On Classical r-Matrix for the Kowalevski Gyrostat on so(4)

We present the trigonometric Lax matrix and classical r-matrix for the Kowalevski gyrostat on so(4) algebra by using the auxiliary matrix algebras so(3,2) or sp(4).


Introduction
The 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 formulae 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 its 5 × 5 Lax matrix and discuss the properties of the r-matrices. Finally, we get the 4 × 4 Lax matrix on the auxiliary sp(4) algebra.
Remind, the Kowalevski top is the third integrable case of motion of rigid body rotating in a constant homogeneous field [5]. 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 Hamiltonian on a generic orbit of the so(4) Lie algebra with the Poisson brackets where ε ijk is the totally skew-symmetric tensor and κ ∈ C (see [6] for references). Fixing values a and b of the Casimir functions one gets a four-dimensional orbit of so(4) which is a reduced phase space for the deformed Kowalevski top.
Because physical quantities y, J in (1) should be real, κ 2 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 κ 2 respectively and to e(3) for κ = 0.
The Hamilton function (1) is fixed up to canonical transformations. For instance, the brackets (2) are invariant with respect to scale transformation y i → cy i and κ → cκ that allows to include scaling parameter c into the Hamiltonian, i.e. to change y 1 by cy 1 . Some other transformations are discussed in [6].
Below we identify Lie algebra g with its dual g * by using invariant inner product and notation g * is used both for the dual Lie algebra and for the corresponding Poisson manifold.

The Kowalevski gyrostat: some known results
The Lax matrices for the Kowalevski gyrostat was found in [9] and [6] at κ = 0 and κ = 0 respectively. The corresponding classical r-matrices have been constructed in [7] and [10]. In these papers different definitions of the classical r-matrix [8,2] were used, which we briefly discuss below.

The Lax matrices
By definition the Lax matrices L and M satisfy the Lax equation with respect to evolution determined by Hamiltonian H. Usually the matrices L and M take values in some auxiliary algebra g (or in its representation), whereas entries of L and M are functions on the phase space of a given integrable system depending on spectral parameter λ. For κ = 0 the Lax matrices for the Kowalevski gyrostat on e(3) algebra were found by Reyman and Semenov-Tian-Shansky [9] L 0 (λ) = and These matrices belong to the twisted loop algebra g λ based on the auxiliary Lie algebra g = so (3,2) in fundamental representation. We have to underline that the phase space of the Kowalevski gyrostat and the auxiliary space of these Lax matrices are essentially different.
Remind the auxiliary Lie algebra so(3, 2) may be defined by all the 5 × 5 matrices satisfying where J = diag (1, 1, 1, −1, −1), and T stands for matrix transposition. The Cartan involution on g = so(3, 2) is given by and g = f + p is the corresponding Cartan decomposition where f = so(3) ⊕ so(2) is the maximal compact subalgebra of so (3,2). The pairing between g and g * is given by invariant inner product that is positively definite on f. We extend the involution σ to the loop algebra g λ by setting (σX)(λ) = σ(X(−λ)). By definition, the twisted loop algebra g λ consists of matrices X(λ) such that The pairing between g λ and g * λ is given by At κ = 0 the Lax matrices for the Kowalevski gyrostat on so(4) were originally found in [6] as a deformation of the matrices L 0 (λ) and M 0 (λ) Algebraic nature of the matrix L(λ) (10) is appeared to be mysterious, because the diagonal matrix Y c does not belong to the fundamental representation of the auxiliary so(3, 2) algebra, hence matrices (10) do not belong to the Reyman-Semenov-Tian-Shansky scheme [8].
In the next section we prove that the Lax matrix L(λ) at κ = 0 is a trigonometric deformation of the rational Lax matrix L 0 (λ) on the same auxiliary space.

Classical r-matrix: operator notations
The classical r-matrix is a linear operator r ∈ End g that determines second Lie bracket on g by the rule 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 In the most common cases r is a skew-symmetric operator such that where P ± are projection operators onto complementary subalgebras g ± of g. In this case there exists a complete classification theory. All details may be found in the book [8] and references therein. Marchall [7] has 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 ∈ g λ where X + (λ) is a Taylor series in λ, X 0 is an independent of λ and X − (λ) is a series in λ −1 . If P ± and P 0 are the projection operators onto g λ parallel to the complementary subalgebras (12), the operator defines the second Lie structure on g λ . According to [7] the r-matrix (13) is the classical r-matrix for the Kowalevski gyrostat. In the standard case (11) operator ̺ is identity, however for the Kowalevski gyrostat ̺ is a difference of projectors in the base g = so(3, 2) (see details in [7]).

Classical r-matrix: tensor notations
Another definition of the classical r-matrix is more familiar in the inverse scattering method [8,1,2]. According to [1], the commutativity of the spectral invariant of the matrix L(λ) is equivalent to existence of a classical r-matrix r 12 (λ, µ) such that the Poisson brackets between the entries of L(λ) may be rewritten in the following commutator form Here and Π is a permutation operator ΠX ⊗ Y = Y ⊗ XΠ for any numerical matrices X, Y . For a given Lax matrix L(λ), r-matrices are far from being uniquely defined. The possible ambiguities are discussed in [8,1,2].
If the Lax matrix takes values in some Lie algebra g (or in its representation), the r-matrix takes values in g × g or its corresponding representation. The matrices r 12 , r 21 may be identified with kernels of the operators r ∈ End g and r * ∈ End g * respectively, using pairing between g and g * (see discussion in [8]).
For the Kowalevski gyrostat the classical r-matrix r 12 (λ, µ) entering (14) has been constructed in [10] by using the auxiliary Lie algebra g = so(3, 2) in fundamental representation. The generating set of this auxiliary space consists of one antisymmetric matrix and three antisymmetric matrices These matrices are orthogonal with respect to the form of trace (7). Four matrices S k form maximal compact subalgebra f = so(3) ⊕ so(2) of so(3, 2) and their norm is 1, whereas six matrices Z i and H i belong to the complementary subspace p in the Cartan decomposition g = f + p and their norms are −1. Operators are projectors onto the orthogonal subspaces f and p respectively. In this basis the Lax matrix L 0 (λ) (5) for the Kowalevski gyrostat on e(3) reads as According to [10] the corresponding r-matrix is equal to We can say that this matrix r 12 (λ, µ) is a specification of the operator r (13) with respect to canonical pairing (7)- (9).
In our case the matrix r 12 (λ, µ) (18) is appeared to be purely numeric matrix, which depends on the ratio λ/µ only. It allows us to change the spectral parameters λ = e u 1 and µ = e u 2 and rewrite this r-matrix in the following form depending on one parameter z = u 1 − u 2 via trigonometric functions. Therefore, the classical r-matrix for the Kowalevski gyrostat on e(3) should be considered as trigonometric r-matrix according to generally accepted classification [2,8]. At the same time it is natural to keep initial rational parameters λ, µ in the Lax matrix. Similar properties holds for the periodic Toda chain, for which N × N Lax matrix depends rationally on spectral parameters, while the corresponding r-matrix is trigonometric.
We have to underline that in contrast with usual cases this r-matrix is non-unitary. Moreover, it has a term (S 3 − S 4 ) ⊗ S 4 , which is independent on spectral parameters and, therefore, the inequality r 12 (z) = −r 12 (−z) takes place.
In order to understand the nature of these items we recall that the Lax matrix L 0 (λ) has been derived in the framework of the Hamiltonian reduction of the so(3, 2) top for which phase space coincides with the auxiliary space. The corresponding classical r-matrix, calculated in [10] r so(3,2) 12 is a trigonometric r-matrix associated with the so(3, 2) Lie algebra [2]. So, the constant term (18) is an immediate result of the Hamilton reduction, which changes the phase space of our integrable system. We recall that classical r-matrices r 12 (λ, µ) are called regular solutions to the Yang-Baxter equation (15) if they pass through the unity at some λ and µ. In our case we have the following counterpart of this property of regularity res r 12 (z)| z=0 = 1 2 P p − P f .

Classical r-matrix for Kowalevski gyrostat on so(4)
Now let us consider Lax matrix L(λ) (10) for the Kowalevski gyrostat on so(4) algebra. After transformation L(λ) → cos φ Y −1/2 c L(λ)Y 1/2 c of the Lax matrix L(λ) (10) and change of the spectral parameter λ = κ/ sin φ one gets a trigonometric Lax matrix on the auxiliary so(3, 2) algebra or In order to consider the real forms so(4, R) or so(3, 1, R) we have to use trigonometric or hyperbolic functions for positive and negative κ 2 , respectively. If we put φ = κλ −1 and take the limit κ → 0 we find the rational Lax matrix L 0 (λ) (5) for the Kowalevski gyrostat on e(3).
The Lax matrices L(φ) and L 0 (λ) are invariant with respect to the following involutions that are compatible with the Cartan involution σ. This simple observation shows that for the Kowalevski so(4) gyrostat the Reyman-Semenov-Tian-Shansky scheme [8] should be extended from rational to trigonometric case. One can prove that the trigonometric Lax matrix L(φ) (19) satisfies relation with the following r-matrix If we put φ = κλ −1 , θ = κµ −1 and take the limit κ → 0 we get classical r-matrix for the Kowalevski gyrostat on e(3) algebra (18). As above the matrix r 12 (φ, θ) satisfies the Yang-Baxter equation (15) and it has the same analog of the property of regularity res r 12 (φ, θ)| φ=θ ≃ P p − P f .
In contrast with r-matrix (18) for the Kowalevski gyrostat on e(3) we can not rewrite this r-matrix (22) as a function depending on the difference of the spectral parameters only. We suggest that it may be possible to present it in terms of elliptic functions of one spectral parameters after a proper similarity transformation and reparametrization. The well known isomorphism between so(3, 2) and sp(4) algebras allows us to consider 4 × 4 Lax matrix instead of 5×5 matrix (20). The generating set Z 1 , Z 2 , Z 3 and S 4 may be represented by different 4 × 4 real or complex matrices, for instance, Other sp(4) generators are constructed by (16)-(17). These matrices are orthogonal with respect to the form of trace (7). Norm of matrices s k is 1/2, whereas six matrices z i and h i have norm −1/2.
If we put φ = κλ −1 and take the limit κ → 0 we find the rational Lax matrix for the Kowalevski gyrostat on e(3) According to [3], this matrix has a mysterious property. Namely, it contains the 3×3 Lax matrix L(λ) for the Goryachev-Chaplygin gyrostat on e(3) algebra as its (1, 1)-minor. Remind that the Goryachev-Chaplygin gyrostat with Hamiltonian It is easy to prove that (1, 1)-minor of the trigonometric Lax matrix (23) cannot be a Lax matrix for any integrable system. It is compatible with the known fact that the Goryachev-Chaplygin gyrostat on e(3) cannot be naturally lifted to so(4) algebra.

Conclusion
There are few Lax matrices obtained for deformations of known integrable systems from their undeformed counterpart in the form (10) (see [6,10] and references within). The important question in construction of these matrices by the Ansatz L = Y c · L 0 (10) is a choice of a proper matrix Y c for a given rational matrix L 0 (λ). In all known cases this transformation destroys the original auxiliary algebra, because the corresponding matrices Y c do not belong it.
In this note we show that if one takes a Lax matrix of the Kowalevski so(4) gyrostat in the symmetric form L = Y 1/2 c · L 0 · Y 1/2 c and makes a trigonometric change of spectral parameter it restores the original auxiliary so(3, 2) algebra and new L respects the trigonometric current involution (21). It means that deformation of the physical space from the orbits of e(3) to that of so(4) algebra is naturally related with transition from rational to trigonometric parametrization of the auxiliary current algebra.
We calculated explicitly the corresponding r-matrices and demonstrated that constant terms in them is due to the Hamiltonian reduction.
The classical r-matrix (22) for the so(4) gyrostat is numeric and the corresponding Lax matrix L(φ) (19) does not contain ordering problem in quantum mechanics. Hence equation (14) holds true in quantum case both for the Lax matrices (19) and (23).