We construct polynomial Poisson algebras of observables for the classical EulerCalogeroMoser (ECM) models. The conserved Hamiltonians and symmetry algebras derived in a previous work are subsets of these algebras. We define their linear, limits, realizing type algebras coupled to current algebras.
PAR LPTHE 9416
1 Introduction
The CalogeroMoser (CM) models [1] and their extensions to internal degrees of freedom known as EulerCalogeroMoser (ECM) models [2] have received a lot of attention in the past year. In particular, remarkable algebraic structures were identified when considering specific sets of invariant functions of the Lax matrices associated with these models. In the case of CM models, polynomial Poissonbracket algebras of observables were constructed [3] for rational and trigonometric potentials; their limit realized respectively the classical linear algebras and [3, 4]. ECM models were shown to have symmetry algebras: exact current algebras [2] — in the rational case — and exact Yangian algebras (quadratic deformations of ) — in the trigonometric case [5]. These algebras were closely connected to the quantum algebra obtained for the Haldanetype spin chain [6].
More recently, it was suggested that a set of observables for the quantum ECM models, realizing a algebra when , could be used to obtain the spectrum of the theory [7]. The finite case is however not so clear. This lead us to consider in detail possible extensions of the previously known classical exact current and Yangian algebras for finite , using the techniques previously developed to build the standard CM observable algebra [3]. We shall here describe general algebras of classical observables, associated both to the rational and trigonometric ECM models; we shall compute the Poissonbracket structures, and describe their limit in connection with the considerations in [7].
At this point, and before delving into the explicit construction of the observable algebras, we need to state a keystone result:
Proposition 1
Given a phase space and a set of matrices on , taking values in a Lia algebra and realizing the Poisson bracket structure:
(1) 
with , the Poisson brackets of invariants
(2) 
vanish exactly.
Proof. The expression (2) takes the form:
by cyclicity of the trace. Eliminating the terms when goes from to leaves us with only the extreme terms:
For fixed values of and , the firstspace terms cancel two by two by obvious cyclicity of their form. Similarly the terms vanish by the exact cyclic structure of the secondspace terms in their contribution.
This result generalizes in a straightforward way the standard theorem for a single Lax matrix with an matrix Poissonstructure [8]. It can be reexpressed in a form which will be more useful for us:
Proposition 2
Given a Lax matrix and a set of dynamical matrices realizing the Poisson algebra:
(3)  
the algebraic structure of the polynomial set is given solely by the extra terms in (3).
2 The rational EulerCalogeroMoser model
The system consists of particles on a line with pairwise interactions depending of their internal degrees of freedom. The phase space is described by conjugate coordinates and internal conjugate coordinates parametrizing a coadjoint orbit in as Integrability of the model requires to restrict oneself to the submanifold The original Hamiltonian is:
(4) 
The equations of motion with spectral parameter take the Lax form with the Lax pair:
(5)  
(6) 
Our algebra of observables will consist of traces of monomials of the set
This choice is the generalization to the ECM model of the observable algebras considered in the CM models [3]. The choice of invariant quantities as observables is natural since the ECM model, reduced to orbits , is a Hamiltonian reduction of a matrixvalued model [2, 9], and the adjointinvariant quantities are precisely those which survive without modification the conjugation which redefines the relevant reduced variables. These matrices realize the following Poisson structure [10]:
(7)  
where
and
is the quadratic Casimir of the algebra.
We are therefore in the situation described by prop.(2), guaranteeing cancellation of the matrix contributions when computing the Poisson structure of the monomial traces of , , . Moreover, as usual, we may also ignore the contribution once we restrict the system to the manifold . Indeed, generates the conjugation of all matrices , and (trivially) by acting on the vector of the basis. Hence the Poisson brackets of with (adjointinvariant) traces always vanish, and it is consistent to compute the algebra of such quantities on the submanifold
Furthermore, the generators and obey the commutation relation:
(8) 
We now describe the algebra of observables, using the following properties.
Proposition 3
Given two matrices , the monomial is rewritten as
It follows that we shall only retain monomials of order or in . The proof of prop.(3) is obvious, relying on the projector structure of as
Proposition 4
All monomials of the form and with can be written as polynomials of normalordered generators and
Proof. As in [3] the commutation relation (8) and the projective property (3) allow a recursive proof of prop.(4). Specifically, denoting by the length of a monomial, i.e., the total number of and generators, one has:

For , normalordering is immediate.

If normalordering is ensured up to the length , consider first of length If it is already normalordered, the procedure stops; if not, normalordering is achieved by commuting ’s through ’s. Each such step eliminates two generators and creates one generator , thereby leaving residual terms of length to which the recursion hypothesis applies. Note that, had we allowed as a generator, the relevant commutation relation prevents the normalordering recursion. This will not occur in the trigonometric case.
Consider now . Each commutation operation eliminates again two generators and creates one generator The projection property (3) allows then factorization of the residual terms into terms with one single generator, of length at most , to which the recursion property then applies.
Note that, as in [3], the normalordering of a given monomial may not be unique; due to the particular form of the matrices and their finite size, degeneracies will occur; in any explicit computation, they will be fixed at every order by the choice of a reordering path for a given monomial.
Proposition 5
The quantities can be rewritten as polynomials of the variables and where .
Proof. Again by recursion on the length .

When the length is 0 or 1,

Assuming that the proposition stands up to the length , we take Then
Reordering terms of contain two ’s and at most terms and . Hence they factorize into terms linear in , following prop.(3), and the factors normalorder following prop.(4). Finally every factor is rewritten as with , to which the recursion hypothesis applies.
It follows that the generators of our observable algebra to be considered are reduced to the set:
(9)  
(10) 
The Poisson algebra (2) is slightly modified to take into account the change of generators:
(12)  
(13) 
the ordering terms being of length
(14)  
The last term in (14) generates a subleading contribution with respect to the third term. The factorization property (3) is now replaced by:
(15)  
which is then converted into a canonical expression in terms of by using prop.(5). From prop.(5) we know that normalorder reexpresses as Hence, keeping only the explicit highest order of each term in (14) we get:
(16)  
and contain the extra terms in (15) due to the redefinition of , plus ordering terms. contains reordering terms, and contains the purely reordering terms arising from the commutator like term in (14), consequence of the Poissonbracket modification (2).
At this point a number of remarks are to be made:
 1. No reordering terms.

A number of Poissonbrackets (12, 13,16) have no extra reordering terms. This is true each time the second term in the Poissonbracket expression contains only one generator at any power and another generator at power 1. One has:
(17) This closed set generates a symmetric Lie algebra. Note that, as a curiosity, one also has:
(18) The current algebra symmetry also belongs to this class [2, 5]:
(19) The Virasoro algebra acts on the current algebra as:
(20) No extra reordering terms also occurs in a different class of Poisson brackets:
(21) This structure is related, as we shall see, to the classical version of the construction of the observables in [7].
 2. Different choices of basis.

Absence of normalordering contributions also occur when computing Poissonbrackets of very specific polynomials of the original generators ; they correspond in fact to relevant nonlinear changes of basis: the first typical example consists of the higher Hamiltonians. Only the first Hamiltonians are simple generators ; the higher ones are polynomials in and , typically Weylordered monomials of and
Since , the conserved higher Hamiltonians are particular polynomials in , necessarily scalar under , and their commutation follows much more easily from direct computation using the spectralparameter Lax operator [10].
Another set of exactly Poissoncommuting Hamiltonians can be obtained by direct construction, following the procedure in [3]. We define:(22) From the Poisson structure (2), and repeating the derivation in [3] one gets:
with
(24) Although not an exact matrix structure, this nevertheless guarantees the commutation of the first Hamiltonians In particular this set includes a natural onebody extension of the rational ECM Hamiltonian:
(25) This extension was indeed considered in [11] and a set of commuting Hamiltonians was constructed. Note that full integrability would require constructing an extra set of commuting Hamiltonians, which we do not see clearly how to get at the moment.
The quantities naturally belong to our algebra of observables. They are however not naturally normalordered in terms of and , hence they are expressed as complicated polynomials of our generators — except Obviously the associated algebra of observables from which to deduce exact eigenfunctions of the Hamiltonian flows should be extracted from the representation. This represents a nonlinear change of variables; but the identical normalordering procedure (in terms of ) and algebra structure will be obtained in a similar way to the case of since the Poisson structure and commutation relations are essentially of the same form. Suitability of the basis depends on which problem — precisely which Hamiltonians— one considers. A similar connection relates the algebras obtained in [3] and [4] for the CM model.  3. The HikamiWadati algebra.

The algebra considered in [7] consisted of generators obtained by successively commuting the quantum Hamiltonians by the operator . These quantum Hamiltonians are in fact obtained from expanding the quantum determinant of the generating transfer matrix [6]. It is to be expected that their classical limit is equivalently described by the Poissoncommuting classical traces . Hence the classical construction of the HikamiWadati algebra is achieved by bracketing by , the first step of which is one of the Poissonbrackets (21). This is easily seen to substitute at each step one term into one term , hence leading to a set of the form:
(26) This follows from the fact that for any matrices ,
Hence the are a set of “Weylordered” generators. However, for finite, they do not close an algebra. Indeed
The last quantity is a typical reordering contribution, not to be obtained from the explicitely symmetric Weylordered generators (26). One can compute its term which is : it is non zero, dynamical and does not contain , hence it lies outside the original algebra .
This third remark deserves elaboration. It is nevertheless possible to define a large limit of the observable algebra in which the two sets and are identified and then close a Poisson algebra. Redefining
(27)  
(28) 
eliminates all reordering terms when , since reordering always decreases the number of and terms —not forgetting that the term , which arises as first step in reordering of and would be of order also, is in fact reexpressed through prop.(5) as polynomials of of length , hence is actually of lower order.
In this limit:
(29)  
(30)  
(31) 
Indeed the first algebra realizes the structure of , classical limit of the quantum algebra defined in [7]; the limit achieves the decoupling of the two algebras and (double loop algebra of