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# Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices

Leonid Faybusovich
Department of Mathematics
University of Notre Dame
Notre Dame, IN 46556
Leonid.F
Research partially supported by the NSF grant DMS98-03191
Michael Gekhtman
Department of Mathematics
University of Notre Dame
Notre Dame, IN 46556
Michael.G
Corresponding author
###### Abstract

We introduce a family of compatible Poisson brackets on the space of rational functions with denominator of a fixed degree and use it to derive a multi-Hamiltonian structure for a family of integrable lattice equations that includes both the standard and the relativistic Toda lattices.

PACS 02.30, 05.45, 11.30.N

Keywords: integrable lattices, compatible Poisson structures

## 1

It has been known since Moser’s work on finite non-periodic Toda lattice () that the space of rational functions plays an important role in solving integrable lattice and constructing action-angle variables. The Moser map that sends the space of tri-diagonal matrices (the phase space for the Toda lattice) into a space of proper rational functions with a denominator of degree was later utilized in a more general context (see, e.g. , , ) and generalized in  for an arbitrary semisimple Lie algebra to be used, mainly, as a tool for linearization for a class of finite systems of differential-difference equations.

The goal of this paper is to show that the Moser map also serves as an effective tool in establishing a multi-Hamiltonian nature of a class of integrable lattices that includes, in particular, the standard and the relativistic Toda lattices. This class was studied in our recent paper . Our approach is in contrast to those of , , , where the search for compatible Poisson structures was conducted in terms of the matrix entries of the Lax operator. Instead, we introduce explicitly the family of compatible Poisson brackets on the space of rational functions and then pull them back to a phase space of a lattice in question via the Moser map. Note that the linear Poisson structure on the phase space then corresponds to the Atiyah-Hitchin Poisson structure on rational functions . (In the case of the finite non-periodic Toda lattice, this correspondence was first explicitly pointed out in .)

In sect. 2 we review the construction of the family of integrable systems introduced in  as restrictions of the full Kostant-Toda flows in to elements of a certain family of symplectic leaves. A construction of compatible Poisson structures on rational functions and a consequent description of multi-Hamiltonian structure for lattices of sect. 2 is given in sect. 3. In the last section we give explicit formulae in terms of the corresponding Lax operators for master symmetries that generate these multi-Hamiltonian structure. In the case of the symmetric Toda lattice such master symmetries were implicitly defined via recursive relations in .

We conclude this introduction with two natural questions that we would like to address in the future. First, it would be interesting to extend our approach to more general (e.g. generic) symplectic leaves of the full Kostant-Toda flows. Second, we would like to understand how results of this paper can be translated into a geometric language for bi-Hamiltonian systems advocated in .

We would like to thank Yu. Suris and referees for valuable remarks.

## 2

Let us first recall the definition of the Kostant-Toda flows. Denote by vectors of the standard basis in , by an elementary matrix and by an matrix with s on the first sub-diagonal and s everywhere else. Let be, resp., algebras of upper triangular, strictly upper triangular, lower triangular and strictly lower triangular matrices. Denote by the set of upper Hessenberg matrices.

For any matrix we write its decomposition into a sum of lower triangular and strictly upper triangular matrices as

 A=A−+A0+A+

and define .

The hierarchy of commuting Kostant-Toda flows on is given by the family of Lax equations

 ˙X=[X,(Xm)≤0] (m=1,…,n). (2.1)

Each of the flows defined by (2.1) is Hamiltonian with respect to a linear Poisson structure on is obtained as a pull-back of the Kirillov-Kostant structure on , the dual of , if one identifies and via the trace form. Then a Poisson bracket of two functions on is

 {f1,f2}(X)=⟨X,[(∇f1(X))≤0,(∇f2(X))≤0]⟩ , (2.2)

where we denote by the trace form and gradients are computed w.r.t. this form. The -th flow of the hierarchy (2.1) is generated by the Hamiltonian

 Hm+1(X)=1m+1Tr(Xm+1)=1m+1n∑i=0λm+1i , (2.3)

where are the eigenvalues of .

The Weyl function

 M(λ)=M(λ,X)=((λ1−X)−1e0,e0)=∞∑j=0sj(X)λj+1 (2.4)

is an important tool in the study of the Toda flows (2.1) (see, e.g., , , ,, ). Here .

If , where consists of elements with simple real spectrum , one can write (2.4) as

 M(λ)=n∑i=0ρi(X)λ−λi(X) (n∑i=0ρi(X)=1) . (2.5)

The Lax equation (2.1) implies the following evolution for

 ˙ρi(X)=(λi(X)m−sm)ρi(X), ˙λi(X)=0 . (2.6)

An identity

 sm=n∑j=0λmjρj(X) (2.7)

allows one to write the system (2.6) in a closed form. The solution of (2.6) is given by

 ρi(X(t))=eλmitρi(X(0))∑nj=0eλmjtρj(X(0)) . (2.8)

As is well-known, symplectic leaves of the bracket (2.2) are orbits of the coadjoint action of the group of lower triangular invertible matrices:

 (2.9)

In  we described a family of integrable lattices associated with orbits of a special kind. This family contains both the standard and relativistic Toda lattices. Its members are parameterized by increasing sequences of natural numbers . To each sequence there corresponds a -parameter family of -dimensional coadjoint orbits , where

 XI=e0i1+k−1∑j=1eijij+1 . (2.10)

Two Darboux parametrizations for were found in . Each of them allows us to lift the first () of the Kostant-Toda flows (2.1) on to an integrable flow on equipped with the standard symplectic structure. In one of these parametrizations the Toda Hamiltonian has a form

 ~HI(Q,P)=12n∑i=0P2i+n∑0≤i

The set of elements of the form

 X=(J+D)(1−Ck)−1(1−Ck−1)−1⋯(1−C1)−1, (2.12)

where

 Cj=ij−1∑α=ij−1cαeα,α+1 , (2.13)

is dense in .

Then the first () of the Kostant-Toda flows (2.1) on is equivalent to the following system

 ˙di=di(ci−ci−1), ˙ci=ci(di+1−di+ci+1−ci−1)(ij

Equations (2.14) can be viewed as particular cases of the constrained KP lattice introduced and discretized in . In fact, all the minimal indecomposable invariant submanifolds for the latter system can be obtained this way. To emphasize a connection to Toda flows and the dependence on we will denote the lattice (2.14) by . If , we recover the relativistic Toda lattice

 ˙ci=ci(di+1−di+ci+1−ci−1), ˙di=di(ci−ci−1) . (2.15)

If, on the other hand, , then is the set of tri-diagonal matrices in with non-zero entries on the super-diagonal and thus, one obtains the standard Toda lattice. In coordinates equations of motion form a system

 ˙di=di(ci−ci−1), ˙ci=ci(di+1−di), (2.16)

which, after relabeling , becomes the Volterra lattice

 ˙ui=ui(ui+1−ui−1).

In  we proved the following

###### Proposition 2.1

For any , there exists a unique birational transformation of the from from to , that preserves the Weyl function and, for , sends the -th Toda flow (2.1) on into the -th Toda flow on . Here is a unipotent upper triangular matrix, whose off-diagonal elements in the first row are all equal to zero.

One of the consequences of Proposition 2.1 is that, for any , on the open dense set in , an element can be uniquely determined by its Weyl function . (This, of course, is well-known in the case of the tri-diagonal and relativistic Toda lattices, see, e.g., , , , .) In the next section, we use this fact to derive the multi-Hamiltonian structure for systems .

## 3

Let denote a space of rational functions of the form

 m(λ)=q(λ)p(λ)=∞∑i=0hiλi+1 , (3.1)

where is a monic polynomial of degree and is a polynomial of degree less than . To define a Poisson bracket on , it is sufficient to specify pairwise brackets for , where and are arbitrary.

For fixed and , let us denote

 q[k](λ)=λkq(λ)(mod p(λ)) (3.2)

and define a skew-symmetric bracket on coefficients of by setting

 {p(λ),p(μ)}k={q(λ),q(μ)}k=0 (3.3)

and

 {p(λ),q(μ)}k=p(λ)q[k](μ)−p(μ)q[k](λ)λ−μ . (3.4)
###### Proposition 3.1

() are compatible Poisson structures on .

Proof. It is sufficient to check the statement on an open dense subset of defined by the assumption that and are co-prime and all roots of are distinct. On this subset

 m(λ)=q(λ)p(λ)=n∑i=0riλ−λi

and the data determines and completely. In particular,

 rk=q(λi)∏j≠i1λi−λj=q(λi)p′(λi) (3.5)

By (3.3),

 {λi,λj}k=0 (3.6)

Since

 0={p(λi),q(μ)}k=p′(λi){λi,q(μ)}k+{p(λ),q(μ)}k↾λ=λi,

one obtains from (3.4)

 {λi,q(μ)}k=p(μ)q[k](λi)p′(λi)(λi−μ)=p(μ)λkiq(λi)p′(λi)(λi−μ)=−λkiq(λi)∏j≠iμ−λjλi−λj , (3.7)

which, together with (3.6), implies

 {λi,q(λj)}k=−λkiq(λi)δji, (3.8)

and, consequently,

 {q(λi),q(λj)}k=0 . (3.9)

It follows from (3.6), (3.8), (3.9), that in coordinates , any linear combination has a form

 {λi,λj}c={qi,qj}c=0 , {λi,qj}c=−c(λi)qiδji , (3.10)

where . It is easy to see that the bracket defined by (3.10) satisfies the Jacobi identity, with canonical coordinates given by

 xi=∫dλic(λi) , yi=lnqi (i=0,…,n). (3.11)

Thus, any linear combination of is a Poisson bracket, which finishes the proof.

Remarks. 1. The expression in the right hand side of (3.4) is called a Bezoutian of polynomials and . For more information on bezoutians and the role they in the control theory we refer the reader to a survey 

2. When , brackets (3.3), (3.4) give Atiyah-Hitchin Poisson structure on .

Poisson structure (3.3), (3.4) can be re-written directly in terms of the elements as follows

 {m(λ),m(μ)}k=((λkm(λ))−−(μkm(μ))−)m(λ)−m(μ)λ−μ , (3.12)

where, for a rational function , denotes the polynomial part of its Laurent decomposition and . It follows from (3.12) that , in terms of coefficients of the Laurent expansion (3.1) of , has a form

 {hi,hj}k=j∑α=ihk+αhi+j−1−α  (i

Now we can restrict brackets to a subset of that contains the image of the map described by (2.4). This subset, denoted by is the set of all with both polynomials and monic. To compute a Poisson bracket induced by on , we first note that, by (3.4), the bracket between , the leading coefficient of , and is . Then, since defines a surjective map from to , a straightforward computation leads to the following

###### Proposition 3.2

The family of compatible Poisson structures on is given by

 {M(λ),M(μ)}k=((λkM(λ))−−(μkM(μ))−)(M(λ)−M(μ)λ−μ +M(λ)M(μ)) . (3.14)

Any linear combination of brackets (3.14) is degenerate. Indeed, it follows from (3.7), that

 {λi, h0}k=−λkiqip′(λi) . (3.15)

Let , where are defined in (3.11). Then, clearly, and

 {q(λ)h0,F}c=n∑i=0q(λi)∏j≠iλ−λjλi−λj−q(λ)h0n∑i=0q(λi)h0p′(λi)

But, by (3.5), and so by the Lagrange interpolation formula. Thus, is a Casimir for on , while canonical coordinates coordinates for this bracket can be easily derived from (3.11):

 xi=∫dλic(λi) , yi=lnqiqn (i=0,…,n−1). (3.16)

Denote and define and . Then (3.8), (3.15) imply

 {ρi,Hj}k=(λk+j−1i−n∑l=0λk+j−1lρl)ρi .

Comparing the last equation with (2.6), (2.7), one sees that equations of motion induced on via the map (2.4) by the -th Toda flow (2.1) coincide with Hamilton equations generated in the Poisson structure by the Hamiltonian . Now, by Proposition 2.1, for any index set , (2.4) defines an almost everywhere invertible map from to . Moreover, the bracket (3.14) is polynomial in terms of the coefficients of the Laurent expansion , the well-known determinantal formulae, expressing the entries of the element of via are rational in (), and, by Proposition 2.1, so are formulae expressing matrix entries of elements of in terms of . Thus each of the brackets in (3.14) uniquely defines a Poisson bracket on which is rational if written in terms of either matrix entries of elements of or in terms of the parameters defined on by (2.12), (2.13). We obtain

###### Theorem 3.3

For any , the restriction of the hierarchy (2.1) to possesses a multi-Hamiltonian structure. Compatible Poisson brackets for this structure are obtained as a pull-back of the Poisson brackets (3.14) via the restriction of the map (2.4) to . The -th flow of the hierarchy (2.1) is generated in the Poisson structure by the Hamiltonian , where

 Hj={1jTr(Xj)j≠0lndet(X)j=0 . (3.17)

Theorem 3.17 provides a uniform way of constructing a multi-Hamiltonian structure for both standard and relativistic Toda lattices, as well as all lattices . Furthermore, since the relativistic Toda hierarchy is connected to the Ablowitz-Ladik one () via a birational transformation (cf., e.g. , ), one can use Theorem 3.17 to construct a multi-Hamiltonian structure for the latter hierarchy too. Note also, that for any , is the restriction of the bracket (2.2) to

We shall now derive a formula that, for fixed and , expresses in terms of , that agrees with the formula conjectured (and proved for ) in  for compatible Poisson brackets for the symmetric Toda lattice. First, recall the definition of the -matrix associated with the Lax equation 2.1. It is defined by (see, e.g. ).

###### Proposition 3.4
 {M(λ,X),M(μ,X)}k=14⟨X,[R(Xk∇λ+∇λXk),∇μ]+[∇λ,R(Xk∇μ+∇μXk)]⟩ , (3.18)

where .

Proof. Denote . It follows from (2.4) that . The following identities are easily checked:

 1λ−μ(Rλ−Rμ)=−RλRμ ,[X,∇λ]=[Rλ,E00] . (3.19)

Note also that . Taking (3.19) into an account, one sees that the second factor in the right-hand side of (3.14) is equal to ,while the first factor is equal to .

Thus,

 {M(λ,X),M(μ,X)}k=Tr(Xk(Rλ−Rμ)E00Rλ(E00−1)RμE00) =Tr(Xk∇λ(E00−1)RμE00−E00Rλ(E00−1)∇μXk) =Tr(Xk∇λ[E00,Rμ]E00−E00[E00,Rλ]∇μXk) (???)=Tr(E00Xk∇λ[∇μ,X]−[∇λ,X]∇μXkE00) (3.20)

Since the Weyl function is invariant under the adjoint action of a subgroup of that consists of matrices whose off-diagonal entries in the first row and column are zero, one concludes that for any matrix , that satisfies this property, . Then (3.20) can be re-written as

 {M(λ,X),M(μ,X)}k=⟨(Xk∇λ)>0,[∇μ,X]⟩−⟨[∇λ,X],(∇μXk)≤0⟩ (3.21)

Next, since (3.14) defines a skew-symmetric bracket on and, hence, , (3.21) implies

 {M(λ,X),M(μ,X)}k=12⟨X,[(Xk∇λ)>0−(∇λXk)≤0,∇μ]+[∇λ,(Xk∇μ)>0−(∇μXk)≤0]⟩ =12⟨X,[(Xk∇λ+∇λXk)>0,∇μ]+[∇λ,(Xk∇μ+∇μXk)>0]−[∇λXk,∇μ]−[∇λ,∇μXk]⟩ =12⟨X,−[(Xk∇λ+∇λXk)≤0,∇μ]−[∇λ,(Xk∇μ+∇μXk)≤]+[Xk∇λ,∇μ]+[∇λ,Xk∇μ]⟩

Taking the average of the last two lines and observing that, due to the Jacobi identity,
, one obtains (3.18).

## 4

In , master symmetries were used to establish a multi-Hamiltonian structure of the symmetric finite non-periodic Toda lattice. (For a definition and examples of master symmetries see, e.g., .) Namely, a family of vector fields was constructed, that satisfies the properties, that, in the context of the non-symmetric Toda lattice, can be described as follows. Let denote the Lie derivative in the direction of the vector field , be the Hamiltonian vector field on defined by the right-hand side of (2.1) and let functions be defined by (3.17). Then (i) ; (ii) ; (iii) and (iv) Lie derivatives of the Poisson bracket obtained as a restriction of the bracket (2.2) to are compatible Poisson brackets on . For , was defined via a differential equation of the form

 ˙X=Xl+1+[X,Bl(X)] , (4.1)

where an auxiliary matrix was chosen in such a way, that the right-hand side of (4.1) is tridiagonal (such choice is not unique). For , vector fields were defined recursively as . Various extensions of the results of  can be found in .

Using results from the previous section, we shall give explicit formulae for master symmetries . For all , they will be described by nonisospectral equations of the form (4.1).

First, we need to modify the results of ,  to describe nonisospectral flows on . Recall that for any there exist uniquely defined vectors , such that and

 XP(λ)=λP(λ)−pn+1(λ)en,~P(λ)TX=λ~P(λ)T−~pn+1(λ)eTn . (4.2)

Necessarily, are polynomials of degree (moreover, are monic). Furthermore, is equal to and is a scalar multiple of the characteristic polynomial of . Consequently,

 ddλP(λ)=DP(λ),ddλ~P(λ)T=~P(λ)T~D, (4.3)

where (resp. ) is a uniquely defined strictly lower (resp. upper) triangular matrix independent of . Then a differentiation of equalities (4.2) w.r.t. leads to the following relations

 [X,D]=1+envT,[X,~D]=−1+~veTn , (4.4)

where are vectors that depend on .

Remark. It is worth mentioning that pairs and belong to a class matrix pairs satisfying a condition rank. This class plays an important role in study of classical and quantum solvable models, see, e.g. , where it was studied in connection with the Calogero-Moser model, and  where it was used to derive a discrete time integrable system for the energies certain solvable quantum models.

For any polynomial , consider now a differential equation

 ˙X=Q(X)+[X,(Q(X)~D)≤0−(DQ(X))>0] , (4.5)
###### Proposition 4.1

The vector field defined by (4.5) is tangent to . If evolves according to (4.5), then the evolution of functions and defined in (2.4), (2.5) is given by equations

 ˙ρi=0, ˙λi=Q(λi) . (4.6)

and

 ˙sj=jm∑k=0Qksk+j−1 (4.7)

Proof. To prove the first statement, we have to show that only diagonal and super-diagonal entries of the right hand side of (4.5) are nonzero. First note, that is upper triangular. Therefore, the lower triangular part of of the right hand side of (4.5) is equal to due to the second equality in (4.4). Similarly, if denotes the part of the matrix strictly above the super-diagonal, then by the first equality in (4.4). Since is upper Hessenberg, the first statement is proved.

Next, (4.5) implies

 ˙Xj=jXj−1Q(X)+[Xj,(Q(X)~D)≤0−(DQ(X))>0] , (4.8)

Since and are, resp., strictly lower and upper triangular, both and have zero first row and zero first column. Thus, it follows from (4.8) that and (4.7) follows. By (2.7), . Then it is easily seen, that (4.6) is consistent with (4.7) and therefore is satisfied by due to the well-known fact that are determined uniquely by .

Consider now vector fields on defined, in coordinates , by

 Vl=n∑i=0λl+1i∂∂λi (4.9)

and let, as before, and and be the Poisson brackets (3.3), (3.4).

###### Proposition 4.2

Vector fields satisfy the following properties:

Proof. (i) is obvious. To prove (ii), it suffices to use (3.6), (3.8), (3.8) and an identity .

To prove (iii), recall from (3.1), (3.5) that

 hj=n∑i=0riλji=n∑i=0qip′(λi)λji (4.10)

Since

 LVllnp′(λi)=∑α≠iLVlln(λi−λα)=∑α≠iλl+1i−λl+1αλi−λα =∑α≠il∑β=0λβiλl−β
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