The Calogero-Moser (CM) models are a collection of exactly integrable or solvable one-dimensional dynamical systems characterized by long-range pairwise interaction with the potential either given by rational, trigonometric, hyperbolic or elliptic function.^[1] It turns out that both classical and quantum versions of these systems are connected with various branches of mathematical physics ranging from random matrix theory to two-dimensional gravity and play an important role in several areas of physical applications such as quantum Hall effect, solid state physics, and particle physics. Some new aspects have been demonstrated extensively in Refs.^[2-3] that there exists profound interrelations between CM integrable models and Seiberg-Witten curve and differential in the framework of four-dimensional $\mathcal{N}=2$ supersymmetric Yang-Mills theory. In addition, a more delicate picture of relationships between integrable models and gauge theories in higher dimensions including five and six can be found in Refs.^[4-5].

The big prominent advance was made by Bordner^[6] who proposed a beautiful and universal formulation of Lax representation or generally called the root type Lax pair for different kinds of models of Calogero-Moser systems, which was closely related to the basis of the root systems of semi-simple Lie algebras.^[7] This technical approach for CM integrable systems exhibits a novel feature that it provides us with a simple and effective method of deriving sufficiently many number of conserved quantities for all models in a unified description and allowing to prove in some cases the algebraic integrability of the relevant systems both at classical and quantum level. More importantly, the significance of this construction lies in its applications cover not only the usual Calogero-Moser systems but also the “twisted models" adopted by D'Hoker and Phong and their generalizations, namely “extended twisted models".^[8-9] Furthermore, as pointed out by Takasaki,^[10] a multi-spin generalization of this construction is also possible which is nothing but the genus-one case of Levin and Olshanetsky' framework.

Spin generalization is a direct and natural extension of the Calogero-Moser many-body system coupled to additional internal degrees of freedom with properties of the integrability related to the Dunkl operators and the presence of an algebra of symmetries which gives rise to a representation of the Yangian Y(gls).^[11] Historically, the simplest sl$(n)$ spin generalization was first investigated by Krichever through introducing creation and annihilation analogue operators^[12] and they discovered that the spin generalization just like the spinless case via the pole dynamics of the matrix KP hierarchy.^[13] These models are slightly different from the ordinary one discussed by Bordner due to such spin generalization is determined by the semi-simple Lie algebras themselves rather than their root systems. Applying the abstract language of symplectic geometry, the classical spin variables can be formulated in the dual space of Lie algebra $\mathfrak{g}$ or a coadjoint orbit of the action of Lie group on the dual space of its Lie algebra. Meanwhile the Poisson brackets of spin variables are defined in terms of Kostant-Kirillov Poisson structure on the space of functions on $\mathfrak{g}^{*}$, which take the similar form as the Lie brackets of the Lie algebra basis.^[14] This point of view is mainly inspired by the work of Kirillov in the study of coadjoint representation theory and later it also suggests us an insight and powerful tool for the studies of equivariant cohomology, localization principle,^[15] and geometric quantization^[16] and so on.

Along this way, Feher and Pusztai^[17] introduced a new integrable model with two types of the interacting spin variables from the Hamiltonian reductions of the free geodesic motion on a non-compact simple Lie group $G$, which is analogous of the classical Calogero-Sutherland system. Their ideas are based mainly on the $G_{+}\times G_{+}$ symmetry of the extension of the symplectic form and Hamiltonian of the geodesic system defined on the cotangent bundle $T^{\ast}G$ and if one chooses generic values of the momentum map, we will get the desired new spin model. Recently, an intuitive geometrical picture was provided by Kharche, Levin, Olshanetsky, and Zotov^[18] who gave an elegant Hitchin type description of this new integrable system. Roughly speaking, they constructed the Lax operators as the Higgs fields defined over a singular rational curve as well as the classical $R$-matrix depending on the spectral parameter. By means of the basis of meromorphic differentials on the curve, they obtained hierarchy of independent integrals of motion and proved that the number of these integrals is equal to the half of dimension of the fixed point set or usually called phase space.

In this paper we first review the basic knowledge of semi-simple Lie algebras and the associated root systems,some definitions and useful notations are given. Then the basic ingredients of the CM system with two types interacting spins via a trigonometric potential were introduced such as the Hamiltonian of the system and the Poisson brackets of the dynamic variables. This model is a straightforward generalization of the ordinary spin CM model and we present the Lax pair with spectral parameter expressed in the form of the root type matrix of semi-simple Lie algebra. Accordingly, we give a proof of the consistency between the Lax equation and the canonical equations of motion generated by Hamiltonian in detail. In order to illustrate the integrability of the CM system we construct the $R$-matrix and check that the $R$-matrix is equivalent to the Poisson brackets of Lax matrix. After these analysis, we come to the main discovery of this paper that we discuss the quantized model of the two type spins CM system and show that the $R$-matrix of this situation is the same as in the classical case. The final section of this paper is devoted to summary and comments.

Let us briefly sketch some aspects of the concepts of semi-simple and simply-laced Lie algebra $\mathfrak{g}$ with rank $n$ which are very useful in the description of Calogero-Moser model. In fact we only need the data of its roots. Following the notations in Ref. ^[6], we denote the set of all roots by $\Delta$ and view the elements of $\Delta$ as real vectors in $n$-dimensional space which are normalised, without loss of generality, to 2, namely

the dynamical variables are canonical coordinates $\left\{q_{i}\right\}$ associated with their canonical conjugate momenta $\left\{p_{i}\right\}$ satisfying the standard Poisson brackets:

it is convenient to denote them by $n$-dimensional vectors $q$ and $p$, that is

naturally the scalar products of $q$ and $p$ as well as the roots $\alpha\cdot q$, $p\cdot\beta$, etc. can be defined in the usual way.

In the subsequent part we intend to review the basic knowledge of simple Lie algebra and the associated root system.Let $\mathfrak{g}$ be a simple Lie algebra of rank $n$, $\mathfrak{b}$ a Cartan subalgebra,and $\Delta$ the associated root system, it is well known that the Cartan subalgebra, gives rise to a root space direct decomposition of the Lie algebra $\mathfrak{g}$,or in other words $\mathfrak{g}=\mathfrak{b}\oplus\bigoplus_{\alpha∈\Delta}\mathfrak{g}_{\alpha}$.We can choose a basis $\left\{E_{\alpha},H_{\mu}|\alpha∈\Delta,\mu=1,\ldots,n\right\}$ of $\mathfrak{g}$ with the following properties:

(i) $H_{\mu}~(\mu=1,\ldots,n)$ constitute an orthonormal basis of Cartan subalgebra $\mathfrak{b}$ equipped with the Killing form $B:\mathfrak{b}\times\mathfrak{b}\rightarrow\mathbb{C}$ induced from $\mathfrak{g}$ which satisfy $B(H_{\mu},H_{\nu})=\delta_{\mu\nu}$. Obviously by means of the Killing form there exists an isomorphism between $\mathfrak{b}^{*}=\mathrm{Hom}(\mathfrak{b},\mathbb{C})$ and $\mathfrak{b}$, hence for each $\alpha∈\mathfrak{b}^{*}$ one is able to determine an element $H_{\alpha}$. Now making use of the basis $H_{\mu}$ of $\mathfrak{b}$, this correspondence can be represented more explicitly in the form

(ii) For each $\alpha∈\Delta$, the root subspace $\mathfrak{g}_{\alpha}$ is one-dimensional vector space together with the basis $E_{\alpha}$ such that $[E_{\alpha},E_{-\alpha}]=H_{\alpha}$ and for simplicity, we suppose that they fulfil the normalization condition $B(E_{\alpha},E_{-\alpha})=1$.

(iii) The Lie brackets of the basis elements take the form

here the structure constants $N_{\alpha,\beta}$ are anti-symmetric with respect to the indices and vanish unless $\alpha+\beta$ is also a root included in $\Delta$. More importantly, it is necessary to stress the fact that these structure constants satisfy the general relations given by

Under these preparations, we are able to formulate the classical spin variables in the context of simple Lie algebra and root system, or more precisely, those spin variables are described as coordinates of the dual space $\mathfrak{g}^{*}=\mathrm{Hom}(\mathfrak{g},\mathbb{C})$. Let us introduce two sets of spin variables $F_{\alpha},F_{\mu}$ and $S_{\alpha},S_{\mu}$ respectively dual to the above basis $E_{\alpha}$ and $H_{\mu}$, which can be served as the coefficients of $E_{\alpha}$ and $H_{\mu}$ in the linear combination

furthermore we assume these spin variables satisfy the following generalized Poisson brackets

here $\alpha(\mu)$ is the short for $\alpha(H_{\mu})$. Comparing Eq.(8) with Eq.(5),

these expressions reveal their resemblances to the Lie brackets of the Lie algebra basis and using Eq.(6), it is easy for us to check that the Jacobi identities of the generalized Poisson bracket holds true.

As illustrated in classical and quantum mechanics, since quantization of a dynamic system requires full information about the canonical Poisson bracket relations between its variables, the Hamiltonian structure of nonlinear equations will be of utmost importance in future discussions. Thus let us restrict ourselves to the classical integrable system and our generalization of the Hamiltonian of the Calogero-Moser with two types of interacting spins is as follows^[18-19]

here $p$ and $q$ represent the positions and conjugate momenta of $n$ identical particles while $F_{\alpha},S_{\alpha}$ are the spin variables as explained above. For the purpose of obtaining the explicit equations of motion of this Hamiltonian which are governed by the generalized Poisson brackets Eq.(8), for $\beta\neq\pm\alpha$, a simple and direct calculation shows

along with

for convenience, we will define functions $u(\alpha\cdot q), v(\alpha\cdot q)$ as

moreover, it should be remarked here that they obey the following crucial identities

Let us now return to the original Hamiltonian Eq.(9) and adopt the convention of coordinates $q_{\mu}=q\cdot H_{\mu}$ as well as momenta $p_{\mu}=p\cdot H_{\mu}$, especially notice that $\left\{F_{\alpha}, F_{-\beta}\right\}=-\left\{F_{\alpha},F_{\beta}\right\}$ and $\left\{F_{\alpha}, F_{-\beta}F_{\beta}\right\}=-\left\{F_{\alpha}, F_{\beta}F_{\beta}\right\}$. Based on these preliminaries, we can derive the equations of motion of the generalized dynamic system straightforwardly

and the similar expression for $\dot{S}_{\alpha}$. Also we have $\dot{F}_{\mu}=\dot{G}_{\mu}=0$, which implies the diagonal elements of the spin variables are conserved quantities.

For further analysis, it is useful to search for an appropriate Lax pair $(L(\lambda),M)$ which reflects the classical system is completely integrable in the Liouville sense. Generally speaking, the Lax pair may depend on an auxiliary parameter $\lambda$ usually referred to as the spectral parameter of the Lax equation. Indeed, due to the motion integrations $\dot{F}_{\mu}=\dot{G}_{\mu}=0$, or on the reduced phase space $F_{\mu}=f_{\mu},G_{\mu}=g_{\mu}$, here $f_{\mu},g_{\mu},\mu=1,2,...,n$ are some constants and if these constants satisfy the constraints

the previous Hamiltonian admits the following elegant Lax matrix representation

here $\lambda$ is the spectral parameter and the functions $u(\alpha\cdot q),v(\alpha\cdot q)$ are defined as above.

${{Proof}}$ To easy of exposition, we define $\tilde{v}(\alpha\cdot q)=v(\alpha\cdot q)+v(\lambda)$.Let us take the time derivative of the matrix $L(\lambda)$ and the result can be divided into three parts $\dot{L}=A+B+C+D$, here symbols $A,B,C,D$ correspond to the terms of $t$-derivative of momentum $p_{\mu}$, $t$-derivative of functions $u(\alpha\cdot q),v(\alpha\cdot q)$ and $t$-derivative of spin variables $F_{\alpha},G_{\alpha}$ in $L(\lambda)$ respectively. Specifically, with the help of $\dot{F}_{\alpha}=-\dot{F}_{-\alpha}$ we find that

on the other hand, the commutation relation of the Lax pair $(L(\lambda), M)$ is expressible in the form of $[L(\lambda), M]=R+T+X+Y$, here $R$ is the commutator of $H_{\mu}$ and $M$, that is

while $T=T_{1}+T_{2}+T_{3}+T_{4}$ stands for the terms of Cartan part ($\alpha+\beta=0$) in the commutator $[L(\lambda), M]$. At first, due to the symmetry with respect to $\alpha$ and $-\alpha$ we conclude

here we have already taken the advantage of $H_{\alpha}+H_{-\alpha}=0$. In addition, since $v'(\alpha\cdot q)$ is an even function, applying the symmetry between $\alpha$ and $-\alpha$ in a similar way we assert

which leads to

analogously we have

as well as

putting Eqs.(22), (23) together, one gets

Now let us turn the attention to $X$ as given in the commutator which are the terms of off-Cartan part ($\alpha+\beta\neq 0$) of $[L(\lambda), M]$, firstly for pure spin variables $F_{\alpha}$ we simply have

making use of the symmetry between $\alpha$ and $\beta$ one obtains

a similar deduction for the other terms in Eq.(25) we arrive at

consequently, combing Eqs.(26), (27) with Eq.(28) we can rewrite Eq.(25) as

now the analogous expression for the single spin variable $S_{\alpha}$ turns out to be

besides, the terms of mixed spin variables $F_{\alpha}$ and $G_{\beta}$ are given by

thus we should exploit the functions $u(\alpha\cdot q)v'(\beta\cdot q),\tilde{v}(\alpha\cdot q)u'(\beta\cdot q)$ mentioned above and pay attention that $v(\beta\cdot q)=u(\beta\cdot q)\text{cos}(\beta\cdot q)$ which immediately leads to

therefore from Eq.(32) one is capable of evaluating the addition of $X_{3}, X_{4}$ as

together with

finally upon substitutions of Eqs.(33) and (34) we get

through the above calculations, we verify that the consistency of Lax pair is essentially the same as for the canonical equations of the Hamiltonian system. That is complete the proof.

We have seen that the Lax pair method is an excellent and conceptual tool for the investigation of soliton theory, or more generally the issue of integrability in the classical mechanic system. Another essential and more particularly heuristic ingredient of the integrable system is inspired from the point of view of the $R$-matrix^[20] from which, like the previously known Lax pair, it means that the system possesses sufficiently many number of conserved quantities in soliton theory. The significance of this method lies in it works not only for the classical case but also is a central object in the modern approach to the studies of quantum integrable systems deriving in particular a basic equation satisfied by it, namely the Yang-Baxter equation and perhaps even their supersymmetric generalizations of the associated system. Now considering the Calogero-Moser with two type spins model discussed above, we should emphasize that it is only the reduced system or in the equivalent form of $\dot{F}_{\mu}=\dot{G}_{\mu}=0$ that the original system is integrable. Actually, according to the condition Eq. (15), we establish the following $R$-matrix, which can be formulated naturally via the linear combinations of tensor product of root matrix

here

obviously we have

With the help of this $R$-matrix, it is possible to generate a hierarchy of integrable equations which clearly indicates a prominent role played in the case of classical integrable systems. For the sake of simplicity, we reformulate the matrix $L(\lambda)=\sum_{\mu}L_{\mu}H_{\mu}+\sum_{\alpha∈\Delta}L_{\alpha}(\lambda)E_{\alpha}$ as obviously we have

and it is easily to deduce the commutation relations ($\alpha\neq\pm\beta$) between dynamic functions $L_{\mu},L_{\alpha}$ from the Poisson bracket (8)

obviously we have

noting $L_{1}=L\otimes1$, $L_{2}=1\otimes L$ together with the antisymmetric property of $N_{\alpha,\beta}$ we have obviously we have

noting that on the reduced phase space (15), the last term in Eq.(41) vanishes.As an aside, we find obviously we have

one can then write obviously we have

now in terms of $b(\alpha\cdot q,\lambda,\eta)$ the commutator assumes the form

and for $c(\alpha\cdot q,\lambda,\eta)$ term the commutator obeys

analogously for $b(-\alpha\cdot q,\eta,\lambda)$ term we get

and for $c(\alpha\cdot q,\lambda,\eta)$ term we also have

now in order to prove $\{L_{1}(\lambda),L_{2}(\eta)\}=[R_{12}(\lambda,\eta),L_{1}(\lambda)]$ $-\left[R_{21}(\eta,\lambda),L_{2}(\eta)\right]$, a comparison of the previous equations Eqs. (41)-(47), it remains to guarantee the identities below

the proofs of these equations are elementary and direct, so we do not want to explain them here any more.

In this section we mainly illustrate the new result of this paper that although the classical and quantum treatments differ in many aspects, the procedure above indeed is applicable to quantized counterpart of Eq.(9), which plays a vital role in the formulation of the quantum inverse scattering problem and to explore the quantum integrability of the corresponding dynamic system, we give the following more detailed description. It is well known that in order to quantize the classical system one should replace the Poisson bracket $\{ , \}$ with the Dirac commutator bracket $-i[\quad]$. In this situation the dynamic functions $L_{\mu},L_{\alpha}$ all being operators and because of the non-commutativity of the operators, the commutation relation between $R$-matrix and Lax matrix $L_{1,2}$ is a slightly different from classical case. To be specific, the commutators in quantized case are expressible in the form of

To proceed we firstly recall that $H_{\mu},E_{\alpha}$ constitute a complete basis of the simple Lie algebra $\mathfrak{g}$ and from $\left[H_{\mu},E_{\alpha}\right]=\alpha(\mu)E_{\alpha}$ we have

which means $H_{\alpha}E_{\alpha}$ belonging to $\mathfrak{g}$, hence for every $\alpha$ one can rewrite $H_{\alpha}E_{\alpha}$ explicitly as

here $C_{\mu},C_{\beta}$ are some coefficients and there must $\exists \nu$ satisfies $\alpha(\nu)\neq 0$ which immediately leads to

combining Eqs. (52) with (53) we find $C_{\mu}=0$ for all $\mu$.

In a similar spirit, for $\forall β∈\Delta,\beta\neq \alpha$, there exist $\nu$ satisfies $(\alpha-\beta)(\nu)\neq 0$, once again applying $[H_{\nu},H_{\alpha}E_{\alpha}]=\sum_{β∈\Delta}C_{\beta}[H_{\nu},E_{\beta}]$ we infer $(\alpha-\beta)(\nu)C_{\beta}=0$, which implies $C_{\beta}=0$. It is then seen that for every $\alpha$ one obtains $H_{\alpha}E_{\alpha}=C_{\alpha}E_{\alpha}$ for some constant $C_{\alpha}$ and imposition of this idea on other cases we confirm

here $\tilde{C}_{\alpha}$ are the corresponding coefficients. Thus if expanding the left hand side of commutation relation $[H_{\alpha}E_{\alpha},E_{-\alpha}]=C_{\alpha}H_{\alpha}$ we have

as well as

putting all these results together one deduces that

with the solution of the constants

now from the previous detailed analysis one claims that

and assemble all these relations together with Eqs. (49), (50) we have

also we get

now by using of these identities, the other terms in Dirac commutator $[L_{1}(\lambda),L_{2}(\eta)]$ and $[R_{12}(\lambda,\eta),L_{1}(\lambda)],$ $[R_{21}(\eta,\lambda),L_{2}(\eta)]$ are the same as in classical Poisson bracket $\{L_{1}(\lambda),L_{2}(\eta)\}$ and $[R_{12}(\lambda,\eta),L_{1}(\lambda)],[R_{21} (\eta,\lambda),$ $L_{2}(\eta)]$, therefore it is necessary to comment on that in the quantized version of the two types spin Calogero-Moser Hamiltonian system, the $R$-matrix coincides with the original form as in classical case.

The constructions of universal Lax pair and $R$-matrix for the Calogero-Moser system with two type interacting spins formulated in terms of root system of any semi-simple Lie algebras are illustrated in this paper. The primary difference compared to the ordinary spin Calogero-Moser model is that we generalize the Poisson brackets of spin dynamic variables, which are deeply related to the structure constants of root system of the semi-simple Lie algebra. Under this postulation, a detailed exposition of consistency of the Lax pair with spectral parameter is given in the case of the trigonometric type of interaction potential in a direct way, of course, one can apply the similar calculations to the hyperbolic potential straightforwardly. Thus based on the notion of Lax pair, we explore the universal R-matrix in classical case and argue that it is equivalent to the Poisson brackets between original Lax matrix with different spectral parameters. For the quantized model as opposed to the classical case, taking into account of the completion of the root system basis, we demonstrate that the $R$-matrix behaves the same manner as in the classical situation, which is the major conclusion of this paper. We should mention here that the essential ingredient of the construction of Lax pair for the two type interacting spins heavily relying on the elegant property of trigonometric function, nevertheless, this idea is not available to the rational and elliptic interactions. Thus it is plausible to pose new expectations of the further work that how to give an analogous and precise description of two type spins Calogero-Moser system with rational or elliptic potential. Along this line, it appears that this viewpoint of universal construction may be employed to the other CM model such as the generalization of supersymmetric spin Calogero-Moser systems with two type interacting spins or deformed quantum Calogero-Moser systems. Further investigation in this direction is a challenging problem.