HUEPHAH 0H3HKA, 2007, moM 70, № 3, c. 582-587
== ELEMENTARY PARTICLES AND FIELDS. THEORY ON CLASSICAL r-MATRIX FOR THE KOWALEVSKI GYROSTAT ON so(4)
© 2007 I. V. Komarov*, A. V. Tsiganov**
St.Petersburg State University, Russia Received May 16, 2006
We present trigonometric Lax matrix and classical r-matrix for the Kowalevski gyrostat on so(4) algebra by using auxiliary matrix algebras so(3, 2) or sp(4).
PACS numbers: 03.65.Fd
1. INTRODUCTION
A classical r-matrix structure is an important tool for investigating integrable systems. It encodes the Hamiltonian structure of the Lax equation, provides the involution of integrals of motion and gives a natural framework for quantizing integrable systems. The aim of this paper is severalfold. First, we present a formula for the classical r-matrices of the Kowalevski gyrostat on Lie algebra so(4), derived in the framework of the Hamiltonian reduction. In the process we shall get new form of the 5 x 5 Lax matrix and discuss the properties of the r-matrices. Finally, we get the 4 x 4 Lax matrix on auxiliary sp(4) algebra.
Remind, the Kowalevski top is the third integrable case of a motion of rigid body rotating in a constant homogeneous field [1]. This is an integrable system on the orbits of the Euclidean Lie algebra e(3) with a quadratic and a quartic in angular momenta integrals of motion.
The Kowalevski top can be generalized in several directions. We can change either initial phase space or the form of the Hamilton function. In this paper we consider the Kowalevski gyrostat with the Hamilto-nian
H = Ji + J22 + 2J32 + 2p.J3 +2yi, p e R, (1)
on a generic orbit of the so(4) Lie algebra with the Poisson brackets
{Jí,Jj} = £ijk Jk, {Jí,yj} = £ijkyk, (2) {yi,yj} = K2£ijk Jk,
where eijk is the totally skew-symmetric tensor and k e C. Fixing values a and b of the Casimir functions
3 3 3
A = 5>? + *2EJ2, b = T,J (3)
i=1 i=1 i=1
E-mail: ivkoma@rambler.ru
E-mail: andrey-ts@yandex.ru
one gets a four-dimensional orbit of so(4)
Oab : {y,J : A = a, B = b},
which is the reduced phase space for the deformed Kowalevski top.
Because physical quantities y,J in (1) should be real, k2 must be real too and algebra (2) is reduced to its two real forms so(4, R) or so(3,1, R) for positive and negative k2, respectfully, and to e(3) for k = 0.
The Hamilton function (1) is fixed up to canonical transformations. For instance, the brackets (2) are invariant with respect to scale transformation yi ^ ^ cyi and k ^ ck, which allows to include scaling parameter c into the Hamiltonian, i.e., change y1 to cy1. Some another transformations are discussed in [2].
2. THE KOWALEVSKI GYROSTAT: SOME KNOWN RESULTS
The Lax matrices for the Kowalevski gyrostat was found in [3] and [2] at k = 0 and k = 0, respectively. The corresponding classical r-matrices have been constructed in [4] and [5]. In these papers were used different definitions of the classical r-matrix [6, 7], which we briefly discuss below.
2.1. The Lax Matrices
By definition the Lax matrices L and M satisfy the Lax equation
|l(a) = {h,l(a)} = [m (A),L(A)] (4)
with respect to evolution determined by Hamiltonian H. Usually the matrices L and M take values in some auxiliary algebra g (or representation of this), whereas entries of L and M are functions on the phase space of a given integrable system and functions on the spectral parameter A.
At k = 0 the Lax matrices for the Kowalevski gyrostat on e(3) algebra was found by Reyman and Semenov-Tian-Shansky [3]:
in T T \ yi n \
Lo(A) =
0 J3 - J2 \ Vl A - T 0
-J3 0 Ji V2 A A
J2 - Ji 0 V3 A 0
VI ' A V2 A V3 A 0 - J3
0 A 0 J3 + P 0
and
Mo(A) = 2
0 -2J3 J2 -A 0
2J3 0 - Ji 0 -A
- J2 Ji 0 0 0
-A 0 0 0 0
0 -A 0 0 0
(X,Y ) = -1 tr(XY)
Y = diag 1,1,1
A2
A2-
1
K2
(5)
(6)
V
These matrices are elements of the twisted loop algebra gA based on auxiliary Lie algebra g = so(3,2) in fundamental representation. We have to underline that the phase space of the Kowalevski gyrostat and auxiliary space of these Lax matrices are different spaces.
Remind the auxiliary Lie algebra so(3,2) consists of all the 5 x 5 matrices satisfying
XT = -JXJ,
where J = diag(1,1,1, -1, -1), and T means a matrix transposition. The Cartan involution on g = = so(3,2) is given by
aX = -XT,
and g = f + p is the corresponding Cartan decomposition. The pairing between g and g* is given by invariant inner product
The algebraic nature of the matrix L(A) (10) was mysterious, because diagonal matrix Y does not belong to fundamental representation of the auxiliary so(3,2) algebra.
In the next section we prove that Lax matrix L(A) at k = 0 is a trigonometric deformation of the rational Lax matrix L0(A) on the same auxiliary space.
2.2. Classical r -Matrix: Operator Notations
The classical r-matrix is the linear operator r e Endg, which determines the second Lie brackets on g
[X,Y]r = [rX,Y] + [X, rY].
The operator r is a classical r-matrix for a given integrable system if the corresponding equations of motion with respect to the r-brackets have the Lax form (4) and the second Lax matrix M is given by
M = 1 r(dH).
(11)
In most abundant cases r is a skew-symmetric operator, such that
r = P+ - P-,
(12)
(7)
which is positive definite on f.
We extend the involution a to the loop algebra gA by setting (aX)(A) = a(X(—A)). By definition, the twisted loop algebra gA consists of matrices X(A) such that
X (A) = -XT (-A). (8)
The pairing between gA and gA is given by
(X,Y) = Res A-1(X,Y). (9)
At k =0 the Lax matrices for the Kowalevski gyrostat on so(4) are deformation of the matrices Lo(A) and Mo(A) [2]:
L = Y ■ L0, M = M0 ■ Y
-i
(10)
where P± are projection operators onto complimentary subalgebras g± of g. In this case there exists a complete classification theory. All details may be found in the book [6] and references within.
In [4] it has been shown that the Lax matrices (5) for the Kowalevski gyrostat on e(3) may be obtained by direct application of this r-matrix approach. Let us introduce the standard decomposition of any element X e gA
X (A) = X+ (A)+Xo + X-(A), (13)
where X+ (A) is a Taylor series in A, X0 is an independent of A, and X- (A) is a series in A-1. If P± and P0 be the projection operators onto gA parallel to the complimentary subalgebras (13), the operator
r = P- + p ◦ Po - P+ (14)
defines the second Lie structure on gA. According to [4] the r-matrix (14) is the classical r-matrix for the Kowalevski gyrostat. In the standard case (12) operator p is identity, however, for the Kowalevski gyrostat p is a difference of projectors in the base g = so(3,2) (see details in [4]).
A
aflEPHAa OH3HKA tom 70 № 3 2007
2.3. Classical r -Matrix: Tensor Notations
matrix
Another definition of the classical r-matrix is more familiar in the inverse scattering method [6—8]. According to [8], the commutativity of the spectral invariant of the matrix L(\) is equivalent to existence of a classical r-matrix r12(X, f) such that the Poisson brackets between the entries of L(X) may be rewritten in the following commutator form:
i 2 L (A),L (ß)} —
(15)
ri2(A,ß),L (A)
r2i(\,ß),L (ß)
Here,
[ri2(A, ß), ri3(A, V) + r23(ß, v)] -- [ri3 (A,v), T32 (v,ß)] = 0
(16)
S4 —
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 -1
0 0 0 1 0
and three symmetric matrices
£1 —
L (X) = L(X) ® 1, L (f) = 1 ® L(f), r2i (X,x) = nri2 (x,X)n,
and n is a permutation operator nx ® Y = Y ® Xn for any numerical matrices X, Y.
For a given Lax matrix L(X), r-matrices are far from being uniquely defined. All the possible ambiguity are discussed in [6—8].
If the Lax matrix takes values in some Lie algebra g (or in its representation), the r-matrix takes values in g x g or its corresponding representation. The matrices r12, r21 may be identified with kernels of the operators r e Endg and r* e Endg*, respectively, using pairing between g and g* (see discussion in [6]).
Generally speaking, the matrix r12 (X, f) is a function of dynamical variables [7, 9]. The best studied one is the case of purely numeric r-matrices satisfying the classical Yang—Baxter equation
0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 , Z2 = 0 0 0 0 0
1 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0
£3 —
which ensures the Jacobi identity for the Poisson brackets (15). If r12(X,f) is unitary numeric matrix depending on the difference of the spectral parameters 2 = X — f, there exists a profound algebraic theory which allows to classify r-matrices in various families [6, 7].
For the Kowalevski gyrostat the classical r-matrix r12(X,f) entering (15) has been constructed in [5] by using the the auxiliary Lie algebra g = so(3,2) in fundamental representation. The generating set of this auxiliary space consists of one antisymmetric
0 0 0 0 0
0 0 0 0 0
0 0 0 1 0
0 0 10 0
\0 0 0 0 0/
which are the generators of the so(3,2) algebra. Other generators are three symmetric matrices
Hi = [S4,Zi] = S4Z - ZiS4, i = 1,2,3, (17)
and three antisymmetric matrices
S1 = [Z2, Z3], S2 = [Z3,Zl}, S3 = [Z1,Z2 ]•
(18)
These matrices are orthogonal with respect to the form of trace (7). Four matrices Sk form maximal compact subalgebra f = so(3) © so(2) of so(3,2) and their norm is 1, whereas six matrices Zi and Hi belong to the complimentary subspace p in the Cartan decomposition g = f + p and their norm are —1. Operators
43
Pf = E Sk 0 Sk, Pp = Y^(Hi 0 Hi + Zi 0 Zi)
k=1
i=1
aflEPHAa OH3HKA TOM 70 № 3 2007
are projectors onto the orthogonal subspaces f and p, respectively.
In this basis the Lax matrix L0(A) (5) for the Kovalewski gyrostat on e(3) reads as
3
Lo = A(Zi + H2) + J2(JiSi - \-1xZi) +
i=1
+ ( J3 + p)S4.
According to [5] the corresponding r-matrix is equal to
ri2 (A,ß) =
Aß
ß2
Aß
A2 - ß2
A2 - ß2 p A2 - ß2
+ (S3 - S4) ® S4 =
3
J2(Hi ® Hi + Zi ® Zi)
Pf + (19)
i=1
ß2
A2 - ß2
^ Sk ® Sk + (S3 - S4) ® S4.
k=1
We can say that this matrix ri2 (A, f) is a specification of the operator r (14) with respect to canonical
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