Why are the Ruijsenaars{Schneider and the Calogero{Moser hierarchies governed by the same {matrix?

Why are the Ruijsenaars{Schneider and the Calogero{Moser hierarchies governed by the same {matrix?

Why are the Ruijsenaars{Schneider
and the Calogero{Moser hierarchies
governed by the same r{matrix?
Yuri B. SURIS
Centre for Complex Systems and Visualization, Unversity of Bremen,
Postfach 330 440, 28334 Bremen, Germany
e-mail: suris @ mathematik.uni-bremen.de
Abstract. We demonstrate that in a certain gauge the Ruijsenaars{Schneider
models admit Lax representations governed by the same dynamical r{matrix as
their non{relativistic counterparts (Calogero{Moser models). This phenomenon is
explained by establishing the quadratic r-matrix Poisson bracket for the Ruijsenaars{
Schneider models.
1 Introduction
In the recent years the interest in the Calogero{Moser type of models [1]{[4]
is considerably revitalized. One of the directions of this recent development
was connected with the notion of the dynamical r{matrices and their interpretation in terms of Hamiltonian reduction [5]{[14]. Very recently, this line
of research touched also the so{called Ruijsenaars{Schneider models [13],[14],
which may be seen as relativistic generalizations of the Calogero{Moser ones
[3],[4].
In the present paper the same subject as in [14] is handled. However, the
results are somewhat dierent, and, as we hope, somewhat more beautiful.
The dierence is due to another gauge we choose for the Lax matrix of the
Ruijsenaars{Schneider models. A striking circumstance comes out when using our gauge, namely that the both classes of models are governed by one
and the same dynamical r{matrix. This seems to pass unnoticed in the existing literature and is explained in the next two sections 2,3, treating rational
and hyperbolic models, respectively. The computations presented there are
hardly new, but we could not nd in the literature the main message following from these computations, namely that they give Lax reprentations for an
arbitrary ow of the corresponding hierarchies, and hence are perfectly suited
for guessing (not proving!) the correct r{matrix ansatz. Section 4 contains
the main result of the paper, namely the quadratic r{matrix structure for
the Ruijsenaars|Schneider models. The proof is purely computational and
is therefore relegated to an Appendix. Section 5 is devoted to some problems
arising from our results.
2 Rational systems
The rational non{relativistic Calogero{Moser hierarchy is described in terms
of two matrices: the Lax matrix
L = L(x; p) =
N
X
k=1
pk Ekk +
and the auxiliary diagonal matrix
k6=
X = diag(x ; : : : ; xN ) =
1
1
E ;
kj
j xk xj
X
N
X
k=1
xk Ekk :
(2.1)
(2.2)
Here (and below) is a parameter, usually supposed to be pure imaginary.
The dynamical variables x = (x1; : : :; xN )T and p = (p1; : : :; pN )T are supposed to be canonically conjugated, i.e. to have canonical Poisson brackets:
fxk ; xj g = fpk ; pj g = 0; fxk ; pj g = kj :
(2.3)
Given the diagonal matrix X , the structure of the Lax matrix L is completely
described by the following fundamental commutation relation:
XL LX = X
k6=j
Ekj = (eeT I );
(2.4)
where I stands for the N N unity matrix, and e = (1; : : :; 1)T .
The rational relativistic Ruijsenaars{Schneider model is also described in
terms of two matrices, the Lax matrix
L=
xj + bj Ekj ;
N
X
k;j =1 xk
(2.5)
and the same diagonal matrix X as before. Here we use an abbreviation
bk = exp(pk )
Y
j 6=k
2
1 (x x )2
k
j
=
!1 2
;
(2.6)
so that in the variables (x; b) the canonical Poisson brackets (2.3) take the
form
fxk ; xj g = 0; fxk ; bj g = bk kj ;
!
1
1
2(1
kj )
fbk ; bj g = bk bj x x + x x + + x x :
(2.7)
j
k
k
j
k
j
Given the diagonal matrix X , the structure of the Lax matrix L is this time
completely described by the fundamental commutation relation:
XL LX + L = ebT :
(2.8)
Let us note that the non{relativistic limit, leading from the Ruijsenaars{
Schneider model to the Calogero{Moser one, is achieved by rescaling p 7! p,
7! and subsequent sending ! 0 (in this limit Lrel = I + Lnonrel +
O( 2)).
2
The fundamental results about the solution of these models are by now
well known and go back to the work by Olshanetsky{Perelomov [1] for the
non{relativistic case, and by Ruijsenaars{Schneider [3] for the relativistic
one. A unied formulation was given in the paper [4] by Ruijsenaars and
may be presented in following lines. Let '(L) be a conjugation{invariant
function on sl(N ), and take its value on one of the Lax matrices (2.1), (2.5)
as a Hamiltonian function of the corresponding model. Then the quantities
xk (t) are nothing else then the eigenvalues of the matrix
X + tf (L );
0
where 1
0
f (L) = r'(L) for the non relativistic case;
f (L) = Lr'(L) = r'(L)L for the relativistic case:
(2.9)
(2.10)
From this statement we want now to derive the Lax equation for the corresponding ows in the form suitable for our present purposes. Dene the
evolution of the matrices X , L by the equations
X = X (t) = V (X + tf (L ))V ;
L = L(t) = V L V :
0
0
1
(2.11)
(2.12)
Let us explain, in which sence these equations dene an evolution. The
matrix X = X (t) consists of eigenvalues of the matrix X0 + tf (L0), and the
matrix V = V (t) is a diagonalizing one (it is easy to see that the matrix
X0 + tf (L0) is similar to a self{adjoint one, if 2 iR). If we x the ordering
of xk (for example, x1 < : : : < xN ), then the only freedom in the denition
of V is a left multiplication by a diagonal matrix. We x now V by the
condition
V e = e;
(2.13)
1
0
Recall that the gradient r'(L) of a smooth function ' on sl(N ) with respect to the
scalar product hU; V i = tr(U V ) is dened by the relation
1
hr'(L); U i =
d
d
( + U )
' L
=0
;
8U 2 sl(N );
and for a conjugation{invariant ' one has [r'(L); L] = 0.
3
and show that this assures that the corresponding requirements (2.4) and
(2.8) are satised for all t, provided they were satised for t = 0. Indeed, we
have for the non{relativistic case:
XL LX = V (X0L0 L0X0)V 1 = V eeT V 1 I :
Since the diagonal entries of the matrix on the left{hand side vanish, we
see that V e = e implies eT V 1 = eT , i.e. the validity of the commutation
relation (2.4) for all t.
For the relativistic case we have:
XL LX + L = V (X L L X + L )V = V ebT V ;
and denoting bT V = bT , we see again that due to (2.13) the commutation
relation (2.8) is satised for all t.
Now we are in a position to derive the evolution equations for L, X . This
0
0
0
0
0
0
1
0
1
1
calculation is identical for both the non{relativistic and the relativistic cases.
Dierentiating (2.12), (2.11), we get:
L_ = [M; L];
(2.14)
(so that the evolution of L is governed by a Lax equation),
X_ = [M; X ] + f (L);
where
(2.15)
M = V_ V :
(2.16)
We want to nd the matrix M more explicitly. From (2.15) we immediately
obtain the o{diagonal entries of the matrix M :
(2.17)
M = f (L)kj ; k 6= j:
1
kj
xk xj
The normalizing condition following from (2.13), (2.16) reads:
Me = 0;
hence
Mkk =
X
j 6=k
Mkj =
4
f (L)kj :
k xk xj
X
j 6=
(2.18)
(2.19)
3 Hyperbolic systems
We want now to describe the analogous results for the more general systems
of the Calogero{Moser class, namely the hyperbolic ones.
The hyperbolic non{relativistic Calogero{Moser hierarchy is described in
terms of the Lax matrix
L = L(x; p) =
N
X
k=1
pk Ekk +
Ekj ;
j sinh(xk xj )
X
k6=
(3.1)
and the auxiliary diagonal matrix
X = diag(exp(x ); : : :; exp(xN )) =
1
N
X
k=1
exp(xk )Ekk :
(3.2)
The dynamical variables x = (x1; : : :; xN )T and p = (p1; : : : ; pN )T are still
supposed to be canonically conjugated. The fundamental commutation relation, describing the structure of the Lax matrix L in terms of a given diagonal
matrix X , reads this time:
X
XLX 1 X 1 LX = 2 Ekj = 2 (eeT I ):
(3.3)
k6=j
The hyperbolic relativistic Ruijsenaars{Schneider model is described in
terms of the Lax matrix
N
X
sinh( )
L = L(x; p) =
bj Ekj ;
(3.4)
k;j =1 sinh(xk xj + )
and the same diagonal matrix X as in (3.2). Here we use an abbreviation
bk = exp(pk )
Y
j 6=k
2
sinh
( )
1
2
sinh (xk xj )
=
!1 2
;
(3.5)
so that in the variables (x; b) the canonical Poisson brackets (2.3) take the
form
fxk ; xj g = 0; fxk ; bj g = bk kj ;
fbk ; bj g = bk bj coth(xj xk + ) coth(xk xj + )+2(1 kj )coth(xk xj ) :
(3.6)
5
Given the diagonal matrix X , the structure of the Lax matrix L is this time
completely described by the fundamental commutation relation:
e XLX
e X LX = 2sinh( )ebT :
1
(3.7)
1
The non{relativistic limit is performed in just the same manner as as in
the rational case. Let us note also that the rational models can be recovered
from the hyperbolic ones after rescaling x 7! x, 7! and sending ! 0
(in this limit Lhyperbolic = Lrational + O()).
The results about the solution of these models obtainable from [1], [3]
may be formulated as follows. Let '(L) be a conjugation{invariant function
on sl(N ), and take its value on one of the Lax matrices as a Hamiltonian
function for the corresponding model. Then the quantities exp(2xk (t)) are
just the eigenvalues of the matrix
X exp(2tf (L ));
2
0
0
where (2.9), (2.10) still hold.
From this statement the Lax equations for the corresponding ows can be
derived. Indeed, dene the evolution of the matrices X , L by the equations
X = X (t) = V X exp(2tf (L ))V ;
L = L(t) = V L V :
2
2
2
0
(3.8)
1
0
(3.9)
As in the previous section, these equations have to be complemented with
a normalizing condition on the matrix V , assuring that the corresponding
commutation relations (3.3), (2.8) remain valid in the time evolution. We x
V by the condition
V X0e = Xe;
(3.10)
and show that this is enough to assure that the corresponding requirements
(3.3), (3.7) are satised for all t, provided they were satised for t = 0.
Indeed, for the non{relativistic case we have:
1
0
XLX
1
X LX = X V X X L X
1
1
0
0
0
0
1
X LX X V X
0
1
0
0
0
1
1
= 2X 1 V X0(eeT I )X0 1V 1X = 2 X 1V X0eeT X0 1 V 1X I :
6
The same argument as for the rational case works: since the diagonal entries of the matrix on the left{hand side vanish, we see that (3.10) implies
eT X0 1 V 1X = eT , which proves the commutation relation (3.3) for all t.
Analogously, for the relativistic case we have:
e XLX
e X LX
= X V X e X L X
e X L X X V X
= 2sinh( )X V X ebT X V X:
so that denoting bT X V X = bT , we see that (3.10) implies the validity
of the commutation relation (3.7) for all t.
From this point the derivation of the evolution equations for L, X is iden1
1
0
0
0
0
1
1
0
1
0
0
1
0
0
1
0
0
1
0
0
1
1
1
1
tical for both the non{relativistic and the relativistic cases. Dierentiating
(3.9), (3.8), we get:
L_ = [M; L];
(3.11)
(so that the evolution of L is governed by a Lax equation),
2X X_ = [M; X 2] + 2X 2f (L);
(3.12)
where
M = V_ V :
(3.13)
In order to nd the matrix M explicitly, consider rst the o{diagonal part
of (3.12), which implies:
exp(xk xj ) f (L) = (1 + coth(x x ))f (L) ; k 6= j: (3.14)
Mkj = sinh(
k
j
kj
xk xj ) kj
The normalizing condition following from (3.10), (3.13) reads:
_
MXe = Xe;
(3.15)
hence
1
Mkk = x_ k
X
j 6=k
Mkj exp(xj xk ):
An expression for x_ k can be read o the diagonal part of (3.12):
x_ k = f (L)kk :
7
Substituting this in the previous formula and using (3.14), we get:
f (L)kj :
k sinh(xk xj )
Mkk = f (L)kk
X
j 6=
(3.16)
Finally, notice that we can redene M f (L) as a new M (this does not
inuence the equations of motion described by the Lax pair), which results
in the more convenient expressions
Mkj = coth(xk xj )f (L)kj ; k 6= j;
f (L)kj :
Mkk =
j 6 k sinh(xk xj )
X
(3.17)
(3.18)
=
4 Dynamical r-matrix formulation
So we see that in both the non{relativistic and the relativistic cases we have
a relation strongly resembling the basic relation of the r{matrix theory (see,
for example, [15]):
M = R(r'(L)) for the non relativistic case;
(4.1)
M = R(Lr'(L)) for the relativistic case;
(4.2)
where R is a linear operator on sl(N ), depending, however, on the dynamical
variables X . It is a striking feature that for both the non{relativistic and the
relativistic cases the operator R is one and the same. Its action is described
by dierent formulas for the rational and hyperbolic systems (the "rational"
operator being a natural limiting case of the "hyperbolic" one).
We represent the operator R as a sum:
R = A + S;
(4.3)
where A is a skew{symmetric operator on sl(N ), and S is a non{skew{
symmetric one, whose image consists of diagonal matrices. In the rational
case
A(Ekj ) = x 1 x Ekj ; k 6= j ; A(Ekk ) = 0;
(4.4)
k
j
8
S (Ekj ) = x 1 x Ekk ; k 6= j ; S (Ekk ) = 0:
k
j
(4.5)
In the hyperbolic case
A(Ekj ) = coth(xk xj )Ekj ; k 6= j ; A(Ekk ) = 0;
(4.6)
S (Ekj ) = sinh(x1 x ) Ekk ; k 6= j ; S (Ekk ) = 0:
(4.7)
k
j
A natural hypothesis for the non{relativistic case would be that the corresponding Lax matrices satisfy the linear r{matrix ansatz
fL ; Lg = [I L; r] [L I; r] ;
(4.8)
where the N 2 N 2 matrix r corresponding to the operator R is dened as
r=
N
rij;kmEij Ekm ;
X
i;j;k;m=1
where
rij;km = hR(Eji ); Emk i = coe : by Ekm in R(Eji):
and r corresponds in the same way to the operator R, so that
r =
N
X
i;j;k;m=1
(4.9)
rkm;ij Eij Ekm :
For the so dened matrix r we have a decomposition
r = a+s
into two matrices corresponding to the operators A, S above, so that in the
rational case
X
1 E E ;
(4.10)
a=
k6=j xk
s=
X
k6=j xk
xj
1
jk
kj
xj Ejk Ekk ;
and in the hyperbolic one
X
a = coth(xk xj )Ejk Ekj ;
k6=j
9
(4.11)
(4.12)
1
E E :
(4.13)
sinh(xk xj ) jk kk
The hypothesis (4.8) turns out to be true, and the proof of it (a direct
verication) was just the subject of the paper by Avan, Talon [5]. In a
subsequent paper [9] a derivation of this formula was given, which is in some
sense a sort of further development of the calculations (for the non{relativistic
case) presented in the sections 2, 3.
A natural hypothesis for the relativistic case would be that the corresponding Lax matrices satisfy the quadratic r{matrix ansatz:
fL ; Lg = (L L)a1 a2(L L)+(I L)s1(L I ) (L I )s2(I L); (4.14)
where the matrices a1; a2; s1; s2 correspond in a canonical way to some linear
(dynamical) operators A1; A2; S1; S2 and satisfy the conditions
a1 = a1; a2 = a2; s2 = s1;
(4.15)
and
a1 + s1 = a2 + s2 = r:
(4.16)
The rst of these conditions assures the skew{symmetry of the Poisson
bracket (4.14), and the second one garantees that the Hamiltonian ows
with invariant Hamiltonian functions '(L) have the Lax form (3.11) with
the M {matrix (4.2).
Such general quadratic r-matrix structures were discovered several times
independently [16], [17], [18] and found their applications to concrete mechanical systems in [18], to the problem of regularization of the monodromy
matrix in [19], and to the constrained KP hierarchies in [20].
The hypothesis (4.14) turns out to be also true, corresponding theorem
constituting the main result of the present paper. To formulate it, we introduce an auxiliary linear (dynamical) skew{symmetric operator W , acting on
the space of diagonal matrices as follows: in the rational case
X
1 E ;
W (E ) =
(4.17)
s=
X
k6=j
kk
j 6=k xk
xj
jj
and in the hyperbolic case
X
W (Ekk ) = coth(xk xj )Ejj :
j 6=k
10
(4.18)
The corresponding matrices are given in the rational case by:
X
1 E E ;
w=
(4.19)
and in the hyperbolic case by:
X
w = coth(xk xj )Ekk Ejj :
(4.20)
k6=j xk
xj
kk
jj
k6=j
Theorem. For the Lax matrices of the relativistic models (2.5), (3.4)
there holds a quadratic r{matrix ansatz (4.14) with the matrices
a1 = a + w; s1 = s w;
a2 = a + s s w; s2 = s + w;
The proof of this Theorem we could nd so far is not very instructive,
since it consists of rather direct (hence somewhat tedious) computations.
They are presented in the Appendix.
Let us discuss the dierences of this result from the existing ones [13],
[14]. The most obvious ones are connected with the dierent gauge for the
Lax matrix we use here. We have not tried to re{express our results in the
gauge used in [13], [14], although it is surely worth doing, at least in order
to clarify, whether the results are in fact equivalent. However, some features
of the result in [14] indicate that it might be not equivalent to our (it would
be not very surprizing because of the well{known non{uniqueness of the r{
matrix). First, although the bracket found in [14] has a linear structure
(4.8) with a linear dependence r on L, we could not see how to put it in the
quadratic form (4.14). Second, the r{matrix found in [14] depends essentially
on the parameter we denote (called in [14]), while in our formulation it
does not play any role. On the other hand, in the gauge used in [13], [14] the
value = i=2 plays an exceptional role for the hyperbolic model, related in
this case to the solitons of the sine{Gordon model. The corresponding Lax
matrix becomes symmetric, which leads to an alternative quadratic Poisson
structure found in [13]. This structure is of the form (4.14) with 2 a1 = a2 = a
and s1 = s2 = a , where denotes the transposition in the rst factor of
tensor products. This structure can be considered as a "quadratization" of a
twisted linear r{matrix bracket with r = a + a . This structure is not covered
by the construction in [14], neither by our construction.
2
The matrix a here is of course dierent from (4.10), (4.12).
11
5 Conclusions
The Theorem formulated in the previous section is surely only one link in
a long chain of future results concerning the Ruijsenaars{Schneider models. The whole work made for their non{relativistic counterparts should be
repeated, or, better, generalized.
First of all, as it became already a tradition in the investigation of the
Calogero{Moser type models, our Theorem should be generalized to
the elliptic case. The corresponding Lax matrices and the matrices a1,
a2, s1, s2 should contain an additional spectral parameter, as in [6], [7].
The spin Ruijsenaars{Schneider models, introduced recently in [21],
should be also put into the r{matrix framework, as it was done for the
spin Calogero{Moser systems in [8].
Another important point is a derivation of our Theorem in the framework of Hamiltonian reduction; to this end the results in [11] should
be used and further developed.
It would be rather important to investigate the dynamical analogs of the
modied Yang{Baxter equation assuring the Poisson bracket properties
of the ansatz (4.14). The investigations of these objects were started in
[6] for the linear ansatz (4.8) and are expected to unveil new interesting
structures in the case of quadratic brackets. Recall [17], [18] that in the
case when the matrices a1, a2, s1, s2 do not depend on the dynamical
variables and (4.16) is fullled, sucient condition for (4.14) to be
a Poisson bracket reads: three operators A1, A2, and R satisfy the
modied Yang{Baxter equation.
Further, our Theorem being established, situation with the Calogero{
Moser type models becomes perfectly analogous to the situation with the
Toda{like ones. Namely, the transition to the relativistic generalization corresponds to the transition from the linear r{matrix Poisson structure to the
quadratic one. (See [22],[18] for the Toda case). Can this analogy be pursued to the further aspects elaborated for the Toda models in [22], [18]? More
concretely:
12
Can the existence of two abstract (dynamical) Poisson structures on
sl(N ) be used to establish a bi{Hamiltonian structure for both the
non{relativistic and relativistic models?
Does there exist a time{discretization of the non{relativistic models
living on the same orbit as the relativistic one?
These two problems are under current investigation.
6 Acknowledgements
The research of the author is nancially supported by the DFG (Deutsche
Forschungsgemeinschaft). It is also a plesure to thank the International Erwin Schroedinger Institute in Vienna for nancial support, as well as for the
hospitality and nice atmosphere at the "Discrete Geometry and Condenced
Matter Physics" workshop, where part of this work was done.
7 Appendix: proof of the Theorem
Since the rational case is a simple limiting case of the hyperbolic one, we can
restrict ourselves with the latter. Let us denote
fLij ; Lkm g = ijkm Lij Lkm;
(
(L L)a1 =
and analogously
a (L L) =
2
N
X
i;j;k;m=1
N
X
i;j;k;m=1
(I L)s1(L I ) =
(L I )s2(I L) =
)
ijkm Lij Lkm Eij Ekm ;
(
1
)
ijkm Lij Lkm Eij Ekm ;
(
2
N
X
i;j;k;m=1
N
X
i;j;k;m=1
13
)
ijkm Lij LkmEij Ekm;
(
1
)
ijkm Lij LkmEij Ekm:
(
2
)
The statement of the Theorem is equivalent to
(ijkm) = (1ijkm) (2ijkm) + 1(ijkm) 2(ijkm):
(7.1)
According to the Poisson brackets (3.6) we have:
(ijkm) = coth(xm xj + ) coth(xj xm + ) + 2(1 jm)coth(xj xm)
+(jk jm)coth(xk xm + ) (im jm)coth(xi xj + ): (7.2)
From the denitions of the matrices a1, a2, s1, s2 we have:
(1ijkm)Lij Lkm = (1 jm)coth(xj xm)LimLkj
+(1 ik )coth(xi xk )Lij Lkm;
ijkm Lij Lkm = (1 ik)coth(xk xi)Lim Lkj
+(1 ik ) sinh(x1 x ) Lkj Lkm
i
k
1
(1 ik ) sinh(x x ) Lij Lim
k
i
(1 jm)coth(xj xm)Lij Lkm;
(
2
)
ijkm Lij Lkm = (1 im) sinh(x1 x ) Lmj Lkm
i
m
(1 im)coth(xi xm)Lij Lkm ;
(
1
)
ijkm Lij Lkm = (1 jk ) sinh(x1 x ) Lij Ljm
k
j
(1 jk )coth(xk xj )Lij Lkm :
Using the expressions for the elements of the matrix L, we get:
sinh(xi xj + )sinh(xk xm + ) coth(x x )
ijkm = (1 jm) sinh(
j
m
xi xm + )sinh(xk xj + )
+(1 ik )coth(xi xk );
(
2
(
1
)
)
xi xj + )sinh(xk xm + ) coth(x x );
ijkm = (1 ik ) sinh(
k
i
sinh(xi xm + )sinh(xk xj + )
(
2
)
14
i xj + )
+(1 ik ) sinh(x sinh(x x+
k
j )sinh(xi xk )
sinh(xk xm + )
(1 ik )
sinh(xi xm + )sinh(xk xi)
(1 jm)coth(xj xm);
j + )
ijkm = (1 im) sinh(x sinh(x x+i x)sinh(
xi xm)
m
j
(1 im)coth(xi xm);
(
1
)
m + )
ijkm = (1 jk ) sinh(x sinh(x xk+ x)sinh(
xk xj )
j
m
(1 jk )coth(xk xj ):
(
2
)
By the simplifying these expressions one uses systematically the identity
sinh(u1 + u2) = coth(u ) + coth(u ):
1
2
sinh(u1)sinh(u2)
In the expressions for (1ijkm), (2ijkm) one uses also the identity
sinh(xi xj + )sinh(xk xm + ) = 1+ sinh(xi xk )sinh(xj xm) :
sinh(xi xm + )sinh(xk xj + )
sinh(xi xm + )sinh(xk xj + )
One obtains following expressions:
ijkm = (1 jm)coth(xj xm) + (1 ik )coth(xi xk )
+(1 jm) sinh(xi xk )cosh(xj xm)
sinh(xi xm + )sinh(xk xj + )
= (1 jm)coth(xj xm) + (1 ik )coth(xi xk )
jm coth(xi xm + ) coth(xk xj + )
+ sinh(xi xk )cosh(xj xm) ;
sinh(xi xm + )sinh(xk xj + )
ijkm = (1 ik )coth(xk xi) (1 jm)coth(xj xm)
(1 ik ) coth(xi xm + ) + coth(xk xj + )
(
1
)
(
2
)
15
xi xk )sinh(xj xm)
+(1 ik ) sinh(cosh(
xi xm + )sinh(xk xj + )
= (1 ik )coth(xi xk ) (1 jm)coth(xj xm)
+ sinh(xi xk )cosh(xj xm) ;
sinh(xi xm + )sinh(xk xj + )
ijkm = (1 im)coth(xm xj + );
(
1
)
ijkm = (1 jk )coth(xj xm + ):
(
2
)
Now it is easy to see that combining the obtained expressions as in (7.1),
one gets exactly the right{hand side of (7.2), which proves the Theorem.
16
References
[1] M.A.Olshanetski, A.M.Perelomov. Classical integrable nite - dimensional systems related to Lie algebras. { Phys. Rep., 1981, v.71, p.313.
[2] D.Kazhdan, B.Kostant, S.Sternberg. Hamiltonian group actions and dynamical systems of Calogero type. { Commun. Pure Appl. Math., 1978,
v.31, p.491{508.
[3] S.N.M.Ruijsenaars, H.Schneider. A new class of integrable systems and
its relation to solitons. { Ann. Phys., 1986, v.170, p.370{405.
[4] S.N.M.Ruijsenaars. Action{angle maps and scattering theory for some
nite{dimensional integrable systems. { Commun. Math. Phys., 1988,
v.115, p.127{165.
[5] J.Avan, M.Talon. Classical R{matrix structure for the Calogero model.
{ Phys. Lett. B, 1993, v.303, p.33{37.
[6] E.K.Sklyanin. Dynamical r{matrices for elliptic Calogero{Moser model.
{ Alg. i Anal., 1994, v.6, p.227.
[7] H.W.Braden, T.Suzuki. R{matrices for elliptic Calogero{Moser models.
{ Lett. Math. Phys., 1994, v.30, p.147.
[8] E.Billey, J.Avan, O.Babelon. The r{matrix structure of the Euler{
Calogero{Moser model. { Phys. Lett. A, 1994, v.186, p.114;
Exact Yangian symmetry in the classical Euler{Calogero{Moser model.
{ Phys. Lett. A, 1994, v.188, p.263.
[9] J.Avan, O.Babelon, M.Talon. Construction of the classical R{matrices
for the Toda and Calogero models. { Alg. i Anal., 1994, v.6, p.67.
[10] A.Gorsky, N.Nekrasov. Elliptic Calogero{Moser system from two dimensional current algebra { hep/th 9401021;
Hamiltonian systems of Calogero type and two{dimensional Yang{Mills
theory. { Nucl. Phys B , 1994, v.414, p.213{238;
Relativistic Calogero{Moser model as gauged WZW. {Nucl. Phys. B,
1995, v.436, p.582{608.
17
[11] A.Gorsky. Integrable many{body systems in the eld theories. { hep/th
9410228.
[12] G.E.Arutyunov, P.B.Medvedev. Geometric construction of the classical
R{matrices for the elliptic and trigonometric Calogero{Moser systems.
{ hep/th 9511070
[13] O.Babelon, D.Bernard. The sine{Gordon solitons as an N {body problem. { Phys. Lett. B, 1993, v.317, p.363{368.
[14] J.Avan, G.Rollet. The classical r{matrix for the relativistic Ruijsenaars{
Schneider system. { Phys. Lett. A, 1996, v.212, p.50{54.
[15] A.Reyman, M.Semenov-Tian-Shansky. Group{theoretic methods in the
theory of nite{dimensional integrable systems. { In: Encyclopaedia of
mathematical sciences, v.16, dynamical systems VII. Springer, 1994,
p.116{225.
[16] L.Freidel, J.M.Maillet. Quadratic algebras and integrable systems. {
Phys. Lett. B, 1991, v.262, p.278{284.
[17] S.Parmentier. Twisted ane Poisson structures, decompositions of Lie
algebras and the classical Yang{Baxter equations. { Preprint Inst. Max
Plank, Bonn, 1991.
[18] Yu.B.Suris. On the bi{Hamiltonian structure of Toda and relativistic
Toda lattices. { Physics Letters A, 1993, v.180, p.419{429.
[19] M.A.Semenov-Tian-Shansky. Monodromy map and classical r{matrices.
{ Zap. Nauch. Semin. POMI, 1993, v.200, see also hep/th 9402054.
[20] W.Oevel. A note on the Poisson brackets associated with Lax operators
{ Physics Lett. A, 1994, v.186, p.79{86.
[21] I.Krichever, A.Zabrodin. Spin generalization of the Ruijsenaars - Schneider model, non{abelian 2D Toda chain and representations of Sklyanin
algebra. - hep/th 9505039.
[22] Yu.B.Suris. Algebraic structure of discrete time and relativistic Toda
lattices. { Phys. Lett. A, 1991, v.156, p.467{474.
18