P̃(μ) ⊗ Q̃(λ))P − 1 ⊗ Q̃(μ)Q̃(λ)P (4.24)
{P (λ) ,^{⊗} P̃(μ)} = [(Q(λ) ⊗ 1)P, 1 ⊗ P̃(μ)]  (4.25)
\{P (λ) ,^{⊗} Q̃(μ)\} = (\frac{1}{λ − μ}(P (λ)Q̃(μ) − Q(λ)P̃(μ)) ⊗ 1 − Q(λ) ⊗ Q̃(μ)) P(4.26)

\{Q(λ) ,^{⊗} P̃(μ)\} = (\frac{1}{λ − μ}(Q(λ)P̃(μ) − P (λ)Q̃(μ)) ⊗ 1 + Q(λ)Q̃(μ) ⊗ 1) P (4.27)
{Q(λ) ,^{⊗} Q̃(μ)} = 0  (4.28)

Proof. Consider {Q(λ) ,^{⊗} M(μ)} = −{P (λ)M(λ) ,^{⊗} M(μ)}. This is a polynomial in λ, therefore, by (4.3), one can see that

((P (λ) ⊗ 1){M(λ) ,^{⊗} M(μ)} + {P (λ) ⊗ M(μ)}  (4.29)

+ {P (λ) ,^{⊗} M(μ)}(M(λ) ⊗ 1))_{−,λ} = 0.

Due to (4.16),

((P (λ) ⊗ 1){M(λ) ,^{⊗} M(μ)})_{−,λ} =

((P (λ) ⊗ 1) (\frac{1}{λ − μ}(M(λ) − M(μ)) ⊗ 1 − M(λ) ⊗ M(μ)) P(M(λ) ⊗ 1))_{−,λ} .

Thus, (4.29) can be rewritten as (X(λ, μ) · (M(λ) ⊗ 1))_{−,λ} = 0, where

\begin{aligned}X(λ, μ) & = \{P (λ) ,^{⊗} M(μ)\} \\ & + (P (λ) ⊗ 1) (\frac{1}{λ − μ}(M(λ) − M(μ)) ⊗ 1 − M(λ) ⊗ M(μ)) P \end{aligned}

is a polynomial in λ of degree N. But then, due to Lemma 2.2, X(λ, μ) = 0 and we have

\{P (λ) ,^{⊗} M(μ)\} = −(P (λ) ⊗ 1) (\frac{1}{λ − μ}(M(λ) − M(μ)) ⊗ 1 − M(λ) ⊗ M(μ)) P.

(4.30)

Similarly, {P (λ) ,^{⊗} P (μ)M(μ)} = −{P (λ) ,^{⊗} Q(μ)} is polynomial in both λ and μ, so

0 = ({P (λ) ,^{⊗} P (μ)}(1 ⊗ M(μ)) + (1 ⊗ P (μ)){P (λ) ,^{⊗} M(μ)})_{−,μ}

= (\{P (λ) ,^{⊗} P (μ)\}(1 ⊗ M(μ)) + \frac{1}{λ − μ}(P (λ) ⊗ P (μ))(M(μ) ⊗ 1)P)_{−,μ}

= ((\{P (λ) ,^{⊗} P (μ)\} + \frac{1}{λ − μ})P (λ) ⊗ P (μ) − P (μ) ⊗ P (λ))P)(1 ⊗ M(μ)))_{−,μ}

and, again by Lemma 2.2,

\{P (λ) ,^{⊗} P (μ)\} = \frac{1}{λ − μ}(P (λ) ⊗ P (μ) − P (μ) ⊗ P (λ))P = [\frac{1}{λ − μ}P, P (λ) ⊗ P (μ)] ,

which proves (4.19).

Equation (4.21) is obtained by substituting (4.30), (4.19) into

{P (λ) ,^{⊗} Q(μ)} = −{P (λ) ,^{⊗} P (μ)}(1 ⊗ M(μ)) − (1 ⊗ P (μ)){P (λ) ,^{⊗} M(μ)},

whereas (4.20) can be derived from the identity

{Q(λ) ,^{⊗} Q(μ)} = (P (λ) ⊗ P (μ))({M(λ) ,^{⊗} M(μ)} − {P^{−1}(λ) ,^{⊗} Q(μ)}(Q(λ) ⊗ 1)

−(1 ⊗ Q(μ)){Q(λ) ,^{⊗} P^{−1}(μ)})(P (λ) ⊗ P (μ)),

which follows from (2.7), (4.2)-(4.4). The rest of the Poisson relations (4.19)-(4.28) can be proven analogously. ■

Remark. The Sklyanin bracket (4.19) is one of the basic Poisson structures in the theory of integrable systems associated with loop algebras (see [FT], [RS]). We see that the Poisson structure (4.6) is equivalent to Sklyanin brackets for matrix polynomials P (λ), Q(λ) coupled by equation (4.21). Note also, that as one sees from (4.18), the standard Lie-Poisson structure on sℓ(n(N + 1)) would have resulted in the Sklyanin bracket for matrix elements of M(λ). Instead, we have obtained, to the best of our knowledge, new Poisson bracket (4.16) for matrix rational function of the form (2.7).

Poisson relations from Proposition 4.4 can be used to prove a complete integrability of the nonabelian Toda lattice (1.2).

In Propositions 3.2, 3.3 we have found a large family of first integrals for (1.2) given by matrix elements of ZP (λ), where Z is an arbitrary constant matrix, and by the coefficients of the polynomial det(P̃(λ) + z1). Note, that matrix elements of ZP (λ) satisfy Poisson relations of the form (4.19):

\{ZP (λ) ,^{⊗} ZP (μ)\} = [\frac{1}{λ − μ}P, ZP (λ) ⊗ ZP (μ)] ,  (4.31)

therefore the family of integrals that we have is not Poisson-commutative. On the other hand, not all of the functions in this family are independent. In particular,

det(P (λ)) = det(P̃(λ)) = det(L − λ1).

Note also, that det(P (λ)) generated a center of the Poisson structure (4.31) considered independently from relations (4.20), (4.21) (see, e.g. [RS]). We can claim even more:

Proposition 4.5.

{det(P̃(λ) + z1), det(P̃(μ) + w1)} = 0,  (4.32)
{P (λ), det(P̃(μ) + z1)} = 0.  (4.33)

Proof. It suffices to prove that T r(P̃^{k}(μ)), T r(P̃^{ℓ}(ν)) and matrix elements of P (λ) Poisson-commute for any k, ℓ, μ, ν, λ.

Due to (4.22) and (4.3),

{T r(P̃^{k}(μ)), T r(P̃^{l}(ν))} = kl T r ((P̃^{k}(μ) ⊗ P̃^{l}(ν)){P̃(μ) ,^{⊗} P̃(ν)}) = 0,

which proves (4.32).

Furthermore, (4.25) implies

{P (λ) ,^{⊗} P̃^{k}(μ)} = [(Q(λ) ⊗ 1)P, 1 ⊗ P̃^{k}(μ)].

Then (4.33) follows from the identity

∑_{i=1}^{n}[X, 1 ⊗ Y]_{αi,βi} = 0

for any X ∈ gℓ(n) ⊗ gℓ(n), Y ∈ gℓ(n), α and β. ■

As an immediate consequence of (4.33), we have

Corollary 4.6. Any function f(P (λ)) that generates a nontrivial Hamiltonian flow w.r.t. the Poisson structure (4.19) is independent with coefficients of det(P̃(λ) + z1).

Let us now evaluate the number of independent on O_{L} functions that appear as coefficients of det(P̃(λ) + z1). It follows from (2.10) that P̃(λ) that corresponds to L has a form

P̃(λ) = λ^{N+1}1 + Λ_{L}λ^{N} + …  (4.34)

On the other hand, given a pair S(λ) = λ^{N+1}1 + S_{N}λ^{N} + …, T (λ) = λ^{N}1 + …, such that T (λ)S^{−1}(λ) satisfies nondegeneracy conditions (2.3) one can construct a block-tridiagonal matrix L ∈ ϵ + g_{−}^{⊥} such that −T (λ)S^{−1}(λ) is its Weyl function. Moreover, due to Proposition 4.2, L belongs to O_{L} if and only if S_{N} belongs to the adjoint orbit of SL(n) through Λ_{L}. Employing condition (4.11), we can state now that the number we are looking for is equal to the number of coefficients of the polynomial det(S(λ) + z1) independent as functions on the affine space {S(λ) = λ^{N+1}1 + S_{N}λ^{N} + … }, where S_{N} is a fixed diagonal matrix with distinct eigenvalues. The latter number is known to be equal to N \frac{n(n+1)}{2} (see, e.g., [RS, AHH1]).

Now, due to Corollary 4.6, any Poisson-commutative family of (N + 1)\frac{n(n−1)}{2} independent functions of P (λ) that generate nontrivial Hamiltonian flows w.r.t. the Sklyanin’s bracket (4.19) (or, equivalently, (4.31)) will yield the complete integrability of the nonabelian Toda lattice. Let us choose Z in (4.31) to be invertible matrix with distinct eigenvalues. Then ([RS]) non-leading coefficients of polynomials p_{1}(λ), …, p_{n−1}(λ) in the expansion

det(ZP (λ) + z1) = z^{n} + p_{n−1}(λ)z^{n−1} + … + p_{0}(λ)

provide the requisite number of integrals in involution.

Alternatively, one can use the following construction proposed independently in [G4, S] (see also [AHH2]). In these papers the generating functions were found for the Darboux coordinates on the manifold {P (λ) = Zλ^{N+1} + …; det P (λ) has distinct fixed zeroes }, equipped with the Sklyanin’s bracket (4.31). Let v ∈ ℂ^{n} be a cyclic vector for Z, i.e. vectors v, Zv, …, Z^{n−1}v are linearly independent. Then (N + 1)\frac{n(n−1)}{2} nonleading coefficients of the polynomial

B(P (λ)) = det[v, P (λ)v, …, P^{n−1}(λ)v]

are in involution and independent. Thus, we have proved Theorem 4.7. The nonabelian Toda lattice is completely integrable on generic symplectic leaves O_{L}.

Acknowledgments. The author is grateful to A. Bloch and M. Shmoish for valuable discussions and to the Max-Planck-Institut für Mathematik for warm hospitality. This work was partially supported by AFOSR grant F49620-96-1-0100.

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Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, D-53225 Bonn, Germany

   E-mail address: gekhtman@mpim-bonn.mpg.de