Generalized model of interacting integrable tops

We introduce a family of classical integrable systems describing dynamics of $M$ interacting ${\rm gl}_N$ integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the ${\rm GL}_N$ $R$-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the spin type Calogero-Moser models with (the classical analogues of) anisotropic spin exchange operators given in terms of the $R$-matrix data. In $N=1$ case the spin Calogero-Moser model is reproduced. Explicit expressions for ${\rm gl}_{NM}$-valued Lax pair with spectral parameter and its classical dynamical $r$-matrix are obtained. Possible applications are briefly discussed.


Introduction
In this paper we describe the classical integrable gl N M model given by the Hamiltonian of the following form: where p i and q j are the canonical variables: where {e ab , a, b = 1...N} is the standard basis in Mat(N, C). They are naturally arranged into NM × NM block-matrix S: S ij ab E ij ⊗ e ab ∈ Mat(NM, C) , (1.4) where {E ij , i, j = 1...M} is the standard basis in Mat(M, C). The Poisson structure is given by the Poisson-Lie brackets on gl * N M Lie coalgebra: {S ij ab , S kl cd } = S kj cb δ il δ ad − S il ad δ kj δ bc (1.5) Integrable tops. In order to clarify the structure of the Hamiltonian (1.1) consider the case M = 1. Then the last term in (1.1) is absent, and we are left with a free particle (with momenta p 1 ) and the Hamiltonian H top (S 11 ) of integrable top of Euler-Arnold type [3]. Here we deal with the models admitting the Lax pairs with spectral parameter on elliptic curves [30]. The general form for equations of motion isṠ = [S, J(S)] , (1.6) where S ∈ Mat(N, C) is the matrix of dynamical variables, while the inverse inertia tensor J is a linear map J ijkl e ij S lk ∈ Mat(N, C) (1.7) In the general case the model (1.6) is not integrable. It is integrable for some special J(S) only. More precisely, here we consider special tops, which were described in [18,34], [16], [1,19] for elliptic, trigonometric and rational cases respectively. All of them can be written [19,21] in the R-matrix form based on a quantum GL N R-matrix (in the fundamental representation) satisfying the associative Yang-Baxter equation [12,26]: R 12 (q 12 )R η 23 (q 23 ) = R η 13 (q 13 )R −η 12 (q 12 ) + R η− 23 (q 23 )R 13 (q 13 ) , q ab = q a − q b . (1.8) Having solution of (1.8) with some additional properties (see the next Section) the inverse inertia tensor comes from the term m 12 (z) in the classical limit expansion: of GL N Lie group, i.e. the space spanned by S ij with some fixed eigenvalues of S (or the Casimir functions C k = trS k ). Its dimension depends on the eigenvalues. The minimal orbit O min N corresponds to N − 1 coincident eigenvalues, i.e the matrix S (up to a matrix proportional to identity matrix) is of rank one: The Lax pair is given in the Appendix C.
Spin Calogero-Moser model. In the case N = 1 the second term in (1.1) is trivial, and the last one boils down to the spin Calogero-Moser model [13,4]:  (1.2). In the general case the spin variables can be parameterized by the set of canonically conjugated variables: The Poisson structure (B.7) is reproduced in this way. Using these notations it is easy to see that 20) and the potential in the Hamiltonian (1.16) takes the form Below we construct anisotropic (in Mat(N, C) space) generalizations of (1.21).
In the special case, when the matrix of spin variables S is of rank The spinless Calogero-Moser models are gauge equivalent to the special top with the minimal orbit (1.15). See [18,16,1] for details.
Interacting tops. Turning back to the gl N M model (1.1) consider the special case when the matrix S is of rank 1: We will see that in this case the last term in (1.1) is rewritten in the form has dimension 2NM, while its "spin part" is of dimension A brief summary of the described models is given on the following scheme: for the general model (1.1) and for the model of interacting tops (1.26). Then we proceed to the classical (dynamical) r-matrix.
It is similar to the one for the spin Calogero-Moser case [4] but this time its matrix elements are R-matrices themselves. The classical exchange relations are verified directly. This guarantees the Poisson commutativity of the Hamiltonians generated by the Lax matrix. Possible applications of the described models are discussed in the end.

Lax equations
In this Section we construct the NM × NM Lax pair L(z), M(z) satisfying the Lax equationṡ

R-matrix properties.
We consider R-matrices satisfying (1.8) and (1.9). Let us also impose the following set of conditions for GL N R-matrices under consideration: Expansion near z = 0: Unitarity: We are also going to use the Fourier symmetry: It is not necessary but convenient property. The following relations on the coefficients of expansions (1.9) and (2.2) follow from the skew-symmetry: (2.7) Similarly, from the Fourier symmetry we have (see details in [35]):  In what follows we use special notation for the R-matrix derivative: It is the R-matrix analogue of the function (A.5) entering the M-matrix of the spin Calogero-Moser model (B.3) likewise R-matrix itself is a matrix analogue of the Kronecker function (A.1) due to similarity of (A.6) and (1.8). See [20]. Then from the classical limit (1.9) we have The latter is the R-matrix analogue of the function −E 2 (q) (A.12) entering the Calogero-Moser potential. Notice also that F 0 12 (q) = F 0 21 (−q) due to (2.7). From (2.9) and (2.2) the local expansion near q = 0 is as follows and, therefore, On the other hand (2.14) In the elliptic case the set of properties is fulfilled by the Baxter-Belavin [5] R-matrix (C.14).
A family of trigonometric R-matrices include the XXZ 6-vertex one, its 7-vertex deformation [8] and GL N generalizations [2,28]. See a brief review and applications to integrable tops in [16]. The rational R-matrices possessing the properties are the XXX Yang's R-matrix, its 11-vertex deformation [8] and higher rank analogues obtained from the elliptic case by special limiting procedure [32]. The final answer for such R-matrix was obtained in [19] through the gauge equivalence between the relativistic top with minimal orbit and the rational Ruijsenaars-Schneider model.

Lax pair and equations of motion
Using coefficients of the expansion of the GL N R-matrix near z = 0 we define NM × NM Lax pair (2. 16) and similarly for M ij (z) ∈ Mat N M ij (z) = δ ij tr 2 (S ii 2 R z,(1) where the entries are defined from (2.2) and (2.9). The tensor notations are similar to those used in (C.3)-(C.6).  19) for diagonal N × N blocks of S: and for momenta:ṗ (2.21) Proof: We imply p i =q i in the formulae above. This follows from the Hamiltonian description, which is given in the next paragraph.
1. Let us begin with the non-diagonal blocks. Consider the one numbered ij (i = j). The l.h.s. of the Lax equations reads The r.h.s. of the Lax equation is as follows: The last sum is computed using identity which follows from (1.8). It is the R-matrix analogue of (A.7). In its turn (A.7) is the key tool underlying ansatz for the Lax pairs with spectral parameter [17]. For k = i, j we have This expression provides the upper line in the equations of motion (2.19). To proceed we need degenerations of the identity (2.24) when y → 0. It comes from the expansions (2.2), (2.11) and (2.13): In the same way in the limit x → 0 (2.24) takes the form Similarly to the ordinary (spin) Calogero-Moser case the terms linear in momenta in the r.h.s. (2.23) (p i − p j )M ij are cancelled out by the last term in the l.h.s. of (2.22). Consider the first and the fourth terms from (2.23) without momenta. Using evaluations similar to (2.25) we get where the relation (2.14) was also used (for the first term in the answer). The first and the second terms in the obtained answer provide the last line in the equations of motion (2.19), while the last term in (2.19) is the "unwanted term".
In the same way, using (2.27) one gets Again, the first two terms provide an input to equations of motion -the second line in (2.19). The last term is the "unwanted term". It is cancelled by the one from (2.28) after taking the trace over the third component and imposing the constraints (2.18).

Consider a diagonal
The r.h.s. of the Lax equation is as follows: The commutator term in (2.31) provides the commutator term in the equations of motion (2.20) since it is the input from the internal ii-th top's dynamics, and this was derived in [21]. See (C.2)-(C.4). In order to simplify expression in the sum we need the following degeneration of (1.8): It corresponds to = η = z. In the scalar case it is the identity (A.9). In the limit x = q = −y from (2.32) we get or, using (2.8) By differentiating (2.34) with respect to q we obtain (2.36) The commutator term in the obtained expression yields the sum term in the equations of motion (2.20), while the last term in (2.36) provides equations of motion (2.21). Indeed,

Hamiltonian description
The Hamiltonian function. Let us compute the Hamiltonian for the model (2.15)-(2.21). It comes from the generating function As before, the numbered tensor components are Mat(N, C)-valued. In order to simplify (2.39) we use the identity (see [20]) which can be treated as a half of the classical Yang-Baxter equation 2 . In the limit w → 0 (2.40) yields Also, we are going to use the following R-matrix property: whereφ(z, q) is the Kronecker function (A.1) but with possibly different normalization factor and normalization of arguments.The property (2.42) holds true in the elliptic case (C.15) as well as for its trigonometric and rational degenerations. From (2.42), expansion (1.9) and (A.10) we also have similar properties for tr 1 r 12 (z) =Ẽ 1 (z) and tr 1 m 12 (z) -they are scalar operators: Return now to (2.39). On the constraints (2.18) the second term is equal to 2p iẼ1 (z)const. After summation over i it provides the Hamiltonian proportional to M i=1 p i . Plugging (2.41) into the last term of (2.39) we get (2.44) Due to (2.42) the first two terms are cancelled out after taking the trace over the component 1. By the same reason the last two terms in (2.44) provide 2tr 1 (m 12 (z))tr 23 (S ii 2 S ii 3 ). These are constants on the constraints (2.18). The rest of the terms are It is a scalar function coming from (2.43) and similar to E 2 (z) (A.4). The factor N in the last term comes from tr 1 . The first term in (2.45) is a part of the Casimir function trS 2 , and the second one is H top (S ii ) from (1.1): (2.46) Next, consider tr L ij (z)L ji (z) = tr 123 R z 12 (q ij )P 12 R z 13 (q ji )P 13 S ij 2 S ji (2.47) Again, the commutator term vanishes after taking the trace over the first tensor component. Therefore,

Interacting tops
Suppose the matrix S is of rank one, i.e. (1.24) is fulfilled. Consider the potential The right multiplication of an element T 12 = N i,j,k,l=1 T ijkl E ij ⊗E kl ∈ Mat(N, C) ⊗2 by permutation operator P 12 yields T ijkl → T ilkj , i.e.
(F 0 12 (q ji )) ad,cb S ij ba S ji dc . (2.57) In the rank 1 case we have (2.58) Therefore, (2.59) The Hamiltonian of interacting tops model acquires the form: (2.60) From the Poisson brackets (2.51), (2.54) we get the corresponding equations of motion: (2.62) In 3. The spin part of the phase space for the model of interacting tops coincides with the phase space of GL N classical spin chain on M sites with the spins described by minimal coadjoint orbits at each site.
Let us also remark that the top like models with matrix-valued variables were studied in [21,35] and [6]. In contrast to these papers here we deal with the models, where the matrix variables have their own internal dynamics.

Classical r-matrix
In this Section we describe the classical r-matrix structure for the Lax matrix (2.16). Since L ∈ Mat(NM, C) then the corresponding classical gl N M r-matrix r ∈ Mat(NM, C) ⊗2 . Recall that for the Lax matrix we use the matrix basis (2.15), in which L ∈ Mat(M, C)⊗Mat(N, C). Let the Mat(M, C)-valued tensor components be numbered by primed numbers, and the Mat(N, C)valued components -without primes (as before). Introduce the following r-matrix: In the case M = 1 we come to a non-dynamical r-matrix describing the top model, while in the N = 1 we reproduce the dynamical r-matrix of the spin Calogero-Moser model (B.9). r-matrices of these type are known in gl N M case and can be extended for arbitrary complex semisimple Lie algebras [10,11,23]. In the elliptic case (3.1) is known in the quantum case as well [24]. At the same time (3.1) includes the cases, which have not been described yet. For instance, the new cases correspond to the rational R z 12 (q)-matrix from [19]. Similarly to the Lax equations the construction of the r-matrix (3.1) is based on the associative Yang-Baxter equation (1.8) and its degenerations.
The rest of the components are verified similarly.

Elliptic models
Let us begin with the elliptic model [33,24,14]. The Lax pair is of the form: where the basis (C.8) in Mat(N, C) is used. Similarly, the M-matrix is of the form where κ α,β are the constants from (C.10).
The Hamiltonian easily follows from 1 2N trL 2 (z) = 1 2N E 2 (z)tr(S 2 ) + H due to (C.11) and (A.8): Let us show how this Hamiltonian is reproduced from the general formula (2.50). In order to get the second term in (4.6) one should substitute m 12 (0) into (2.50) from (C.17) and use relation (A.12). For evaluation of the last sum in (2.50) we need to calculate F 0 12 (q)P 12 . The answer for F 0 12 (q) is given in (C.18). Multiply it by (4.7) Let us redefine the summation index b → b − a in the last sum. Since κ a,b = κ a,b−a we have (4.8) Finally, In the rank 1 case the answer for the Hamiltonian is given by (2.60). Plugging (C.18) into (2.60) we get (4.10) Let us show how the latter expression appears from (4.6). In the rank one case using (C.11) (so that S ij α = tr(S ij T −α )/N) we get In this way the Hamiltonian (4.6) acquires the form which is the model of interacting tops of (1.26) type. The last terms in (4.12) can be simplified in the following way. Substitute S ii = γ S ii γ T γ and S jj = γ S jj µ T µ into (4.12). It follows from (C.10)-(C.11) that tr(T γ T α T µ T −α ) = Nκ 2 α,µ δ µ+γ . (4.13) Therefore, (4.14) Using (C.20)-(C.21) and summing up over α we obtain the last term in (4.10).

Trigonometric models
The general classification of the unitary trigonometric R-matrices satisfying associative Yang-Baxter equation was given in [28]. It includes the 7-vertex deformation [8] of the 6-vertex R-matrix and its GL N generalizations such as the non-standard R-matrix [2]. The integrable tops and related structures based on these R-matrices were described in [16].
Here we restrict ourselves to the case N = 2. The 7-vertex R-matrix is of the following form: where C is a constant. In the limit C → 0 the lower left-hand corner vanishes and we get the 6-vertex XXZ R-matrix. For the classical r-matrix and its derivative (F 0 12 (z) = ∂ z r 12 (z)) we have respectively. The Fourier transformed F 0 matrix is of the form: From the latter matrix using (1.32) we obtain Similarly, using (1.33) and (4.17) we get the potential for the model of interacting tops: (4.20) (4.21)

Rational models
The rational R-matrices satisfying the required properties are represented by the 11-vertex deformation [8] of the 6-vertex XXX (Yang's) R-matrix. Its higher rank analogues were derived in [32] and [19]. As in trigonometric case here we restrict ourselves to the case N = 2. The 11-vertex R-matrix is of the following form: In order to get the XXX R-matrix one may take the limit lim The classical r-matrix, the F 0 12 matrix and its Fourier dual are of the form: From (4.25) using (1.32) we obtain (4.26) Similarly, from (4.24) using (1.33) we obtain (4.27) (4.28)

Discussion
Applications of the obtained models are related to the so-called R-matrix-valued Lax pairs for the (classical) spinless Calogero-Moser model [20,29,14]. These are the Lax pairs in a large space Mat(M, C) ⊗ Mat(N, C) ⊗M : and similarly for the M-matrix  [29] of the Haldane-Shastry-Inozemtsev type chains [15]. An open question is which F 0 provide integrable spin chains? To confirm integrability we need to construct higher Hamiltonians, which commute with each other and with F 0 (q i = i/N). Taking into account all the above we guess that the model of interacting tops together with the freezing trick (the quantum version of the equilibrium position) can be used to calculate higher spin chain Hamiltonians. For this purpose we need to construct a quantization for the model of interacting tops. It is the subject of our next paper. One of the most intriguing questions is to construct relativistic generalization of the models discussed above. While the classical model of relativistic interacting tops is expected to be relatively simple (the block L ij in (2.16) should be replaced by tr 2 (S ij 2 R z 12 (q ij + η)P 12 )) its quantum version and the related long-range spin chain remains to be an interesting open problem.
Let us also mention that there is another class of integrable models with the Hamiltonian of type (1.26). These are the Gaudin type models [25]. The corresponding Lax matrix is of size M × M. It has simple poles at n points on elliptic curve (or its degenerations) with the classical spin variables matrices attached to each point. The number of points is not necessarily equal to M. It is an interesting task to find interrelations between the Gaudin models and interacting tops.

A: Definitions and identities
The following set of functions is used in this paper [31]. The first one is the Kronecker function: Its elliptic version is given in terms of the odd theta-function on elliptic curve with moduli τ (Im(τ ) > 0). Next are the first Eisenstein (odd) function and the Weierstrass (even) ℘-function: We also need the derivatives The one (A.4) is the second Eisenstein function.
The main relation is the Fay trisecant identity: The following degenerations of (A.6) are necessary for the Lax equations and r-matrix structures: .
The local behavior of the Kronecker function and the first Eisenstein function near its simple pole at z = 0 is as follows: provide (after restriction on the constraints (1.17)) equations of motioṅ The classical r-matrix structure is as follows: Here the linear Poisson brackets (B.7) are assumed as well. The Dirac reduction is not yet performed. However, we can see that the restriction on the constraints (1.17) kills the last term in (B.8), and we are left with the standard linear classical r-matrix structure. It is enough for Poisson commutativity {tr(L k (z)), tr(L n (w))} = 0 , ∀ k, n ∈ Z + , z, w ∈ C (B. l.h.s. of (B.8): r.h.s. of (B.8): Expressions (B.11) and (B.12) coincide due to (A.6).