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From the Toda lattice to the Volterra lattice and back
REPORTS
Vol. 50 (2002)
Oh’ MATHEMATICAL
No. 3
PHYSICS
FROM THE TODA LATTICE TO THE VOLTERRA LATTICE AND BACK PANTELIS
A. DAMIANOU
Department of Mathematics and Statistics, University of Cyprus, P. 0. Box 537, Nicosia, Cyprus (e-mail: [email protected])
and RUI LOJA FERNANDES Departamento de Matem&tica, Instituto Superior Tecnico, 1049-001 Lisboa, Portugal (e-mail: [email protected]) (Received January 21, 2002-Revised
July 2, 2002)
We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices. Keywords:
Integrable systems, Toda lattices, Volterra lattices.
2000 Mathematics
Subject
Classification.
35Q58,
37J35,
58172, 7OHO6.
1. Introduction The Volterra lattice is the system of ordinary differential equations tii = &(Ui-1 - &+I),
i = 1,...,n,
(1) where a,+1 = aa = 0. This system was first studied by Kac, van-Moerbeke and Moser in two fundamental papers [ 17, 201 for the modem theory of integrable systems. They showed, for example, that this system arises as a finite-dimensional approximation of the famous KdV equation. In this paper we shall refer to system (1) as the A,-Volterru system. Another well-known discretization of the KdV equation is the Toda lattice [25]. The Toda system can be written in the form: ai = Ui(bi - bi+l), (
hi
=
Ui-1
-
i = l,...,n
Ui,
(2)
with a0 = b,+l = 0. We shall refer to this system as the A,_l-Todu lattice. Generalizations of both equations to root systems other than A, were obtained by Bogoyavlenskij in [4, 51. The second author was supported in part by FCT-Portugal through program POCTI and grant POCTJ/l999/MAT/3308 1. Presented during the conference “Multi-Hamiltonian Structures: Geometric and Algebraic Aspects”, 9-17 August 2001, in Mathematical Conference Center in Bpdlewo, Poland. [3611
362
P. A. DAMIANOU
and R. LOJA FFXNANDES
For the A,-Toda lattice, quadratic and cubic brackets were constructed, respectively, by Adler [l] and by Kupershmidt [ 191. Multiple Hamiltonian structures for these systems were introduced by the first author [8] in Flaschka coordinates and by the second author [14] in natural (4, p) coordinates. The analogous results for B,,-Toda were computed in Flaschka coordinates [lo], and in (4, p) coordinates [7]. The C, case is in [l l] in Flaschka coordinates, and in [7] in natural (4, p) coordinates. Finally, the D,, case can be found in [12]. The construction of these multiple Hamiltonian structures for the exceptional root systems is an open problem. For the Volterra system, multiple Hamiltonian structures were constructed recently by Kouzaris [ 181 for B,,, C,, and D, systems, which generalize the An-case. For the A, case, it was proved in [15] that the (periodic) Volterra lattice is an algebraic completely integrable (a.c.i.) system. In [16], the hyperelliptic systems were introduced similar to the even and odd Mumford systems (see [21, 22]), and the following link was established: there is a commutative diagram
in which M, P, I, K are (in that order) the phase spaces of the (even) Mumford system, the hyperelliptic Prym system (odd or even), the periodic A,-Toda lattice and the periodic A,-Volterra system. The vertical arrows are natural inclusion maps exhibiting for both spaces the subspace as fixed points varieties and the horizontal arrows are injective maps that map every fiber of the momentum map on the left injectively into (but not onto) a fiber of the momentum map on the right. In addition, Poisson structures were constructed so that this diagram has a meaning in the Poisson category. These give a precise geometric description, in the A,, case, of the well-known connection between the Toda and Volterra systems (see also [23, 241). In this paper we initiate the study of the relationship between the Toda and Volterra systems, for root systems other than A,. Here we shall restrict our attention mainly to the B, case, for which we give a complete description. The analysis for this case is simplified since we can view this system as a subsystem of the AZ,-system. We exhibit the phase space of the B,-system as a fixed point set of a finite group action by Poisson automorphisms. We prove a general result which allows us to reduce the Poisson structures in such situations, and hence we can relate the multiple Hamiltonian structures of the Volterra and Toda systems. The plan of this paper is as follows: In Section 2, we discuss several conditions under which the fixed point set of a Poisson action inherits a Poisson bracket, generalizing the Poisson involution theorem (see [16, 271). In Section 3, we recall the definition of the Toda systems, and give the multiple Hamiltonian structure for the B,, system. These were discovered in [lo] (see also [7]) and we use here a new approach based on the Poisson involution theorem to deduce them. In Section 4, we first recall the construction of the Volterra system and its generalizations for any root
FROM THE TODA LATTICE TO THE VOLTEFNA LATTICE AND BACK
363
system associated with a simple Lie algebra. Then we give the multiple Hamiltonian structure for the B,, (and C,) cases and, by applying a symmetry approach, we explain the relation between the Poisson structures for Volterra and Toda lattices. In the final section, we describe the connections we have been able to find between the two systems, including a generalization of Moser’s recipe (see [20]) to pass from the Volterra to the Toda lattices. 2.
Fixed point sets of Poisson actions
We shall see that the relation between the Toda and Volterra systems relies on special symmetries of the phase spaces. In this section, we study conditions under which the fixed point set of a Poisson action inherits a Poisson bracket. Although we will be interested mainly in finite symmetries, we give the following general result. THEOREM 2.1. Suppose that (M, {.,.}) is a Poisson manifold, and G a compact group acting on M by Poisson automorphisms. Let N = MG be the submanifold of M consisting of the fixed points of the action and let L : N C, M be the inclusion. Then N carries a (unique) Poisson structure {e, .JN such that
I* IF19 for all G-invariant functions
F21 =
{z*fi,
z*F2}N
(3)
Fl, F2 E C?(M).
Proof: For fi , f2 E C”(N) we choose Fl , F2 E COO(M) such that fi = z* Fi. We may assume that Fl and F2 are G-invariant by replacing Fi by
where p is the Haar measure
on G, so that jG ldp = 1. We set Ifi, f2IN = I* {Fl,
F21
and show that this definition is independent of the choice of Fi. For this we observe that, since the action of G is Poisson, the Hamiltonian vector field XF associated with a G-invariant function F : M + W is tangent to N. It follows that the Poisson bracket {Fl , Fz} (N, where FI and F2 are G-invariant functions, depends only on the restrictions Fi 1N . It is obvious from its definition that {a, .JN is bilinear, and satisfies the Jacobi and Leibniz identities, as these identities hold for {., .}. Hence, we obtain a Poisson 0 bracket on N, and it is the only Poisson bracket satisfying (3). &MARK 2.1. The previous result can be seen as a particular case of Dirac reduction. For the general theorem on Dirac reduction see Weinstein ([26], Proposition 1.4) and Courant ([6], Theorem 3.2.1). The case where G is a reductive algebraic group is discussed in [22] using a different method.
Since this result applies in particular corollary,
when G is a finite group, we have the
P. A. DAMIANOUand R. LOJAFERNANDES
364
COROLLARY2.1. Suppose that (M, {., .}) is a Poisson manifold, and G is a finite group acting on M by Poisson automorphisms. Then the fixed point set N = MG curries a (unique) Poisson structure {s, -}u satisfying Eq. (3). Let us consider the special case G = Z 2. Then G = {I, $}, where is a Poisson involution. We conclude that N = MG = {x : 4(x) = x} Poisson bracket satisfying (3). So we see that Theorem 2.1 contains case the following result, which is known as the Poisson involution [16, 271).
4 :M + M has a unique as a special theorem
(see
COROLLARY2.2. Suppose that (M, {a, s}) is a Poisson manifold, and # : M -+ M is a Poisson involution. Then the fied point set N = {x E M : d(x) = x} carries a (unique) Poisson structure {., .}u such that t* IFlY F2) = {z*F,, for all junctions
FI, F2 E Cm(M)
I*F2}N
invariant under 4.
One should note that, in general, the fixed point set MG is not a Poisson submanifold of M, since it is not a union of symplectic leaves of M. In other words, underlying Theorem 2.1 there is a true Dirac type reduction, rather than a restriction or Poisson reduction. 3, 3.1.
The Toda lattices The A,-Toda system
The phase space 7, of the (non-periodic) of all Lax operators in 51, of the form
L=
bl
al
0
1
b2
~2
0
..
0
lattice is the affine variety
0 0
1 ..
(4)
..
0 0
A,_l-Toda
h-1 0
...
..
1
G-I
bn
The reader will notice that we adopt a set of variables (due to Kostant) that differ slightly from the usual Flaschka variables (see r31, Chapter V). This Lax pair has an-obkous Lie algebraic interpretation which leads to the generalized Toda -Lattices (see next paragraph). The Hamiltonians I&, k = 1, . . . , n, defined by Hk = ;
tl-(Lk)
FROM THE TODA LA’lTICE TO THE VOLTERRA LA’l-l-ICE AND BACK
are in involution
with respect to the linear Poisson bracket {., .)&, defined by (5)
{bi9bj}:=09
{Ui,Uj}L
=
(49 bj};
= &C&j - &+l,j)*
vector fields Xk = (. , I&}:
The commuting
365
XG)
= [L,
admit the Lax representation
(Lk+9+l,
(6)
where the subscript + denotes projection into the Lie subalgebra of aI, generated by tbe positive roots. In [20], it was proved that the nonperiodic Toda lattice is a completely integrable system, by applying a finite-dimensional analogue of the inverse scattering method. For the periodic case, the flows are linear on the generic fibers of the momentum map K : 7, + C[x], and since these fibers are afIine parts of hyperelliptic Jacobians, the periodic Toda lattice is an a.c.i. system (see [2] for details). The Toda lattice is also Hamiltonian with respect to a quadratic Poisson bracket {., .}& defined by {ai, bj}:
{bi, bj}:
= Qibj(h,j
- &+l,j),
(7)
= ui(Ji,j+l - &+l,j)*
The quadratic Toda bracket appeared in a paper of Adler [l] in 1979. The brackets {., .}: and {a, .}t are compatible, and are in fact part of a full hierarchy of higher-order Poisson brackets [8, 141. Denote by xl and 1r2 the Poisson tensors associated with the linear and quadratic brackets. Also, let ZO be the Euler vector field Zo=paia+ebiL. i=l aui The following
i=l
(8)
abi
result is proved in [14].
PROPOSITION 3.1. There exists a symmetries Zk, k = 0, 1,2, . . . , such (i) The nk are compatible Poisson with respect to all of the nk. (ii) The vector fields xk admit the
.&t-l
sequence of Poisson tensors nk and muster that tensors and the functions Hk are in involution multiple Humiltoniun formulation = nkdHt = nk_ldHt+1.
(iii) The Poisson tensors, the integrals und the Humiltoniun vector fields satisfy the deformation relations: &XI
= (1 - k -
cz,
=
Xl
(1 -
2bk+lt
l)Xk+l,
Zk(H,) = (k + l)Hk+t, [zk,
zzl
=
(1 -
k)Zk+l.
(9)
P.A. DAMIANOU and R. LOJA FERNANDES
366
These relations will be important later to deduce some properties of the Volterra lattices. Using these relations we can find explicit formulae for the higher-order Poisson brackets. For example, we have that n-1 21
=
(bi(l - 2i) + bi+l(l + 2i)) fg
ELIi
I
i=l
(10)
+e(ai-1(2-2i)+ai(2+2i)+bf)&, I
i=l
so we find that the cubic bracket 7~3= -Cz,1r2 Iai,
&+I)+
=
244+lbi+l9
{Ui,
bi+l};
=
Uibj’,l
=
-UiUi+l~
{ai+1
9h
1;
+
is given by (see [lo]):
{Ui, Uf,
bi}:
=
-Uib,2
IUi
7bi+21$
=
UiUi+l,
Ibi
Tbi+lI$
=
Ui (h
-
Uf,
(11) +
&+I).
Similarly, one can compute any higher-order Poisson bracket. The cubic bracket ~r3 was found by Kupershmidt [19] via the infinite lattice. 3.2.
Toda
The B,-Toda system
Generalizations of the Toda lattice for other root systems are well known and were first given in [4]. We consider here only the case of B, . The phase space for the B,-Toda lattice is the affine variety of all Lax operators in 51zn+t of the form: b,
a,
0
1
b2
~2
..
L=
...
...
0
0
...
0 0
*. bn-1
an-1
1
b,
an
1
0
--a,
-1
-b,
-u,_~
-1
-b,-1
.
.. 0
0
.. -1
-b2
--al
0
-1
-bl
(12)
FROM THE TODA LAlTICE TO THE VOLTERRA LATTICE AND BACK
361
This matrix actually lies in the real simple Lie algebra g = so (n, n + 1). We would like to have a multiple Hamiltonian formulation for the B,-Toda systems fg
= [L,
(Lk+‘)+].
(13)
Note that, in this case, only the even powers of L give nontrivial 1 H2k = -tr(L2k), 2k
k=
integrals
1,2,...
In order to write down a Hamiltonian formulation for these systems we recall that the B,-Toda is a subsystem of the Ah-system. This is well known, and in fact more is true: we can find a symmetry such that the B,-Toda is obtained through a symmetry reduction from the Ah-Toda. To see this, observe that conjugation by a diagonal matrix takes the Lax matrix (12) to the Lax matrix bl
a1
0
1
b2
~2 *.
L=
...
-..
*. bn-1
an-1
1
bn
an
1
0
an
(14)
-b, 1
4-l
-b,-i *.
*.
0 0
0 0
1
0
0
**I
1
-b2
~1
0
1
-bl
and this is just a matrix in ‘&+I satisfying u&+1-i = ai and bzn+z-i = -bi. Therefore, if we consider the involution C#J: 7h+l + 72n+l defined by
(b(ai, bi) = (a2n+l-i, -hn+2-i), then the phase space of the B,-Toda can be identified with the fixed point set of this involution. On the other hand the &-action generated by this involution is a Poisson action for the odd Poisson brackets. More precisely, we have the following result. PROPOSITION
3.2. The map C#J : ‘T&+1 + I&+1 satisjies @k
= (-l)k+lrtk.
368
P. A. DAMJANOU and R. LOJAFERNANDES
Proof: For lower-order Poisson brackets one can check this relation by direct computation. For higher-order Poisson brackets, we note that expression (10) for the master symmetry ZJ shows after some tedious computation that 4*z, = -z1
+ x,
where X is a multiple of the first Hamiltonian &,nk so
it follows by induction
=
(k
-
vector field Xt. Now Eq. (9) gives
%nk+l,
that
1 = -&,(-l)k+l~k k-3
= (-l)k+2nk+i.
El
Using Corollary 2.2, we conclude that the odd Poisson brackets for the AZ,,-Toda induce Poisson brackets for the B,-Toda lattices, which therefore possesses a multiple Hamiltonian formulation. In this way we have deduced the following result of [lo]. COROLLARY3.1. There exists a sequence of Poisson tensors nl, ~3, n5, . . . such that the B,,-Toda lattices (13) possess a multiple Hamiltonian formulation: nk,zdHzl It should be noted that Theorem the Poisson brackets, as is illustrated
= %dHz+z. 2.1 gives an effective procedure by the following example.
to compute
EXAMPLE 3.1. In this example we compute the cubic B,-bracket. We denote by (ai, bj), where i, j = 1, . . . , n, the coordinates on the phase space of the B,-Toda lattice, and by (Gil 6j), where i = 1, . . . ,2n, j = 1, . . . ,2n + 1, the coordinates on the phase space of the Azn-Toda lattice. Suppose we would like to compute the cubic B,-brackt {ai, bi}$. First we extend the function ai and bi to invariant functions zi and bi on the phase space of the Ati-Toda lattice. For example, we can take the functions zi =
iii + Zizn+l-i 2 ’
&
6i - &nf2_i 2
Then we compute the AZ,-cubic bracket of these functions: for i c n we have
.
we find using (11) that
FROM THE TODA LATTICE TO THE VOLTERRA LATTICE AND BACK
369
while for i = n we obtain
{%a}; =-i (ii.4+ii,’ + iin+1i,2+2+ ii;,, Finally, we restrict the brackets to the &-phase
space, to obtain the cubic &-bracket
i
{~i,hi}~=-~(oib?+.z),
{~.,b,}:=-~(a.~~+2a,Z). The other brackets are computed
+ 2ii.ii”+l).
(15) (16)
in a similar fashion and they give the following
expressions:
REMARK 3.1. It is not hard to check that the (nonperiodic) C,-Toda system is a subsystem of the Ah-i-Toda system. The reader can check that the multiple Hamiltonian structure for the C,-Toda lattices found in [ 11, 71 can be obtained from the multiple Hamiltonian structure for the Ati- -Toda systems, as in the &-case. In fact, the involution 4 : I$, + 72, defined by $(ai,
satisfies &?rk = (-l)k?rk, 4.
h)
=
(a2n+l-i9
-b2n+2-i)
so the same method applies.
The Volterra Lattices
4.1. The A,-Volterra system
We now turn to the (nonperiodic) A,-Volterra system. Its phase space Ic, is the subspace of 7, consisting of all Lax operators (4) with zeros on the diagonal:
L=
0
a1
0
1
0
u2
0
1 ..,
**.
0
. . .
0 0
(18)
...
0 0
0
. . .
0
4l-1
1
()
370
P. A. DAMIANOU
and R. LOJA FERNANDES
Note that K, is not a Poisson subspace of 7,. However, K, is the fixed manifold of the involution $ : 7, -+ 7, defined by
((al,
a2..
. , a,),
@I, b2. . . , b,))
I+
((al,
~2..
.,
a,), C-h, 42..
. , -b,)),
and we have the following result, see [ 161. F’ROPOSWION
4.1. @ : I, +
is a Poisson
I,
automorphism
of (I,, {., +}“,), if k
is even. Proof: Again, one checks by direct computation
for k < 4 that
+.,?rk = (-l)k,rk. On the other hand, we have that $*Zt = -Zt , so by induction @,~k+l
we see that
* k:‘jqL
=
-
=
k-&Z,
=
j-&L-Z,(-l)‘nk
Z,nk
1 @*rk
=
(-l)k+‘nk+,.
0
and the result follows.
Therefore, by Corollary 2.2, Kc, inherits a family of Poisson brackets n2,7r4, . . . . For example, the quadratic bracket can be computed from Eqs. (7), in a form entirely analogous to Example 3.1, and is given by {ai~aj}L =
aiaj(h,j+l
-
Ji+l,j),
(19)
while the Poisson bracket rc4 is found to be given by the formulae {ai9 4,114,
{ai
=
Q&+1(%
+ ai+
9ai+21”, = &4+14+21
i=l,...,n-1
(20)
i = l,...,n-2.
The formulae for the brackets rr2 and n4 appeared in [9], and the existence of this hierarchy of Poisson brackets is proved in [16]. The first three Poisson structures of this system appeared first in [ 131 where the Volterra lattice is treated in detail, however the quadratic and cubic brackets do not coincide with the brackets in [9]. It follows that the restriction of the integrals & to K, gives a set of commuting integrals, with respect to these Poisson brackets. Also, for i odd, the Lax equations (6) lead to Lax equations for the corresponding flows, merely by putting all bi equal to zero. Taking i = 1, we recover the system
aj = aj(ai-1
- Ui+1),
i = 1,. . . ,n,
(21)
FROM THE TODA LA-l-l-ICE TO THE VOLTERRA LA-l-l-ICE AND BACK
371
which we called the A,-Volterra lattice in the introduction. More generally, taking i odd we find a family of commuting Hamiltonian vector fields on K, which are restrictions of the Toda vector fields, while for i even the Toda vector fields Xi are not tangent to Ic,. Hence, we obtain a family of integrable systems admitting a multiple Hamiltonian formulation: X2(k+l)-1
=
n2k+2dfh
=
k=
mkdH21+2r
1,2,...
This system was shown to be a completely integrable system in [20], while the periodic version of this system was proved to be an a.c.i. system in [15, 161. Generalized Volterra systems
4.2.
We now describe the construction of the generalized Volterra systems of Bogoyavlensky (see [5]). Let g be a simple Lie algebra, with rankg = n, and let lI = {or, 02,. . . , w,} be a Cartan-Weyl basis for the simple roots in g. There exist unique positive integers ki such that ’ kowo + klwl + -. e + kno,, = 0,
where ko = 1 and ~0 is the minimal negative root. We consider i = [B, L]
the Lax pair
)
where
L(t)-=
2
bi(t)eoi
+
e,
C
+
i=l
B(t)
=
[e,,
ewj],
lg
2 -e_,i ki h(t)
+ e-,.
i=l
Let h c g be the Cartan subalgebra.
For every root o, E h* there is a unique
Hmu E f~ such that w(h) = fi
(H,,,h),for all h E h, where B denotes the Killing form. Also, B induces an inner product on h* by setting (ma, wb) = @ (H,,,H,,), and we define
Cij
=
and i
1
if (Wi,Wj)#O
0
if (Wit Oj) = 0 or
I -1
and i > j,
if (wi,aj)#O
With these choices, the Lax pair above is equivalent
&=_$5( j=l
i = j,
bj
to the system of o.d.e.‘s
(22)
372
P. A. DAMIANOU and R. LOJA FERNANDES
To obtain a Lotka-Volterra
type system one introduces
a new set of variables
bv
Xji
=
-Xij,
Xjj
=
0.
Note that Xij # 0 iff there exists an edge in the Dynkin diagram for the Lie algebra g connecting the vertices mi and wj. System (22) in the variables xij, takes the form Xij = Xij
2 ks (Xis+ Xjs),
(23)
s=l
which is a Lotka-Volterra type system. We call (23) the Bogoyavlensky-Volterra system associated with g (or the g-Volterra system for short). EXAMPLE
4.1.
Let us take g = A,. Then we have the Dynkin diagram .
0
-
.
l -
.
-
l
with ki = 1, i = 0, . . . , n. If we label the edges of this diagram by ui, . . . , a,, then system (23) takes precisely the form of the A,-Volterra lattice. We now turn our attention to the B,-case. 4.3.
The B,-Volterra
system
For g = B,, the Dynkin diagram has the form 0
. ~
.
. ~
l
--_m
and for the obvious labeling (and after a linear change of variables) B, -Volterra system
we obtain the
ai = -uiu2, tii =
Ui(Ui_1 -Ui+l),
i=2,...,n-1,
(24)
I a, = a, (G-1 + a,). The multiple Hamiltonian structure for the B,-Volterra system can be obtained from the AZn-Volterra system in the same fashion as the multiple Hamiltonian formulation for the B,-Toda system was obtained from the Azn-Toda system (cf. Section 3.2). In fact, we notice that Eqs. (24) for the B,-Volterra system can be obtained from Eqs. (21) for the AZ,,-Volterra system by setting azn+i_i = -ai (this defines an invariant submanifold). Hence, we take as phase space of the B,,-Volterra system the Lax operators of the form (18) satisfying these additional restrictions:
373
FROM THE TODA LATTICE TO THE VOLTERFU LA’ITICE AND BACK
0
a1
0
1
0
a2
...
..*
0
0 0
*. .
... an-1
0
1 L=
0
a,
1
0
-a,
1
0
-G-1
1
0
(25)
..
.. 1
0
-a1
0
1
0
Notice that this subspace appears as the fixed point set of the involution &,, defined by V(ai)
=
(p : K2, +
-QZn+l-i.
We will see below that this involution can be used to obtain the multiple tonian structure of the I?,-Volterra system. REMARK 4.1. There is a diagonal matrix D which conjugates given above for the B, system, to the following Lax matrix
0
a1
0
1
0
a2 ...
L=
..
***
0
0
***
the Lax matrix
0 0
... 0
G-1
1
0
a,
1
0
an
-1
0
aa-1
-1
0
(26)
*. 0
0
Hamil-
*.
-1
0
a1
0
-1
0
P. A. DAMIANOUand R. LOJAFERNANDES
374
Notice that these Lax matrices are not obtained as Lax matrices for the B,-Volterra system (matrices of the form (12)) with zeros on the diagonal. We now have the following proposition The proof is similar and will be omitted.
analogous
to Propositions
3.2 and 4.1.
PROPOSITION4.2. Let nk, k = 2,4, . . . be the Poisson tensors of the A2n-Volterra system. Then q,nk = (-l)k’2nk. In this way, by Corollary 2.2, the phase space of the B,-Volterra system carries Poisson brackets IQ, ns, . . . . For example, from (20) we can compute, using the same technique as in Example 3.1, the following explicit formulae for the bracket ~r4: 1%9ai+1 1; =
1 (ai +
-ai&+
2
Iatl-l? a?& = ;a.-la.(an-I
ai+
+ 2a,),
i = l,...,n-2,
(27)
1 IQ7 %+21”,
The restriction
=
-aiai+lai+2*
2
of the invariant functions 1 H4k = - tI(L4k) 4k
to this space defines formulation: X4(k+l)-1
a hierarchy
=
n4k+4dH41
of systems,
=
possessing
n4kdH41+4,
k=
a multiple
Hamiltonian
1,2,....
The function 14 = i CyI;(2af + aiai+i) gives the B,-Volterra system (24). The brackets above, as well as the master symmetries for these systems (including the C, and D, cases) appear in a recent preprint of Kouzaris [ 181. However, the Lax matrices above lead to Lax pairs which are di$erent from the Lax pairs given by Kouzaris. Finally, we remark that the C,-Volterra system can be identified with the B,-Volterra systems (see [IS]), and hence admits the same description. 5.
Epilogue:
From Toda to Volterra and back
In this concluding section we would like to explain what we know so far about the connection between the Toda and Volterra systems, including the results obtained
FROM THE TODA LATTICE TO THE VOLTERRA LAlTICE AND BACK
375
above. The relation between the two systems relies on special symmetries of the phase spaces. We have shown above that the phase space of the A,-Volterra system appears as the fixed point set of a Poisson involution of the A,-Toda phase space. On the other hand, we have also shown that the multiple Hamiltonian structure of the B,-Toda system can be obtained from the AZ,-Toda system. Hence, to get to the B,-Volterra system, there are a priori two distinct ways to proceed, as explained by the following diagram: A%-Toda y/y Ah -Volterra \
B,, -Toda
z4 1
/
I
B, -Volterra We have seen that the correct way to proceed is to choose the left side of this diagram. In fact, we saw above that we can get from Ah-Toda to AZn-Volterra by a Poisson involution * : 72, + 72, (i.e. a Z$synunetry). Also, we can get from A&-Volterra to B,-Volterra using another Poisson involution cp : /cb + lch (and, hence, again a &-symmetry). Note also that we can go straight from AZ,-Toda to B,-Volterra using a Z4-symmetry: if one defines the map ~0: ‘Tz,, + Zn by ;l;
a
b2n+2-i),
then one checks that the group G = {Z,&?2,?3} acts by Poisson automorphisms on (Ih, 7r4, k = 1,2, . . . The fixed point set (i.e. the phase space for B,-Volterra) inherits Poisson structures z$k and the even flows reduce to this space. Notice also that &cz,, = ~0 and F2 = +, so this reduction in one stage (the middle line in the diagram) coincides with the reduction in two stages (the left part of the diagram). On the other hand, there seems to be no such symmetry reduction from the B,-Toda lattice to the B,-Volterra lattice: the various Poisson structures and Hamiltonian functions we have for the B,-Toda systems do not restrict to the phase space of the B,-Volterra systems. We could however still find a connection between these two systems, which goes in the opposite direction. To explain this connection, we perform the change of variable, ai = -2x?, and we consider the following equivalent (i.e. conjugate) form of the Lax matrix (25) (or, equivalently, (26)):
376
P. A. DAMIANOU and R. LOJA FWUWNDES
0
x1
0
xl
0
x2
0
...
...
0
...
...
L=
0
&I
%I
0
G&l
axn
0
0
0 0
...
0
In these new variables the equations x’l =
I
0
0
for the B,-Volterra
&ixxl
ax, 0
lattice (24) become:
x1x;,
Xi = Xi(Xf_1 - XF+l),
i=2,...,n-1.
x, = -Xn(X,2 +x,2-t),
The relation between the B,-Volterra system and the corresponding Toda system of types B,, and C, is similar to the relation observed by Moser (see [20]) between the A,-Volterra and the A,-Toda lattices. One starts by taking the square of the Lax matrix above and notices that L2 leaves certain subspaces invariant. Moreover, L2 reduces on each of these invariant spaces to a symmetric Jacobi matrix. More precisely, assume that L2 is an N x N matrix. Then there are two distinct cases: l N = 4n + 1 : Removing all odd columns and all odd rows we end up with a 2n x 2n matrix and a Toda system of type C,,. On the other hand, removing all even columns and all even rows we end up with a 2n + 1 x 2n + 1 matrix and a Toda system of type B,. l N = 4n + 3 : Removing all odd columns and all odd rows we end up with a 2n + 1 x 2n + 1 matrix and a Toda system of type B,, . On the other hand, removing all even columns and all even rows we end up with a 2(n + 1) x 2(n + 1) matrix and a Toda system of type Cn+i. In this way, we have a procedure which takes us from a Volterra systems of type B, (or C,) to either a Toda system of type B, or a Toda system of type C,. EXAMPLE 5.1. Take N = 9 and n = 2. Omitting columns of L2 we obtain the 5 x 5 matrix
all even rows and all even
FROM THE TODA LATTICE TO THE VOLTERRA LATTICE AND BACK
0
0
x3x4
0
x1x2 xi
We identify
+
xi
x3x4
0
0
-x3x4
0
0
--x3x4 -x;
and the equations
-
x;
-x1x2 -Xf
-x1x2
this matrix with a symmetric
B1 =xf,
377
Jacobi matrix of type B2, i.e. we let
B2 = x; + x&
A1 =
~1x2,
A2
=
~3x4,
satisfied by Al, AZ, Bl, and B2 are & = Al
(B2
-
Bl) ,
A2 = -A&, BI = 2A;, B2
These are precisely
=
the Toda equations
2A; - 2A:. of type B2.
REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20]
M. Adler: Invent Math. 50 (1979), 219-248. M. Adler and P. van Moerbeke: Invent Math. 103 (1991). 223-278. M. Audin: Spinning Tops, Cambridge University Press, Cambridge 1996. 0. Bogoyavlenskij: Commun. Math. Phys. 51 (1976), 201-209. 0. Bogoyavlenskij: Phys. Z&r. A 134 (1988), 34-38. T. Courant: Trans. Amer. Math. Sot. 319 (1990), 631-661. J. N. da Costa, P. A. Damianou: Bull. Sci. Math. 125 (2001), 49-69. P. A. Damianou: Z.&t. Math. Phys. 20 (1990), 101-112. P. A. Damianou: Phys. Lat. A155 (1991), 126-132. P. A. Damianou: .Z. Math. Phys. 35 (1994), 5511-5541. P. A. Damianou: Regul. Chaotic. Dyn. 5 (2000). 17-32. P. A. Damianou and S. P. Kouzaris: Multiple Hamiltonian Structures of Bogoyavlensky-Toda systrnems of type DN, Technical Report TRZlOZ2001,Department of Mathematics and Statistics, University of Cyprus. L. D. Fadeev and L. A. Takhtajan: Hamiltonian Methods in the Theory of Solitons, Springer, Berlin 1986. R. L. Fernandes: J. Phys. A: Math. Gen. 26 (1993). 3797-3803. R. L. Fernandes and J. P Santos: Rep. Math. Phys. 40 (1997), 475-484. R. L. Femandes and P. Vanhaecke: Commun. Math. Phys. 221 (2001), 169-196. M. Kac and P. van Moe&eke: Advances in Math. 3 (1975), 160-169. S. P. Kouzaris: Multiple Hamiltonian Structures for Bogoyavlensky-Volterra systems, University of Cyprus preprint TRZO3/2001. B. Kupershmidt: Discrete Lax equations and differential-difference calculus, Asterisque 123 (1985), 1-212. J. Moser: Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math. 16 (1975), 197-220.
378
P. A. DAMIANOU and R. LOJA FERNANDES
[21] D. Mumford: Tara Lectures on Theta, II, Birkhluser, Boston 1984. [22] P. Vanhaecke: Zntegmble Systems in he Realm of Algebraic Geometry, second edition, Springer, Berlin 2001. [23] A. P. Veselov and A. V. Penskdi: Regul. Chaotic Dyn. 3 (1998). 3-9. [24] A. Volkov: J. Soviet Math. 46 (1989). (15761581). [25] M. Toda: Theory of nonlinear laftices, Springer Series in Solid-State Sciences, 20, Springer-Verlag. Berlin, 1989. [26] A. Weinstein: J. Di$erentiul Geometry 18 (1983), 523-557. [27] P. Xu: Dirac submanifolds and Poisson involutions, preprint math.SG/O110326
Vol. 50 (2002)
Oh’ MATHEMATICAL
No. 3
PHYSICS
FROM THE TODA LATTICE TO THE VOLTERRA LATTICE AND BACK PANTELIS
A. DAMIANOU
Department of Mathematics and Statistics, University of Cyprus, P. 0. Box 537, Nicosia, Cyprus (e-mail: [email protected])
and RUI LOJA FERNANDES Departamento de Matem&tica, Instituto Superior Tecnico, 1049-001 Lisboa, Portugal (e-mail: [email protected]) (Received January 21, 2002-Revised
July 2, 2002)
We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices. Keywords:
Integrable systems, Toda lattices, Volterra lattices.
2000 Mathematics
Subject
Classification.
35Q58,
37J35,
58172, 7OHO6.
1. Introduction The Volterra lattice is the system of ordinary differential equations tii = &(Ui-1 - &+I),
i = 1,...,n,
(1) where a,+1 = aa = 0. This system was first studied by Kac, van-Moerbeke and Moser in two fundamental papers [ 17, 201 for the modem theory of integrable systems. They showed, for example, that this system arises as a finite-dimensional approximation of the famous KdV equation. In this paper we shall refer to system (1) as the A,-Volterru system. Another well-known discretization of the KdV equation is the Toda lattice [25]. The Toda system can be written in the form: ai = Ui(bi - bi+l), (
hi
=
Ui-1
-
i = l,...,n
Ui,
(2)
with a0 = b,+l = 0. We shall refer to this system as the A,_l-Todu lattice. Generalizations of both equations to root systems other than A, were obtained by Bogoyavlenskij in [4, 51. The second author was supported in part by FCT-Portugal through program POCTI and grant POCTJ/l999/MAT/3308 1. Presented during the conference “Multi-Hamiltonian Structures: Geometric and Algebraic Aspects”, 9-17 August 2001, in Mathematical Conference Center in Bpdlewo, Poland. [3611
362
P. A. DAMIANOU
and R. LOJA FFXNANDES
For the A,-Toda lattice, quadratic and cubic brackets were constructed, respectively, by Adler [l] and by Kupershmidt [ 191. Multiple Hamiltonian structures for these systems were introduced by the first author [8] in Flaschka coordinates and by the second author [14] in natural (4, p) coordinates. The analogous results for B,,-Toda were computed in Flaschka coordinates [lo], and in (4, p) coordinates [7]. The C, case is in [l l] in Flaschka coordinates, and in [7] in natural (4, p) coordinates. Finally, the D,, case can be found in [12]. The construction of these multiple Hamiltonian structures for the exceptional root systems is an open problem. For the Volterra system, multiple Hamiltonian structures were constructed recently by Kouzaris [ 181 for B,,, C,, and D, systems, which generalize the An-case. For the A, case, it was proved in [15] that the (periodic) Volterra lattice is an algebraic completely integrable (a.c.i.) system. In [16], the hyperelliptic systems were introduced similar to the even and odd Mumford systems (see [21, 22]), and the following link was established: there is a commutative diagram
in which M, P, I, K are (in that order) the phase spaces of the (even) Mumford system, the hyperelliptic Prym system (odd or even), the periodic A,-Toda lattice and the periodic A,-Volterra system. The vertical arrows are natural inclusion maps exhibiting for both spaces the subspace as fixed points varieties and the horizontal arrows are injective maps that map every fiber of the momentum map on the left injectively into (but not onto) a fiber of the momentum map on the right. In addition, Poisson structures were constructed so that this diagram has a meaning in the Poisson category. These give a precise geometric description, in the A,, case, of the well-known connection between the Toda and Volterra systems (see also [23, 241). In this paper we initiate the study of the relationship between the Toda and Volterra systems, for root systems other than A,. Here we shall restrict our attention mainly to the B, case, for which we give a complete description. The analysis for this case is simplified since we can view this system as a subsystem of the AZ,-system. We exhibit the phase space of the B,-system as a fixed point set of a finite group action by Poisson automorphisms. We prove a general result which allows us to reduce the Poisson structures in such situations, and hence we can relate the multiple Hamiltonian structures of the Volterra and Toda systems. The plan of this paper is as follows: In Section 2, we discuss several conditions under which the fixed point set of a Poisson action inherits a Poisson bracket, generalizing the Poisson involution theorem (see [16, 271). In Section 3, we recall the definition of the Toda systems, and give the multiple Hamiltonian structure for the B,, system. These were discovered in [lo] (see also [7]) and we use here a new approach based on the Poisson involution theorem to deduce them. In Section 4, we first recall the construction of the Volterra system and its generalizations for any root
FROM THE TODA LATTICE TO THE VOLTEFNA LATTICE AND BACK
363
system associated with a simple Lie algebra. Then we give the multiple Hamiltonian structure for the B,, (and C,) cases and, by applying a symmetry approach, we explain the relation between the Poisson structures for Volterra and Toda lattices. In the final section, we describe the connections we have been able to find between the two systems, including a generalization of Moser’s recipe (see [20]) to pass from the Volterra to the Toda lattices. 2.
Fixed point sets of Poisson actions
We shall see that the relation between the Toda and Volterra systems relies on special symmetries of the phase spaces. In this section, we study conditions under which the fixed point set of a Poisson action inherits a Poisson bracket. Although we will be interested mainly in finite symmetries, we give the following general result. THEOREM 2.1. Suppose that (M, {.,.}) is a Poisson manifold, and G a compact group acting on M by Poisson automorphisms. Let N = MG be the submanifold of M consisting of the fixed points of the action and let L : N C, M be the inclusion. Then N carries a (unique) Poisson structure {e, .JN such that
I* IF19 for all G-invariant functions
F21 =
{z*fi,
z*F2}N
(3)
Fl, F2 E C?(M).
Proof: For fi , f2 E C”(N) we choose Fl , F2 E COO(M) such that fi = z* Fi. We may assume that Fl and F2 are G-invariant by replacing Fi by
where p is the Haar measure
on G, so that jG ldp = 1. We set Ifi, f2IN = I* {Fl,
F21
and show that this definition is independent of the choice of Fi. For this we observe that, since the action of G is Poisson, the Hamiltonian vector field XF associated with a G-invariant function F : M + W is tangent to N. It follows that the Poisson bracket {Fl , Fz} (N, where FI and F2 are G-invariant functions, depends only on the restrictions Fi 1N . It is obvious from its definition that {a, .JN is bilinear, and satisfies the Jacobi and Leibniz identities, as these identities hold for {., .}. Hence, we obtain a Poisson 0 bracket on N, and it is the only Poisson bracket satisfying (3). &MARK 2.1. The previous result can be seen as a particular case of Dirac reduction. For the general theorem on Dirac reduction see Weinstein ([26], Proposition 1.4) and Courant ([6], Theorem 3.2.1). The case where G is a reductive algebraic group is discussed in [22] using a different method.
Since this result applies in particular corollary,
when G is a finite group, we have the
P. A. DAMIANOUand R. LOJAFERNANDES
364
COROLLARY2.1. Suppose that (M, {., .}) is a Poisson manifold, and G is a finite group acting on M by Poisson automorphisms. Then the fixed point set N = MG curries a (unique) Poisson structure {s, -}u satisfying Eq. (3). Let us consider the special case G = Z 2. Then G = {I, $}, where is a Poisson involution. We conclude that N = MG = {x : 4(x) = x} Poisson bracket satisfying (3). So we see that Theorem 2.1 contains case the following result, which is known as the Poisson involution [16, 271).
4 :M + M has a unique as a special theorem
(see
COROLLARY2.2. Suppose that (M, {a, s}) is a Poisson manifold, and # : M -+ M is a Poisson involution. Then the fied point set N = {x E M : d(x) = x} carries a (unique) Poisson structure {., .}u such that t* IFlY F2) = {z*F,, for all junctions
FI, F2 E Cm(M)
I*F2}N
invariant under 4.
One should note that, in general, the fixed point set MG is not a Poisson submanifold of M, since it is not a union of symplectic leaves of M. In other words, underlying Theorem 2.1 there is a true Dirac type reduction, rather than a restriction or Poisson reduction. 3, 3.1.
The Toda lattices The A,-Toda system
The phase space 7, of the (non-periodic) of all Lax operators in 51, of the form
L=
bl
al
0
1
b2
~2
0
..
0
lattice is the affine variety
0 0
1 ..
(4)
..
0 0
A,_l-Toda
h-1 0
...
..
1
G-I
bn
The reader will notice that we adopt a set of variables (due to Kostant) that differ slightly from the usual Flaschka variables (see r31, Chapter V). This Lax pair has an-obkous Lie algebraic interpretation which leads to the generalized Toda -Lattices (see next paragraph). The Hamiltonians I&, k = 1, . . . , n, defined by Hk = ;
tl-(Lk)
FROM THE TODA LA’lTICE TO THE VOLTERRA LA’l-l-ICE AND BACK
are in involution
with respect to the linear Poisson bracket {., .)&, defined by (5)
{bi9bj}:=09
{Ui,Uj}L
=
(49 bj};
= &C&j - &+l,j)*
vector fields Xk = (. , I&}:
The commuting
365
XG)
= [L,
admit the Lax representation
(Lk+9+l,
(6)
where the subscript + denotes projection into the Lie subalgebra of aI, generated by tbe positive roots. In [20], it was proved that the nonperiodic Toda lattice is a completely integrable system, by applying a finite-dimensional analogue of the inverse scattering method. For the periodic case, the flows are linear on the generic fibers of the momentum map K : 7, + C[x], and since these fibers are afIine parts of hyperelliptic Jacobians, the periodic Toda lattice is an a.c.i. system (see [2] for details). The Toda lattice is also Hamiltonian with respect to a quadratic Poisson bracket {., .}& defined by {ai, bj}:
{bi, bj}:
= Qibj(h,j
- &+l,j),
(7)
= ui(Ji,j+l - &+l,j)*
The quadratic Toda bracket appeared in a paper of Adler [l] in 1979. The brackets {., .}: and {a, .}t are compatible, and are in fact part of a full hierarchy of higher-order Poisson brackets [8, 141. Denote by xl and 1r2 the Poisson tensors associated with the linear and quadratic brackets. Also, let ZO be the Euler vector field Zo=paia+ebiL. i=l aui The following
i=l
(8)
abi
result is proved in [14].
PROPOSITION 3.1. There exists a symmetries Zk, k = 0, 1,2, . . . , such (i) The nk are compatible Poisson with respect to all of the nk. (ii) The vector fields xk admit the
.&t-l
sequence of Poisson tensors nk and muster that tensors and the functions Hk are in involution multiple Humiltoniun formulation = nkdHt = nk_ldHt+1.
(iii) The Poisson tensors, the integrals und the Humiltoniun vector fields satisfy the deformation relations: &XI
= (1 - k -
cz,
=
Xl
(1 -
2bk+lt
l)Xk+l,
Zk(H,) = (k + l)Hk+t, [zk,
zzl
=
(1 -
k)Zk+l.
(9)
P.A. DAMIANOU and R. LOJA FERNANDES
366
These relations will be important later to deduce some properties of the Volterra lattices. Using these relations we can find explicit formulae for the higher-order Poisson brackets. For example, we have that n-1 21
=
(bi(l - 2i) + bi+l(l + 2i)) fg
ELIi
I
i=l
(10)
+e(ai-1(2-2i)+ai(2+2i)+bf)&, I
i=l
so we find that the cubic bracket 7~3= -Cz,1r2 Iai,
&+I)+
=
244+lbi+l9
{Ui,
bi+l};
=
Uibj’,l
=
-UiUi+l~
{ai+1
9h
1;
+
is given by (see [lo]):
{Ui, Uf,
bi}:
=
-Uib,2
IUi
7bi+21$
=
UiUi+l,
Ibi
Tbi+lI$
=
Ui (h
-
Uf,
(11) +
&+I).
Similarly, one can compute any higher-order Poisson bracket. The cubic bracket ~r3 was found by Kupershmidt [19] via the infinite lattice. 3.2.
Toda
The B,-Toda system
Generalizations of the Toda lattice for other root systems are well known and were first given in [4]. We consider here only the case of B, . The phase space for the B,-Toda lattice is the affine variety of all Lax operators in 51zn+t of the form: b,
a,
0
1
b2
~2
..
L=
...
...
0
0
...
0 0
*. bn-1
an-1
1
b,
an
1
0
--a,
-1
-b,
-u,_~
-1
-b,-1
.
.. 0
0
.. -1
-b2
--al
0
-1
-bl
(12)
FROM THE TODA LAlTICE TO THE VOLTERRA LATTICE AND BACK
361
This matrix actually lies in the real simple Lie algebra g = so (n, n + 1). We would like to have a multiple Hamiltonian formulation for the B,-Toda systems fg
= [L,
(Lk+‘)+].
(13)
Note that, in this case, only the even powers of L give nontrivial 1 H2k = -tr(L2k), 2k
k=
integrals
1,2,...
In order to write down a Hamiltonian formulation for these systems we recall that the B,-Toda is a subsystem of the Ah-system. This is well known, and in fact more is true: we can find a symmetry such that the B,-Toda is obtained through a symmetry reduction from the Ah-Toda. To see this, observe that conjugation by a diagonal matrix takes the Lax matrix (12) to the Lax matrix bl
a1
0
1
b2
~2 *.
L=
...
-..
*. bn-1
an-1
1
bn
an
1
0
an
(14)
-b, 1
4-l
-b,-i *.
*.
0 0
0 0
1
0
0
**I
1
-b2
~1
0
1
-bl
and this is just a matrix in ‘&+I satisfying u&+1-i = ai and bzn+z-i = -bi. Therefore, if we consider the involution C#J: 7h+l + 72n+l defined by
(b(ai, bi) = (a2n+l-i, -hn+2-i), then the phase space of the B,-Toda can be identified with the fixed point set of this involution. On the other hand the &-action generated by this involution is a Poisson action for the odd Poisson brackets. More precisely, we have the following result. PROPOSITION
3.2. The map C#J : ‘T&+1 + I&+1 satisjies @k
= (-l)k+lrtk.
368
P. A. DAMJANOU and R. LOJAFERNANDES
Proof: For lower-order Poisson brackets one can check this relation by direct computation. For higher-order Poisson brackets, we note that expression (10) for the master symmetry ZJ shows after some tedious computation that 4*z, = -z1
+ x,
where X is a multiple of the first Hamiltonian &,nk so
it follows by induction
=
(k
-
vector field Xt. Now Eq. (9) gives
%nk+l,
that
1 = -&,(-l)k+l~k k-3
= (-l)k+2nk+i.
El
Using Corollary 2.2, we conclude that the odd Poisson brackets for the AZ,,-Toda induce Poisson brackets for the B,-Toda lattices, which therefore possesses a multiple Hamiltonian formulation. In this way we have deduced the following result of [lo]. COROLLARY3.1. There exists a sequence of Poisson tensors nl, ~3, n5, . . . such that the B,,-Toda lattices (13) possess a multiple Hamiltonian formulation: nk,zdHzl It should be noted that Theorem the Poisson brackets, as is illustrated
= %dHz+z. 2.1 gives an effective procedure by the following example.
to compute
EXAMPLE 3.1. In this example we compute the cubic B,-bracket. We denote by (ai, bj), where i, j = 1, . . . , n, the coordinates on the phase space of the B,-Toda lattice, and by (Gil 6j), where i = 1, . . . ,2n, j = 1, . . . ,2n + 1, the coordinates on the phase space of the Azn-Toda lattice. Suppose we would like to compute the cubic B,-brackt {ai, bi}$. First we extend the function ai and bi to invariant functions zi and bi on the phase space of the Ati-Toda lattice. For example, we can take the functions zi =
iii + Zizn+l-i 2 ’
&
6i - &nf2_i 2
Then we compute the AZ,-cubic bracket of these functions: for i c n we have
.
we find using (11) that
FROM THE TODA LATTICE TO THE VOLTERRA LATTICE AND BACK
369
while for i = n we obtain
{%a}; =-i (ii.4+ii,’ + iin+1i,2+2+ ii;,, Finally, we restrict the brackets to the &-phase
space, to obtain the cubic &-bracket
i
{~i,hi}~=-~(oib?+.z),
{~.,b,}:=-~(a.~~+2a,Z). The other brackets are computed
+ 2ii.ii”+l).
(15) (16)
in a similar fashion and they give the following
expressions:
REMARK 3.1. It is not hard to check that the (nonperiodic) C,-Toda system is a subsystem of the Ah-i-Toda system. The reader can check that the multiple Hamiltonian structure for the C,-Toda lattices found in [ 11, 71 can be obtained from the multiple Hamiltonian structure for the Ati- -Toda systems, as in the &-case. In fact, the involution 4 : I$, + 72, defined by $(ai,
satisfies &?rk = (-l)k?rk, 4.
h)
=
(a2n+l-i9
-b2n+2-i)
so the same method applies.
The Volterra Lattices
4.1. The A,-Volterra system
We now turn to the (nonperiodic) A,-Volterra system. Its phase space Ic, is the subspace of 7, consisting of all Lax operators (4) with zeros on the diagonal:
L=
0
a1
0
1
0
u2
0
1 ..,
**.
0
. . .
0 0
(18)
...
0 0
0
. . .
0
4l-1
1
()
370
P. A. DAMIANOU
and R. LOJA FERNANDES
Note that K, is not a Poisson subspace of 7,. However, K, is the fixed manifold of the involution $ : 7, -+ 7, defined by
((al,
a2..
. , a,),
@I, b2. . . , b,))
I+
((al,
~2..
.,
a,), C-h, 42..
. , -b,)),
and we have the following result, see [ 161. F’ROPOSWION
4.1. @ : I, +
is a Poisson
I,
automorphism
of (I,, {., +}“,), if k
is even. Proof: Again, one checks by direct computation
for k < 4 that
+.,?rk = (-l)k,rk. On the other hand, we have that $*Zt = -Zt , so by induction @,~k+l
we see that
* k:‘jqL
=
-
=
k-&Z,
=
j-&L-Z,(-l)‘nk
Z,nk
1 @*rk
=
(-l)k+‘nk+,.
0
and the result follows.
Therefore, by Corollary 2.2, Kc, inherits a family of Poisson brackets n2,7r4, . . . . For example, the quadratic bracket can be computed from Eqs. (7), in a form entirely analogous to Example 3.1, and is given by {ai~aj}L =
aiaj(h,j+l
-
Ji+l,j),
(19)
while the Poisson bracket rc4 is found to be given by the formulae {ai9 4,114,
{ai
=
Q&+1(%
+ ai+
9ai+21”, = &4+14+21
i=l,...,n-1
(20)
i = l,...,n-2.
The formulae for the brackets rr2 and n4 appeared in [9], and the existence of this hierarchy of Poisson brackets is proved in [16]. The first three Poisson structures of this system appeared first in [ 131 where the Volterra lattice is treated in detail, however the quadratic and cubic brackets do not coincide with the brackets in [9]. It follows that the restriction of the integrals & to K, gives a set of commuting integrals, with respect to these Poisson brackets. Also, for i odd, the Lax equations (6) lead to Lax equations for the corresponding flows, merely by putting all bi equal to zero. Taking i = 1, we recover the system
aj = aj(ai-1
- Ui+1),
i = 1,. . . ,n,
(21)
FROM THE TODA LA-l-l-ICE TO THE VOLTERRA LA-l-l-ICE AND BACK
371
which we called the A,-Volterra lattice in the introduction. More generally, taking i odd we find a family of commuting Hamiltonian vector fields on K, which are restrictions of the Toda vector fields, while for i even the Toda vector fields Xi are not tangent to Ic,. Hence, we obtain a family of integrable systems admitting a multiple Hamiltonian formulation: X2(k+l)-1
=
n2k+2dfh
=
k=
mkdH21+2r
1,2,...
This system was shown to be a completely integrable system in [20], while the periodic version of this system was proved to be an a.c.i. system in [15, 161. Generalized Volterra systems
4.2.
We now describe the construction of the generalized Volterra systems of Bogoyavlensky (see [5]). Let g be a simple Lie algebra, with rankg = n, and let lI = {or, 02,. . . , w,} be a Cartan-Weyl basis for the simple roots in g. There exist unique positive integers ki such that ’ kowo + klwl + -. e + kno,, = 0,
where ko = 1 and ~0 is the minimal negative root. We consider i = [B, L]
the Lax pair
)
where
L(t)-=
2
bi(t)eoi
+
e,
C
+
i=l
B(t)
=
[e,,
ewj],
lg
2 -e_,i ki h(t)
+ e-,.
i=l
Let h c g be the Cartan subalgebra.
For every root o, E h* there is a unique
Hmu E f~ such that w(h) = fi
(H,,,h),for all h E h, where B denotes the Killing form. Also, B induces an inner product on h* by setting (ma, wb) = @ (H,,,H,,), and we define
Cij
=
and i
1
if (Wi,Wj)#O
0
if (Wit Oj) = 0 or
I -1
and i > j,
if (wi,aj)#O
With these choices, the Lax pair above is equivalent
&=_$5( j=l
i = j,
bj
to the system of o.d.e.‘s
(22)
372
P. A. DAMIANOU and R. LOJA FERNANDES
To obtain a Lotka-Volterra
type system one introduces
a new set of variables
bv
Xji
=
-Xij,
Xjj
=
0.
Note that Xij # 0 iff there exists an edge in the Dynkin diagram for the Lie algebra g connecting the vertices mi and wj. System (22) in the variables xij, takes the form Xij = Xij
2 ks (Xis+ Xjs),
(23)
s=l
which is a Lotka-Volterra type system. We call (23) the Bogoyavlensky-Volterra system associated with g (or the g-Volterra system for short). EXAMPLE
4.1.
Let us take g = A,. Then we have the Dynkin diagram .
0
-
.
l -
.
-
l
with ki = 1, i = 0, . . . , n. If we label the edges of this diagram by ui, . . . , a,, then system (23) takes precisely the form of the A,-Volterra lattice. We now turn our attention to the B,-case. 4.3.
The B,-Volterra
system
For g = B,, the Dynkin diagram has the form 0
. ~
.
. ~
l
--_m
and for the obvious labeling (and after a linear change of variables) B, -Volterra system
we obtain the
ai = -uiu2, tii =
Ui(Ui_1 -Ui+l),
i=2,...,n-1,
(24)
I a, = a, (G-1 + a,). The multiple Hamiltonian structure for the B,-Volterra system can be obtained from the AZn-Volterra system in the same fashion as the multiple Hamiltonian formulation for the B,-Toda system was obtained from the Azn-Toda system (cf. Section 3.2). In fact, we notice that Eqs. (24) for the B,-Volterra system can be obtained from Eqs. (21) for the AZ,,-Volterra system by setting azn+i_i = -ai (this defines an invariant submanifold). Hence, we take as phase space of the B,,-Volterra system the Lax operators of the form (18) satisfying these additional restrictions:
373
FROM THE TODA LATTICE TO THE VOLTERFU LA’ITICE AND BACK
0
a1
0
1
0
a2
...
..*
0
0 0
*. .
... an-1
0
1 L=
0
a,
1
0
-a,
1
0
-G-1
1
0
(25)
..
.. 1
0
-a1
0
1
0
Notice that this subspace appears as the fixed point set of the involution &,, defined by V(ai)
=
(p : K2, +
-QZn+l-i.
We will see below that this involution can be used to obtain the multiple tonian structure of the I?,-Volterra system. REMARK 4.1. There is a diagonal matrix D which conjugates given above for the B, system, to the following Lax matrix
0
a1
0
1
0
a2 ...
L=
..
***
0
0
***
the Lax matrix
0 0
... 0
G-1
1
0
a,
1
0
an
-1
0
aa-1
-1
0
(26)
*. 0
0
Hamil-
*.
-1
0
a1
0
-1
0
P. A. DAMIANOUand R. LOJAFERNANDES
374
Notice that these Lax matrices are not obtained as Lax matrices for the B,-Volterra system (matrices of the form (12)) with zeros on the diagonal. We now have the following proposition The proof is similar and will be omitted.
analogous
to Propositions
3.2 and 4.1.
PROPOSITION4.2. Let nk, k = 2,4, . . . be the Poisson tensors of the A2n-Volterra system. Then q,nk = (-l)k’2nk. In this way, by Corollary 2.2, the phase space of the B,-Volterra system carries Poisson brackets IQ, ns, . . . . For example, from (20) we can compute, using the same technique as in Example 3.1, the following explicit formulae for the bracket ~r4: 1%9ai+1 1; =
1 (ai +
-ai&+
2
Iatl-l? a?& = ;a.-la.(an-I
ai+
+ 2a,),
i = l,...,n-2,
(27)
1 IQ7 %+21”,
The restriction
=
-aiai+lai+2*
2
of the invariant functions 1 H4k = - tI(L4k) 4k
to this space defines formulation: X4(k+l)-1
a hierarchy
=
n4k+4dH41
of systems,
=
possessing
n4kdH41+4,
k=
a multiple
Hamiltonian
1,2,....
The function 14 = i CyI;(2af + aiai+i) gives the B,-Volterra system (24). The brackets above, as well as the master symmetries for these systems (including the C, and D, cases) appear in a recent preprint of Kouzaris [ 181. However, the Lax matrices above lead to Lax pairs which are di$erent from the Lax pairs given by Kouzaris. Finally, we remark that the C,-Volterra system can be identified with the B,-Volterra systems (see [IS]), and hence admits the same description. 5.
Epilogue:
From Toda to Volterra and back
In this concluding section we would like to explain what we know so far about the connection between the Toda and Volterra systems, including the results obtained
FROM THE TODA LATTICE TO THE VOLTERRA LAlTICE AND BACK
375
above. The relation between the two systems relies on special symmetries of the phase spaces. We have shown above that the phase space of the A,-Volterra system appears as the fixed point set of a Poisson involution of the A,-Toda phase space. On the other hand, we have also shown that the multiple Hamiltonian structure of the B,-Toda system can be obtained from the AZ,-Toda system. Hence, to get to the B,-Volterra system, there are a priori two distinct ways to proceed, as explained by the following diagram: A%-Toda y/y Ah -Volterra \
B,, -Toda
z4 1
/
I
B, -Volterra We have seen that the correct way to proceed is to choose the left side of this diagram. In fact, we saw above that we can get from Ah-Toda to AZn-Volterra by a Poisson involution * : 72, + 72, (i.e. a Z$synunetry). Also, we can get from A&-Volterra to B,-Volterra using another Poisson involution cp : /cb + lch (and, hence, again a &-symmetry). Note also that we can go straight from AZ,-Toda to B,-Volterra using a Z4-symmetry: if one defines the map ~0: ‘Tz,, + Zn by ;l;
a
b2n+2-i),
then one checks that the group G = {Z,&?2,?3} acts by Poisson automorphisms on (Ih, 7r4, k = 1,2, . . . The fixed point set (i.e. the phase space for B,-Volterra) inherits Poisson structures z$k and the even flows reduce to this space. Notice also that &cz,, = ~0 and F2 = +, so this reduction in one stage (the middle line in the diagram) coincides with the reduction in two stages (the left part of the diagram). On the other hand, there seems to be no such symmetry reduction from the B,-Toda lattice to the B,-Volterra lattice: the various Poisson structures and Hamiltonian functions we have for the B,-Toda systems do not restrict to the phase space of the B,-Volterra systems. We could however still find a connection between these two systems, which goes in the opposite direction. To explain this connection, we perform the change of variable, ai = -2x?, and we consider the following equivalent (i.e. conjugate) form of the Lax matrix (25) (or, equivalently, (26)):
376
P. A. DAMIANOU and R. LOJA FWUWNDES
0
x1
0
xl
0
x2
0
...
...
0
...
...
L=
0
&I
%I
0
G&l
axn
0
0
0 0
...
0
In these new variables the equations x’l =
I
0
0
for the B,-Volterra
&ixxl
ax, 0
lattice (24) become:
x1x;,
Xi = Xi(Xf_1 - XF+l),
i=2,...,n-1.
x, = -Xn(X,2 +x,2-t),
The relation between the B,-Volterra system and the corresponding Toda system of types B,, and C, is similar to the relation observed by Moser (see [20]) between the A,-Volterra and the A,-Toda lattices. One starts by taking the square of the Lax matrix above and notices that L2 leaves certain subspaces invariant. Moreover, L2 reduces on each of these invariant spaces to a symmetric Jacobi matrix. More precisely, assume that L2 is an N x N matrix. Then there are two distinct cases: l N = 4n + 1 : Removing all odd columns and all odd rows we end up with a 2n x 2n matrix and a Toda system of type C,,. On the other hand, removing all even columns and all even rows we end up with a 2n + 1 x 2n + 1 matrix and a Toda system of type B,. l N = 4n + 3 : Removing all odd columns and all odd rows we end up with a 2n + 1 x 2n + 1 matrix and a Toda system of type B,, . On the other hand, removing all even columns and all even rows we end up with a 2(n + 1) x 2(n + 1) matrix and a Toda system of type Cn+i. In this way, we have a procedure which takes us from a Volterra systems of type B, (or C,) to either a Toda system of type B, or a Toda system of type C,. EXAMPLE 5.1. Take N = 9 and n = 2. Omitting columns of L2 we obtain the 5 x 5 matrix
all even rows and all even
FROM THE TODA LATTICE TO THE VOLTERRA LATTICE AND BACK
0
0
x3x4
0
x1x2 xi
We identify
+
xi
x3x4
0
0
-x3x4
0
0
--x3x4 -x;
and the equations
-
x;
-x1x2 -Xf
-x1x2
this matrix with a symmetric
B1 =xf,
377
Jacobi matrix of type B2, i.e. we let
B2 = x; + x&
A1 =
~1x2,
A2
=
~3x4,
satisfied by Al, AZ, Bl, and B2 are & = Al
(B2
-
Bl) ,
A2 = -A&, BI = 2A;, B2
These are precisely
=
the Toda equations
2A; - 2A:. of type B2.
REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20]
M. Adler: Invent Math. 50 (1979), 219-248. M. Adler and P. van Moerbeke: Invent Math. 103 (1991). 223-278. M. Audin: Spinning Tops, Cambridge University Press, Cambridge 1996. 0. Bogoyavlenskij: Commun. Math. Phys. 51 (1976), 201-209. 0. Bogoyavlenskij: Phys. Z&r. A 134 (1988), 34-38. T. Courant: Trans. Amer. Math. Sot. 319 (1990), 631-661. J. N. da Costa, P. A. Damianou: Bull. Sci. Math. 125 (2001), 49-69. P. A. Damianou: Z.&t. Math. Phys. 20 (1990), 101-112. P. A. Damianou: Phys. Lat. A155 (1991), 126-132. P. A. Damianou: .Z. Math. Phys. 35 (1994), 5511-5541. P. A. Damianou: Regul. Chaotic. Dyn. 5 (2000). 17-32. P. A. Damianou and S. P. Kouzaris: Multiple Hamiltonian Structures of Bogoyavlensky-Toda systrnems of type DN, Technical Report TRZlOZ2001,Department of Mathematics and Statistics, University of Cyprus. L. D. Fadeev and L. A. Takhtajan: Hamiltonian Methods in the Theory of Solitons, Springer, Berlin 1986. R. L. Fernandes: J. Phys. A: Math. Gen. 26 (1993). 3797-3803. R. L. Fernandes and J. P Santos: Rep. Math. Phys. 40 (1997), 475-484. R. L. Femandes and P. Vanhaecke: Commun. Math. Phys. 221 (2001), 169-196. M. Kac and P. van Moe&eke: Advances in Math. 3 (1975), 160-169. S. P. Kouzaris: Multiple Hamiltonian Structures for Bogoyavlensky-Volterra systems, University of Cyprus preprint TRZO3/2001. B. Kupershmidt: Discrete Lax equations and differential-difference calculus, Asterisque 123 (1985), 1-212. J. Moser: Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math. 16 (1975), 197-220.
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P. A. DAMIANOU and R. LOJA FERNANDES
[21] D. Mumford: Tara Lectures on Theta, II, Birkhluser, Boston 1984. [22] P. Vanhaecke: Zntegmble Systems in he Realm of Algebraic Geometry, second edition, Springer, Berlin 2001. [23] A. P. Veselov and A. V. Penskdi: Regul. Chaotic Dyn. 3 (1998). 3-9. [24] A. Volkov: J. Soviet Math. 46 (1989). (15761581). [25] M. Toda: Theory of nonlinear laftices, Springer Series in Solid-State Sciences, 20, Springer-Verlag. Berlin, 1989. [26] A. Weinstein: J. Di$erentiul Geometry 18 (1983), 523-557. [27] P. Xu: Dirac submanifolds and Poisson involutions, preprint math.SG/O110326