Dynamical R-Matrices for Calogero Models

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

Dynamical R-matrices for Calogero models Michael Forger 1, Axel Winterhalder 2 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 as well as elliptic, associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras, where “integrability” is understood to encompass not only the existence of a Lax 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, we show that integrability of these models, in this sense, can be reduced to the existence of a certain function, denoted here by F , deﬁned on the relevant root system and taking values in the respective Cartan subalgebra, subject to a rather simple set of algebraic constraints: these ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, all given by explicit formulas. We also show that among the simple Lie algebras, only those belonging to the A-series admit a solution of these constraints, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p, q)/S(U(p) × U(q)), allows non-trivial solutions when |p − q| 1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our method provides new ones, namely for the degenerate models when |p − q|=1 and for the elliptic models when |p − q|=1orp = q.  2002 Elsevier Science B.V. All rights reserved.

PACS: 02.30.Ik Keywords: Integrable systems; Dynamical R-matrices; Calogero models

1. Introduction

The Calogero–Moser–Sutherland models, or Calogero models, for short, constitute an important class of completely integrable Hamiltonian systems, intimately related to the theory of (semi-)simple Lie algebras [1–6]. Unfortunately, the group-theoretical underpinnings 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)00506-5 524 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 root systems of ordinary (semi-)simple Lie algebras and/or affine Lie algebras. Indeed, it has proven surprisingly difficult to extend the results obtained for the Calogero model based on sl(n, C), such as the Lax pair found in Ref. [2] for the degenerate model and in Ref. [7] for the elliptic model or the R-matrix given in Ref. [8] for the degenerate model and in Ref. [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. One of these is based on Hamiltonian reduction, starting out from the geodesic flow on the corresponding group and leading to the construction of Lax pairs and dynamical R- matrices [11]. However, the approach is rather indirect and the results do not really crystallize 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 simple Lie algebras such as sl(n, C) but fails for others. Other authors have addressed the entire issue from a different point of view, namely by searching for new Lax pairs by taking values not in the (semi-)simple Lie algebra or symmetric pair used in the traditional approach [4–6] but rather in the algebra of matrices over a representation space for some other 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 algebra but in an arbitrary representation (not just the adjoint), while others [13–19] propose to modify not only the representation but also the structure of the algebraic object that is being 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 situations where the traditional approach [4–6] fails, they do not fit into the standard Lie algebraic framework, so the problem of how to construct R-matrices in this more general context remains open. A priori, it is not even clear what would be the meaning of the concept of an R-matrix in a theory of integrable systems based on more general Lax representations such 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, our approach in the present paper has originally been motivated by a different question, namely the quest for a better understanding of the concept of a dynamical R-matrix. One of the central features that distinguishes the Calogero models from the (much simpler) Toda models is that their R-matrix is not a numerical object but has a dynamical character. Generally speaking, the main role of the R-matrix associated with a given Lax equation L˙ =[L,M], (1) is to control the Poisson brackets between the components of the Lax matrix L [20], according to the following well-known formula:

{L1,L2}=[R12,L1]−[R21,L2]. (2)

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

Numerical R-matrices are well understood mathematical objects that play an important role in the theory of quantum groups, or rather its classical counterpart, the theory of Lie bialgebras [21,22]. They can be viewed as providing the mathematical expression for the concept of algebraic integrability, which is considerably stronger than that of Liouville integrability. Indeed, the latter is really just a statement about the possibility to separate the dynamical variables of a given model into mutually independent conjugate pairs, whereas the former actually allows to integrate the equations of motion in terms of explicitly known functions. (For example, any system with only one degree of freedom is Liouville integrable, and so is the motion of a particle in ordinary three-dimensional space under the influence of an arbitrary central potential, but only the Kepler problem is also algebraically integrable, due to the presence of an additional symmetry generated by the conserved Runge–Lenz vector.) On the other hand, the mathematical status of dynamical R-matrices is not nearly as clear: there does not even seem to exist any generally accepted definition. The first examples of dynamical R-matrices appeared in the study of the nonlinear sigma models on spheres [23–26] and, more generally, on Riemannian symmetric spaces [27]. Unfortunately, the analysis of the underlying algebraic structure is hampered by technical problems due to the fact that these models are not ultralocal [28] and hence Poisson brackets between components of the transition matrix T constructed from the Lax matrix L show discontinuities that must be removed by regularization. Therefore, it seems much more promising to undertake such an analysis not in the context of two-dimensional field theory but in the context of mechanics, where such technical problems are avoided due to finite-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 dynamical R-matrices make their appearance. In view of this situation, we have performed a systematic study of the integrability of the Calogero models, within the traditional formulation outlined long ago by Olshanetsky and Perelomov [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 encompass the construction not only of a Lax representation but also of a dynamical R-matrix, thus characterizing these models as examples of Hamiltonian systems that apparently occupy an intermediate status between Liouville integrable and algebraically integrable. As usual, we shall require the R-matrix to depend only on the coordinates but not on the momenta: this can be motivated (a) by realizing that the standard L-matrix [2,4–7] is the sum of a term linear in the momenta with constant coefficients and a term depending only on the coordinates, forcing the l.h.s. of Eq. (2) to be momentum independent, and (b) by noting that the simplest way to guarantee momentum independence for the r.h.s. of Eq. (2) is to assume R itself to be momentum independent and chosen such that the two contributions to 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 further restrictions on the R-matrix; in particular, they force its root–root part to have the standard form given by the last term in Eq. (35) or Eq. (38) below and to be accompanied by an appropriate 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 526 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 have a definite transformation law under the pertinent involutive automorphism θ,sogiven the fact [4–6] that the standard L-matrix is odd under θ while the standard M-matrix is even under θ, we require R to be even under θ ⊗ 1 and odd under 1 ⊗ θ. (In the elliptic case, each of these automorphisms must be combined with a change of sign in the corresponding spectral parameter(s), as suggested by constructions for twisted affine Kac– Moody algebras.) Again, this strongly restricts the form of R, forcing its root–root part to have the form given by the last term in Eq. (93) or Eq. (96) below and to be accompanied by an appropriate Cartan–root correction term plus, in the elliptic case, a Casimir type Cartan–Cartan correction term. In both cases, we write down a general Ansatz for the Cartan–root correction term in terms of an appropriate vector-valued function F defined on the pertinent root system. In this way, we arrive at a general Ansatz for the Lax pair and for the R-matrix through which (a) the proof of equivalence between the equation of motion and the Lax equation (1) and (b) the proof of the Poisson bracket relation (2) can both be reduced to one and the same set of algebraic constraints on the function F .These state that for any two roots α and β

gαα(Fβ ) − gβ β(Fα) = Γα,β, (3) in the case of (semi-)simple Lie algebras g and − = θ gαα(Fβ ) gβ β(Fα) Γα,β, (4) θ in the case of symmetric pairs (g,θ), where the coefficients Γα,β and Γα,β are defined in terms of the structure constants of g and the coupling constants of the model; see Eq. (9) and Eq. (88) below. It is worth noting that this is a highly overdetermined system of equations which will admit non-trivial solutions only in special circumstances; its deeper algebraic meaning is yet to be discovered. The paper is divided into two main sections, dealing with the Calogero models defined in terms 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 to specifying the Lax pair and the dynamical R-matrix, in terms of the function F .Themain goal is to show how (a) the proof of equivalence between the equation of motion and the Lax equation (1) and (b) the proof of the Poisson bracket relation (2) can both be reduced to the combinatoric identity given above. Moreover, we show in Section 2 that the only simple Lie algebras for which a function F with the desired property exists are those belonging to the A-series, i.e., the Lie algebras sl(n, C); the resulting R-matrix coincides with that given in Ref. [8] for the degenerate Calogero model and with that given in Ref. [9] for the elliptic Calogero model. In Section 3, we study as an example the AIII-series of symmetric pairs, corresponding to the complex Grassmannians SU(p, q)/S(U(p) × U(q)), and we prove by explicit calculations that there are non-trivial solutions, depending on free parameters, when |p−q| 1. For p = 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 model is new. For |p − q|=1, all R-matrices found seem to be new. Together, these also provide R-matrices for all Calogero models, degenerate and elliptic, associated with any one of the four classical root systems An, Bn, Cn and Dn. For the convenience of the reader, M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 527 the main conventions from the theory of semisimple Lie algebras and symmetric spaces that 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 understanding integrability of the Calogero models that have recently been proposed [11–19], as well as on 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 root system ∆ of a (semi-)simple Lie algebra g, referring to Appendix A for a summary of the conventions and notation used. The configuration space Q of the model is a Weyl chamber or a Weyl alcove within the real form hR of the Cartan subalgebra h, depending on whether we are dealing with the degenerate or elliptic model, respectively. In any case, it is an open subset of hR, so the phase space of the model is given by the associated cotangent bundle ∗ ∗ T Q = Q × hR, which for the sake of simplicity will often be identified with the tangent bundle TQ= Q × hR. The Hamiltonian for the degenerate Calogero model reads 1 r H(q,p)= p2 + g2 w α(q) 2 , (5) 2 j α j=1 α∈∆ while that for the elliptic Calogero model reads r 1 H(q,p)= p2 + g2 ℘ α(q) , (6) 2 j α j=1 α∈∆ where w is a smooth, real-valued function on R\{0} to be specified soon, while ℘ denotes the doubly periodic Weierstraß function (see below). The coefficients gα (α ∈ ∆)are positive real coupling constants, supposed to be symmetric:

g−α = gα. (7) A stronger assumption that we shall always make is that they are invariant under the action of the Weyl group W(g) of g:

gwα = gα, for all w ∈ W(g). (8) For later use we introduce the following combination of coupling constants and structure constants:

Γα,β = gα+β Nα,β . (9) Using the abbreviations r r p = pj Hj ,q= qj Hj , (10) j=1 j=1 528 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 we can write the Hamiltonian equations of motion as vector equations in hR: they read ˙ = ˙ =− 2 q p, p gαw α(q) w α(q) Hα, (11) α∈∆ for the degenerate Calogero model and 1 2 q˙ = p, p˙ =− g ℘ α(q) Hα, (12) 2 α α∈∆ for the elliptic Calogero model. Concerning the choice of the potential functions w in Eq. (5) and ℘ in Eq. (6), it is well- known that these functions must satisfy a certain set of functional equations that we shall discuss briefly in order to make our presentation self-contained. An elementary property is that w is supposed to be odd while ℘ is even: w(−t) =−w(t), ℘(−t)= ℘(t). (13) The basic functional equation imposed on w is the following: w(s) w(t) + w(s + t)+ w(s)w(t) = 0. (14) w(s) w(t) Differentiating with respect to s and to t and subtracting the results gives a second functional equation: w (s) w (t) − w(s + t)= w(s)w (t) − w (s)w(t), (15) 2w(s) 2w(t) The solutions are derived in Refs. [4–6]: there are essentially three different ones, satisfying the relation w(t) k − w(t)2 =− , (16) 2w(t) 2 where k is a numerical constant that allows to distinguish between them, namely 1 w(t) = , with k = 0, t 1 w(t) = , with k =+1, sin t 1 w(t) = , with k =−1, (17) sinh t which explains the terminology “rational” (for the first case), “trigonometric” (for the second case) and “hyperbolic” (for the third case). On the other hand, as explained in Ref. [31], ℘ is a meromorphic function on the complex plane with second order poles at the points of its lattice of periodicity Λ, consisting of the integer linear combinations n1ω1 + n2ω2 of its two basic periods ω1 and ω2, which are two arbitrarily chosen but fixed complex numbers; explicitly, it can be defined by the series expansion 1 1 1 ℘(z)= − . 2 2 2 z (z − n1ω1 − n2ω2) (n1ω1 + n2ω2) n1,n2∈Z (n1,n2)=(0,0) M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 529

For later use, it will be convenient to also introduce the Weierstraß sigma function, an entire function on the complex plane that can be defined by the infinite product expansion z σ(z)= z 1 − n1ω1 + n2ω2 n1,n2∈Z (n ,n )=(0,0) 1 2 2 × z + 1 z exp 2 , n1ω1 + n2ω2 2 (n1ω1 + n2ω2) and the Weierstraß zeta function, another meromorphic function on the complex plane with first order poles at the points of the same lattice of periodicity, which can be defined by the series expansion 1 1 z + n ω + n ω ζ(z)= + 1 1 2 2 . z z − n1ω1 − n2ω2 n1ω1 + n2ω2 n1,n2∈Z (n1,n2)=(0,0) Obviously, σ (z) ζ(z)= ,℘(z)=−ζ (z). (18) σ(z) Finally, we set

σ(z1 + z2) Φ(z1,z2) = . (19) σ(z1)σ(z2) Clearly, these functions have the following symmetry properties under reﬂection at the origin: σ(−z) =−σ(z), ζ(−z) =−ζ(z), ℘(−z) = ℘(z), (20)

Φ(−z1, −z2) =−Φ(z1,z2). (21) Moreover, they satisfy a set of functional equations that can all be derived from a single quartic 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 the following calculations: Φ(s,u)Φ(−s,u)=− ℘(s)− ℘(u) , (22) Φ(s,u)Φ (−s,u)− Φ (s, u)Φ(−s,u)= ℘ (s), (23) Φ(s,u)Φ (t, u) − Φ (s, u)Φ(t, u) = ℘(s)− ℘(t) Φ(s + t,u), (24) Φ(−s,v− u)Φ(s + t,v)+ Φ(−t,u− v)Φ(s + t,u)=−Φ(s, u)Φ(t, v), (25) Φ(−s,u− v)Φ(s,u) + ζ(v − u) + ζ(u) Φ(s,v)= Φ (s, v), (26) where Φ denotes the derivative of Φ with respect to the first argument and all arguments must be different from zero mod Λ. Note that these equations are not independent: for 530 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 example, Eq. (23) can be immediately derived from Eq. (22) by differentiation with respect to the first argument and from Eq. (24) by passing to the limit s + t → 0. Note also that these properties do not depend on the choice of the periods ω1 and ω2, but of course the latter should be taken so that ℘ assumes positive real values on the real axis, in order for the Hamiltonian (6) to be real and positive: this is the case, for example, when ω1 is real and τ = ω2/ω1 is purely imaginary [29]. With these preliminaries out of the way, we can pass to discuss the Lax pair and the R-matrix of the Calogero model. It is at this point that an important difference between the degenerate and the elliptic Calogero models appears, because the Lax pair and the R- 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 show up explicitly in the coefficient functions. Within the present framework, the Lax pair consists of two mappings 4 ∗ L : T Q → g and M : Q → g, (27) each of which will in the elliptic case depend on an additional spectral parameter u,such that the Hamiltonian equations of motion can be rewritten in the Lax form, namely

L˙ =[L,M], (28) for the degenerate Calogero model and L(u)˙ = L(u), M(u) , (29) for the elliptic Calogero model. Similarly, the dynamical R-matrix is a mapping 4

R : Q → g ⊗ g, (30) which will in the elliptic case depend on two additional spectral parameters u and v,such that the Poisson brackets of L can be written in the form

{L1,L2}=[R12,L1]−[R21,L2], (31) for the degenerate Calogero model and L1(u), L2(v) = R12(u, v), L1(u) − R21(v, u), L2(v) , (32) for the elliptic Calogero model, where as usual L1 = L ⊗ 1,L2 = 1 ⊗ L and R12 = R, T R21 = R , with “T ” denoting transposition of the two factors in the tensor product. As is well-known, the Lax equation (28) or (29) implies that the ad(g)-invariant polynomials on 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 the formulas found in the literature for special cases, we postulate a Lax pair and an R-matrix

4 As we shall see, M and R depend only on the position variables and not on the momentum variables, i.e., ∗ they are functions on T Q that descend to functions on Q. M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 531 given by r L = pj Hj + igαw α(q) Eα, (33) j=1 α∈∆ w (α(q)) M =− igα Fα + igαw α(q) Eα, (34) 2w(α(q)) α∈∆ α∈∆ w(α(q)) R = w α(q) F ⊗ E + E ⊗ E− , (35) α α w(α(q)) α α α∈∆ α∈∆ for the degenerate Calogero model (see [2] for the Lax pair and [8,10] for the R-matrix when g = sl(r + 1, C) and [4–6] for the Lax pair in the general case) and by r L(u) = pj Hj + igαΦ α(q),u Eα, (36) j=1 α∈∆ M(u)=− igα℘ α(q) Fα + igαΦ α(q),u Eα, (37) α∈∆ α∈∆ r R(u,v) =− ζ(u− v) + ζ(v) Hj ⊗ Hj + Φ α(q),v Fα ⊗ Eα j=1 α∈∆ − Φ α(q),u − v Eα ⊗ E−α, (38) α∈∆ for the elliptic Calogero model (see [7] for the Lax pair and [9,10] for the R-matrix when g = sl(r + 1, C)). Here, the Fα form a collection of generators which are supposed to belong to hR and can be viewed as a (vector valued) function

F : ∆ → hR, (39) which we shall decompose into its even part F + and its odd part F −:

+ 1 − 1 F = (Fα + F−α), F = (Fα − F−α). (40) α 2 α 2 Note that only the even part F + of F is used in the deﬁnition of M. In the sequel we shall derive a combinatoric equation which determines F and, together with the functional equations discussed above, turns out to be the basic ingredient for proving the equivalence between the Hamiltonian equations of motion and the Lax equation, as well as the validity of 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 sides of the Lax equation (29): L(u)˙ − L(u), M(u) =˙p + igαΦ α(q),u α(q)E˙ α − igαΦ α(q),u α(p)Eα ∈ ∈ α ∆ α ∆ + gαgγ ℘ α(q) Φ γ(q),u γ(Fα)Eγ α,γ ∈∆ 532 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 1 2 + g Φ α(q),u Φ −α(q),u − Φ −α(q),u Φ α(q),u Hα 2 α α∈∆ 1 + gαgβ Φ α(q),u Φ β(q),u − Φ α(q),u Φ β(q),u Nα,β Eγ . 2 α,β,γ ∈∆ α+β=γ

Here, the double sum over commutators [Eα,Eβ ] appearing in the commutator between L(u) and M(u) has, after antisymmetrization of the coefficients in α and β, been split into two contributions, one corresponding to terms where α + β = 0 and the other to terms where α + β = 0, which are non-zero only if α + β ∈ ∆. Inserting the functional equations (23), (24) and rearranging terms, we get L(u)˙ − L(u), M(u) =˙+ 1 2 + ˙ − p gα℘ α(q) Hα igαΦ α(q),u α(q p)Eα 2 ∈ ∈ α ∆ α ∆ + gαgγ ℘ α(q) Φ γ(q),u γ(Fα)Eγ α,γ ∈∆ 1 + gαgγ −α℘ α(q) Φ γ(q),u Nα,γ −αEγ 2 α,γ ∈∆ γ −α∈∆ 1 − gγ −β gβ ℘ β(q) Φ γ(q),u Nγ −β,βEγ . 2 β,γ∈∆ γ −β∈∆ Renaming summation indices in the last two sums (α →−α in the first, β → α in the second) and using Eqs. (A.9), (A.10), (7) and (13), we finally obtain L(u)˙ − L(u), M(u) 1 2 =˙p + g ℘ α(q) Hα + igαΦ α(q),u α(q˙ − p)Eα 2 α α∈∆ α∈∆ 1 + gα 2gγ γ(Fα) − gγ +αNγ,α − gγ −αNγ,−α ℘ α(q) Φ γ(q),u Eγ . 2 α,γ ∈∆ The vanishing of the sum of the first two terms and of the third term are precisely the equations of motion, so the last double sum must vanish, which requires that for all roots γ ∈ ∆, gα 2gγ γ(Fα) − Γγ,α − Γγ,−α ℘ α(q) = 0. (41) α∈∆ Since these relations must hold identically in q but the coefficient functions are invariant under the substitution α →−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 and only if the combinatoric identity + = + 2gβ β Fα Γβ,α Γβ,−α, (42) M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 533 holds for any two roots α, β ∈ ∆. For the degenerate Calogero model, the same calculation with Φ(s,u) replaced by w(s) and ℘(s) replaced by w(s)/2w(s) leads to the same conclusion (with Eq. (29) replaced by Eq. (28)). Passing to the Poisson bracket relation (32) in the elliptic case, we use Eq. (36) to compute 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): L1(u), L2(v) = igα Φ α(q),v Hα ⊗ Eα − Φ α(q),u Eα ⊗ Hα . (43) α∈∆ 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(p) Φ α(q),u − v Eα ⊗ E−α − Φ α(q),v − u E−α ⊗ Eα = 0. α∈∆ The remaining terms are, according to Eq. (A.11), R12(u, v), L1(u) − p1 − R21(v, u), L2(v) − p2 =− 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∆ + igαα(Fβ ) − igβ β(Fα) Φ α(q),u Φ β(q),v Eα ⊗ Eβ α,β∈∆ + igαΦ −α(q),u − v Φ α(q),u Hα ⊗ Eα ∈ α∆ − igαΦ −α(q),v − u Φ α(q),v Eα ⊗ Hα ∈ α ∆ − igγ Φ −β(q),u− v Φ γ(q),u N−β,γ Eα ⊗ Eβ α,β,γ ∈∆ α+β=γ + igγ Φ −α(q),v − u Φ γ(q),v N−α,γ Eα ⊗ Eβ , α,β,γ ∈∆ α+β=γ where the last four terms have been obtained by splitting each of the two terms containing tensor products of a root generator with the commutator of two other root generators, say 534 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

[Eα,Eβ ]⊗Eγ or Eγ ⊗[Eα,Eβ ], into two contributions: one corresponding to terms where α + β = 0 and one corresponding to terms where α + β ∈ ∆; moreover, some renaming of summation indices has been performed. 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 =+ igα ζ(v − u) + ζ(u) Φ α(q),v α∈∆ + Φ −α(q),u − v Φ α(q),u Hα ⊗ Eα − igα ζ(u− v) + ζ(v) Φ α(q),u α∈∆ + Φ −α(q),v − u Φ α(q),v Eα ⊗ Hα + igαα(Fβ ) − igβ β(Fα) Φ α(q),u Φ β(q),v Eα ⊗ Eβ α,β∈∆ + igγ Nα,β Φ −β(q),u− v Φ γ(q),u α,β,γ ∈∆ α+β=γ + 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 = igα Φ α(q),v Hα ⊗ Eα − Φ α(q),u Eα ⊗ Hα ∈ α ∆ + igαα(Fβ ) − igβ β(Fα) − igα+β Nα,β ∈ α,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 bracket relation (32) will be satisfied if and only if the combinatoric identity (3) stated in the introduction holds for any two roots α, β ∈ ∆. For the degenerate Calogero model, repeating the same calculation with Φ(s,u)and Φ(s,v)both replaced by w(s), Φ(s,u−v) and Φ(s,v − u) both replaced by −w(s)/w(s) and ζ replaced by zero leads to the same conclusion (with Eq. (32) replaced by Eq. (31)). Moreover, decomposing F in the combinatoric identity (3) into its even part F + and odd part F − and using symmetry properties of the coefficients Γα,β derived from Eq. (7) and Eq. (A.9), we see that Eq. (3) is equivalent to the following set of combinatorical identities + + gαα F − gβ β F = Γα,β , (44) β α − − − = gαα Fβ gβ β Fα 0, (45) the first of which is easily shown to be equivalent to the previously imposed identity (42). We have thus proved the following M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 535

Theorem 1. The Calogero model associated with the root system of a (semi-)simple Lie algebra g, degenerate as well as elliptic, with Hamiltonian given by Eq. (5) and by Eq. (6), respectively, is integrable in the sense of admitting the Lax representation and the dynamical R-matrix explicitly given by Eqs. (33), (34) and (35) and by Eqs. (36), (37) and (38), respectively, if and only if the function F appearing in these equations satisﬁes the basic combinatoric identity (3), whose even and odd parts are the identities (44) (or equivalently, (42)) and (45), respectively.

Note that the two parts of this theorem, namely that concerning the Lax representation and that concerning the dynamical R-matrix, are of course not independent because the Hamiltonian is a quadratic function of the L-matrix whereas the M-matrix is essentially just the composition of the R-matrix with the L-matrix. More precisely, using the invariant bilinear form (. , .) on g to identify g with its dual space g∗ and to reinterpret R as a linear mapping from g to g rather than as a tensor in g ⊗ g, 5 we can form the composition R · L and R(u,v) · L(u), respectively; then 1 H = (L, L), (46) 2 and, due to Eq. (16), M = R · L, (47) for the degenerate Calogero model while 1 H = Res| = L(u), L(u) , (48) u 0 2u and, due to Eqs. (22) and (26), R(u,v) · L(v) = M(u)− ζ(u− v) + ζ(v) L(u) + i℘(v) gαFα, α∈∆ and hence 1 M(u)= Res| = R(u,v) · L(v) + ζ(u)L(u), (49) v u v − u for the elliptic Calogero model, provided that the function F mentioned above satisﬁes the simple constraint equation + = gαFα 0. (50) α∈∆ (We have dropped the contributions from the odd part F −, 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 substitutes R · L for M and R(u,v) · L(u) for M(u), respectively, shows that with the above choices the Lax equation follows from the Poisson bracket relation.

∗ 5 More precisely, the convention used is to apply the identiﬁcation between g and g to the second factor in the tensor product g ⊗ g. 536 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

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

Nwα,wβ = Nα,β , for all w ∈ W(g), (51) together with that of the coupling constants gα, as required in Eq. (8), it is reasonable to demand that the function F be covariant (equivariant) under this group:

Fwα = w(Fα), for all w ∈ W(g). (52) Moreover, it is interesting to note that the condition (45) on the odd part F − of F is easily solved: it sufﬁces to take − = Fα λgαHα, (53) where λ is some constant. Therefore, we concentrate on the condition (44) to be satisﬁed by the even part F + 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 coupling constants gα, there exists a (vector valued) function F as in Eq. (39), satisfying the required combinatoric identity (42). In the following, we want to answer this question in full generality, for all semisimple Lie algebras. To begin with, observe that the problem for semisimple Lie algebras is easily reduced to that for simple Lie algebras. In fact, when a semisimple Lie algebra is decomposed into the direct sum of its simple ideals, its root system will be decomposed orthogonally into irreducible root systems, and since the sum of any two roots belonging to different subsystems 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 for each of them. To analyze the problem for simple Lie algebras, we remark ﬁrst of all that, due to the fact that all roots of a simple Lie algebra having the same length constitute a single Weyl group 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 constant g,which should be non-zero. As a result, the coupling constants drop out from Eq. (42), which reduces to + = + 2β Fα Nβ,α Nβ,−α. (54)

• 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 precisely two coupling constants, gl for the long roots and gs for the short roots, at least one of which should be non-zero. In fact, we can be somewhat more precise, requiring that for the simple Lie algebras so(2r + 1, C) of the B-series, gl should be non-zero while for the simple Lie algebras sp(2r, C) of the C-series, gs should be non-zero; when M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 537

r = 2, both should be non-zero. Otherwise, the Hamiltonian of the corresponding Calogero model (as given by Eq. (5) or by Eq. (6)) would decouple, that is, would decompose into the sum of r copies of the same Hamiltonian for a system with only one degree of freedom, and such a system is trivially (Liouville) integrable. Note also that the function F +, if it exists, must be unique, since otherwise all coupling constants would have to vanish. (The argument relies on the fact that even when there are ∗ roots of different length, the space hR is 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 set F + ≡ 0 because there is just a single positive root, so that the r.h.s. of the combinatoric identity (42) vanishes identically. More generally, a particular role is played by the simple Lie algebras sl(n, C) of the A- series. To handle this case, we first fix some notation. The correctly normalized invariant bilinear form on sl(n, C) is the trace form in the defining representation, which can in fact be extended to a non-degenerate invariant bilinear form on gl(n, C):

(X, Y ) = trace(XY ), for X, Y ∈ gl(n, C). (55) Letting indices a,b,..., run from 1 to n, we introduce the standard basis of gl(n, C) consisting of the matrices Eab with 1 at the position where the ath row and the bth column meet and with 0 everywhere else; they satisfy the multiplication rule

EabEcd = δbcEad, (56) and hence the commutation relation

[Eab,Ecd]=δbcEad − δdaEcb. (57) The Cartan subalgebra h of g = sl(n, C) will be the usual one, consisting of all traceless diagonal matrices; this can be extended to the standard Cartan subalgebra hˆ of gˆ = gl(n, C), ˆ consisting of all diagonal matrices; then hR and hR consist of all real traceless diagonal and of all real diagonal matrices, respectively. Obviously, the matrices Ha ≡ Eaa form an ˆ orthonormal basis of hR and the linear functionals ea given by projection of a diagonal matrix H to its entries, ˆ ea(H ) = Haa, for H ∈ hR, (58) ˆ∗ form the orthonormal basis of hR dual to the previous one. Moreover, setting

e = e1 +···+en, (59) ∗ ˆ∗ we see that hR can be identiﬁed with the orthogonal complement of e in hR and r = n − 1. Next, the root system of sl(n, C) is given by

∆ ={αab = ea − eb | 1 a = b n}, (60) ≡ = with root generators Eαab Eab (1 a b n) and with = − Hαab Eaa Ebb. (61) 538 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

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

Nab,cd = δbc − δad, (62) valid for 1 a,b,c,d n with a = b and c = d. Moreover, using the aforementioned ˆ ˆ ∗ Rn orthonormal bases to identify hR and hR with , we see that the Weyl reﬂection sαab n along the root αab operates on a vector in R by simply permuting the ath and the bth component, so the Weyl group W(sl(n, C)) of sl(n, C) is just the permutation group in n letters. With these preliminaries out of the way, we proceed to search for a collection of matrices + + + + F ≡ F ( a = b n) R F = F αab ab 1 that belong to h , with ba ab, satisfying the property (52) of covariance under the Weyl group and the combinatoric identity + − + = + = + = − + − 2 Fab cc Fab dd 2αcd Fab Ncd,ab Ncd,ba δda δcb δdb δca, (63) valid for 1 a,b,c,d n with a = b and c = 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. This + means that the ath and bth entry of the matrix Fab have to be equal and also that all other + entries of the matrix Fab have to be equal among themselves. Thus we can write

+ 1 1 F = λab(Eaa + Ebb) + τab1, ab 2 n with real coefficients λab and τab to be determined. If on the other hand we choose c and d so that the intersection of the sets {c,d} and {a,b} contains precisely one element, the r.h.s. is equal to ±1, and the equation is solved by putting λab =−1 whereas the other = + coefficient is fixed to be τab 1 by the condition that Fab should be traceless. The result is the following

Proposition 1. For the complex simple Lie algebras sl(n, C) of the A-series, there exists a non-trivial solution to the combinatoric identity (42) which is given by

+ 1 1 F =− (Eaa + Ebb) + 1. (64) ab 2 n This solution also satisﬁes the constraint equation (50).

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

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

In the proof, we shall for the sake of simplicity use the pertinent invariant bilinear form ∗ (. , .) to identify the space hR with its dual hR and introduce a basis {e1,...,en} which is orthonormal except possibly for an overall normalization factor; then the root system ∆ M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 539 will be considered as a ﬁnite subset of Rn and F + will be a map from ∆ to Rn required to satisfy the combinatoric identity + = + ∈ 2gβ β,Fα Γβ,α Γβ,−α, for α, β ∆. (65) As observed before, when g is simply laced, this relation reduces to + = + ∈ 2 β,Fα Nβ,α Nβ,−α, for α, β ∆. (66)

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

+ 1 1 F =− (ea + eb) + e(1 a = b n). (67) ab 2 n We begin with the case of the complex simple Lie algebras so(2n, C) of the D-series, with n 4, which are simply laced and for which ∆ consists of the following roots:

±αkl =±(ek − el), ±βkl =±(ek + el)(1 k

±αkl =±˜αkl, ±βkl =∓9klmα˜ m4 (1 k

Thus the solution for A3 given by Eq. (67) can be translated into a solution for D3:

+ 1 + 1 F = 9klmem,F=− 9klmem (1 k

• Bn (n 4): Bn contains Dn, • Cn (n 4): Cn contains Dn, • E8: E8 contains D8, • F4: F4 contains D4.

Therefore, all that remains is to analyze a few isolated cases, namely B2 = C2, G2, B3 and C3, E6 and E7. This can be done by applying similar arguments to the ones already used before. ∼ • B2 (so(5, C)) = C2 (sp(4, C)): The root system consists of four long roots and four short roots, which can be written in the form

long roots: ±e1 ± e2,

short roots: ±e1, ±e2, (72)

when g is realized as so(5, C) (B2)orintheform

long roots: ±2e1, ±2e2,

short roots: ±e1 ± e2, (73)

when g is realized as sp(4, C) (C2). Either way, analyzing the combinatoric identity (65) with α any long root results in a contradiction (provided we take into account the condition that neither of the two coupling constants gl and gs should vanish): when β is a long root, neither of the two expressions β + α and β − α will + be a root, so the r.h.s. must vanish, leading to the conclusion that Fα must vanish, but

6 If the index of the embedding of g into g is not equal to 1, some formulas will be modiﬁed because it will appear as a normalization factor relating the invariant bilinear forms for g and for g. M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 541

when β is a short root, then one and only one of the two expressions β + α and β − α + will be a root, so the r.h.s. cannot vanish, leading to the conclusion that Fα cannot vanish.

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

long roots: ±ek ± el (1 k

short roots: ±ek (1 k 3), (74)

while the root system for C3 consists of six long roots and twelve short roots, which can be written in the form

long roots: ±2ek (1 k 3),

short roots: ±ek ± el (1 k

Note that both systems contain the system D3 as a subsystem: it is generated by the + roots of the form ±ek ± el (1 k

long roots: ±βkl =±(29jklej − ek − el)(1 k

short roots: ±αkl =±(ek − el)(1 k

spanning the subspace orthogonal to the vector e = e1 + e2 + e3. Note that this system is the disjoint union of two subsystems that are copies of the system A2. In particular, + the value of F on any short root αkl must, according to Eq. (67), be proportional to the long root βkl. But then analyzing the combinatoric identity (65) with α any short root and β the corresponding long root results in a contradiction (provided we take into account the condition that neither of the two coupling constants gl and gs should vanish because otherwise, we would really be dealing not with a G2 model but with an A2 model in disguise): since neither of the two expressions βkl + αkl and βkl − αkl 542 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

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

• E7: The root system consists of 126 roots, all of the same length, spanning a 7- dimensional subspace of R8, which can be written in the form

±αkl =±(ek − el)(1 k

±α78 =±(e7 − e8), ±β =±(e + e )(1 k

Obviously, these generators are also orthogonal to all roots of the form γ±,...,±.But then analyzing the combinatoric identity (66) with α = αkl or α = βkl and β an appropriate root of the form γ±,...,± results in a contradiction. For example, letting γ12 be any root of the form 1 e − e − (e − e ) ± e ± e ± e ± e , 2 8 7 1 2 3 4 5 6

with an even number of minus signs, we see that γ12 − α12 is not a root whereas γ12 + α12 is a root, so the r.h.s. cannot vanish, whereas the l.h.s. is necessarily equal to zero. • E6: The root system consists of 72 roots, all of the same length, spanning a 6-dimensional subspace of R8, which can be written in the form

±αkl =±(ek − el)(1 k

To show that these generators are also orthogonal to the vector e8 − e7 − e6 and hence to all roots of the form γ±,...,±, we use the combinatoric identity (66) with α = αkl = − or α βkl and β an appropriate root of the form γ±,...,±. For example, letting γ12 be any root of the form

1 e − e − e + (e + e ) ± e ± e ± e , 2 8 7 6 1 2 3 4 5 + with an even number of minus signs and letting γ12 be any root of the form 1 e − e − e + (e − e ) ± e ± e ± e , 2 8 7 6 1 2 3 4 5 − + − − with an odd number of minus signs, we see that neither γ12 α12 nor γ12 α12 is a + + + − + − root and neither γ β12 nor γ β12 is a root, forcing Fα to be orthogonal to γ + 12 +12 12 12 and F to be orthogonal to γ , so both must be orthogonal to the vector e − e − e β12 12 8 7 6 and hence to all roots of the form γ±,...,±. But then analyzing the combinatoric identity (66) with α = αkl or α = βkl and β another appropriate root of the form γ±,...,± results in a contradiction. For example, letting γ12 be any root of the form 1 e − e − e − (e − e ) ± e ± e ± e , 2 8 7 6 1 2 3 4 5

with an odd number of minus signs, we see that γ12 − α12 is not a root whereas γ12 + α12 is a root, so the r.h.s. cannot vanish, whereas the l.h.s. is necessarily equal to 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 dynamical R-matrix, directly in terms of simple Lie algebras, are the ones based on the simple Lie algebras sl(n, C) of the A-series. Using the abbreviation 1 w(t)2 for the degenerate case V(t)= 2 , (79) 1 2 ℘(t) for the elliptic case for the potential, their Hamiltonian (5) or (6) is given by

n 1 2 2 H(q,p)= p + g V(qk − ql), (80) 2 k k=1 1k=ln which is the Calogero Hamiltonian associated to the root system An−1. Inserting the solution (64) of the combinatoric identity (42) reproduces the known results, both for the Lax pair [2,4–6] and the R-matrix [8,9]. Of course, this result comes as no surprise. The main new aspect of the systematic analysis presented here is to show why the standard approach works only for the A-series and why it is not possible to extend this technique to other simple Lie algebras. 544 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

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 Lie algebra, requires only minor modifications; again, we refer to Appendix A for a summary of the conventions and notation used. As before, the configuration space Q of the theory is a Weyl chamber or Weyl alcove in the appropriate sense; in particular, it is again an open subset of a vector space denoted by a0, and the phase space of the model is given by ∗ = × ∗ the associated cotangent bundle T Q Q a0, often identified with the tangent bundle TQ= Q × a0. The Hamiltonian for the degenerate Calogero model then reads r 1 2 2 2 H(q,p)= p + mα¯ g ¯ w α(q)¯ , (81) 2 j α j=1 α¯ ∈∆ or equivalently 1 r H(q,p)= p2 + g2 w α(q) 2 , (82) 2 j α j=1 α∈∆ while that for the elliptic Calogero model reads r 1 2 2 H(q,p)= p + mα¯ g ¯ ℘ α(q)¯ , (83) 2 j α j=1 α¯ ∈∆ or equivalently r 1 H(q,p)= p2 + g2 ℘ α(q) , (84) 2 j α j=1 α∈∆ where w and ℘ are as before. The coefficients gα¯ = gα (α ∈ ∆) are positive real coupling constants whose definition will be extended from ∆to all of ∆ by setting

gα = 0, for α ∈ ∆0. (85) They are supposed to be symmetric (cf. Eq. (7)) and θ-invariant:

g−α = gα = gθα. (86) A stronger assumption that we shall always make is that they are invariant under the action of the subgroup Wθ (g) of W(g) associated with the symmetric pair (g,θ):

gwα = gα, for all w ∈ Wθ (g). (87) For later use we introduce the following combination of coupling constants and structure constants: θ 1 Γ = gα+β Nα,β + gθα+β Nθα,β + gα+θβNα,θβ + gθα+θβNθα,θβ . (88) α,β 4 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 545

Using the abbreviations (10) as before, we can write the Hamiltonian equations of motion as vector equations in a0: they read ˙ = ˙ =− 2 q p, p gαw α(q) w α(q) (Hα)a, (89) α∈∆ for the degenerate Calogero model and 1 2 q˙ = p, p˙ =− g ℘ α(q) (Hα)a, (90) 2 α α∈∆ for the elliptic Calogero model. The general considerations on the structure of the Lax pair and the R-matrix remain unaltered, so we may pass directly to postulate a Lax pair and an R-matrix which, in accordance with the discussion in the introduction, are given by

r L = pj Hj + igαw α(q) Eα, (91) j=1 α∈∆ w (α(q)) M =− igα Fα + igαw α(q) Eα, (92) 2w(α(q)) α∈∆ α∈∆ 1 w(α(q)) R = w α(q) Fα ⊗ Eα + (Eα ⊗ E−α + Eθα ⊗ E−α), (93) 2 w(α(q)) α∈∆ α∈∆ for the degenerate Calogero model (see [4–6] for the Lax pair) and by

r L(u) = pj Hj + igαΦ α(q),u Eα, (94) j=1 α∈∆ M(u)=− igα℘ α(q) Fα + igαΦ α(q),u Eα, (95) α∈∆ α∈∆ r 1 R(u,v) =− ζ(u− v) + ζ(u+ v) Hj ⊗ Hj 2 j=1 1 − ζ(u− v) − ζ(u+ v) + 2ζ(v) Cz + Φ α(q),v Fα ⊗ Eα 2 α∈∆ 1 − Φ α(q),u − v Eα ⊗ E−α + Φ α(q),−u − v Eθα ⊗ E−α , 2 α∈∆ (96) for the elliptic Calogero model. Here, Cz is the quadratic Casimir element for the centralizer z of a in k (see Eq. (A.26) for an explicit deﬁnition), i.e.,

r+s Cz = Hj ⊗ Hj + Eα ⊗ E−α, (97) j=r+1 α∈∆0 546 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 and the Fα form a collection of generators which are now supposed to belong to ib0 and can be viewed as a (vector valued) function F : ∆ → ib0, (98) which we shall assume to be θ-invariant:

Fα = Fθα. (99) As before, it is convenient to decompose F into its even part F + and its odd part F −:

+ 1 − 1 F = (Fα + F−α), F = (Fα − F−α). (100) α 2 α 2 Note that only the even part F + of F is used in the deﬁnition of M. Once again, we shall derive a combinatoric equation which determines F and, together with the functional equations discussed in the previous section, turns out to be the basic ingredient for proving the equivalence between the Hamiltonian equations of motion and the Lax equation, as well as the validity of the Poisson bracket relation. Basically, the proof goes as previously but it presents some additional subtleties. As before, 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 Lax equation (29): L(u)˙ − L(u), M(u) =˙p + igαΦ α(q),u α(q)E˙ α − igαΦ α(q),u α(p)Eα α∈∆ α∈∆ + gαgγ ℘ α(q) Φ γ(q),u γ(Fα)Eγ , α,γ ∈∆ 1 2 + g Φ α(q),u Φ −α(q),u − Φ −α(q),u Φ α(q),u Hα 2 α α∈∆ 1 + gαgβ Φ α(q),u Φ β(q),u − Φ α(q),u Φ β(q),u 2 α,β∈∆ α+β∈∆0

× Nα,β Eα+β 1 + gαgβ Φ α(q),u Φ β(q),u − Φ α(q),u Φ β(q),u 2 α,β,γ ∈∆ α+β=γ

× Nα,β Eγ .

Here, the double sum over commutators [Eα,Eβ ] appearing in the commutator between L(u) and M(u) has, after antisymmetrization of the coefﬁcients in α and β, been split into three contributions, one corresponding to terms where α + β = 0, one corresponding to terms where α + β ∈ ∆0 and one corresponding to terms where α + β ∈ ∆. Inserting the functional equation (23) to transform the ﬁrst two of these contributions (note that M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 547

α + β ∈ ∆0 implies α(q) + β(q) = 0forallq ∈ Q) and the functional equation (24) to transform the third contribution (note that α + β ∈ ∆ implies α(q) + β(q) = 0modΛ for all q ∈ Q) and rearranging terms, we get L(u)˙ − L(u), M(u) 1 2 =˙p + g ℘ α(q) Hα + igαΦ α(q),u α(q˙ − p)Eα 2 α α∈∆ α∈∆ 1 + gαgβ ℘ α(q) − ℘ β(q) Nα,β Eα+β 4 α,β∈∆ α+β∈∆ 0 + gαgγ ℘ α(q) Φ γ(q),u γ(Fα)Eγ α,γ ∈∆ 1 + gαgγ −α℘ α(q) Φ γ(q),u Nα,γ −αEγ 2 α,γ ∈∆ γ −α∈∆ 1 − gγ −β gβ ℘ β(q) Φ γ(q),u Nγ −β,βEγ . 2 β,γ∈∆ γ −β∈∆ Renaming summation indices in the last two sums (α →−α in the first, β → α in the second) and using Eqs. (A.9), (A.10), (7) and (13), together with 2 1 2 g ℘ α(q) Hα = g ℘ α(q) Hα + ℘ (θα)(q) Hθα α 2 α α∈∆ α∈∆ 1 2 = g ℘ α(q) (Hα − θHα) 2 α ∈ α ∆ = 2 gα℘ α(q) (Hα)a, α∈∆ and the fact that, according to Eq. (A.28) and Eq. (A.29), gαgβ ℘ α(q) − ℘ β(q) Nα,β Eα+β α,β∈∆ α+β∈∆0 1 = gαgβ ℘ α(q) − ℘ β(q) Nα,β Eα+β 2 α,β∈∆ + ∈ α β ∆0 + ℘ (θα)(q) − ℘ (θβ)(q) Nθα,θβEθα+θβ 1 = gαgβ ℘ α(q) − ℘ β(q) Nα,β (Eα+β − θEα+β) 2 α,β∈∆ α+β∈∆0 = 0, 548 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 we finally obtain L(u)˙ − L(u), M(u) 1 2 =˙p + g ℘ α(q) (Hα)a + igαΦ α(q),u α(q˙ − p)Eα 2 α α∈∆ α∈∆ 1 + gα 2gγ γ(Fα) − gγ +αNγ,α − gγ −αNγ,−α 2 ∈ α,γ ∆ × ℘ α(q) Φ γ(q),u Eγ . The vanishing of the sum of the first two terms and of the third term are precisely the equations of motion, so the last double sum must vanish, which requires that for all roots γ ∈ ∆, gα 2gγ γ(Fα) − Γγ,α − Γγ,−α ℘ α(q) = 0. (101) α∈∆ Since these relations must hold identically in q but the coefficient functions are invariant under the substitutions α →−α and α → θα, we are led to conclude that, for the elliptic Calogero model, the Lax equation (29) will be equivalent to the Hamiltonian equations of motion (90) if and only if the combinatoric identity + = θ + θ 2gβ β Fα Γβ,α Γβ,−α, (102) holds for any two roots α, β ∈ ∆. For the degenerate Calogero model, the same calculation with Φ(s,u) replaced by w(s) and ℘(s) replaced by w(s)/2w(s) leads to the same conclusion (with Eq. (29) replaced by Eq. (28)). Passing to the Poisson bracket relation (32) in the elliptic case, we use Eq. (94) to compute the l.h.s., starting out from the same canonical Poisson brackets as before but now 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) = igα Φ α(q),v (Hα)a ⊗ Eα − Φ α(q),u Eα ⊗ (Hα)a . α∈∆ (103) 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 = α(p) +Φ α(q),u − v Eα ⊗ E−α − Φ α(q),−u − v Eθα ⊗ E−α 2 α∈∆ − Φ α(q),v − u E−α ⊗ Eα + Φ α(q),−v − u E−α ⊗ Eθα = 0. M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 549

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 =− igα ζ(u− v) + ζ(u+ v) Φ α(q),u Eα ⊗ (Hα)a 2 α∈∆ 1 + ig ζ(v − u) + ζ(v + u) Φ α(q),v (H ) ⊗ E 2 α α a α α∈∆ 1 − igα ζ(u− v) − ζ(u+ v) + 2ζ(v) Φ α(q),u Eα ⊗ (Hα) 2 b α∈∆ 1 + igα ζ(v − u) − ζ(v + u) + 2ζ(u) Φ α(q),v (Hα) ⊗ Eα 2 b α∈∆ 1 − igγ ζ(u− v) − ζ(u+ v) + 2ζ(v) Φ γ(q),u 2 β∈∆0 γ ∈∆

× N−β,γ E−β+γ ⊗ Eβ 1 + igγ ζ(v − u) − ζ(v + u) + 2ζ(u) Φ γ(q),v 2 α∈∆0 γ ∈∆

× N−α,γ Eα ⊗ E−α+γ + igαα(Fβ ) − igβ β(Fα) Φ α(q),u Φ β(q),v Eα ⊗ Eβ α,β∈∆ 1 + igαΦ −α(q),u − v Φ α(q),u Hα ⊗ Eα 2 α∈∆ 1 − igαΦ −α(q),−u − v Φ α(q),−u θHα ⊗ Eα 2 α∈∆ 1 − igαΦ −α(q),v − u Φ α(q),v Eα ⊗ Hα 2 α∈∆ 1 + igαΦ −α(q),−v − u Φ α(q),−v Eα ⊗ θHα 2 α∈∆ 1 − igγ Φ −β(q),u− v Φ γ(q),u N−β,γ E−β+γ ⊗ Eβ 2 β,γ∈∆ β−γ ∈∆0 1 + igγ Φ −β(q),−u − v Φ γ(q),−u N−β,γ θE−β+γ ⊗ Eβ 2 β,γ∈∆ β−γ ∈∆0 550 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

1 + igγ Φ −α(q),v − u Φ γ(q),v N−α,γ Eα ⊗ E−α+γ 2 α,γ ∈∆ α−γ ∈∆0 1 − igγ Φ −α(q),−v − u Φ γ(q),−v N−α,γ Eα ⊗ θE−α+γ 2 α,γ ∈∆ α−γ ∈∆0 1 − igγ Φ −β(q),u− v Φ γ(q),u N−β,γ E−β+γ ⊗ Eβ 2 β,γ∈∆ β−γ ∈∆ 1 + igγ Φ −β(q),−u − v Φ γ(q),−u N−β,γ θE−β+γ ⊗ Eβ 2 β,γ∈∆ β−γ ∈∆ 1 + igγ Φ −α(q),v − u Φ γ(q),v N−α,γ Eα ⊗ E−α+γ 2 α,γ ∈∆ α−γ ∈∆ 1 − igγ Φ −α(q),−v − u Φ γ(q),−v N−α,γ Eα ⊗ θE−α+γ , 2 α,γ ∈∆ α−γ ∈∆ where the last twelve terms have been obtained by splitting each of the four terms containing tensor products of a root generator with the commutator of two other root generators, say [Eα,Eβ ]⊗Eγ or Eγ ⊗[Eα,Eβ ], into three contributions: one corresponding to terms where α + β = 0, one corresponding to terms where α + β ∈ ∆0 and one corresponding to terms where α + β ∈ ∆; moreover, some renaming of summation indices 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 =+ igα ζ(v − u) + ζ(v + u) Φ α(q),v 2 ∈ α ∆ + Φ −α(q),u − v Φ α(q),u + Φ −α(q),−u − v Φ α(q),−u (Hα)a ⊗ Eα 1 − igα ζ(u− v) + ζ(u+ v) Φ α(q),u 2 ∈ α ∆ + Φ −α(q),v − u Φ α(q),v + Φ −α(q),−v − u Φ α(q),−v Eα ⊗ (Hα)a M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 551 1 + igα ζ(v − u) − ζ(v + u) + 2ζ(u) Φ α(q),v 2 ∈ α ∆ + Φ −α(q),u − v Φ α(q),u − Φ −α(q),−u − v Φ α(q),−u (Hα)b ⊗ Eα 1 − igα ζ(u− v) − ζ(u+ v) + 2ζ(v) Φ α(q),u 2 ∈ α ∆ + Φ −α(q),v − u Φ α(q),v − Φ −α(q),−v − u Φ α(q),−v Eα ⊗ (Hα)b 1 + igγ Nα,β ζ(u− v) − ζ(u+ v) + 2ζ(v) Φ γ(q),u 2 α∈∆,β∈∆0 = + ∈ γ α β ∆ + Φ −α(q),v − u Φ γ(q),v − Φ −α(q),−v − u Φ γ(q),−v Eα ⊗ Eβ 1 + igγ Nα,β ζ(v − u) − ζ(v + u) + 2ζ(u) Φ γ(q),v 2 α∈∆0,β∈∆ = + ∈ γ α β ∆ + Φ −β(q),u− v Φ γ(q),u − Φ −β(q),−u − v Φ γ(q),−u Eα ⊗ Eβ + igαα(Fβ ) − igβ β(Fα) Φ α(q),u Φ β(q),v Eα ⊗ Eβ α,β∈∆ 1 + igγ Nα,β Φ −β(q),u− v Φ γ(q),u 2 α,β,γ ∈∆ α+β=γ + Φ −α(q),v − u Φ γ(q),v Eα ⊗ Eβ 1 − igγ Nα,β Φ −β(q),−u − v Φ γ(q),−u 2 α,β,γ ∈∆ α+β=γ + 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 to Eα ⊗ (Hα)b with α ∈ ∆, as well as the terms proportional to Eα ⊗ Eβ with α ∈ ∆ and β ∈ ∆0 or with α ∈ ∆0 and β ∈ ∆ 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 = igα Φ α(q),v (Hα)a ⊗ Eα − Φ α(q),u Eα ⊗ (Hα)a α∈∆ + igαα(Fβ ) − igβ β(Fα) Φ α(q),u Φ β(q),v Eα ⊗ Eβ α,β∈∆ 1 − igγ Nα,β Φ α(q),u Φ β(q),v Eα ⊗ Eβ 2 α,β∈∆ − Φ α(q),−u Φ β(q),v θEα ⊗ Eβ = igα Φ α(q),v (Hα)a ⊗ Eα − Φ α(q),u Eα ⊗ (Hα)a α∈∆ + − − θ ⊗ igαα(Fβ ) igβ β(Fα) iΓα,β Φ α(q),u Φ β(q),v Eα Eβ . α,β∈∆ 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 bracket relation (32) will be satisfied if and only if the combinatoric identity (4) stated in the introduction holds for any two roots α, β ∈ ∆. For the degenerate Calogero model, the same calculation with Φ(s,u) and Φ(s,v) both replaced by w(s), Φ(s,u− v) and Φ(s,v − u) both replaced by −w(s)/w(s) and ζ replaced by zero leads to the same conclusion (with Eq. (32) replaced by Eq. (31)). Moreover, decomposing F in the combinatoric identity (4) into its even part F + and odd part F − and using symmetry properties of the coefficients θ Γα,β derived from Eq. (86) and Eq. (A.9), we see that Eq. (4) is equivalent to the following set of combinatorical identities + + θ gαα F − gβ β F = Γ , (104) β α α,β − − − = gαα Fβ gβ β Fα 0, (105) 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 Lax representation and the dynamical R-matrix explicitly given by Eqs. (91), (92) and (93) and by Eqs. (94), (95) and (96), respectively, if and only if the function F appearing in these equations satisﬁes the basic combinatoric identity (4), whose even and odd parts are the identities (104) (or equivalently, (102)) and (105), respectively.

Note that, once again, the two parts of this theorem, namely that concerning the Lax representation and that concerning the dynamical R-matrix, are not independent because the Hamiltonian is a quadratic function of the L-matrix whereas the M-matrix is essentially M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 553 just the composition of the R-matrix with the L-matrix. More precisely, 1 H = (L, L), (106) 2 and, due to Eq. (16),

M = R · L, (107) for the degenerate Calogero model while 1 H = Res|u= L(u), L(u) , (108) 0 2u and, due to Eq. (22) and Eq. (26), 1 R(u,v) · L(v) = M(u)− ζ(u− v) + ζ(u+ v) L(u) + i℘(v) gαFα, 2 α∈∆ and hence 1 1 M(u)= Res|v=u R(u,v) · L(v) + ζ(2u)L(u), (109) v − u 2 for the elliptic Calogero model, provided that the function F mentioned above satisﬁes the simple constraint equation + = gαFα 0. (110) α∈∆ (As before, we have dropped the contributions from the odd part F −, since they add up to zero.) Thus once again, we may conclude that the Lax equation follows from the Poisson bracket relation. Concluding, we mention the fact that in view of the invariance of the structure constants Nα,β under the action of the entire Weyl group W(g) of g,

Nwα,wβ = Nα,β , for all w ∈ W(g), (111) together with the invariance of the coupling constants gα under the action of the subgroup Wθ (g) of W(g) associated with the symmetric pair (g,θ), as required in Eq. (87), it is reasonable to demand that the function F be covariant (equivariant) under this subgroup:

Fwα = w(Fα), for all w ∈ Wθ (g). (112) Moreover, it is interesting to note that the condition on the odd part F − of F is easily solved: it sufﬁces to take − = Fα λgα(Hα)b, (113) where λ is some constant. Therefore, we concentrate on the condition (104) to be satisﬁed by the even part F + of F , rewritten in the form (102). 554 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

3.2. Solution for the AIII-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 the AIII-series, which are given by the choice

g = sl(n, C), g0 = su(p, q), gk = su(n), (114) where n = p + q and p q. In order to take advantage of the calculations performed in Section 2.2 for the sl(n, C) case, we choose a representation in which the Cartan subalgebra h of g is the standard one, consisting of purely diagonal matrices; this forces us to deviate from another standard convention, adopted, e.g., in Ref. [30], according to which the involution θ is given by conjugation with a matrix having p entries equal to +1andq entries equal to −1 on the diagonal. Instead, we shall represent all matrices in sl(n, C) in the (3 × 3) block form   (...) (...) (...) p n ... =  (...) (...) (...) q − p (115) ←→ (...) (...) (...) p n ←→ ←→ ←→ pq− pp and suppose the automorphism θ to be given by conjugation

θX = Jp,qXJp,q, for X ∈ sl(n, C), (116) with the matrix   001p   Jp,q = 0 1q−p 0 . (117) 1p 00

Taking into account that the conjugation τ with respect to the compact real form gk = su(n) should be given by τX=−X†, for X ∈ sl(n, C), (118) where “†” denotes Hermitean adjoint, in order to guarantee that the matrices constituting gk = su(n) continue to be the traceless anti-Hermitean matrices, we conclude that the conjugation σ with respect to the real form g0 = su(p, q) must in this representation be given by † σX=−Jp,qX Jp,q, for X ∈ sl(n, C), (119) so the matrices constituting g0 = su(p, q) are the traceless matrices of the form   AC E  DB−C†  , (120) F −D† −A† where A, C and D are arbitrary complex matrices whereas B, E and F are anti-Hermitean matrices. Similarly, the matrices constituting the intersection k0 of the two real forms are M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 555 the traceless matrices of the form   A1 C1 E1  − † − †  C1 B1 C1 , (121) E1 C1 A1 where C1 is an arbitrary complex matrix whereas A1, B1 and E1 are anti-Hermitean matrices. Finally, the matrices belonging to the complementary space m0 appearing in the decompositions (A.13) and (A.14) are the traceless matrices of the form   A2 C2 E2  † − †  C2 0 C2 , (122) −E2 −C2 −A2 where C2 is an arbitrary complex matrix whereas A2 is a Hermitean matrix and E2 is an anti-Hermitean matrix. In particular, the p-dimensional subspace a0 that appears as the ambient space for the conﬁguration space of the Calogero models consists of the real diagonal matrices of the form   P 00  00 0, (123) 00−P whereas its (q − 1)-dimensional orthogonal complement ib0 in hR consists of the real traceless diagonal matrices of the form   P 00  0 Q 0  . (124) 00P For later use, we also introduce the diagonal (n × n)-matrices     1p 00 000     I2p = 000,Iq = 01q 0 . (125) 001p 000 Passing to the explicit index calculations, we continue to use the notation and the conventions introduced in Section 2.2 for handling the sl(n, C) case. In particular, we continue to let indices a,b,...,run from 1 to n, but we shall use the following terminology to characterize the subdivision of this range indicated by the (3 × 3) block form introduced above: an index a will be said to belong to the ﬁrst block if 1 a p, to the second block if p + 1 a q and to the third block if q + 1 a n; moreover, we introduce the following abbreviation to characterize the action of the involution θ on these indices:

θ(a)= a + q, for 1 a p, θ(a)= a, for p + 1 a q, θ(a)= a − q, for q + 1 a n. (126) This allows for a considerable simpliﬁcation of the notation. For example, we have

θαab = αθ(a)θ(b), (127) 556 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 with

θEab = Eθ(a)θ(b). (128) In particular, the two parts of the root system ∆ of sl(n, C) appearing in the decomposition (A.23) are given by ∆ ={αab ∈ ∆ | a or b belongs to the ﬁrst or third block}, (129)

∆0 ={αab ∈ ∆ | a and b belong to the second block}. (130) For the sake of completeness and for later use, we specify explicitly the elements of ∆and the resulting restricted root system ∆, with the corresponding multiplicities. Introducing { } ∗ an orthonormal basis e1,...,ep of a0 by     P 00 P 00     ek 00 0 = Pkk, for 00 0 ∈ a0, (131) 00−P 00−P C ∗ − we see that restricting the roots of sl(n, ) to a0 provide 2p(p 1) restricted roots of multiplicity 2 deﬁned by

α¯kl =¯αθ(l)θ(k) = ek − el,

α¯lk =¯αθ(k)θ(l) = el − ek,

α¯kθ(l) =¯αlθ(k) =+ek + el,

α¯θ(k)l =¯αθ(l)k =−ek − el, (132) where 1 k

α¯kθ(k) =+2ek,

α¯θ(k)k =−2ek, (133) where 1 k p and ﬁnally 2p restricted roots of multiplicity 2(q − p) deﬁned by

α¯km =¯αmθ(k) =+ek (p + 1 m q),

α¯mk =¯αθ(k)m =−ek (p + 1 m q), (134) ≡ where 1 k p. The corresponding root generators continue to be given by Eαab Eab where 1 a = b n, with a or b belonging to the first or third block, since these also satisfy the constraints listed in Eq. (A.28). However, in contrast to the sl(n, C) case studied in Section 2.2, we must now deal with the possibility of encountering non-trivial coupling ≡ constants gαab gab (1 a,b n). A priori, these are only defined when a or b belong to the first or third block, but their definition will be extended to all values of a and b by setting

gab = 0, if a and b belong to the second block, (135) as postulated in Eq. (85), and by adding the convention

gaa = 0. (136) M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 557

Using Eq. (62), we can then derive the following simple expressions for the combinations Γ ≡ Γ and Γ θ ≡ Γ θ introduced in Eq. (9) and Eq. (88), respectively, αab,αcd ab,cd αab,αcd ab,cd

Γab,cd = gadδbc − gbcδad, (137) θ = 1 − + − Γab,cd gadδbc gbcδad gθ(a)dδθ(b)c gθ(b)cδθ(a)d 4 + gaθ(d)δbθ(c) − gbθ(c)δaθ(d) + gθ(a)θ(d)δθ(b)θ(c) − gθ(b)θ(c)δθ(a)θ(d) , (138) valid for 1 a,b,c,d n with a = b and c = d. Next, we have for the Weyl reﬂections

θsabθ = sθ(a)θ(b). (139) This relation implies that the Weyl group W(sl(n, C), θ) associated with this symmetric pair can be realized not only as the quotient of a subgroup but actually as a subgroup of the Weyl group W(sl(n, C)) of sl(n, C), namely the subgroup generated by the reflections saθ(a) = sθ(a)a with index a in the first block (or equivalently in the third block) and by the products of reflections sabsθ(a)θ(b) = sθ(a)θ(b)sab with both indices a and b in the first block (or equivalently in the third block). Under the action of this group, ∆ decomposes into three distinct orbits, namely α a and b belong to the first or third block ab , (140) with b = a,b = θ(a)

{αaθ(a) | a belongs to the first or third block}, (141)    a belongs to the first or third block     and b to the second block  αab or , (142)    a belongs to the second block    and b to the first or third block and so there will be three independent coupling constants which, following Refs. [4–6], we shall denote by g, g1 and g2. Explicitly,

• gab = g if a and b belong to the ﬁrst or third block, with b = 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 and b to the second block or if a belongs to the second block and b to the first or third block. Note also that at least the coupling constant g should be non-zero, since otherwise, the Hamiltonian of the corresponding Calogero model (as given in Eqs. (81), (82) or (83), (84)) would decouple, that is, would decompose into the sum of p copies of the same Hamiltonian 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 matrices + F + ≡ F (1 a = b n, with a or b belonging to the first or third block) that lie in αab ab + = + = + ib0, with Fba Fab Fθ(a)θ(b) as required in Eq. (99), satisfying the property (112) of 558 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 covariance under the Weyl group and the combinatoric identity + + 2g F − F cd ab cc ab dd = + = θ + θ 2gcd αcd Fab Γcd,ab Γcd,ba 1 = gcbδda − gdaδcb + gcaδdb − gdbδca 4 + 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) + g δ − g δ θ(c)θ(b) θ(d)θ(a) θ(d)θ(a) θ(c)θ(b) + gθ(c)θ(a)δθ(d)θ(b) − gθ(d)θ(b)δθ(c)θ(a) , (143) valid for 1 a,b,c,d n with a = b and c = d, corresponding to Eq. (102). Note also + ∈ + that the condition Fab ib0 means that Fab must be a real traceless diagonal matrix whose entries satisfy + = + Fab cc Fab θ(c)θ(c), (144) whereas the invariance under the reflections saθ(a) = sθ(a)a with index a in the first block (or equivalently in the third block), which belong to the Weyl group W(sl(n, C), θ) and act trivially on the space ib0, means that + = + = + = + Fab Fθ(a)b Faθ(b) Fθ(a)θ(b). (145) + Therefore, it suffices to determine Fab in the following three cases: when a and b both belong to the first block, with a = b,whena belongs to the first block while b belongs to the third block, with b = θ(a), and finally when a belongs to the first block while b belongs to the second block. Moreover, the system of equations (143) is obviously invariant under the substitution c → θ(c), d → θ(d) and under the exchange of c and d,aswellas under that of a and b. 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, independently + of the choice of the coupling constants, if we assume all entries of the matrix Fab except + + + + (Fab)aa, (Fab)bb, (Fab)θ(a)θ(a) and (Fab)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 relations + between the remaining entries of the matrix Fab. To find these, we observe that, due to the invariance mentioned above, it is in this case sufficient to evaluate Eq. (143) with c = a and d = b, d = θ(b), d = θ(a).Nowwhena and b both belong to the first block, with a = b, the first two cases (d = b and d = θ(b)) lead to the relation + − + = 2g Fab aa Fab bb 0, while the third (d = θ(a)) gives a trivial identity. Similarly, when a belongs to the first block while b belongs to the third block, with b = 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. Thus taking into account the fact that g = 0, we can cover both situations in one stroke by writing

+ 1 1 F = λab(Eaa + E + Ebb + E ) + τab1, ab 4 θ(a)θ(a) θ(b)θ(b) n M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 559 with real coefficients λab and τab to be determined. Finally, when a belongs to the first block while b belongs to the second block, the first two cases coincide (d = b = θ(b))and lead to the relation

+ + 1 2g F − F = g , 1 ab aa ab bb 2 2 while the third (d = θ(a)) gives a trivial identity, suggesting that we should write

+ 1 1 F = λab(Eaa + E ) + µabEbb + τab1, ab 2 θ(a)θ(a) n with real coefﬁcients λab, µab and τab 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 invariance mentioned above, it is in this case sufficient to evaluate Eq. (143) with c = a and with c = b, supposing that d/∈{a,b,θ(a),θ(b)}.Whena and b both belong to the first block and also when a belongs to the first block while b belongs to the third block, with b = θ(a), this equation is solved identically, independently of the choice of the coupling constants, by putting λab =−1. On the other hand, when a belongs to the first block while b belongs to the second block, we obtain non-trivial relations between the coupling constants, which are the following. If c = a = b, choosing d/∈{a,b,θ(a)} to belong to the first or third block gives

2gλab =−g1, while choosing d/∈{a,b,θ(a)} to belong to the second block (which is only possible when q p + 2) gives

g1λab = 0. Similarly, if c = b = a, choosing d/∈{a,b,θ(a)} to belong to the ﬁrst or third block gives

2g1µab =−g, while choosing d/∈{a,b,θ(a)} to belong to the second block (which is only possible when q p + 2) gives

g1 = 0. + Finally, ﬁxing the coefﬁcients τab by requiring that Fab should be traceless, we arrive at the following

Proposition 3. For the symmetric pairs associated with the complex Grassmannians, which are the symmetric spaces of the AIII-series corresponding to the choice (114) with n = p + q and p q, a non-trivial solution to the combinatoric identity (102) exists if and only if either q = p or q = p + 1 and is then given as follows: 560 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

• q = p: The function F + is given by

+ 1 1 F =− (E + E + E + E ) + 1; (146) ab 4 aa θ(a)θ(a) bb θ(b)θ(b) n • q = p + 1: The function F + is given by

+ 1 1 F =− (Eaa + E + Ebb + E ) + 1, (147) ab 4 θ(a)θ(a) θ(b)θ(b) n when a and b both belong to the ﬁrst or third block and by + =−g1 + − g + 1 g1 + g Fab (Eaa Eθ(a)θ(a)) Ebb 1, (148) 4g 2g1 n 2g 2g1 when a belongs to the ﬁrst or third block while b belongs to the second block, provided that the coupling constants g, g1 and g2 satisfy the relations

g = 0,g1 = 0, (149) together with 2 − 2 + = g1 2g gg2 0. (150) This solution also satisﬁes the constraint equation (110).

The only statement that has not yet been proved is the last one, referring to the identity (110). This is easily done by computing separately the sum over the different Weyl group orbits. Using the matrices introduced in Eq. (125), we see that the first gives 4 times a contribution of the form p + p − 1 p(p − 1) g F = g − I p + 1 , ab 2 2 n a,b=1 a=b while the second gives 2 times a contribution of the form p + 1 p g F = g − I p + 1 , 2 aθ(a) 2 2 2 n a=1 and finally the third (which only exists if q>p) gives 4 times a contribution of the form p q−p + = − g1 − − g + 1 g1 + g − g1 Fab g1 (q p)I2p pIq p(q p)1 . 4g 2g1 n 2g 2g1 a=1 b=1 Summing up everything and using that q − 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 which q − p should be either 0 or 1, admits a natural interpretation: it characterizes exactly those Grassmannians for which the root system ∆0 of the centralizer z of a in k is empty, which means that z should be Abelian. Therefore, a glance at Eq. (97) shows that the first two terms in the elliptic R-matrix (96) for symmetric pairs M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 561 combine into a pure Cartan–Cartan type contribution, in the same way as for the elliptic R-matrix (38) for Lie algebras. In more geometric terms, the criterion states that the symmetric space in question should be such that the centralizers of its maximal flat totally geodesic 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 the restricted root system to a subsystem. In fact, Eqs. (132)–(134) reveal that the system of restricted roots associated with the complex Grassmannians SU(n, n)/S(U(n) × U(n)) and 7 SU(n, n + 1)/S(U(n) × U(n + 1)) is of type Cn and of type BCn, respectively. Inserting into the Hamiltonian (81) or (83) and using the abbreviation (79) for the potential, we arrive at the following special cases:

• SU(n, n)/S(U(n) × U(n)), g2 = 0: n 1 2 2 H(q,p)= p + g V(qk − ql) + V(qk + ql) . (151) 2 k k=1 1k=ln

This is the Calogero Hamiltonian associated to the root system Dn.

• SU(n, n)/S(U(n) × U(n)), g2 arbitrary: n 1 2 2 H(q,p)= p + g V(qk − ql) + V(qk + ql) 2 k k=1 1k=ln n + 2 g2 V(2qk). (152) k=1

This is the Calogero Hamiltonian associated to the root system Cn. • SU(n, n + 1)/S(U(n) × U(n + 1)), g2 = 0: 1 n H(q,p)= p2 + g2 V(q − q ) + V(q + q ) 2 k k l k l k=1 1k=ln n + 2 2g1 V(qk). (153) k=1

This is the Calogero Hamiltonian associated to the root system Bn, subject to the 2 = 2 = constraints (149) and (150), which reduce to g1 2g 0. • SU(n, n + 1)/S(U(n) × U(n + 1)), g2 arbitrary (the general case): n 1 2 2 H(q,p)= p + g V(qk − ql) + V(qk + ql) 2 k k=1 1k=ln n n + 2 + 2 g2 V(2qk) 2g1 V(qk). (154) k=1 k=1

7 We now use n instead of p, as before. 562 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

This is the Calogero Hamiltonian associated to the non-reduced root system BCn, subject to the constraints (149) and (150). Thus all Calogero Hamiltonians associated with the classical root systems are completely integrable, admitting a Lax pair formulation with a dynamical R-matrix.

4. Conclusions and outlook

In the present paper, we have investigated the question of integrability of the Calogero models, systematizing the traditional approach of Olshanetsky and Perelomov [4–6] based on Lax representations using (semi-)simple Lie algebras or symmetric pairs and at the same time extending it to provide dynamical R-matrices (a concept that did not exist at the time when the approach of Olshanetsky and Perelomov was originally developed). We find that the existence of a Lax representation and that of a dynamical R-matrix both hinge on one 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 function F on the pertinent root system that appears both in the Lax pair and the R-matrix and whose deeper meaning is yet to be discovered: more precisely, the question is what might be the additional algebraic structure on the pertinent (semi-)simple Lie algebra or symmetric pair provided by the function F and whose structure equations are the constraints (3) or (4), respectively. We also show that among the simple Lie algebras, those belonging to the classical A-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 models based on sl(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 rigorous proof of this assertion. For the case of symmetric pairs, our analysis is not yet complete, but for the time being is restricted to the classical AIII-series of complex Grassmannians SU(p, q)/S(U(p) × U(q)): we show that among these, the ones where p = q or p = q + 1 or q = p + 1 are the only ones to admit a solution of these constraints. However, this is enough to provide symmetric pair type Lax representations and dynamical R-matrices for Calogero models based on all of the other classical root systems (Bn, Cn, Dn and even BCn). It seems worth mentioning that functions analogous to our function F also appear in various other contexts, for example, in the construction of Lax pairs and dynamical R- matrices for Calogero models through Hamiltonian reduction [11]. However, no mention is made in Ref. [11] of the constraints that this function must satisfy in order for the method to work. We believe that analyzing the problem from this point of view would greatly help to unveil the aforementioned deeper meaning of these constraints. Regarding the various alternative Lax representations for the Calogero models that have been presented recently in the literature [12–19], it must be stressed first of all that none of these proposals deals with the question of how to construct R-matrices. This is certainly a challenge for future work, but it should be emphasized that the algebraic context is somewhat different. A common denominator of all these approaches is that they replace M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 563 the semisimple Lie algebra g—or rather its universal enveloping algebra U(g)—by a more general associative algebra A, assuming that L and M take values in A whereas R takes values in A ⊗ A (see Eq. (27) and Eq. (30)). This can certainly be done without doing harm to the mathematical status of the basic equations, namely the Lax equation (1) and the Poisson bracket relation (2); in fact, even in the traditional Lie algebraic framework, a correct mathematical interpretation of the latter requires passing from g to U(g).The question is as to what is the correct choice for A, which is usually taken to be a matrix algebra defined in terms of generators and relations similar to but not identical with those of ordinary Lie algebras. Thus although these approaches do provide Lax representations in situations where the traditional approach [4–6] fails, we still think that the results reported in 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 of an earlier manuscript, we became aware of important recent work [32,33] where it is shown that, within the traditional Lie algebraic framework, the Calogero models based on sl(n, C), degenerate as well as elliptic, admit a “gauge transformation” taking the dynamical R-matrix into a numerical one: this is achieved by explicitly constructing a group-valued function on the configuration space which is used to conjugate the standard Lax pair and dynamical R-matrix of this model into a new Lax pair and a numerical R- matrix. Should that turn out to be possible in general, it would provide an answer to the question that was the original motivation of our work: it would mean that the appearance of dynamical R-matrices is merely a gauge artifact. This question is presently under investigation.

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 discrete geometry, partly because of the central role they play in various areas of algebra. The most traditional application of root systems is to the classiﬁcation of (semi-)simple Lie algebras and of symmetric pairs of Lie algebras, due to Cartan. In the ﬁrst case, one obtains only the 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 Cartan subalgebra h of g,leth∗ denote its dual, ∆ ⊂ h∗ the root system of g with respect to g and ∗ ∗ hR the real subspace of h spanned by the roots α (α ∈ ∆). Furthermore, let (. , .) be the invariant bilinear form on g, normalized so that with respect to the induced bilinear forms on h and h∗, also denoted by (. , .) and related to each other by 564 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

(α, β) = (Hα,Hβ), (A.1) with

(Hα,H)= α(H), for all H ∈ h, (A.2) √ the long roots have length 2. The form (. , .) is positive definite on the real subspace of h spanned by the generators Hα (α ∈ ∆), which is denoted by hR.WealsofixaWeyl chamber C and a Weyl alcove A; these are open subsets of hR with boundaries formed by hyperplanes on which some root α ∈ ∆ vanishes and by hyperplanes on which some root α ∈ ∆ assumes integer values, respectively. Recall that the choice of a Weyl chamber C establishes an ordering in ∆, the positive roots being the ones that assume strictly positive values on C. The different Weyl chambers are permuted by the elements of the Weyl group W(g) of g, which is the finite group generated by the reflections sα in hR along the roots α ∈ ∆. Finally we choose a basis {Eα | α ∈ ∆} of generators in g normalized so that

(Eα,Eβ ) = δα+β,0, (A.3) and therefore obeying the standard Cartan–Weyl commutation relations

[H,Eα]=α(H)Eα, for all H ∈ h, (A.4)

[Eα,E−α]=Hα, (A.5)

[Eα,Eβ ]=Nα,β Eα+β, (A.6) with structure constants Nα,β which satisfy Nβ,α =−Nα,β and which, by deﬁnition, are supposed to vanish whenever α + β is not a root—in particular when α ± β = 0. These conditions determine uniquely the Hα (according to Eq. (A.2)) but not the Eα, because it is always possible to rescale the root generators Eα according to → = = Eα Eα aαEα, with aαa−α 1, (A.7) which entails a rescaling of the structure constants according to a a → = α β Nα,β Nα,β Nα,β . (A.8) aα+β This freedom can be used in order to impose the additional normalization conditions

N−α,−β =−Nα,β . (A.9)

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

Nα,β = Nβ,γ = Nγ,α. (A.10) In addition, assuming that r is the rank of g, we may choose an orthonormal basis {H1,...,Hr } of hR;then r α(Hj )Hj = Hα. (A.11) j=1 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 565

In the computations made in this paper, we make extensive use of these relations, often without further mention. For proofs, see, for example, [30]. In the symmetric space situation, we assume more speciﬁcally that g is the complexiﬁ- cation of a real semisimple Lie algebra g0 and that we are given a direct decomposition g = k ⊕ m, (A.12) of g induced from a Cartan decomposition

g0 = k0 ⊕ m0, (A.13) of g0;then

gk = k0 ⊕ im0, (A.14) deﬁnes a compact real form gk of g. Writing σ for the conjugation in g with respect to g0 and τ for the conjugation in g with respect to gk (note that these commute), the corresponding Cartan involution θ is simply their product: θ = στ = τσ.Thisisan involutive automorphism of g which is +1onk and −1onm; moreover, it preserves the invariant bilinear form (. , .) on g introduced before, which means that the direct sums in 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 type and of the compact type, respectively. The decomposition of elements corresponding to the direct decompositions (A.12)–(A.14) into eigenspaces under θ is written as

X = Xk + Xm, (A.15) where obviously 1 1 X = (X + θX), Xm = (X − θX). (A.16) k 2 2 Next, let a0 denote a maximal Abelian subalgebra of m0, h0 a maximal Abelian subalgebra of g0 containing a0, b0 the orthogonal complement of a0 in h0 and let a, b and h denote the corresponding complexiﬁcations; then h is a Cartan subalgebra of g and h = b ⊕ a, with b = h ∩ k, a = h ∩ m,

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

The real subspace hR of h introduced before then splits according to

hR = ib0 ⊕ a0, (A.18) and the decomposition of elements corresponding to the direct decompositions (A.17) and (A.18) into eigenspaces under θ is written as

H = Hb + Ha, (A.19) where obviously 1 1 H = (H + θH), Ha = (H − θH). (A.20) b 2 2 Next, we note that the conjugations σ , τ and the involution θ induce bijective transforma- tions of the root system ∆ onto itself that for the sake of simplicity will again be denoted 566 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 by σ , τ and θ, respectively; they are characterized by the condition σ gα = gσα,τgα = gτα,θgα = gθα, α where g is the one-dimensional subspace of g generated by Eα, and are explicitly given by

(σα)(H ) = α(σH ), (τα)(H ) = α(τH ), (θα)(H ) = α(θH), for all H ∈ h, (A.21) In the main body of the paper we write α¯ for the linear functional on a obtained by restricting a given linear functional α on h to a. In this way the root system ∆ ⊂ h∗ gives rise to the restricted root system ∆⊂ a∗: ∆ = α¯ = α|a | α ∈ ∆ . (A.22) Accordingly the root system ∆ itself decomposes into two parts, ∆ = ∆ ∪ ∆0, (A.23) defined by ∆= α ∈ ∆ | θα = α = α ∈ ∆ |¯α = 0 , (A.24) ∆0 = α ∈ ∆ | θα = α = α ∈ ∆ |¯α = 0 , (A.25) where obviously ∆0 is the root system of the centralizer z of a in k: z = Z ∈ k |[Z,X]=0, for all X ∈ a . (A.26) In contrast to ordinary roots, restricted roots will in general have non-trivial multiplicities, defined as follows: mλ = card α ∈ ∆ |¯α = λ . (A.27) As before, we also fix a Weyl chamber C and a Weyl alcove A, but these are now open subsets of a0 (rather than of hR) with boundaries formed by hyperplanes on which some root α ∈ ∆ vanishes and by hyperplanes on which some root α ∈ ∆ assumes integer values, respectively. The choice of such a Weyl chamber C now establishes an ordering only in ∆, the positive roots in ∆being the ones that assume strictly positive values on C, but this can of course be extended to an ordering in ∆. It is interesting and useful to note the behavior of the conjugations σ , τ and of the involution θ 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 group W(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 involution θ, modulo those elements that act trivially on a0. Finally we assume the basis {Eα | α ∈ ∆} of generators in g to be chosen so that, apart from the normalization condition (A.3), the Cartan–Weyl M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 567 commutation relations (A.4), the additional normalization condition (A.9) and the cyclic identity (A.10), we have the following simple behavior under the conjugations σ , τ and the involution θ:

σEα =−Eσα,τEα =−E−α,θEα = Eθα. (A.28) In particular, the structure constants are θ-invariant:

Nθα,θβ = Nα,β . (A.29) (That such a choice is always possible will be shown below.) In addition, assuming that r is the rank of the symmetric pair (g,θ)and r + s is the rank of the Lie algebra g (which by definition means that r = dima and r + s = dim h), we may also choose an orthonormal basis {H1,...,Hr } of a0 together with an orthonormal basis {Hr+1,...,Hr+s} of ib0 to form an orthonormal basis {H1,...,Hr+s} of hR;then r r+s α(Hj )Hj = (Hα)a, α(Hj )Hj = (Hα)b. (A.30) j=1 j=r+1 We conclude this appendix with a proof of the statement made in Eq. (A.28) and Eq. (A.29) above, given the fact that this proof is non-trivial and apparently cannot be found in the literature. The first step consists in going through the proof of Theorem 5.5 in [30, pp. 176,177] which shows how to extend the transformation −id on the Cartan subalgebra h of g to an automorphism ϕ˜ of g (which is by no means uniquely determined by these conditions alone, but this will soon turn out to be an advantage rather than a α drawback), and how to choose a basis {Eα | α ∈ ∆} of generators Eα ∈ g associated with the roots α 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 constants Nα,β defined by Eα, Eβ = Nα,β Eα+β, satisfy the desired condition (A.9). In a second step, we consider the behavior of this basis under the conjugations σ and τ as well as under the involutive automorphism θ. Suppose that we have σEα =−c−αEσα,τEα =−dαE−α, with complex constants cα,dα = 0. Then since σ and τ are antilinear and also involutive (σ 2 = 1 = τ 2), we get ¯ c¯−αc−σα = 1, dαd−α = 1. 568 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570

On the other hand, σ and τ preserve the commutator and hence also the scalar product (. , .) in g, which means that for all X, Y ∈ g, (σX, σY ) = (X, Y ), (τX, τY ) = (X, Y ). (The proof is analogous to that of the well-known statement that automorphisms of Lie algebras always preserve the Killing form.) Hence we also get

c−αcα = 1,dαd−α = 1. Moreover, σ and τ commute, since we have σ = τ on k and σ =−τ on m; this implies ¯ dαcαE−σα = στEα = τσEα =¯c−αdσαE−σα, which by using the previously derived relations can be brought into the form

cσαdσα = cαdα. Note also that the coefficients dα =−(Eα,τEα) are all not just real but positive, since the map g × g → C,(X,Y)→ (X, τY ), defines a negative definite Hermitean sesquilinear form on g. Thus θ is given by θEα = fαEθα, with

fα = cαdα, and we have f + N E + = N θE + = θ E , E = θE ,θE α β θα,θβ θα θβ θα,θβ α β α β α β = fαfβ Eθα, Eθβ = fαfβ Nθα,θβEθα+θβ.

Let us summarize the conditions on the coefﬁcients cα, dα and fα that we have derived. For all α ∈ ∆, cα is a phase factor (|cα|=1), dα is a scale factor (dα > 0) and fα ,their product, is a non-zero complex number (fα = 0) satisfying = −1 = = −1 f−α fα ,fσα fα,fθα fα , and such that for α, β ∈ ∆ with α + β ∈ ∆,

fα+β = fαfβ .

Therefore, there exists a uniquely determined generator Hθ ∈ a such that for all α ∈ ∆, fα = exp α(Hθ ) .

(For the proof, choose any ordering in ∆ and let the αj (j = 1,...,r + s)bethe = corresponding simple roots; then deﬁne Hθ by fαj exp(αj (Hθ )).) We deﬁne α(Hθ ) aα = exp − , 2 M. Forger, A. Winterhalder / Nuclear Physics B 621 [PM] (2002) 523–570 569 so that = −1 = = −1 a−α aα ,aσα aα,aθα aα , and 2 = −1 aα fα , and put Eα = aαEα. In this new basis, we have = = = −1 = θEα aαθEα aαfαEθα aαaθα fαEθα Eθα, =¯ =−¯ =−¯ −1 =− τEα aατEα aαdαE−α aαa−αdαE−α E−α, =¯ =−¯ =−¯ −1 =− σEα aασEα aαc−αEσα aαaσαc−αEσα Eσα, whereas the structure constants Nα,β remain unaltered under the transition from the previous basis to the new one, because it is achieved by means of an inner automorphism of g, namely exp(ad(−Hθ /2)): Hθ Eα = exp ad − Eα. 2

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