Dynamical R-matrices for Calogero models - IME-USPforger/pdffiles/NPB2002.pdf · Dynamical...

Nuclear Physics B 621 [PM] (2002) 523–570www.elsevier.com/locate/npe

DynamicalR-matrices for Calogero models

Michael Forger1, Axel Winterhalder2

Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo,Caixa Postal 66281, BR-05315-970 São Paulo, SP, Brazil

Received 17 July 2001; accepted 18 September 2001

Abstract

We present a systematic study of the integrability of the Calogero models, degenerate aswell as elliptic, associated with arbitrary (semi-)simple Lie algebras and with symmetric pairsof Lie algebras, where “integrability” is understood to encompass not only the existence of aLax representation for the equations of motion but also the—more far-reaching—existence of a(dynamical)R-matrix. Using the standard group-theoretical machinery available in this context, weshow that integrability of these models, in this sense, can be reduced to the existence of a certainfunction, denoted here byF , defined on the relevant root system and taking values in the respectiveCartan subalgebra, subject to a rather simple set of algebraic constraints: these ensure, in one stroke,the existence of a Lax representation and of a dynamicalR-matrix, all given by explicit formulas. Wealso show that among the simple Lie algebras, only those belonging to theA-series admit a solutionof these constraints, whereas theAIII-series of symmetric pairs of Lie algebras, corresponding to thecomplex Grassmannians SU(p, q)/S(U(p)×U(q)), allows non-trivial solutions when|p − q|� 1.Apart from reproducing all presently known dynamicalR-matrices for Calogero models, our methodprovides new ones, namely for the degenerate models when|p − q| = 1 and for the elliptic modelswhen|p − q| = 1 orp = q. 2002 Elsevier Science B.V. All rights reserved.

PACS:02.30.IkKeywords:Integrable systems; DynamicalR-matrices; Calogero models

1. Introduction

The Calogero–Moser–Sutherland models, or Calogero models, for short, constitutean important class of completely integrable Hamiltonian systems, intimately relatedto the theory of (semi-)simple Lie algebras [1–6]. Unfortunately, the group-theoreticalunderpinnings of their integrability are far less understood than in the case of the Toda

E-mail addresses:[email protected] (M. Forger), [email protected] (A. Winterhalder).1 Partially supported by CNPq, Brazil.2 Supported by DFG, Germany and by FAPESP, Brazil.

0550-3213/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0550-3213(01)00506-5

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524 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

models, which allow for a completely general and uniform treatment in terms of rootsystems of ordinary (semi-)simple Lie algebras and/or affine Lie algebras. Indeed, it hasproven surprisingly difficult to extend the results obtained for the Calogero model basedonsl(n,C), such as the Lax pair found in Ref. [2] for the degenerate model and in Ref. [7]for the elliptic model or theR-matrix given in Ref. [8] for the degenerate model and inRef. [9] for the elliptic model (see also Ref. [10]), to other simple Lie algebras.3

In recent years, a number of attempts has been made to improve this situation. Oneof these is based on Hamiltonian reduction, starting out from the geodesic flow on thecorresponding group and leading to the construction of Lax pairs and dynamicalR-matrices [11]. However, the approach is rather indirect and the results do not reallycrystallize into concrete formulas, except for a few examples that are worked out explicitly;moreover, it remains unclear why the traditional approach [4–6] works for certain simpleLie algebras such assl(n,C) but fails for others. Other authors have addressed the entireissue from a different point of view, namely by searching for new Lax pairs by takingvalues not in the (semi-)simple Lie algebra or symmetric pair used in the traditionalapproach [4–6] but rather in the algebra of matrices over a representation space for someother algebraic object associated with the root system that appears in the Hamiltonian.For example, the authors of Ref. [12] continue to use the same (semi-)simple Lie algebrabut in an arbitrary representation (not just the adjoint), while others [13–19] propose tomodify not only the representation but also the structure of the algebraic object that isbeing represented: the only obvious common point is that it must contain the Weyl group(or Coxeter group) of the root system, which is an invariance group of the Hamiltonian.While it is true that these new methods provide Lax representations in a variety of situationswhere the traditional approach [4–6] fails, they do not fit into the standard Lie algebraicframework, so the problem of how to constructR-matrices in this more general contextremains open. A priori, it is not even clear what would be the meaning of the concept ofanR-matrix in a theory of integrable systems based on more general Lax representationssuch as these; we shall further comment on this question in our conclusions.

In contrast to most of the recent literature on integrability of the Calogero models, ourapproach in the present paper has originally been motivated by a different question, namelythe quest for a better understanding of the concept of a dynamicalR-matrix.

One of the central features that distinguishes the Calogero models from the (muchsimpler) Toda models is that theirR-matrix is not a numerical object but has a dynamicalcharacter. Generally speaking, the main role of theR-matrix associated with a given Laxequation

(1)L= [L,M],is to control the Poisson brackets between the components of the Lax matrixL [20],according to the following well-known formula:

(2){L1,L2} = [R12,L1] − [R21,L2].3 We shall collectively refer to the rational, trigonometric and hyperbolic Calogero models as the “degenerate”

ones—as opposed to the elliptic ones.

M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 525

NumericalR-matrices are well understood mathematical objects that play an importantrole in the theory of quantum groups, or rather its classical counterpart, the theory of Liebialgebras [21,22]. They can be viewed as providing the mathematical expression for theconcept of algebraic integrability, which is considerably stronger than that of Liouvilleintegrability. Indeed, the latter is really just a statement about the possibility to separate thedynamical variables of a given model into mutually independent conjugate pairs, whereasthe former actually allows to integrate the equations of motion in terms of explicitlyknown functions. (For example, any system with only one degree of freedom is Liouvilleintegrable, and so is the motion of a particle in ordinary three-dimensional space under theinfluence of an arbitrary central potential, but only the Kepler problem is also algebraicallyintegrable, due to the presence of an additional symmetry generated by the conservedRunge–Lenz vector.) On the other hand, the mathematical status of dynamicalR-matricesis not nearly as clear: there does not even seem to exist any generally accepted definition.The first examples of dynamicalR-matrices appeared in the study of the nonlinear sigmamodels on spheres [23–26] and, more generally, on Riemannian symmetric spaces [27].Unfortunately, the analysis of the underlying algebraic structure is hampered by technicalproblems due to the fact that these models are not ultralocal [28] and hence Poissonbrackets between components of the transition matrixT constructed from the Lax matrixL show discontinuities that must be removed by regularization. Therefore, it seems muchmore promising to undertake such an analysis not in the context of two-dimensional fieldtheory but in the context of mechanics, where such technical problems are avoided due tofinite-dimensionality of the phase space. This leads us naturally to the Calogero models—the first known class of models with a finite number of degrees of freedom where dynamicalR-matrices make their appearance.

In view of this situation, we have performed a systematic study of the integrability of theCalogero models, within the traditional formulation outlined long ago by Olshanetsky andPerelomov [4–6], based on the use of (semi-)simple Lie algebras and, more generally,of symmetric pairs. Our main goal is to extend their formulation so as to encompassthe construction not only of a Lax representation but also of a dynamicalR-matrix, thuscharacterizing these models as examples of Hamiltonian systems that apparently occupyan intermediate status between Liouville integrable and algebraically integrable. As usual,we shall require theR-matrix to depend only on the coordinates but not on the momenta:this can be motivated (a) by realizing that the standardL-matrix [2,4–7] is the sum of aterm linear in the momenta with constant coefficients and a term depending only on thecoordinates, forcing the l.h.s. of Eq. (2) to be momentum independent, and (b) by notingthat the simplest way to guarantee momentum independence for the r.h.s. of Eq. (2) is toassumeR itself to be momentum independent and chosen such that the two contributionsto the r.h.s. of Eq. (2) which are linear in the momenta (one from each commutator)cancel. As shown in Ref. [10] for the Lie algebra case, these requirements impose furtherrestrictions on theR-matrix; in particular, they force its root–root part to have the standardform given by the last term in Eq. (35) or Eq. (38) below and to be accompanied by anappropriate Cartan–root correction term plus, in the elliptic case, a Casimir type Cartan–Cartan correction term. For symmetric pairs, the idea is that all terms in Eq. (2) should


have a definite transformation law under the pertinent involutive automorphismθ , so giventhe fact [4–6] that the standardL-matrix is odd underθ while the standardM-matrixis even underθ , we requireR to be even underθ ⊗ 1 and odd under 1⊗ θ . (In theelliptic case, each of these automorphisms must be combined with a change of sign in thecorresponding spectral parameter(s), as suggested by constructions for twisted affine Kac–Moody algebras.) Again, this strongly restricts the form ofR, forcing its root–root part tohave the form given by the last term in Eq. (93) or Eq. (96) below and to be accompaniedby an appropriate Cartan–root correction term plus, in the elliptic case, a Casimir typeCartan–Cartan correction term. In both cases, we write down a general Ansatz for theCartan–root correction term in terms of an appropriate vector-valued functionF definedon the pertinent root system. In this way, we arrive at a general Ansatz for the Lax pairand for theR-matrix through which (a) the proof of equivalence between the equation ofmotion and the Lax equation (1) and (b) the proof of the Poisson bracket relation (2) canboth be reduced to one and the same set of algebraic constraints on the functionF . Thesestate that for any two rootsα andβ

(3)gαα(Fβ)− gββ(Fα)= Γα,β,

in the case of (semi-)simple Lie algebrasg and

(4)gαα(Fβ)− gββ(Fα)= Γ θα,β,

in the case of symmetric pairs(g, θ), where the coefficientsΓα,β andΓ θα,β are defined

in terms of the structure constants ofg and the coupling constants of the model; seeEq. (9) and Eq. (88) below. It is worth noting that this is a highly overdetermined system ofequations which will admit non-trivial solutions only in special circumstances; its deeperalgebraic meaning is yet to be discovered.

The paper is divided into two main sections, dealing with the Calogero models defined interms of root systems for (semi-)simple Lie algebras and for symmetric pairs, respectively.In both cases, we begin by briefly summarizing the definition of the model and then pass tospecifying the Lax pair and the dynamicalR-matrix, in terms of the functionF . The maingoal is to show how (a) the proof of equivalence between the equation of motion and theLax equation (1) and (b) the proof of the Poisson bracket relation (2) can both be reduced tothe combinatoric identity given above. Moreover, we show in Section 2 that the only simpleLie algebras for which a functionF with the desired property exists are those belonging totheA-series, i.e., the Lie algebrassl(n,C); the resultingR-matrix coincides with that givenin Ref. [8] for the degenerate Calogero model and with that given in Ref. [9] for the ellipticCalogero model. In Section 3, we study as an example theAIII-series of symmetric pairs,corresponding to the complex Grassmannians SU(p, q)/S(U(p) × U(q)), and we proveby explicit calculations that there are non-trivial solutions, depending on free parameters,when|p−q|� 1. Forp = q , theR-matrix thus found coincides with that given in Ref. [11]for the degenerate Calogero model, while the one derived for the elliptic Calogero modelis new. For|p− q| = 1, all R-matrices found seem to be new. Together, these also provideR-matrices for all Calogero models, degenerate and elliptic, associated with any one ofthe four classical root systemsAn, Bn, Cn andDn. For the convenience of the reader,


the main conventions from the theory of semisimple Lie algebras and symmetric spacesthat are needed in our calculations are summarized in Appendix A. Finally, in Section 4,we comment on the relation between our work and other approaches to understandingintegrability of the Calogero models that have recently been proposed [11–19], as well ason perspectives for future work.

2. Calogero models for semisimple Lie algebras

2.1. Definition of the models, Lax pairs and R-matrices

We begin by recalling the definition of the Calogero model associated with the rootsystem∆ of a (semi-)simple Lie algebrag, referring to Appendix A for a summary of theconventions and notation used. The configuration spaceQ of the model is a Weyl chamberor a Weyl alcove within the real formhR of the Cartan subalgebrah, depending on whetherwe are dealing with the degenerate or elliptic model, respectively. In any case, it is an opensubset ofhR, so the phase space of the model is given by the associated cotangent bundleT ∗Q=Q× h∗

R, which for the sake of simplicity will often be identified with the tangent

bundleTQ=Q× hR. The Hamiltonian for the degenerate Calogero model reads

(5)H(q,p)= 1

2

(r∑

j=1

p2j +

∑α∈∆

g2αw(α(q)

)2),

while that for the elliptic Calogero model reads

(6)H(q,p)= 1

2

(r∑

j=1

p2j +

∑α∈∆

g2α℘(α(q)

)),

wherew is a smooth, real-valued function onR\{0} to be specified soon, while℘ denotesthe doubly periodic Weierstraß function (see below). The coefficientsgα (α ∈ ∆) arepositive real coupling constants, supposed to be symmetric:

(7)g−α = gα.

A stronger assumption that we shall always make is that they are invariant under the actionof the Weyl group W(g) of g:

(8)gwα = gα, for all w ∈W(g).

For later use we introduce the following combination of coupling constants and structureconstants:

(9)Γα,β = gα+βNα,β .

Using the abbreviations

(10)p =r∑

j=1

pjHj, q =r∑

j=1

qjHj ,


we can write the Hamiltonian equations of motion as vector equations inhR: they read

(11)q = p, p=−∑α∈∆

g2αw(α(q)

)w′(α(q)

)Hα,

for the degenerate Calogero model and

(12)q = p, p=−1

2

∑α∈∆

g2α℘′(α(q)

)Hα,

for the elliptic Calogero model.Concerning the choice of the potential functionsw in Eq. (5) and℘ in Eq. (6), it is well-

known that these functions must satisfy a certain set of functional equations that we shalldiscuss briefly in order to make our presentation self-contained. An elementary property isthatw is supposed to be odd while℘ is even:

(13)w(−t)=−w(t), ℘ (−t)= ℘(t).

The basic functional equation imposed onw is the following:

(14)

(w′(s)w(s)

+ w′(t)w(t)

)w(s + t)+w(s)w(t)= 0.

Differentiating with respect tos and to t and subtracting the results gives a secondfunctional equation:

(15)

(w′′(s)2w(s)

− w′′(t)2w(t)

)w(s + t)=w(s)w′(t)−w′(s)w(t),

The solutions are derived in Refs. [4–6]: there are essentially three different ones, satisfyingthe relation

(16)w′′(t)2w(t)

−w(t)2=−k

2,

wherek is a numerical constant that allows to distinguish between them, namely

w(t)= 1

t, with k = 0,

w(t)= 1

sint, with k =+1,

(17)w(t)= 1

sinht, with k =−1,

which explains the terminology “rational” (for the first case), “trigonometric” (for thesecond case) and “hyperbolic” (for the third case). On the other hand, as explained inRef. [31], ℘ is a meromorphic function on the complex plane with second order polesat the points of its lattice of periodicityΛ, consisting of the integer linear combinationsn1ω1+n2ω2 of its two basic periodsω1 andω2, which are two arbitrarily chosen but fixedcomplex numbers; explicitly, it can be defined by the series expansion

℘(z)= 1

z2

∑n1,n2∈Z

(n1,n2) �=(0,0)

(1

(z− n1ω1− n2ω2)2− 1

(n1ω1+ n2ω2)2

).


For later use, it will be convenient to also introduce the Weierstraß sigma function, anentire function on the complex plane that can be defined by the infinite product expansion

σ(z)= z∏

n1,n2∈Z

(n1,n2) �=(0,0)

(1− z

n1ω1+ n2ω2

)

× exp

{z

n1ω1+ n2ω2+ 1

2

z2

(n1ω1+ n2ω2)2

},

and the Weierstraß zeta function, another meromorphic function on the complex plane withfirst order poles at the points of the same lattice of periodicity, which can be defined by theseries expansion

ζ(z)= 1

z

∑n1,n2∈Z

(n1,n2) �=(0,0)

(1

z− n1ω1− n2ω2+ z+ n1ω1+ n2ω2

n1ω1+ n2ω2

).

Obviously,

(18)ζ(z)= σ ′(z)σ (z)

, ℘ (z)=−ζ ′(z).

Finally, we set

(19)Φ(z1, z2)= σ(z1+ z2)

σ (z1)σ (z2).

Clearly, these functions have the following symmetry properties under reflection at theorigin:

(20)σ(−z)=−σ(z), ζ(−z)=−ζ(z), ℘ (−z)= ℘(z),

(21)Φ(−z1,−z2)=−Φ(z1, z2).

Moreover, they satisfy a set of functional equations that can all be derived from a singlequartic identity for the Weierstraß sigma function, namely

σ(z1+ z2)σ (z1− z2)σ (z3+ z4)σ (z3− z4)

+ σ(z2+ z3)σ (z2− z3)σ (z1+ z4)σ (z1− z4)

+ σ(z3+ z1)σ (z3− z1)σ (z2+ z4)σ (z2− z4)= 0.

We shall restrict ourselves to listing the identities that will actually be needed in thefollowing calculations:

(22)Φ(s,u)Φ(−s, u)=−(℘(s)−℘(u)),

(23)Φ(s,u)Φ ′(−s, u)−Φ ′(s, u)Φ(−s, u)= ℘ ′(s),(24)Φ(s,u)Φ ′(t, u)−Φ ′(s, u)Φ(t, u)= (℘(s)−℘(t)

)Φ(s + t, u),

(25)Φ(−s, v − u)Φ(s + t, v)+Φ(−t, u− v)Φ(s + t, u)=−Φ(s,u)Φ(t, v),

(26)Φ(−s, u− v)Φ(s,u)+ (ζ(v − u)+ ζ(u))Φ(s, v)=Φ ′(s, v),

whereΦ ′ denotes the derivative ofΦ with respect to the first argument and all argumentsmust be different from zero modΛ. Note that these equations are not independent: for


example, Eq. (23) can be immediately derived from Eq. (22) by differentiation with respectto the first argument and from Eq. (24) by passing to the limits + t → 0. Note also thatthese properties do not depend on the choice of the periodsω1 andω2, but of course thelatter should be taken so that℘ assumes positive real values on the real axis, in order forthe Hamiltonian (6) to be real and positive: this is the case, for example, whenω1 is realandτ = ω2/ω1 is purely imaginary [29].

With these preliminaries out of the way, we can pass to discuss the Lax pair and theR-matrix of the Calogero model. It is at this point that an important difference betweenthe degenerate and the elliptic Calogero models appears, because the Lax pair and theR-matrix of the latter depend on spectral parameters that are absent in the former. However,these do not enter through the root system, as happens in the Toda models, but rather showup explicitly in the coefficient functions.

Within the present framework, the Lax pair consists of two mappings4

(27)L :T ∗Q→ g and M :Q→ g,

each of which will in the elliptic case depend on an additional spectral parameteru, suchthat the Hamiltonian equations of motion can be rewritten in the Lax form, namely

(28)L= [L,M],for the degenerate Calogero model and

(29)L(u)= [L(u),M(u)],

for the elliptic Calogero model. Similarly, the dynamicalR-matrix is a mapping4

(30)R :Q→ g⊗ g,

which will in the elliptic case depend on two additional spectral parametersu andv, suchthat the Poisson brackets ofL can be written in the form

(31){L1,L2} = [R12,L1] − [R21,L2],for the degenerate Calogero model and

(32){L1(u),L2(v)

}= [R12(u, v),L1(u)]− [R21(v,u),L2(v)

],

for the elliptic Calogero model, where as usualL1 = L ⊗ 1,L2 = 1⊗ L andR12= R,R21= RT , with “T ” denoting transposition of the two factors in the tensor product. Asis well-known, the Lax equation (28) or (29) implies that the ad(g)-invariant polynomialson g are conserved under the Hamiltonian flow, whereas the Poisson bracket relation (31)or (32) implies that they are pairwise in involution.

In accordance with the discussion in the introduction and generalizing some of theformulas found in the literature for special cases, we postulate a Lax pair and anR-matrix

4 As we shall see,M andR depend only on the position variables and not on the momentum variables, i.e.,they are functions onT ∗Q that descend to functions onQ.


given by

(33)L=r∑

j=1

pjHj +∑α∈∆

igαw(α(q)

)Eα,

(34)M =−∑α∈∆

igαw′′(α(q))

2w(α(q))Fα +

∑α∈∆

igαw′(α(q)

)Eα,

(35)R =∑α∈∆

w(α(q)

)Fα ⊗Eα +

∑α∈∆

w′(α(q))

w(α(q))Eα ⊗E−α,

for the degenerate Calogero model (see [2] for the Lax pair and [8,10] for theR-matrixwheng= sl(r + 1,C) and [4–6] for the Lax pair in the general case) and by

(36)L(u)=r∑

j=1

pjHj +∑α∈∆

igαΦ(α(q),u

)Eα,

(37)M(u)=−∑α∈∆

igα℘(α(q)

)Fα +

∑α∈∆

igαΦ′(α(q),u

)Eα,

(38)

R(u, v)=−r∑

j=1

(ζ(u− v)+ ζ(v)

)Hj ⊗Hj +

∑α∈∆

Φ(α(q), v

)Fα ⊗Eα

−∑α∈∆

Φ(α(q),u− v

)Eα ⊗E−α,

for the elliptic Calogero model (see [7] for the Lax pair and [9,10] for theR-matrix wheng= sl(r + 1,C)). Here, theFα form a collection of generators which aresupposed to belong tohR and can be viewed as a (vector valued) function

(39)F :∆→ hR,

which we shall decompose into its even partF+ and its odd partF−:

(40)F+α =1

2(Fα + F−α), F−α =

1

2(Fα −F−α).

Note that only the even partF+ of F is used in the definition ofM. In the sequel weshall derive a combinatoric equation which determinesF and, together with the functionalequations discussed above, turns out to be the basic ingredient for proving the equivalencebetween the Hamiltonian equations of motion and the Lax equation, as well as the validityof the Poisson bracket relation.

For the elliptic Calogero model, the calculation goes as follows. First, we use Eq. (36)and Eq. (37) to compute the following expression for the difference between the two sidesof the Lax equation (29):

L(u)− [L(u),M(u)]

= p+∑α∈∆

igαΦ′(α(q),u

)α(q)Eα −

∑α∈∆

igαΦ′(α(q),u

)α(p)Eα

+∑

α,γ∈∆gαgγ ℘

(α(q)

)Φ(γ (q),u

)γ (Fα)Eγ


+ 1

2

∑α∈∆

g2α

(Φ(α(q),u

)Φ ′(−α(q),u

)−Φ(−α(q),u

)Φ ′(α(q),u

))Hα

+ 1

2

∑α,β,γ∈∆α+β=γ

gαgβ

(Φ(α(q),u

)Φ ′(β(q),u

)−Φ ′(α(q),u

)Φ(β(q),u

))Nα,βEγ .

Here, the double sum over commutators[Eα,Eβ ] appearing in the commutator betweenL(u) andM(u) has, after antisymmetrization of the coefficients inα andβ , been split intotwo contributions, one corresponding to terms whereα + β = 0 and the other to termswhereα+β �= 0, which are non-zero only ifα+β ∈∆. Inserting the functional equations(23), (24) and rearranging terms, we get

L(u)− [L(u),M(u)]

= p+ 1

2

∑α∈∆

g2α℘′(α(q)

)Hα +

∑α∈∆

igαΦ′(α(q),u

)α(q − p)Eα

+∑


(α(q)

)Φ(γ (q),u

)γ (Fα)Eγ

+ 1

2

∑α,γ∈∆γ−α∈∆

gαgγ−α℘(α(q)

)Φ(γ (q),u

)Nα,γ−αEγ

− 1

2

∑β,γ∈∆γ−β∈∆

gγ−βgβ℘(β(q)

)Φ(γ (q),u

)Nγ−β,βEγ .

Renaming summation indices in the last two sums (α→−α in the first,β → α in thesecond) and using Eqs. (A.9), (A.10), (7) and (13), we finally obtain

L(u)− [L(u),M(u)]

= p+ 1

2

∑α∈∆

g2α℘′(α(q)

)Hα +

∑α∈∆

igαΦ′(α(q),u

)α(q − p)Eα

+ 1

2

∑α,γ∈∆

gα

(2gγ γ (Fα)− gγ+αNγ,α − gγ−αNγ,−α

)℘(α(q)

)Φ(γ (q),u

)Eγ .

The vanishing of the sum of the first two terms and of the third term are precisely theequations of motion, so the last double sum must vanish, which requires that for all rootsγ ∈∆,

(41)∑α∈∆

gα

(2gγ γ (Fα)− Γγ,α − Γγ,−α

)℘(α(q)

)= 0.

Since these relations must hold identically inq but the coefficient functions are invariantunder the substitutionα→−α, we are led to conclude that, for the elliptic Calogero model,the Lax equation (29) will be equivalent to the Hamiltonian equations of motion (12) if andonly if the combinatoric identity

(42)2gββ(F+α

)= Γβ,α + Γβ,−α,


holds for any two rootsα,β ∈∆. For the degenerate Calogero model, the same calculationwith Φ(s,u) replaced byw(s) and ℘(s) replaced byw′′(s)/2w(s) leads to the sameconclusion (with Eq. (29) replaced by Eq. (28)).

Passing to the Poisson bracket relation (32) in the elliptic case, we use Eq. (36) tocompute the l.h.s., starting out from the canonical Poisson brackets

{pj , qk} = δjk,

which give{pj ,Φ

(α(q),u

)}=Φ ′(α(q),u

)α(Hj ),

and hence, due to the identities (A.11) and (10),{p,Φ

(α(q),u

)}=Φ ′(α(q),u

)Hα.

This leads to the following momentum independent expression for the l.h.s. of Eq. (32):

(43){L1(u),L2(v)

}=∑α∈∆

igα

(Φ ′(α(q), v

)Hα ⊗Eα −Φ ′

(α(q),u

)Eα ⊗Hα

).

To compute the r.h.s. of Eq. (32) we observe first that it is also momentum independent,since the only possibly momentum dependent terms cancel:[

R12(u, v),p1]− [R21(v,u),p2

]=∑α∈∆

α(p)(Φ(α(q),u− v

)Eα ⊗E−α −Φ

(α(q), v − u

)E−α ⊗Eα

)= 0.

The remaining terms are, according to Eq. (A.11),[R12(u, v),L1(u)− p1

]− [R21(v,u),L2(v)− p2]

=−∑α∈∆

igα

(ζ(u− v)+ ζ(v)

)Φ(α(q),u

)Eα ⊗Hα

+∑α∈∆

igα

(ζ(v − u)+ ζ(u)

)Φ(α(q), v

)Hα ⊗Eα

+∑

α,β∈∆

(igαα(Fβ)− igββ(Fα)

)Φ(α(q),u

)Φ(β(q), v

)Eα ⊗Eβ

+∑α∈∆

igαΦ(−α(q),u− v

)Φ(α(q),u

)Hα ⊗Eα

−∑α∈∆

igαΦ(−α(q), v− u

)Φ(α(q), v

)Eα ⊗Hα

−∑

α,β,γ∈∆α+β=γ

igγ Φ(−β(q),u− v

)Φ(γ (q),u

)N−β,γ Eα ⊗Eβ

+∑


igγ Φ(−α(q), v − u

)Φ(γ (q), v

)N−α,γ Eα ⊗Eβ,

where the last four terms have been obtained by splitting each of the two terms containingtensor products of a root generator with the commutator of two other root generators, say


[Eα,Eβ ]⊗Eγ orEγ ⊗[Eα,Eβ ], into two contributions: one corresponding to terms whereα+ β = 0 and one corresponding to terms whereα+ β ∈∆; moreover, some renaming ofsummation indices has been performed. Using Eq. (A.9) and Eq. (A.10) and rearrangingterms, we get[

R12(u, v),L1(u)− p1]− [R21(v,u),L2(v)− p2

]=+

∑α∈∆

igα

((ζ(v − u)+ ζ(u)

)Φ(α(q), v

)+Φ

(−α(q),u− v)Φ(α(q),u

))Hα ⊗Eα

−∑α∈∆

igα

((ζ(u− v)+ ζ(v)

)Φ(α(q),u

)+Φ

(−α(q), v − u)Φ(α(q), v

))Eα ⊗Hα

+∑

α,β∈∆


)Φ(α(q),u

)Φ(β(q), v

)Eα ⊗Eβ

+∑


igγ Nα,β

(Φ(−β(q),u− v

)Φ(γ (q),u

)+Φ

(−α(q), v− u)Φ(γ (q), v

))Eα ⊗Eβ.

Inserting the functional equations (25) and (26), we are finally left with[R12(u, v),L1(u)− p1

]− [R21(v,u),L2(v)− p2]

=∑α∈∆

igα

(Φ ′(α(q), v

)Hα ⊗Eα −Φ ′

(α(q),u

)Eα ⊗Hα

)+

∑α,β∈∆

(igαα(Fβ)− igββ(Fα)− igα+βNα,β

)×Φ

(α(q),u

)Φ(β(q), v

)Eα ⊗Eβ.

The first term gives precisely the r.h.s. of Eq. (43), so the last double sum must vanish.Thus we are led to conclude that, for the elliptic Calogero model, the Poisson bracketrelation (32) will be satisfied if and only if the combinatoric identity (3) stated in theintroduction holds for any two rootsα,β ∈ ∆. For the degenerate Calogero model,repeating the same calculation withΦ(s,u) andΦ(s, v) both replaced byw(s), Φ(s,u−v)

and Φ(s, v − u) both replaced by−w′(s)/w(s) and ζ replaced by zero leads to thesame conclusion (with Eq. (32) replaced by Eq. (31)). Moreover, decomposingF in thecombinatoric identity (3) into its even partF+ and odd partF− and using symmetryproperties of the coefficientsΓα,β derived from Eq. (7) and Eq. (A.9), we see that Eq. (3)is equivalent to the following set of combinatorical identities

(44)gαα(F+β

)− gββ(F+α

)= Γα,β,

(45)gαα(F−β

)− gββ(F−α

)= 0,

the first of which is easily shown to be equivalent to the previously imposed identity (42).We have thus proved the following


Theorem 1. The Calogero model associated with the root system of a(semi-)simpleLie algebrag, degenerate as well as elliptic, with Hamiltonian given by Eq.(5) and byEq. (6), respectively, is integrable in the sense of admitting the Lax representation and thedynamicalR-matrix explicitly given by Eqs.(33), (34)and (35) and by Eqs.(36), (37)and (38), respectively, if and only if the functionF appearing in these equations satisfiesthe basic combinatoric identity(3), whose even and odd parts are the identities(44) (orequivalently,(42))and (45), respectively.

Note that the two parts of this theorem, namely that concerning the Lax representationand that concerning the dynamicalR-matrix, are of course not independent because theHamiltonian is a quadratic function of theL-matrix whereas theM-matrix is essentiallyjust the composition of theR-matrix with theL-matrix. More precisely, using the invariantbilinear form(. , .) ong to identifyg with its dual spaceg∗ and to reinterpretR as a linearmapping fromg to g rather than as a tensor ing⊗ g, 5 we can form the compositionR ·LandR(u, v) ·L(u), respectively; then

(46)H = 1

2(L,L),

and, due to Eq. (16),

(47)M =R ·L,

for the degenerate Calogero model while

(48)H =Res|u=0

(1

2u

(L(u),L(u)

)),

and, due to Eqs. (22) and (26),

R(u, v) ·L(v)=M(u)− (ζ(u− v)+ ζ(v))L(u)+ i℘(v)

∑α∈∆

gαFα,

and hence

(49)M(u)=Res|v=u

(1

v − uR(u, v) ·L(v)

)+ ζ(u)L(u),

for the elliptic Calogero model, provided that the functionF mentioned above satisfies thesimple constraint equation

(50)∑α∈∆

gαF+α = 0.

(We have dropped the contributions from the odd partF−, since they add up to zero.)Therefore, using the (by now standard) fact [20] that the Poisson bracket relation (31)or (32) implies the Lax equation (28) or (29) if one simply substitutesR · L for M andR(u, v) ·L(u) for M(u), respectively, shows that with the above choices the Lax equationfollows from the Poisson bracket relation.

5 More precisely, the convention used is to apply the identification betweeng andg∗ to the second factor in thetensor productg⊗ g.


Concluding, we mention the fact that in view of the invariance of the structure constantsNα,β under the action of the Weyl group,

(51)Nwα,wβ =Nα,β, for all w ∈W(g),

together with that of the coupling constantsgα , as required in Eq. (8), it is reasonable todemand that the functionF be covariant (equivariant) under this group:

(52)Fwα =w(Fα), for all w ∈W(g).

Moreover, it is interesting to note that the condition (45) on the odd partF− of F is easilysolved: it suffices to take

(53)F−α = λgαHα,

whereλ is some constant. Therefore, we concentrate on the condition (44) to be satisfiedby the even partF+ of F , rewritten in the form (42).

2.2. Solution of the combinatoric identity

Up to now it is not clear whether and eventually for what choice of the couplingconstantsgα , there exists a (vector valued) functionF as in Eq. (39), satisfying therequired combinatoric identity (42). In the following, we want to answer this questionin full generality, for all semisimple Lie algebras.

To begin with, observe that the problem for semisimple Lie algebras is easily reducedto that for simple Lie algebras. In fact, when a semisimple Lie algebra is decomposedinto the direct sum of its simple ideals, its root system will be decomposed orthogonallyinto irreducible root systems, and since the sum of any two roots belonging to differentsubsystems is not a root, the combinatoric identity (42) will split into separate identities,one for each of the simple ideals, and with an independent choice of coupling constants foreach of them.

To analyze the problem for simple Lie algebras, we remark first of all that, due to thefact that all roots of a simple Lie algebra having the same length constitute a single Weylgroup orbit, the choice of coupling constants is severely restricted:

• g is simply laced—Ar (r � 1), Dr (r � 4),E6, E7, E8:All roots have the same length, and there is only a single coupling constantg, whichshould be non-zero. As a result, the coupling constants drop out from Eq. (42), whichreduces to

(54)2β(F+α

)=Nβ,α +Nβ,−α.

• g is not simply laced—Br (r � 2),Cr (r � 3),F4, G2:The root system splits into precisely two Weyl group orbits, and there are preciselytwo coupling constants,gl for the long roots andgs for the short roots, at least one ofwhich should be non-zero. In fact, we can be somewhat more precise, requiring thatfor the simple Lie algebrasso(2r+1,C) of theB-series,gl should be non-zero whilefor the simple Lie algebrassp(2r,C) of the C-series,gs should be non-zero; when


r = 2, both should be non-zero. Otherwise, the Hamiltonian of the correspondingCalogero model (as given by Eq. (5) or by Eq. (6)) would decouple, that is, woulddecompose into the sum ofr copies of the same Hamiltonian for a system with onlyone degree of freedom, and such a system is trivially (Liouville) integrable.

Note also that the functionF+, if it exists, must be unique, since otherwise all couplingconstants would have to vanish. (The argument relies on the fact that even when there areroots of different length, the spaceh∗

Ris already generated both by the long roots and by

the short roots alone.)A trivial case is that of the simplest of all simple Lie algebras, namely, the unique one

of rank 1,sl(2,C). Here, we may simply setF+ ≡ 0 because there is just a single positiveroot, so that the r.h.s. of the combinatoric identity (42) vanishes identically.

More generally, a particular role is played by the simple Lie algebrassl(n,C) of theA-series. To handle this case, we first fix some notation. The correctly normalized invariantbilinear form onsl(n,C) is the trace form in the defining representation, which can in factbe extended to a non-degenerate invariant bilinear form ongl(n,C):

(55)(X,Y )= trace(XY ), for X,Y ∈ gl(n,C).

Letting indicesa, b, . . . , run from 1 ton, we introduce the standard basis ofgl(n,C)

consisting of the matricesEab with 1 at the position where theath row and thebth columnmeet and with 0 everywhere else; they satisfy the multiplication rule

(56)EabEcd = δbcEad,

and hence the commutation relation

(57)[Eab,Ecd ] = δbcEad − δdaEcb.

The Cartan subalgebrah of g = sl(n,C) will be the usual one, consisting of all tracelessdiagonal matrices; this can be extended to the standard Cartan subalgebrah of g= gl(n,C),consisting of all diagonal matrices; thenhR and hR consist of all real traceless diagonaland of all real diagonal matrices, respectively. Obviously, the matricesHa ≡ Eaa form anorthonormal basis ofhR and the linear functionalsea given by projection of a diagonalmatrixH to its entries,

(58)ea(H)=Haa, for H ∈ hR,

form the orthonormal basis ofh∗R

dual to the previous one. Moreover, setting

(59)e= e1+ · · · + en,

we see thath∗R

can be identified with the orthogonal complement ofe in h∗R

andr = n− 1.Next, the root system ofsl(n,C) is given by

(60)∆= {αab = ea − eb | 1� a �= b � n},with root generatorsEαab ≡Eab (1 � a �= b � n) and with

(61)Hαab =Eaa −Ebb.


The commutation relations (57) then lead to the following explicit expression for thestructure constantsNαab,αcd ≡Nab,cd :

(62)Nab,cd = δbc − δad,

valid for 1 � a, b, c, d � n with a �= b and c �= d . Moreover, using the aforementionedorthonormal bases to identifyhR and h∗

Rwith Rn, we see that the Weyl reflectionsαab

along the rootαab operates on a vector inRn by simply permuting theath and thebthcomponent, so the Weyl group W(sl(n,C)) of sl(n,C) is just the permutation group innletters.

With these preliminaries out of the way, we proceed to search for a collection of matricesF+αab

≡ F+ab (1� a �= b � n) that belong tohR, with F+ba = F+ab, satisfying the property (52)of covariance under the Weyl group and the combinatoric identity

(63)2((

F+ab)cc− (F+ab)dd

)= 2αcd

(F+ab

)=Ncd,ab +Ncd,ba = δda − δcb + δdb − δca,

valid for 1� a, b, c, d � n with a �= b andc �= d , corresponding to Eq. (54). Obviously,the r.h.s. vanishes if the sets{c, d} and{a, b} coincide and also if they are disjoint. Thismeans that theath andbth entry of the matrixF+ab have to be equal and also that all otherentries of the matrixF+ab have to be equal among themselves. Thus we can write

F+ab =1

2λab(Eaa +Ebb)+ 1

nτab1,

with real coefficientsλab andτab to be determined. If on the other hand we choosec andd so that the intersection of the sets{c, d} and{a, b} contains precisely one element, ther.h.s. is equal to±1, and the equation is solved by puttingλab = −1 whereas the othercoefficient is fixed to beτab = 1 by the condition thatF+ab should be traceless. The resultis the following

Proposition 1. For the complex simple Lie algebrassl(n,C) of theA-series, there existsa non-trivial solution to the combinatoric identity(42)which is given by

(64)F+ab =−1

2(Eaa +Ebb)+ 1

n1.

This solution also satisfies the constraint equation(50).

The question whether other complex simple Lie algebras admit a similar solution has anegative answer:

Proposition 2. Let g be a complex simple Lie algebra not belonging to theA-series, i.e.,not isomorphic to any of the complex simple Lie algebrassl(n,C). Then the combinatoricidentity(42)has no solution.

In the proof, we shall for the sake of simplicity use the pertinent invariant bilinear form(. , .) to identify the spacehR with its dualh∗

Rand introduce a basis{e1, . . . , en} which is

orthonormal except possibly for an overall normalization factor; then the root system∆


will be considered as a finite subset ofRn andF+ will be a map from∆ to Rn required tosatisfy the combinatoric identity

(65)2gβ

(β,F+α

)= Γβ,α + Γβ,−α, for α,β ∈∆.

As observed before, wheng is simply laced, this relation reduces to

(66)2(β,F+α

)=Nβ,α +Nβ,−α, for α,β ∈∆.

In particular, the solution forAn given by Eq. (64) is recast into the form

(67)F+ab =−1

2(ea + eb)+ 1

ne (1 � a �= b � n).

We begin with the case of the complex simple Lie algebrasso(2n,C) of theD-series, withn � 4, which are simply laced and for which∆ consists of the following roots:

(68)±αkl =±(ek − el), ±βkl =±(ek + el) (1 � k < l � n).

Expanding the generatorsF+kl ≡ F+±αklandF ′+kl ≡ F+±βkl

in their components with respectto the orthonormal basis{e1, . . . , en} of Rn, we now argue as follows.

• When the sets{p,q} and {k, l} are disjoint (this possibility exists because we areassumingn � 4), none of the expressions±αpq ± αkl , ±βpq ± αkl , ±αpq ± βkl ,±βpq ± βkl is a root, so using Eq. (66) withβ = αpq as well asβ = βpq andα = αkl

as well asα = βkl , we infer that bothF+kl andF ′+kl must be orthogonal to bothαpq

andβpq , i.e., toep and toeq . In other words, the components ofF+kl and ofF ′+kl alongany basis vector exceptek andel must vanish, i.e., bothF+kl andF ′+kl must be linearcombinations ofek andel .

• When the sets{p,q} and {k, l} are equal, then once again none of the expressions±αpq ± αkl , ±βpq ± αkl , ±αpq ± βkl , ±βpq ± βkl is a root, so using Eq. (66) withβ = αpq as well asβ = βpq andα = αkl as well asα = βkl , we infer that bothF+klandF ′+kl must be orthogonal to bothαkl andβkl , i.e., toek and toel . In other words,the components ofF+kl and ofF ′+kl along the basis vectorsek andel must vanish.

Thus we arrive at the conclusion that Eq. (66) forcesF+ to be identically zero, whichis of course a contradiction because the r.h.s. of Eq. (66) is not identically zero: morespecifically, it does not vanish when we setβ = αpq or β = βpq andα = αkl or α = βkl ,assuming that the intersection of the sets{p,q} and{k, l} contains precisely one element.

It is interesting to note that the first part of the above argument fails whenn= 3 and asolution indeed exists in this case, withF+kl andF ′+kl proportional to the basis vectorεklmem.This is to be expected due to the isomorphism betweenso(6,C) andsl(4,C), which at thelevel of root systems can be implemented by expressing the basis vectorse1, e2, e3 for theroot system ofD3 in terms of the basis vectorse1, e2, e3, e4 for the root system ofA3, aswell as their sume, by setting

(69)ek = ek + e4− 1

2e (1 � k � 3),

which induces an isomorphism between the corresponding root systems given by

(70)±αkl =±αkl , ±βkl =∓εklmαm4 (1 � k < l � 3).


Thus the solution forA3 given by Eq. (67) can be translated into a solution forD3:

(71)F+kl =1

2εklmem, F ′+kl =−

1

2εklmem (1� k < l � 3).

The proof of the theorem for the other complex simple Lie algebras is greatly simplifiedby comparing the eventual solution for a complex simple Lie algebrag with that for acomplex simple Lie algebrag′ when the latter, say, is a subalgebra of the former of thesame rank. In this case, a Cartan subalgebra of the latter also serves as a Cartan subalgebraof the former, so that the root system∆′ of g′ can be identified with a subset of the rootsystem∆ of g. It is then clear that the solution forg must restrict to the solution forg′. Morespecifically, this means that forα ∈∆′ ⊂∆, the generatorF+α for g must (possibly up to auniversal normalization factor6) coincide with the generatorF+α for g′. In particular, thismakes it obvious that the non-existence of a solution forg′ will automatically imply thenon-existence of a solution forg.

This argument can be immediately applied choosingg′ to be one of the complex simpleLie algebrasso(2n,C) of the D-series, withn � 4, to prove the theorem for almost allother complex simple Lie algebras:

• Bn (n � 4): Bn containsDn,• Cn (n � 4): Cn containsDn,• E8: E8 containsD8,• F4: F4 containsD4.

Therefore, all that remains is to analyze a few isolated cases, namelyB2= C2, G2, B3 andC3, E6 andE7. This can be done by applying similar arguments to the ones already usedbefore.

• B2 (so(5,C)) ∼= C2 (sp(4,C)):The root system consists of four long roots and four short roots, which can be writtenin the form

long roots: ±e1± e2,

(72)short roots: ±e1,±e2,

wheng is realized asso(5,C) (B2) or in the form

long roots: ±2e1,±2e2,

(73)short roots: ±e1± e2,

when g is realized assp(4,C) (C2). Either way, analyzing the combinatoricidentity (65) withα any long root results in a contradiction (provided we take intoaccount the condition that neither of the two coupling constantsgl andgs shouldvanish): whenβ is a long root, neither of the two expressionsβ + α andβ − α willbe a root, so the r.h.s. must vanish, leading to the conclusion thatF+α must vanish, but

6 If the index of the embedding ofg′ into g is not equal to 1, some formulas will be modified because it willappear as a normalization factor relating the invariant bilinear forms forg′ and forg.


whenβ is a short root, then one and only one of the two expressionsβ + α andβ − α

will be a root, so the r.h.s. cannot vanish, leading to the conclusion thatF+α cannotvanish.

• B3 (so(7,C)) andC3 (sp(6,C)):The root system forB3 consists of twelve long roots and six short roots, which can bewritten in the form

long roots: ±ek ± el (1 � k < l � 3),

(74)short roots: ±ek (1 � k � 3),

while the root system forC3 consists of six long roots and twelve short roots, whichcan be written in the form

long roots: ±2ek (1 � k � 3),

(75)short roots: ±ek ± el (1 � k < l � 3).

Note that both systems contain the systemD3 as a subsystem: it is generated by theroots of the form±ek ± el (1 � k < l � 3). Therefore, the values ofF+ on anyroot α of this form must, according to Eq. (71), be proportional toεklmem. But thenanalyzing the combinatoric identity (65) withα any root of this form andβ any rootof the form±em (for B3) or ±2em (for C3) results in a contradiction (provided wetake into account the condition that the coupling constantsgl for B3 andgs for C3

should not vanish): whenm is equal tok or l, then one and only one of the twoexpressionsβ + α andβ − α will be a root, so the r.h.s. cannot vanish whereas thel.h.s. is obviously equal to zero; similarly, whenm is different from bothk and l,then neither of the two expressionsβ + α and β − α will be a root, so the r.h.s.must vanish, whereas the l.h.s. is typically different from zero (except when the othercoupling constant (gs for B3 andgl for C3) vanishes).

• G2:The root system consists of six long roots and six short roots, which can be written inthe form

long roots: ±βkl =±(2εjklej − ek − el) (1 � k < l � 3),

(76)short roots: ±αkl =±(ek − el) (1 � k < l � 3),

spanning the subspace orthogonal to the vectore= e1+ e2+ e3. Note that this systemis the disjoint union of two subsystems that are copies of the systemA2. In particular,the value ofF+ on any short rootαkl must, according to Eq. (67), be proportional tothe long rootβkl . But then analyzing the combinatoric identity (65) withα any shortroot andβ the corresponding long root results in a contradiction (provided we takeinto account the condition that neither of the two coupling constantsgl andgs shouldvanish because otherwise, we would really be dealing not with aG2 model but withanA2 model in disguise): since neither of the two expressionsβkl +αkl andβkl −αkl


will be a root, the r.h.s. must vanish, whereas the l.h.s. is typically different from zero(except when the coupling constantgl vanishes).

• E7:The root system consists of 126 roots, all of the same length, spanning a 7-dimensional subspace ofR8, which can be written in the form

±αkl =±(ek − el) (1 � k < l � 6),

±α78=±(e7− e8),

±βkl =±(ek + el) (1� k < l � 6),

(77)±γ±,...,± =±1

2

(e8− e7+

6∑k=1

(−1)s(k)ek

),

where thes(k) (k = 1, . . . ,6) are 0 or 1 and such that∑6

k=1 s(k) is odd. Applying thesame arguments as in theDn case, we see thatF+αkl

andF+βkl(1 � k < l � 6) must be

orthogonal toek (k = 1, . . . ,6) as well as toe7− e8 and can therefore be expressed inthe form

F+αkl= fαkl (e7+ e8), F+βkl

= fβkl (e7+ e8).

Obviously, these generators are also orthogonal to all roots of the formγ±,...,±. Butthen analyzing the combinatoric identity (66) withα = αkl or α = βkl and β anappropriate root of the formγ±,...,± results in a contradiction. For example, lettingγ12 be any root of the form

1

2

(e8− e7− (e1− e2)± e3± e4± e5± e6

),

with an even number of minus signs, we see thatγ12− α12 is not a root whereasγ12+ α12 is a root, so the r.h.s. cannot vanish, whereas the l.h.s. is necessarily equalto zero.

• E6:The root system consists of 72 roots, all of the same length, spanning a 6-dimensionalsubspace ofR8, which can be written in the form

±αkl =±(ek − el) (1 � k < l � 5),

±βkl =±(ek + el) (1� k < l � 5),

(78)±γ±,...,± =±1

2

(e8− e7− e6+

5∑k=1

(−1)s(k)ek

),

where thes(k) (k = 1, . . . ,5) are 0 or 1 and such that∑5

k=1 s(k) is even. Applyingthe same arguments as in theDn case, we see thatF+αkl

andF+βklmust be orthogonal

to ek with k = 1, . . . ,5 and can therefore be expressed in the form

F+αkl= fαkl e6+ gαkl e7+ hαkl e8, F+βkl

= fβkl e6+ gβkl e7+ hβkl e8.


To show that these generators are also orthogonal to the vectore8− e7− e6 and henceto all roots of the formγ±,...,±, we use the combinatoric identity (66) withα = αkl

or α = βkl andβ an appropriate root of the formγ±,...,±. For example, lettingγ−12 beany root of the form

1

2

(e8− e7− e6+ (e1+ e2)± e3± e4± e5

),

with an even number of minus signs and lettingγ+12 be any root of the form

1

2

(e8− e7− e6+ (e1− e2)± e3± e4± e5

),

with an odd number of minus signs, we see that neitherγ−12+ α12 nor γ−12− α12 is aroot and neitherγ+12+β12 norγ+12−β12 is a root, forcingF+α12

to be orthogonal toγ−12andF+β12

to be orthogonal toγ+12, so both must be orthogonal to the vectore8− e7− e6

and hence to all roots of the formγ±,...,±. But then analyzing the combinatoric identity(66) withα = αkl orα = βkl andβ another appropriate root of the formγ±,...,± resultsin a contradiction. For example, lettingγ12 be any root of the form

1

2

(e8− e7− e6− (e1− e2)± e3± e4± e5

),

with an odd number of minus signs, we see thatγ12− α12 is not a root whereasγ12+ α12 is a root, so the r.h.s. cannot vanish, whereas the l.h.s. is necessarily equalto zero.

This completes the proof of our no-go theorem, Proposition 2.To conclude this section, we briefly summarize the main results. The only integrable

Calogero models that allow a Lax formulation with a dynamicalR-matrix, directly interms of simple Lie algebras, are the ones based on the simple Lie algebrassl(n,C) oftheA-series. Using the abbreviation

(79)V (t)={

12w(t)2 for the degenerate case12℘(t) for the elliptic case

},

for the potential, their Hamiltonian (5) or (6) is given by

(80)H(q,p)= 1

2

n∑k=1

p2k + g2

∑1�k �=l�n

V (qk − ql),

which is the Calogero Hamiltonian associated to the root systemAn−1. Inserting thesolution (64) of the combinatoric identity (42) reproduces the known results, both for theLax pair [2,4–6] and theR-matrix [8,9].

Of course, this result comes as no surprise. The main new aspect of the systematicanalysis presented here is to showwhy the standard approach works only for theA-seriesandwhy it is not possible to extend this technique to other simple Lie algebras.


3. Calogero models for symmetric pairs

3.1. Definition of the models, Lax pairs and R-matrices

The formulation of the Calogero model associated with the restricted root system�∆of a symmetric pair(g, θ), rather than the ordinary root system of a (semi-)simple Liealgebra, requires only minor modifications; again, we refer to Appendix A for a summaryof the conventions and notation used. As before, the configuration spaceQ of the theoryis a Weyl chamber or Weyl alcove in the appropriate sense; in particular, it is again anopen subset of a vector space denoted bya0, and the phase space of the model is given bythe associated cotangent bundleT ∗Q =Q× a∗0, often identified with the tangent bundleT Q=Q× a0. The Hamiltonian for the degenerate Calogero model then reads

(81)H(q,p)= 1

2

(r∑

j=1

p2j +

∑α∈ �∆

mαg2αw(α(q)

)2),

or equivalently

(82)H(q,p)= 1

2

(r∑

j=1

p2j +

∑α∈∆

g2αw(α(q)

)2),

while that for the elliptic Calogero model reads

(83)H(q,p)= 1

2

(r∑

j=1

p2j +

∑α∈ �∆

mαg2α℘(α(q)

)),

or equivalently

(84)H(q,p)= 1

2

(r∑

j=1

p2j +

∑α∈∆

g2α℘(α(q)

)),

wherew and℘ are as before. The coefficientsgα = gα (α ∈ ∆) are positive real couplingconstants whose definition will be extended from∆ to all of ∆ by setting

(85)gα = 0, for α ∈∆0.

They are supposed to be symmetric (cf. Eq. (7)) andθ -invariant:

(86)g−α = gα = gθα.

A stronger assumption that we shall always make is that they are invariant under the actionof the subgroup Wθ (g) of W(g) associated with the symmetric pair(g, θ):

(87)gwα = gα, for all w ∈Wθ (g).

For later use we introduce the following combination of coupling constants and structureconstants:

(88)Γ θα,β =

1

4

(gα+βNα,β + gθα+βNθα,β + gα+θβNα,θβ + gθα+θβNθα,θβ

).


Using the abbreviations (10) as before, we can write the Hamiltonian equations of motionas vector equations ina0: they read

(89)q = p, p=−∑α∈∆

g2αw(α(q)

)w′(α(q)

)(Hα)a,

for the degenerate Calogero model and

(90)q = p, p=−1

2

∑α∈∆

g2α℘′(α(q)

)(Hα)a,

for the elliptic Calogero model.The general considerations on the structure of the Lax pair and theR-matrix remain

unaltered, so we may pass directly to postulate a Lax pair and anR-matrix which, inaccordance with the discussion in the introduction, are given by

(91)L=r∑

j=1

pjHj +∑α∈∆

igαw(α(q)

)Eα,

(92)M =−∑α∈∆

igαw′′(α(q))

2w(α(q))Fα +

∑α∈∆

igαw′(α(q)

)Eα,

(93)R =∑α∈∆

w(α(q)

)Fα ⊗Eα + 1

2

∑α∈∆

w′(α(q))

w(α(q))(Eα ⊗E−α +Eθα ⊗E−α),

for the degenerate Calogero model (see [4–6] for the Lax pair) and by

(94)L(u)=r∑

j=1

pjHj +∑α∈∆

igαΦ(α(q),u

)Eα,

(95)M(u)=−∑α∈∆

igα℘(α(q)

)Fα +

∑α∈∆

igαΦ′(α(q),u

)Eα,

(96)

R(u, v)=−1

2

r∑j=1

(ζ(u− v)+ ζ(u+ v)

)Hj ⊗Hj

− 1

2

(ζ(u− v)− ζ(u+ v)+ 2ζ(v)

)Cz +

∑α∈∆

Φ(α(q), v

)Fα ⊗Eα

− 1

2

∑α∈∆

(Φ(α(q),u− v

)Eα ⊗E−α +Φ

(α(q),−u− v

)Eθα ⊗E−α

),

for the elliptic Calogero model. Here,Cz is the quadratic Casimir element for thecentralizerz of a in k (see Eq. (A.26) for an explicit definition), i.e.,

(97)Cz =r+s∑

j=r+1

Hj ⊗Hj +∑α∈∆0

Eα ⊗E−α,


and theFα form a collection of generators which are now supposed to belong to ib0 andcan be viewed as a (vector valued) function

(98)F : ∆→ ib0,

which we shall assume to beθ -invariant:

(99)Fα = Fθα.

As before, it is convenient to decomposeF into its even partF+ and its odd partF−:

(100)F+α =1

2(Fα + F−α), F−α =

1

2(Fα −F−α).

Note that only the even partF+ of F is used in the definition ofM. Once again, weshall derive a combinatoric equation which determinesF and, together with the functionalequations discussed in the previous section, turns out to be the basic ingredient for provingthe equivalence between the Hamiltonian equations of motion and the Lax equation, aswell as the validity of the Poisson bracket relation.

Basically, the proof goes as previously but it presents some additional subtleties. Asbefore, we concentrate on the elliptic Calogero model. First, we use Eq. (94) and Eq. (95)to compute the following expression for the difference between the two sides of the Laxequation (29):

L(u)− [L(u),M(u)]

= p+∑α∈∆

igαΦ′(α(q),u

)α(q)Eα −

∑α∈∆

igαΦ′(α(q),u

)α(p)Eα

+∑


(α(q)

)Φ(γ (q),u

)γ (Fα)Eγ ,

+ 1

2

∑α∈∆

g2α

(Φ(α(q),u

)Φ ′(−α(q),u

)−Φ(−α(q),u

)Φ ′(α(q),u

))Hα

+ 1

2

∑α,β∈∆

α+β∈∆0

gαgβ

(Φ(α(q),u

)Φ ′(β(q),u

)−Φ ′(α(q),u

)Φ(β(q),u

))

×Nα,βEα+β

+ 1

2


gαgβ

(Φ(α(q),u

)Φ ′(β(q),u

)−Φ ′(α(q),u

)Φ(β(q),u

))

×Nα,βEγ .

Here, the double sum over commutators[Eα,Eβ ] appearing in the commutator betweenL(u) andM(u) has, after antisymmetrization of the coefficients inα andβ , been splitinto three contributions, one corresponding to terms whereα + β = 0, one correspondingto terms whereα + β ∈ ∆0 and one corresponding to terms whereα + β ∈ ∆. Insertingthe functional equation (23) to transform the first two of these contributions (note that


α + β ∈ ∆0 impliesα(q) + β(q) = 0 for all q ∈Q) and the functional equation (24) totransform the third contribution (note thatα + β ∈ ∆ impliesα(q)+ β(q) �= 0 modΛ forall q ∈Q) and rearranging terms, we get

L(u)− [L(u),M(u)]

= p+ 1

2

∑α∈∆

g2α℘′(α(q)

)Hα +

∑α∈∆

igαΦ′(α(q),u

)α(q − p)Eα

+ 1

4

∑α,β∈∆

α+β∈∆0

gαgβ

(℘ ′(α(q)

)−℘ ′(β(q)

))Nα,βEα+β

+∑


(α(q)

)Φ(γ (q),u

)γ (Fα)Eγ

+ 1

2

∑α,γ∈∆γ−α∈∆

gαgγ−α℘(α(q)

)Φ(γ (q),u

)Nα,γ−αEγ

− 1

2

∑β,γ∈∆γ−β∈∆

gγ−βgβ℘(β(q)

)Φ(γ (q),u

)Nγ−β,βEγ .

Renaming summation indices in the last two sums (α→−α in the first,β → α in thesecond) and using Eqs. (A.9), (A.10), (7) and (13), together with∑

α∈∆g2α℘′(α(q)

)Hα = 1

2

∑α∈∆

g2α

(℘ ′(α(q)

)Hα +℘ ′

((θα)(q)

)Hθα

)= 1

2

∑α∈∆

g2α℘′(α(q)

)(Hα − θHα)

=∑α∈∆

g2α℘′(α(q)

)(Hα)a,

and the fact that, according to Eq. (A.28) and Eq. (A.29),∑α,β∈∆

α+β∈∆0

gαgβ

(℘ ′(α(q)

)−℘ ′(β(q)

))Nα,βEα+β

= 1

2

∑α,β∈∆

α+β∈∆0

gαgβ

((℘ ′(α(q)

)−℘ ′(β(q)

))Nα,βEα+β

+ (℘ ′((θα)(q))−℘ ′

((θβ)(q)

))Nθα,θβEθα+θβ

)= 1

2

∑α,β∈∆

α+β∈∆0

gαgβ

(℘ ′(α(q)

)−℘ ′(β(q)

))Nα,β(Eα+β − θEα+β)

= 0,


we finally obtain

L(u)− [L(u),M(u)]

= p+ 1

2

∑α∈∆

g2α℘′(α(q)

)(Hα)a +

∑α∈∆

igαΦ′(α(q),u

)α(q − p)Eα

+ 1

2

∑α,γ∈∆

gα

(2gγ γ (Fα)− gγ+αNγ,α − gγ−αNγ,−α

)×℘

(α(q)

)Φ(γ (q),u

)Eγ .

The vanishing of the sum of the first two terms and of the third term are precisely theequations of motion, so the last double sum must vanish, which requires that for all rootsγ ∈ ∆,

(101)∑α∈∆

gα

(2gγ γ (Fα)− Γγ,α − Γγ,−α

)℘(α(q)

)= 0.

Since these relations must hold identically inq but the coefficient functions are invariantunder the substitutionsα→−α andα→ θα, we are led to conclude that, for the ellipticCalogero model, the Lax equation (29) will be equivalent to the Hamiltonian equations ofmotion (90) if and only if the combinatoric identity

(102)2gββ(F+α

) = Γ θβ,α + Γ θ

β,−α,

holds for any two rootsα,β ∈ ∆. For the degenerate Calogero model, the same calculationwith Φ(s,u) replaced byw(s) and ℘(s) replaced byw′′(s)/2w(s) leads to the sameconclusion (with Eq. (29) replaced by Eq. (28)).

Passing to the Poisson bracket relation (32) in the elliptic case, we use Eq. (94) tocompute the l.h.s., starting out from the same canonical Poisson brackets as before butnow using the identities (A.30) and (10) to conclude that{

p,Φ(α(q),u

)}=Φ ′(α(q),u

)(Hα)a,

leading to the following momentum independent expression for the l.h.s. of Eq. (32):{L1(u),L2(v)

}(103)

=∑α∈∆

igα

(Φ ′(α(q), v

)(Hα)a ⊗Eα −Φ ′

(α(q),u

)Eα ⊗ (Hα)a

).

To compute the r.h.s. of Eq. (32) we observe first that it is also momentum independent,since once again, the only possibly momentum dependent terms cancel:[

R12(u, v),p1]− [R21(v,u),p2

]= 1

2

∑α∈∆

α(p)(+Φ

(α(q),u− v

)Eα ⊗E−α −Φ

(α(q),−u− v

)Eθα ⊗E−α

−Φ(α(q), v − u

)E−α ⊗Eα +Φ

(α(q),−v − u

)E−α ⊗Eθα

)= 0.


The remaining terms are, according to Eqs. (A.28), (A.29) and (A.30),[R12(u, v),L1(u)− p1

]− [R21(v,u),L2(v)− p2]

=−1

2

∑α∈∆

igα

(ζ(u− v)+ ζ(u+ v)

)Φ(α(q),u

)Eα ⊗ (Hα)a

+ 1

2

∑α∈∆

igα

(ζ(v − u)+ ζ(v + u)

)Φ(α(q), v

)(Hα)a ⊗Eα

− 1

2

∑α∈∆

igα

(ζ(u− v)− ζ(u+ v)+ 2ζ(v)

)Φ(α(q),u

)Eα ⊗ (Hα)b

+ 1

2

∑α∈∆

igα

(ζ(v − u)− ζ(v + u)+ 2ζ(u)

)Φ(α(q), v

)(Hα)b ⊗Eα

− 1

2

∑β∈∆0

γ∈∆

igγ

(ζ(u− v)− ζ(u+ v)+ 2ζ(v)

)Φ(γ (q),u

)

×N−β,γ E−β+γ ⊗Eβ

+ 1

2

∑α∈∆0

γ∈∆

igγ

(ζ(v − u)− ζ(v + u)+ 2ζ(u)

)Φ(γ (q), v

)

×N−α,γ Eα ⊗E−α+γ

+∑

α,β∈∆


)Φ(α(q),u

)Φ(β(q), v

)Eα ⊗Eβ

+ 1

2

∑α∈∆

igαΦ(−α(q),u− v

)Φ(α(q),u

)Hα ⊗Eα

− 1

2

∑α∈∆

igαΦ(−α(q),−u− v

)Φ(α(q),−u

)θHα ⊗Eα

− 1

2

∑α∈∆

igαΦ(−α(q), v− u

)Φ(α(q), v

)Eα ⊗Hα

+ 1

2

∑α∈∆

igαΦ(−α(q),−v− u

)Φ(α(q),−v

)Eα ⊗ θHα

− 1

2

∑β,γ∈∆

β−γ∈∆0


)Φ(γ (q),u

)N−β,γ E−β+γ ⊗Eβ

+ 1

2

∑β,γ∈∆

β−γ∈∆0

igγ Φ(−β(q),−u− v

)Φ(γ (q),−u

)N−β,γ θE−β+γ ⊗Eβ


+ 1

2

∑α,γ∈∆

α−γ∈∆0

igγ Φ(−α(q), v − u

)Φ(γ (q), v

)N−α,γ Eα ⊗E−α+γ

− 1

2

∑α,γ∈∆

α−γ∈∆0

igγ Φ(−α(q),−v− u

)Φ(γ (q),−v

)N−α,γ Eα ⊗ θE−α+γ

− 1

2

∑β,γ∈∆β−γ∈∆


)Φ(γ (q),u

)N−β,γ E−β+γ ⊗Eβ

+ 1

2

∑β,γ∈∆β−γ∈∆

igγ Φ(−β(q),−u− v

)Φ(γ (q),−u

)N−β,γ θE−β+γ ⊗Eβ

+ 1

2

∑α,γ∈∆α−γ∈∆

igγ Φ(−α(q), v− u

)Φ(γ (q), v

)N−α,γ Eα ⊗E−α+γ

− 1

2

∑α,γ∈∆α−γ∈∆

igγ Φ(−α(q),−v− u

)Φ(γ (q),−v

)N−α,γ Eα ⊗ θE−α+γ ,

where the last twelve terms have been obtained by splitting each of the four termscontaining tensor products of a root generator with the commutator of two otherroot generators, say[Eα,Eβ ] ⊗ Eγ or Eγ ⊗ [Eα,Eβ ], into three contributions: onecorresponding to terms whereα + β = 0, one corresponding to terms whereα + β ∈∆0

and one corresponding to terms whereα+β ∈ ∆; moreover, some renaming of summationindices has been performed and relations of the form

Φ((θγ )(q),w

)=Φ(−γ (q),w

)=−Φ(γ (q),−w

),

have been employed. Using Eq. (A.9) and Eq. (A.10) and rearranging terms, we get[R12(u, v),L1(u)− p1

]− [R21(v,u),L2(v)− p2]

=+1

2

∑α∈∆

igα

((ζ(v − u)+ ζ(v + u)

)Φ(α(q), v

)+Φ


)+Φ

(−α(q),−u− v)Φ(α(q),−u

))(Hα)a ⊗Eα

− 1

2

∑α∈∆

igα

((ζ(u− v)+ ζ(u+ v)

)Φ(α(q),u

)+Φ

(−α(q), v − u)Φ(α(q), v

)+Φ

(−α(q),−v− u)Φ(α(q),−v

))Eα ⊗ (Hα)a


+ 1

2

∑α∈∆

igα

((ζ(v − u)− ζ(v + u)+ 2ζ(u)

)Φ(α(q), v

)+Φ


)−Φ

(−α(q),−u− v)Φ(α(q),−u

))(Hα)b ⊗Eα

− 1

2

∑α∈∆

igα

((ζ(u− v)− ζ(u+ v)+ 2ζ(v)

)Φ(α(q),u

)+Φ

(−α(q), v − u)Φ(α(q), v

)−Φ

(−α(q),−v− u)Φ(α(q),−v

))Eα ⊗ (Hα)b

+ 1

2

∑α∈∆,β∈∆0

γ=α+β∈∆

igγ Nα,β

((ζ(u− v)− ζ(u+ v)+ 2ζ(v)

)Φ(γ (q),u

)

+Φ(−α(q), v− u

)Φ(γ (q), v

)−Φ

(−α(q),−v− u)Φ(γ (q),−v

))Eα ⊗Eβ

+ 1

2

∑α∈∆0,β∈∆γ=α+β∈∆

igγ Nα,β

((ζ(v − u)− ζ(v + u)+ 2ζ(u)

)Φ(γ (q), v

)

+Φ(−β(q),u− v

)Φ(γ (q),u

)−Φ

(−β(q),−u− v)Φ(γ (q),−u

))Eα ⊗Eβ

+∑

α,β∈∆


)Φ(α(q),u

)Φ(β(q), v

)Eα ⊗Eβ

+ 1

2


igγ Nα,β

(Φ(−β(q),u− v

)Φ(γ (q),u

)

+Φ(−α(q), v− u

)Φ(γ (q), v

))Eα ⊗Eβ

− 1

2


igγ Nα,β

(Φ(−β(q),−u− v

)Φ(γ (q),−u

)

+Φ(−α(q), v+ u

)Φ(γ (q), v

))θEα ⊗Eβ.

Inserting the functional equations (25) and (26), we see that the terms proportional to(Hα)b⊗Eα and toEα ⊗ (Hα)b with α ∈ ∆, as well as the terms proportional toEα ⊗Eβ

with α ∈ ∆ andβ ∈∆0 or with α ∈∆0 andβ ∈ ∆ cancel, and using the convention (85),we are finally left with

552 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570[R12(u, v),L1(u)− p1

]− [R21(v,u),L2(v)− p2]

=∑α∈∆

igα

(Φ ′(α(q), v


(α(q),u

)Eα ⊗ (Hα)a

)+

∑α,β∈∆


)Φ(α(q),u

)Φ(β(q), v

)Eα ⊗Eβ

− 1

2

∑α,β∈∆

igγ Nα,β

(Φ(α(q),u

)Φ(β(q), v

)Eα ⊗Eβ

−Φ(α(q),−u

)Φ(β(q), v

)θEα ⊗Eβ

)=∑α∈∆

igα

(Φ ′(α(q), v


(α(q),u

)Eα ⊗ (Hα)a

)+

∑α,β∈∆

(igαα(Fβ)− igββ(Fα)− iΓ θ

α,β

)Φ(α(q),u

)Φ(β(q), v

)Eα ⊗Eβ.

The first term gives precisely the r.h.s. of Eq. (103), so the last double sum must vanish.Thus we are led to conclude that, for the elliptic Calogero model, the Poisson bracketrelation (32) will be satisfied if and only if the combinatoric identity (4) stated in theintroduction holds for any two rootsα,β ∈ ∆. For the degenerate Calogero model, the samecalculation withΦ(s,u) andΦ(s, v) both replaced byw(s), Φ(s,u− v) andΦ(s, v − u)

both replaced by−w′(s)/w(s) andζ replaced by zero leads to the same conclusion (withEq. (32) replaced by Eq. (31)). Moreover, decomposingF in the combinatoric identity (4)into its even partF+ and odd partF− and using symmetry properties of the coefficientsΓ θ

α,β derived from Eq. (86) and Eq. (A.9), we see that Eq. (4) is equivalent to the followingset of combinatorical identities

(104)gαα(F+β

)− gββ(F+α

)= Γ θα,β,

(105)gαα(F−β

)− gββ(F−α

)= 0,

the first of which is easily shown to be equivalent to the previously imposed identity (102).We have thus proved the following

Theorem 2. The Calogero model associated with the root system of a symmetric pair(g, θ), degenerate as well as elliptic, with Hamiltonian given by Eq.(81) or Eq. (82)and by Eq.(83) or Eq. (84), respectively, is integrable in the sense of admitting the Laxrepresentation and the dynamicalR-matrix explicitly given by Eqs.(91), (92)and(93)andby Eqs.(94), (95)and (96), respectively, if and only if the functionF appearing in theseequations satisfies the basic combinatoric identity(4), whose even and odd parts are theidentities(104) (or equivalently,(102))and(105), respectively.

Note that, once again, the two parts of this theorem, namely that concerning the Laxrepresentation and that concerning the dynamicalR-matrix, are not independent becausethe Hamiltonian is a quadratic function of theL-matrix whereas theM-matrix is essentially


just the composition of theR-matrix with theL-matrix. More precisely,

(106)H = 1

2(L,L),

and, due to Eq. (16),

(107)M =R ·L,

for the degenerate Calogero model while

(108)H =Res|u=0

(1

2u

(L(u),L(u)

)),

and, due to Eq. (22) and Eq. (26),

R(u, v) ·L(v)=M(u)− 1

2

(ζ(u− v)+ ζ(u+ v)

)L(u)+ i℘(v)

∑α∈∆

gαFα,

and hence

(109)M(u)=Res|v=u

(1

v − uR(u, v) ·L(v)

)+ 1

2ζ(2u)L(u),

for the elliptic Calogero model, provided that the functionF mentioned above satisfies thesimple constraint equation

(110)∑α∈∆

gαF+α = 0.

(As before, we have dropped the contributions from the odd partF−, since they add up tozero.) Thus once again, we may conclude that the Lax equation follows from the Poissonbracket relation.

Concluding, we mention the fact that in view of the invariance of the structure constantsNα,β under the action of the entire Weyl group W(g) of g,

(111)Nwα,wβ =Nα,β, for all w ∈W(g),

together with the invariance of the coupling constantsgα under the action of the subgroupWθ (g) of W(g) associated with the symmetric pair(g, θ), as required in Eq. (87), it isreasonable to demand that the functionF be covariant (equivariant) under this subgroup:

(112)Fwα =w(Fα), for all w ∈Wθ (g).

Moreover, it is interesting to note that the condition on the odd partF− of F is easilysolved: it suffices to take

(113)F−α = λgα(Hα)b,

whereλ is some constant. Therefore, we concentrate on the condition (104) to be satisfiedby the even partF+ of F , rewritten in the form (102).


3.2. Solution for theAIII -series

As an example that provides non-trivial solutions to the combinatoric identity (102)derived above, we consider the symmetric pairs corresponding to the complex Grassman-nians, the symmetric spaces of theAIII-series, which are given by the choice

(114)g= sl(n,C), g0= su(p, q), gk = su(n),

wheren = p + q andp � q . In order to take advantage of the calculations performed inSection 2.2 for thesl(n,C) case, we choose a representation in which the Cartan subalgebrah of g is the standard one, consisting of purely diagonal matrices; this forces us to deviatefrom another standard convention, adopted, e.g., in Ref. [30], according to which theinvolution θ is given by conjugation with a matrix havingp entries equal to+1 andq

entries equal to−1 on the diagonal. Instead, we shall represent all matrices insl(n,C) inthe(3× 3) block form

(115)n � ( . . . )←→n

= (. . .) (. . .) (. . .)

(. . .) (. . .) (. . .)

(. . .) (. . .) (. . .)

←→←→←→p q −p p

� p

� q − p

� p

and suppose the automorphismθ to be given by conjugation

(116)θX = Jp,qXJp,q, for X ∈ sl(n,C),

with the matrix

(117)Jp,q = 0 0 1p

0 1q−p 01p 0 0

.

Taking into account that the conjugationτ with respect to the compact real formgk = su(n)

should be given by

(118)τX =−X†, for X ∈ sl(n,C),

where “†” denotes Hermitean adjoint, in order to guarantee that the matrices constitutinggk = su(n) continue to be the traceless anti-Hermitean matrices, we conclude that theconjugationσ with respect to the real formg0 = su(p, q) must in this representation begiven by

(119)σX =−Jp,qX†Jp,q, for X ∈ sl(n,C),

so the matrices constitutingg0= su(p, q) are the traceless matrices of the form

(120)

A C E

D B −C†

F −D† −A†

,

whereA, C andD are arbitrary complex matrices whereasB, E andF are anti-Hermiteanmatrices. Similarly, the matrices constituting the intersectionk0 of the two real forms are


the traceless matrices of the form

(121)

A1 C1 E1

−C†1 B1 −C

†1

E1 C1 A1

,

whereC1 is an arbitrary complex matrix whereasA1, B1 and E1 are anti-Hermiteanmatrices. Finally, the matrices belonging to the complementary spacem0 appearing in thedecompositions (A.13) and (A.14) are the traceless matrices of the form

(122)

A2 C2 E2

C†2 0 −C

†2

−E2 −C2 −A2

,

whereC2 is an arbitrary complex matrix whereasA2 is a Hermitean matrix andE2 isan anti-Hermitean matrix. In particular, thep-dimensional subspacea0 that appears asthe ambient space for the configuration space of the Calogero models consists of the realdiagonal matrices of the form

(123)

P 0 00 0 00 0 −P

,

whereas its(q − 1)-dimensional orthogonal complement ib0 in hR consists of the realtraceless diagonal matrices of the form

(124)

P 0 00 Q 00 0 P

.

For later use, we also introduce the diagonal(n× n)-matrices

(125)I2p =1p 0 0

0 0 00 0 1p

, Iq =0 0 0

0 1q 00 0 0

.

Passing to the explicit index calculations, we continue to use the notation and theconventions introduced in Section 2.2 for handling thesl(n,C) case. In particular, wecontinue to let indicesa, b, . . . , run from 1 ton, but we shall use the following terminologyto characterize the subdivision of this range indicated by the(3×3) block form introducedabove: an indexa will be said to belong to the first block if 1� a � p, to the secondblock if p+ 1� a � q and to the third block ifq + 1 � a � n; moreover, we introduce thefollowing abbreviation to characterize the action of the involutionθ on these indices:

θ(a)= a + q, for 1 � a � p,

θ(a)= a, for p+ 1 � a � q,

(126)θ(a)= a − q, for q + 1� a � n.

This allows for a considerable simplification of the notation. For example, we have

(127)θαab = αθ(a)θ(b),


with

(128)θEab =Eθ(a)θ(b).

In particular, the two parts of the root system∆ of sl(n,C) appearing in the decomposi-tion (A.23) are given by

(129)∆= {αab ∈∆ | a or b belongs to the first or third block},(130)∆0= {αab ∈∆ | a andb belong to the second block}.

For the sake of completeness and for later use, we specify explicitly the elements of∆ andthe resulting restricted root system�∆, with the corresponding multiplicities. Introducingan orthonormal basis{e1, . . . , ep} of a∗0 by

(131)ek

P 0 00 0 00 0 −P

= Pkk, for

P 0 00 0 00 0 −P

∈ a0,

we see that restricting the roots ofsl(n,C) to a∗0 provide 2p(p − 1) restricted roots ofmultiplicity 2 defined by

αkl = αθ(l)θ(k) = ek − el,

αlk = αθ(k)θ(l) = el − ek,

αkθ(l) = αlθ(k) =+ek + el,

(132)αθ(k)l = αθ(l)k =−ek − el,

where 1� k < l � p, 2p restricted roots of multiplicity 1 defined by

αkθ(k) =+2ek,

(133)αθ(k)k =−2ek,

where 1� k � p and finally 2p restricted roots of multiplicity 2(q − p) defined by

αkm = αmθ(k) =+ek (p+ 1 � m � q),

(134)αmk = αθ(k)m =−ek (p+ 1 � m � q),

where 1� k � p. The corresponding root generators continue to be given byEαab ≡ Eab

where 1� a �= b � n, with a or b belonging to the first or third block, since these alsosatisfy the constraints listed in Eq. (A.28). However, in contrast to thesl(n,C) case studiedin Section 2.2, we must now deal with the possibility of encountering non-trivial couplingconstantsgαab ≡ gab (1 � a, b � n). A priori, these are only defined whena or b belongto the first or third block, but their definition will be extended to all values ofa andb bysetting

(135)gab = 0, if a andb belong to the second block,

as postulated in Eq. (85), and by adding the convention

(136)gaa = 0.


Using Eq. (62), we can then derive the following simple expressions for the combinationsΓαab,αcd ≡ Γab,cd andΓ θ

αab,αcd≡ Γ θ

ab,cd introduced in Eq. (9) and Eq. (88), respectively,

(137)Γab,cd = gadδbc − gbcδad,

(138)

Γ θab,cd =

1

4

(gadδbc − gbcδad + gθ(a)dδθ(b)c − gθ(b)cδθ(a)d

+ gaθ(d)δbθ(c) − gbθ(c)δaθ(d)+ gθ(a)θ(d)δθ(b)θ(c)− gθ(b)θ(c)δθ(a)θ(d)

),

valid for 1� a, b, c, d � n with a �= b andc �= d . Next, we have for the Weyl reflections

(139)θsabθ = sθ(a)θ(b).

This relation implies that the Weyl group W(sl(n,C), θ) associated with this symmetricpair can be realized not only as the quotient of a subgroup but actually as a subgroup ofthe Weyl group W(sl(n,C)) of sl(n,C), namely the subgroup generated by the reflectionssaθ(a) = sθ(a)a with index a in the first block (or equivalently in the third block) and bythe products of reflectionssabsθ(a)θ(b) = sθ(a)θ(b)sab with both indicesa andb in the firstblock (or equivalently in the third block). Under the action of this group,∆ decomposesinto three distinct orbits, namely

(140)

{αab a andb belong to the first or third block

with b �= a, b �= θ(a)

},

(141){αaθ(a) | a belongs to the first or third block},

(142)

a belongs to the first or third blockandb to the second block

αab ora belongs to the second blockandb to the first or third block

,

and so there will be three independent coupling constants which, following Refs. [4–6], weshall denote byg, g1 andg2. Explicitly,

• gab = g if a andb belong to the first or third block, withb �= a, b �= θ(a),

• gaθ(a) = gθ(a)a = g2 if a belongs to the first or third block,• gab = g1 if a belongs to the first or third block andb to the second block or ifa

belongs to the second block andb to the first or third block.

Note also that at least the coupling constantg should be non-zero, since otherwise, theHamiltonian of the corresponding Calogero model (as given in Eqs. (81), (82) or (83),(84)) would decouple, that is, would decompose into the sum ofp copies of the sameHamiltonian for a system with only one degree of freedom, and such a system is trivially(Liouville) integrable.

With these preliminaries out of the way, we proceed to search for a collection of matricesF+αab

≡ F+ab (1 � a �= b � n, with a or b belonging to the first or third block) that lie inib0, with F+ba = F+ab = F+θ(a)θ(b) as required in Eq. (99), satisfying the property (112) of


covariance under the Weyl group and the combinatoric identity

2gcd

((F+ab

)cc− (F+ab)dd

)= 2gcdαcd

(F+ab

)= Γ θcd,ab + Γ θ

cd,ba

(143)

= 1

4

(gcbδda − gdaδcb + gcaδdb − gdbδca

+ gθ(c)bδθ(d)a − gθ(d)aδθ(c)b + gθ(c)aδθ(d)b − gθ(d)bδθ(c)a

+ gcθ(b)δdθ(a)− gdθ(a)δcθ(b) + gcθ(a)δdθ(b)− gdθ(b)δcθ(a)

+ gθ(c)θ(b)δθ(d)θ(a)− gθ(d)θ(a)δθ(c)θ(b)

+ gθ(c)θ(a)δθ(d)θ(b)− gθ(d)θ(b)δθ(c)θ(a)),

valid for 1� a, b, c, d � n with a �= b andc �= d , corresponding to Eq. (102). Note alsothat the conditionF+ab ∈ ib0 means thatF+ab must be a real traceless diagonal matrix whoseentries satisfy

(144)(F+ab

)cc= (F+ab)θ(c)θ(c),

whereas the invariance under the reflectionssaθ(a) = sθ(a)a with indexa in the first block(or equivalently in the third block), which belong to the Weyl group W(sl(n,C), θ) and acttrivially on the space ib0, means that

(145)F+ab = F+θ(a)b = F+aθ(b) = F+θ(a)θ(b).

Therefore, it suffices to determineF+ab in the following three cases: whena andb bothbelong to the first block, witha �= b, whena belongs to the first block whileb belongsto the third block, withb = θ(a), and finally whena belongs to the first block whilebbelongs to the second block. Moreover, the system of equations (143) is obviously invariantunder the substitutionc→ θ(c), d→ θ(d) and under the exchange ofc andd , as well asunder that ofa andb. Now obviously, the r.h.s. of Eq. (143) vanishes if the sets{c, d}and{a, b, θ(a), θ(b)} are disjoint, so Eq. (143) will in this case be satisfied, independentlyof the choice of the coupling constants, if we assume all entries of the matrixF+ab except(F+ab)aa, (F+ab)bb, (F+ab)θ(a)θ(a) and(F+ab)θ(b)θ(b) to be equal among themselves. Similarly,the case when the set{c, d} is contained in the set{a, b, θ(a), θ(b)} will provide relationsbetween the remaining entries of the matrixF+ab. To find these, we observe that, due to theinvariance mentioned above, it is in this case sufficient to evaluate Eq. (143) withc = a

andd = b, d = θ(b), d = θ(a). Now whena andb both belong to the first block, witha �= b, the first two cases (d = b andd = θ(b)) lead to the relation

2g((

F+ab)aa− (F+ab)bb)= 0,

while the third (d = θ(a)) gives a trivial identity. Similarly, whena belongs to the firstblock while b belongs to the third block, withb = θ(a), the first and last case coincide(d = b = θ(a)) and give a trivial identity, while the second (d = θ(b)) is excluded. Thustaking into account the fact thatg �= 0, we can cover both situations in one stroke by writing

F+ab =1

4λab(Eaa +Eθ(a)θ(a)+Ebb +Eθ(b)θ(b))+ 1

nτab1,


with real coefficientsλab andτab to be determined. Finally, whena belongs to the firstblock whileb belongs to the second block, the first two cases coincide (d = b = θ(b)) andlead to the relation

2g1((

F+ab)aa− (F+ab)bb)= 1

2g2,

while the third (d = θ(a)) gives a trivial identity, suggesting that we should write

F+ab =1

2λab(Eaa +Eθ(a)θ(a))+µabEbb + 1

nτab1,

with real coefficientsλab, µab andτab to be determined, subject to the constraint

2g1(λab − 2µab)= g2.

It remains to check Eq. (143) for the cases where the intersection of the sets{c, d}and {a, b, θ(a), θ(b)} contains precisely one element. Once again, due to the invariancementioned above, it is in this case sufficient to evaluate Eq. (143) withc = a and withc = b, supposing thatd /∈ {a, b, θ(a), θ(b)}. Whena andb both belong to the first blockand also whena belongs to the first block whileb belongs to the third block, withb= θ(a),this equation is solved identically, independently of the choice of the coupling constants,by puttingλab =−1. On the other hand, whena belongs to the first block whileb belongsto the second block, we obtain non-trivial relations between the coupling constants, whichare the following. Ifc = a �= b, choosingd /∈ {a, b, θ(a)} to belong to the first or thirdblock gives

2gλab =−g1,

while choosingd /∈ {a, b, θ(a)} to belong to the second block (which is only possible whenq � p+ 2) gives

g1λab = 0.

Similarly, if c= b �= a, choosingd /∈ {a, b, θ(a)} to belong to the first or third block gives

2g1µab =−g,

while choosingd /∈ {a, b, θ(a)} to belong to the second block (which is only possible whenq � p+ 2) gives

g1= 0.

Finally, fixing the coefficientsτab by requiring thatF+ab should be traceless, we arrive atthe following

Proposition 3. For the symmetric pairs associated with the complex Grassmannians,which are the symmetric spaces of theAIII -series corresponding to the choice(114)withn= p+ q andp � q , a non-trivial solution to the combinatoric identity(102)exists if andonly if eitherq = p or q = p+ 1 and is then given as follows:


• q = p: The functionF+ is given by

(146)F+ab =−1

4(Eaa +Eθ(a)θ(a)+Ebb +Eθ(b)θ(b))+ 1

n1;

• q = p+ 1: The functionF+ is given by

(147)F+ab =−1

4(Eaa +Eθ(a)θ(a)+Ebb +Eθ(b)θ(b))+ 1

n1,

whena andb both belong to the first or third block and by

(148)F+ab =−g1

4g(Eaa +Eθ(a)θ(a))− g

2g1Ebb + 1

n

(g1

2g+ g

2g1

)1,

whena belongs to the first or third block whileb belongs to the second block, providedthat the coupling constantsg, g1 andg2 satisfy the relations

(149)g �= 0, g1 �= 0,

together with

(150)g21 − 2g2+ gg2= 0.

This solution also satisfies the constraint equation(110).

The only statement that has not yet been proved is the last one, referring to theidentity (110). This is easily done by computing separately the sum over the different Weylgroup orbits. Using the matrices introduced in Eq. (125), we see that the first gives 4 timesa contribution of the form

g

p∑a,b=1a �=b

F+ab = g

(−p− 1

2I2p + p(p− 1)

n1

),

while the second gives 2 times a contribution of the form

g2

p∑a=1

F+aθ(a) = g2

(−1

2I2p + p

n1

),

and finally the third (which only exists ifq > p) gives 4 times a contribution of the form

g1

p∑a=1

q−p∑b=1

F+ab = g1

(− g1

4g(q − p)I2p − g

2g1pIq + 1

n

(g1

2g+ g

2g1

)p(q − p)1

).

Summing up everything and using thatq − p is either 0 or 1 and that in the second case,Eq. (150) has to be taken into account, we get zero, as desired.

It is interesting to note that the condition derived during the proof of Proposition 3,according to whichq − p should be either 0 or 1, admits a natural interpretation: itcharacterizes exactly those Grassmannians for which the root system∆0 of the centralizerz of a in k is empty, which means thatz should be Abelian. Therefore, a glance atEq. (97) shows that the first two terms in the ellipticR-matrix (96) for symmetric pairs


combine into a pure Cartan–Cartan type contribution, in the same way as for the ellipticR-matrix (38) for Lie algebras. In more geometric terms, the criterion states that thesymmetric space in question should be such that the centralizers of its maximal flat totallygeodesic submanifolds (which are all conjugate) should be Abelian.

To conclude this section, it seems worthwhile to briefly discuss a few limiting cases,noting that the vanishing of certain coupling constants has the effect of reducing therestricted root system to a subsystem. In fact, Eqs. (132)–(134) reveal that the system ofrestricted roots associated with the complex Grassmannians SU(n,n)/S(U(n)×U(n)) andSU(n,n+ 1)/S(U(n)×U(n+ 1)) is of typeCn and of typeBCn, respectively.7 Insertinginto the Hamiltonian (81) or (83) and using the abbreviation (79) for the potential, we arriveat the following special cases:

• SU(n,n)/S(U(n)×U(n)), g2= 0:

(151)H(q,p)= 1

2

n∑k=1

p2k + g2

∑1�k �=l�n

(V (qk − ql)+ V (qk + ql)

).

This is the Calogero Hamiltonian associated to the root systemDn.

• SU(n,n)/S(U(n)×U(n)), g2 arbitrary:

H(q,p)= 1

2

n∑k=1

p2k + g2

∑1�k �=l�n


)(152)+ g2

2

n∑k=1

V (2qk).

This is the Calogero Hamiltonian associated to the root systemCn.• SU(n,n+ 1)/S(U(n)×U(n+ 1)), g2= 0:

H(q,p)= 1

2

n∑k=1

p2k + g2

∑1�k �=l�n


)(153)+ 2g2

1

n∑k=1

V (qk).

This is the Calogero Hamiltonian associated to the root systemBn, subject to theconstraints (149) and (150), which reduce tog2

1 = 2g2 �= 0.• SU(n,n+ 1)/S(U(n)×U(n+ 1)), g2 arbitrary (the general case):

H(q,p)= 1

2

n∑k=1

p2k + g2

∑1�k �=l�n


)(154)+ g2

2

n∑k=1

V (2qk)+ 2g21

n∑k=1

V (qk).

7 We now usen instead ofp, as before.


This is the Calogero Hamiltonian associated to the non-reduced root systemBCn,subject to the constraints (149) and (150).

Thus all Calogero Hamiltonians associated with the classical root systems are completelyintegrable, admitting a Lax pair formulation with a dynamicalR-matrix.

4. Conclusions and outlook

In the present paper, we have investigated the question of integrability of the Calogeromodels, systematizing the traditional approach of Olshanetsky and Perelomov [4–6] basedon Lax representations using (semi-)simple Lie algebras or symmetric pairs and at thesame time extending it to provide dynamicalR-matrices (a concept that did not exist at thetime when the approach of Olshanetsky and Perelomov was originally developed). We findthat the existence of a Lax representation and that of a dynamicalR-matrix both hinge onone and the same set of algebraic constraints that are here written down for the first time.For the time being, these take the form of a combinatoric identity for a certain vector-valued functionF on the pertinent root system that appears both in the Lax pair and theR-matrix and whose deeper meaning is yet to be discovered: more precisely, the questionis what might be the additional algebraic structure on the pertinent (semi-)simple Liealgebra or symmetric pair provided by the functionF and whose structure equations are theconstraints (3) or (4), respectively. We also show that among the simple Lie algebras, thosebelonging to the classicalA-series are the only ones to admit a solution of these constraints.Of course, this comes as no surprise since it is general wisdom that the Calogero modelsbased onsl(n,C) are the only ones to admit a Lax representation of the traditional form(see Eq. (33) and Eq. (34)). However, our method apparently provides the first rigorousproof of this assertion. For the case of symmetric pairs, our analysis is not yet complete,but for the time being is restricted to the classicalAIII-series of complex GrassmanniansSU(p, q)/S(U(p)×U(q)): we show that among these, the ones wherep= q or p = q+1or q = p + 1 are the only ones to admit a solution of these constraints. However, this isenough to provide symmetric pair type Lax representations and dynamicalR-matrices forCalogero models based on all of the other classical root systems (Bn, Cn, Dn and evenBCn).

It seems worth mentioning that functions analogous to our functionF also appear invarious other contexts, for example, in the construction of Lax pairs and dynamicalR-matrices for Calogero models through Hamiltonian reduction [11]. However, no mentionis made in Ref. [11] of the constraints that this function must satisfy in order for the methodto work. We believe that analyzing the problem from this point of view would greatly helpto unveil the aforementioned deeper meaning of these constraints.

Regarding the various alternative Lax representations for the Calogero models that havebeen presented recently in the literature [12–19], it must be stressed first of all that none ofthese proposals deals with the question of how to constructR-matrices. This is certainlya challenge for future work, but it should be emphasized that the algebraic context issomewhat different. A common denominator of all these approaches is that they replace


the semisimple Lie algebrag—or rather its universal enveloping algebra U(g)—by a moregeneral associative algebraA, assuming thatL andM take values inA whereasR takesvalues inA ⊗ A (see Eq. (27) and Eq. (30)). This can certainly be done without doingharm to the mathematical status of the basic equations, namely the Lax equation (1) andthe Poisson bracket relation (2); in fact, even in the traditional Lie algebraic framework,a correct mathematical interpretation of the latter requires passing fromg to U(g). Thequestion is as to what is the correct choice forA, which is usually taken to be a matrixalgebra defined in terms of generators and relations similar to but not identical with thoseof ordinary Lie algebras. Thus although these approaches do provide Lax representations insituations where the traditional approach [4–6] fails, we still think that the results reportedin this paper do shed new light on the integrability of the Calogero models.

Finally, while preparing the final version of this paper, which is a revised version ofan earlier manuscript, we became aware of important recent work [32,33] where it isshown that, within the traditional Lie algebraic framework, the Calogero models basedon sl(n,C), degenerate as well as elliptic, admit a “gauge transformation” taking thedynamicalR-matrix into a numerical one: this is achieved by explicitly constructing agroup-valued function on the configuration space which is used to conjugate the standardLax pair and dynamicalR-matrix of this model into a new Lax pair and a numericalR-matrix. Should that turn out to be possible in general, it would provide an answer to thequestion that was the original motivation of our work: it would mean that the appearanceof dynamicalR-matrices is merely a gauge artifact. This question is presently underinvestigation.

Acknowledgements

The authors would like to thank the referee for his critical comments and suggestions,which have helped to improve the manuscript.

Appendix A. Conventions and notation

Root systems belong to the most important and intensively studied objects of discretegeometry, partly because of the central role they play in various areas of algebra. The mosttraditional application of root systems is to the classification of (semi-)simple Lie algebrasand of symmetric pairs of Lie algebras, due to Cartan. In the first case, one obtains onlythe so-called reduced root systems, whereas the second case leads to general root systems,reduced as well as non-reduced ones, and to the appearance of non-trivial multiplicities.

Throughout this paper,g denotes a complex semisimple Lie algebra. Fixing a Cartansubalgebrah of g, let h∗ denote its dual,∆⊂ h∗ the root system ofg with respect tog andh∗

Rthe real subspace ofh∗ spanned by the rootsα (α ∈ ∆). Furthermore, let(. , .) be the

invariant bilinear form ong, normalized so that with respect to the induced bilinear formsonh andh∗, also denoted by(. , .) and related to each other by


(A.1)(α,β)= (Hα,Hβ),

with

(A.2)(Hα,H)= α(H), for all H ∈ h,

the long roots have length√

2. The form(. , .) is positive definite on the real subspace ofh spanned by the generatorsHα (α ∈ ∆), which is denoted byhR. We also fix a WeylchamberC and a Weyl alcoveA; these are open subsets ofhR with boundaries formed byhyperplanes on which some rootα ∈∆ vanishes and by hyperplanes on which some rootα ∈ ∆ assumes integer values, respectively. Recall that the choice of a Weyl chamberC

establishes an ordering in∆, the positive roots being the ones that assume strictly positivevalues onC. The different Weyl chambers are permuted by the elements of the Weyl groupW(g) of g, which is the finite group generated by the reflectionssα in hR along the rootsα ∈∆. Finally we choose a basis{Eα | α ∈∆} of generators ing normalized so that

(A.3)(Eα,Eβ)= δα+β,0,

and therefore obeying the standard Cartan–Weyl commutation relations

(A.4)[H,Eα] = α(H)Eα, for all H ∈ h,

(A.5)[Eα,E−α] =Hα,

(A.6)[Eα,Eβ ] =Nα,βEα+β,

with structure constantsNα,β which satisfyNβ,α = −Nα,β and which, by definition, aresupposed to vanish wheneverα + β is not a root—in particular whenα ± β = 0. Theseconditions determine uniquely theHα (according to Eq. (A.2)) but not theEα , because itis always possible to rescale the root generatorsEα according to

(A.7)Eα →E′α = aαEα, with aαa−α = 1,

which entails a rescaling of the structure constants according to

(A.8)Nα,β →N ′α,β =aαaβ

aα+β

Nα,β.

This freedom can be used in order to impose the additional normalization conditions

(A.9)N−α,−β =−Nα,β .

As it it turns out, this determines the structure constantsNα,β uniquely up to signs. Anotherimportant relation between the structure constants that we use frequently is the followingcyclic identity: if α,β, γ ∈∆ are any three roots that add up to zero, then

(A.10)Nα,β =Nβ,γ =Nγ,α.

In addition, assuming thatr is the rank ofg, we may choose an orthonormal basis{H1, . . . ,Hr} of hR; then

(A.11)r∑

j=1

α(Hj )Hj =Hα.


In the computations made in this paper, we make extensive use of these relations, oftenwithout further mention. For proofs, see, for example, [30].

In the symmetric space situation, we assume more specifically thatg is the complexifi-cation of a real semisimple Lie algebrag0 and that we are given a direct decomposition

(A.12)g= k⊕m,

of g induced from a Cartan decomposition

(A.13)g0= k0⊕m0,

of g0; then

(A.14)gk = k0⊕ im0,

defines a compact real formgk of g. Writing σ for the conjugation ing with respectto g0 and τ for the conjugation ing with respect togk (note that these commute), thecorresponding Cartan involutionθ is simply their product:θ = στ = τσ . This is aninvolutive automorphism ofg which is +1 on k and−1 on m; moreover, it preservesthe invariant bilinear form(. , .) on g introduced before, which means that the direct sumsin Eqs. (A.12)–(A.14) are orthogonal, characterizing the direct decompositions (A.13) and(A.14) as being associated with a Riemannian symmetric space of the non-compact typeand of the compact type, respectively. The decomposition of elements corresponding to thedirect decompositions (A.12)–(A.14) into eigenspaces underθ is written as

(A.15)X =Xk +Xm,

where obviously

(A.16)Xk = 1

2(X+ θX), Xm = 1

2(X− θX).

Next, leta0 denote a maximal Abelian subalgebra ofm0, h0 a maximal Abelian subalgebraof g0 containinga0, b0 the orthogonal complement ofa0 in h0 and leta, b andh denotethe corresponding complexifications; thenh is a Cartan subalgebra ofg and

h= b⊕ a, with b= h∩ k, a= h∩m,

(A.17)h0= b0⊕ a0, with b0= h ∩ k0, a0= h∩m0.

The real subspacehR of h introduced before then splits according to

(A.18)hR = ib0⊕ a0,

and the decomposition of elements corresponding to the direct decompositions (A.17) and(A.18) into eigenspaces underθ is written as

(A.19)H =Hb+Ha,

where obviously

(A.20)Hb = 1

2(H + θH), Ha = 1

2(H − θH).

Next, we note that the conjugationsσ , τ and the involutionθ induce bijective transforma-tions of the root system∆ onto itself that for the sake of simplicity will again be denoted


by σ , τ andθ , respectively; they are characterized by the condition

σ(gα)= g

σα, τ(gα)= g

τα, θ(gα)= g

θα,

wheregα is the one-dimensional subspace ofg generated byEα , and are explicitly givenby

(σα)(H)= α(σH), (τα)(H)= α(τH), (θα)(H)= α(θH),

(A.21)for all H ∈ h,

In the main body of the paper we writeα for the linear functional ona obtained byrestricting a given linear functionalα on h to a. In this way the root system∆⊂ h∗ givesrise to the restricted root system�∆⊂ a∗:

(A.22)�∆= {α = α|a | α ∈∆}.

Accordingly the root system∆ itself decomposes into two parts,

(A.23)∆= ∆∪∆0,

defined by

(A.24)∆= {α ∈∆ | θα �= α}= {α ∈∆ | α �= 0

},

(A.25)∆0={α ∈∆ | θα = α

}= {α ∈∆ | α = 0},

where obviously∆0 is the root system of the centralizerz of a in k:

(A.26)z= {Z ∈ k | [Z,X] = 0, for all X ∈ a}.

In contrast to ordinary roots, restricted roots will in general have non-trivial multiplicities,defined as follows:

(A.27)mλ = card{α ∈∆ | α = λ

}.

As before, we also fix a Weyl chamberC and a Weyl alcoveA, but these are now opensubsets ofa0 (rather than ofhR) with boundaries formed by hyperplanes on which somerootα ∈∆ vanishes and by hyperplanes on which some rootα ∈∆ assumes integer values,respectively. The choice of such a Weyl chamberC now establishes an ordering only in∆,the positive roots in∆ being the ones that assume strictly positive values onC, but this canof course be extended to an ordering in∆. It is interesting and useful to note the behaviorof the conjugationsσ , τ and of the involutionθ with respect to this ordering:

α ∈ ∆± ⇒ σα ∈ ∆±, τα =−α, θα ∈ ∆∓,α ∈∆0⇒ σα =−α, τα =−α, θα = α.

Again, the different Weyl chambers are permuted by the elements of the Weyl groupW(g, θ) of (g, θ), which is the quotient of the subgroup Wθ (g) of the Weyl group W(g)of g consisting of those elements that commute with the involutionθ , modulo thoseelements that act trivially ona0. Finally we assume the basis{Eα | α ∈ ∆} of generatorsin g to be chosen so that, apart from the normalization condition (A.3), the Cartan–Weyl


commutation relations (A.4), the additional normalization condition (A.9) and the cyclicidentity (A.10), we have the following simple behavior under the conjugationsσ , τ andthe involutionθ :

(A.28)σEα =−Eσα, τEα =−E−α, θEα =Eθα.

In particular, the structure constants areθ -invariant:

(A.29)Nθα,θβ =Nα,β.

(That such a choice is always possible will be shown below.) In addition, assuming thatr

is the rank of the symmetric pair(g, θ) andr + s is the rank of the Lie algebrag (which bydefinition means thatr = dima andr + s = dimh), we may also choose an orthonormalbasis{H1, . . . ,Hr} of a0 together with an orthonormal basis{Hr+1, . . . ,Hr+s} of ib0 toform an orthonormal basis{H1, . . . ,Hr+s} of hR; then

(A.30)r∑

j=1

α(Hj )Hj = (Hα)a,

r+s∑j=r+1

α(Hj )Hj = (Hα)b.

We conclude this appendix with a proof of the statement made in Eq. (A.28) andEq. (A.29) above, given the fact that this proof is non-trivial and apparently cannot befound in the literature. The first step consists in going through the proof of Theorem 5.5in [30, pp. 176,177] which shows how to extend the transformation−id on the Cartansubalgebrah of g to an automorphismϕ of g (which is by no means uniquely determinedby these conditions alone, but this will soon turn out to be an advantage rather than adrawback), and how to choose a basis{Eα | α ∈∆} of generatorsEα ∈ gα associated withthe rootsα normalized according to(

Eα, E−α

)= 1, i.e.,[Eα, E−α

]=Hα,

and satisfying

ϕ(Eα

)=−E−α,

which also guarantees that the structure constantsNα,β defined by[Eα, Eβ

]=Nα,βEα+β,

satisfy the desired condition (A.9). In a second step, we consider the behavior of this basisunder the conjugationsσ andτ as well as under the involutive automorphismθ . Supposethat we have

σEα =−c−αEσα, τ Eα =−dαE−α,

with complex constantscα, dα �= 0. Then sinceσ andτ are antilinear and also involutive(σ 2= 1= τ2), we get

c−αc−σα = 1, dαd−α = 1.


On the other hand,σ andτ preserve the commutator and hence also the scalar product(. , .) in g, which means that for allX,Y ∈ g,

(σX,σY )= (X,Y ), (τX, τY )= (X,Y ).

(The proof is analogous to that of the well-known statement that automorphisms of Liealgebras always preserve the Killing form.) Hence we also get

c−αcα = 1, dαd−α = 1.

Moreover,σ andτ commute, since we haveσ = τ on k andσ =−τ onm; this implies

dαcαE−σα = στEα = τσ Eα = c−αdσαE−σα,

which by using the previously derived relations can be brought into the form

cσαdσα = cαdα.

Note also that the coefficientsdα =−(Eα, τ Eα) are all not just real but positive, since themap

g× g→C, (X,Y ) %→ (X, τY ),

defines a negative definite Hermitean sesquilinear form ong. Thusθ is given by

θEα = fαEθα,

with

fα = cαdα,

and we have

fα+βNθα,θβEθα+θβ =Nθα,θβθEα+β = θ([

Eα, Eβ

])= [θEα, θEβ

]= fαfβ

[Eθα, Eθβ

]= fαfβNθα,θβEθα+θβ.

Let us summarize the conditions on the coefficientscα , dα andfα that we have derived.For all α ∈ ∆, cα is a phase factor (|cα| = 1), dα is a scale factor (dα > 0) andfα , theirproduct, is a non-zero complex number (fα �= 0) satisfying

f−α = f−1α , fσα = fα, fθα = f−1

α ,

and such that forα,β ∈∆ with α+ β ∈∆,

fα+β = fαfβ .

Therefore, there exists a uniquely determined generatorHθ ∈ a such that for allα ∈∆,

fα = exp(α(Hθ)

).

(For the proof, choose any ordering in∆ and let theαj (j = 1, . . . , r + s) be thecorresponding simple roots; then defineHθ by fαj = exp(αj (Hθ)).) We define

aα = exp

(−α(Hθ )

2

),


so that

a−α = a−1α , aσα = aα, aθα = a−1

α ,

and

a2α = f−1

α ,

and put

Eα = aαEα.

In this new basis, we have

θEα = aαθEα = aαfαEθα = aαa−1θα fαEθα =Eθα,

τEα = aατ Eα =−aαdαE−α =−aαa−1−αdαE−α =−E−α,

σEα = aασ Eα =−aαc−αEσα =−aαa−1σα c−αEσα =−Eσα,

whereas the structure constantsNα,β remain unaltered under the transition from theprevious basis to the new one, because it is achieved by means of an inner automorphismof g, namely exp(ad(−Hθ/2)):

Eα = exp

(ad

(−Hθ

2

))Eα.

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