3 Spectral properties of the Lax operator

Real Hamiltonian Forms of Af f ine Toda Models Related to Exceptional Lie Algebras

Vladimir S. GERDJIKOV ^† and Georgi G. GRAHOVSKI ^†‡

† Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, 1784 Sof ia, Bulgaria

E-mail: [email protected], [email protected]

‡ Laboratoire de Physique Th´eorique et Mod´elisation, Universit´e de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, F-95302 Cergy-Pontoise Cedex, France

Received December 19, 2005, in final form February 05, 2006; Published online February 17, 2006 Original article is available athttp://www.emis.de/journals/SIGMA/2006/Paper022/

Abstract. The construction of a family of real Hamiltonian forms (RHF) for the special class of affine 1 + 1-dimensional Toda field theories (ATFT) is reported. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinite- dimensional Hamiltonian systems. The construction method is illustrated on the explicit nontrivial example of RHF of ATFT related to the exceptional algebras E6 and E7. The involutions of the local integrals of motion are proved by means of the classical R-matrix approach.

Key words: solitons; affine Toda field theories; Hamiltonian systems

2000 Mathematics Subject Classification: 37K15; 17B70; 37K10; 17B80

1 Introduction

To each simple Lie algebra rankg one can relate Toda field theory (TFT) in 1 + 1 dimensions.

It allows Lax representation: [L, M] = 0, where L and M are first order ordinary differential operators, see e.g. [2,3,4,5,6,7]:

Lψ≡

i d

dx−iqx(x, t)−λJ0

ψ(x, t, λ) = 0, qx(x, t) =

r

X

k=1

q_k,xH_k, (1)

M ψ ≡

id dt − 1

λI(x, t)

ψ(x, t, λ) = 0.

whose potentials take values ing. Hereq(x, t)∈his the Cartan subalgebra ofg,~q(x, t) =

r

P

k=1

q_k~e_k is its dualr-component vector,r= rankg. Hkare the Cartan generators dual to the orthonormal basis elements~e_k in the root space, and

J0 =X

α∈π

Eα, I(x, t) =X

α∈π

e^{−(α,~q(x,t))}E−α.

By π_g we denote the set of admissible roots of g, i.e.π_g ={α₀, α₁, . . . , α_r}whereα₁, . . . , α_r are the simple roots of gand α0 is the minimal root of g. The corresponding TFT is known as the affine TFT. The Dynkin graph that corresponds to the set of admissible roots of g is called

extended Dynkin diagrams (EDD). The equations of motion are of the form:

∂²~q

∂x∂t =

r

X

j=0

α_je^{−(αj,~q(x,t))}.

The present paper extends the ideas of [8] and [9] to the ATFT related to the exceptional simple Lie algebra E₆; for finite Toda chains see [10,11,1,12].

2 The reduction group

The operators L and M are invariant with respect to the reduction group G_R ' Dh where h is the Coxeter number of g. It is generated by two elements satisfyingg₁^h = g²₂ = (g₁g₂)² = 11 which allow realizations both as elements in Autg and in Conf C. The invariance condition has the form [2]:

C1(U(x, t, κ1(λ))) =U(x, t, λ), C1(V(x, t, κ1(λ))) =V(x, t, λ),

C_k(U^†(x, t, κ_k(λ))) =U(x, t, λ), C_k(V^†(x, t, κ_k(λ))) =V(x, t, λ), k= 2,3. (2) where U(x, t, λ) = −iq_x(x, t)−λJ0 and V(x, t, λ) = −¹_λI(x, t). Here Ck are automorphisms of finite order of g, i.e. C₁^h = C₂² = (C₁C₂)² = 11 while κ_k(λ) are conformal mappings of the complexλ-plane:

κ₁(λ) =ωλ, κ₂(λ) =λ^∗, κ₃(λ) = (ωλ)^∗,

where ω = exp(2πi/h). The algebraic constraints (2) are automatically compatible with the evolution. A number of nontrivial reductions of nonlinear evolution equations can be found in [13,14].

3 Spectral properties of the Lax operator

The reduction conditions (2) lead to rather special properties of the operatorL. Along withL we will use also the equivalent system:

Lm(x, t, λ)e ≡idm

dx −iq_xm(x, t, λ)−λ[J₀, m(x, t, λ)] = 0, (3) where m(x, t, λ) = ψ(x, t, λ)e^iJ0xλ. Here ~qx is the potential which we choose to be a Schwartz- type function taking values in the complexified Cartan subalgebra h_C⊂g. This means that the boundary conditions for~q are determined up to the constant vectorρ~₀:

~

ρ₀= lim

x→±∞~q(x, t).

The change of the variables~q⁰ =~q−~ρ0 does not affect the potential and the spectral data ofL,e but they change the right hand side of the ATFT equations into:

∂²~q⁰

∂x∂t =

r

X

j=0

sjαje^{−(αj,~q0(x,t))},

where s_j = exp[−(α_j, ~ρ₀)] obviously satisfy the consistency conditions:

r

Y

j=0

sⁿ_j^j = exp



−

r

X

j=0

(njαj, ~ρ0)



= 1.

where nj are the minimal positive integers for which

r

P

j=0

njαj = 0. In the literature there are two canonical ways of fixing up the vector~ρ0. The first one is to putsj = 1 for allj, the second is to havesj =nj for all j.

The spectral properties of the Lax operator are rather involved due to the fact that bothJ₀ and I(x, t) have complex-valued eigenvalues. In fact all these eigenvalues are constant and are proportional to ω^k,k= 0,1, . . . , h−1, whereω =e^2πi/h and h is the Coxeter number ofg. As a result one finds that the continuous spectrum of L(λ) fills up 2h rays passing through the origin:

λ∈lν : argλ= (ν−1)π

h .

For such operators one can introduce Jost solutions only for potentials on a compact sup- port [15,16]:

x→∞lim Ψ(x, t, λ)e^{iλJ0x+(i/λ)I0t}=11, lim

x→−∞Φ(x, t, λ)e^{iλJ0x+(i/λ)I0t}=11, where

I₀ = lim

x→∞I(x, t) =

r

X

k=0

E−α_k, α_k∈π(g).

Then the corresponding scattering matrix T(t, λ) is defined by T(t, λ) = (Ψ(x, t, λ))⁻¹Φ(x, t, λ).

The compactness of the potential ensures that the Jost solutions Φ(x, t, λ), Ψ(x, t, λ) and the scattering matrix T(t, λ) are meromorphic functions of λ.

The fact that Φ(x, t, λ) and Ψ(x, t, λ) are also Jost solutions of M(λ) means that T(t, λ) evolves according to:

idT dt − 1

λ[I0, T(t, λ)] = 0. (4)

The limiting procedure to non-compact potentials can be done only for the so-called fundamental analytical solutions (FAS) m_ν(x, t, λ), λ∈Ω_ν, i.e. (ν−1)π/h≤argλ≤νπ/h. Their construc- tion is outlined in [15, 16] for generic complex-valued J0. Our Lax pair is special, because it satisfies the reduction condition (2) with the Coxeter automorphism [17]:

C¯₁(X)≡C₁⁻¹XC₁, C₁=e^2πiHρ/h, ρ= 1 2

X

α>0

α;

obviously C₁^h = 11 and ¯C₁(J₀) = ω⁻¹J₀. The FAS m_ν(x, t, λ) of (3) are defined only in the sector Ω3 and satisfy:

C¯₁(m_ν(x, t, ωλ)) =mν−2(x, t, λ), λ∈lν−2.

Thus the inverse scattering problem forL(λ) can be formulated as a Riemann–Hilbert problem form_ν(x, t, λ) on the continuous spectrum Σ≡S2h

ν=1l_ν.

Skipping the details, we remark that equation (4) allows one to show that ATFT possess generating functionals of the integrals of motion. This can be shown easily if the potentialq(x, t) is such that T(t, λ) can be diagonalized:

T(t, λ) =u⁻¹₀ (t, λ)D(λ)u0(t, λ). (5)

All factors in the above equation take values in the Lie groupG with the Lie algebragandD(λ) is a diagonal matrix. Then there exist a set of functions τ_k(λ),k= 1, . . . , r such that:

D(λ) = exp

r

X

k=1

2τk(λ) (α_k, α_k)Hαk

!

, (6)

where H_αk are the Cartan generators of g corresponding to the simple root α_k. Inserting (5) and (6) into (4) one finds that

dD

dt = 0, i.e. dτ_k dt = 0,

for allk= 1, . . . , r. Obviously all eigenvalues of the scattering matrix T(t, λ) will be expressed in terms of τ_k. Each τ_k(λ) can be expanded over the negative powers of λ:

τk(λ) =

∞

X

s=0

I_s^(k)λ^−s,

and all the coefficientsIs^(k) will be integrals of motion.

4 Real Hamiltonian forms

The Lax representations of the ATFT models (see e.g. [2,3,4,18,6] and the references therein) are related mostly to the normal real form of the Lie algebrag, see [20].

Our aim here is to:

1) generalize the ATFT to complex-valued fields ~q^C=~q⁰+i~q¹, and to 2) describe the family of RHF of these ATFT models.

We also provide a tool generalizing of the one in [1] for the construction of new inequivalent RHF’s of the ATFT. The ATFT for the algebra sl(n) can be written down as an infinite- dimensional Hamiltonian system as follows:

dqk

dt ={q_k, HATFT}, dpk

dt ={p_k, HATFT}, HATFT=

Z ∞

−∞

dx 1

2(~p(x, t), ~p(x, t)) +

r

X

k=0

e^{−(~q(x,t),αk)}

! ,

where ~q(x, t) and ~p = ∂~q/∂x are the canonical coordinates and momenta satisfying canonical Poisson brackets:

{p_k(x), q_j(y)}=δ_jkδ(x−y). (7)

Next we define the involutionC acting on the phase space Mas follows:

1) C(F(p_k, q_k)) =F(C(p_k),C(q_k)),

2) C({F(p_k, q_k), G(p_k, q_k)}) ={C(F),C(G)}, 3) C(H(p_k, q_k)) =H(p_k, q_k).

Here F(pk, qk), G(pk, qk) and the Hamiltonian H(pk, qk) are functionals on M depending ana- lytically on the fields q_k(x, t) and p_k(x, t).

The complexification of the ATFT is rather straightforward. The resulting complex ATFT (CATFT) can be written down as standard Hamiltonian system with twice as many fields

~

q^a(x, t), ~p^a(x, t),a= 0,1:

~

p^C(x, t) =~p⁰(x, t) +i~p¹(x, t), ~q^C(x, t) =~q⁰(x, t) +i~q¹(x, t), p⁰_k(x, t), q_j⁰(y, t) =−

p¹_k(x, t), q_j¹(y, t) =δkjδ(x−y).

The densities of the corresponding Hamiltonian and symplectic form equal H^C_ATFT≡ReH_ATFT(~p⁰+i~p¹, ~q⁰+i~q¹)

= 1

2(~p⁰, ~p⁰)−1

2(~p¹, ~p¹) +

r

X

k=0

e^−(~q0,αk)cos((~q¹, α_k)), ω^C= (d~p⁰∧

0 d~q⁰)−(d~p¹∧

0 d~q¹).

The family of RHF then are obtained from the CATFT by imposing an invariance condition with respect to the involution ˜C ≡ C ◦ ∗ where by ∗ we denote the complex conjugation. The involution ˜C splits the phase spaceM^C into a direct sum M^C≡ M^C₊⊕ M^C₋ where

M^C₊=M₀⊕iM₁, M^C₋=iM₀⊕ M₁,

The phase space of the RHF is M_R≡ M^C₊. ByM₀ andM₁ we denote the eigensubspaces ofC, i.e. C(u_a) = (−1)^au_a for any u_a∈ M_a.

Then extracting of Real Hamiltonian forms (RHF’s) is similar to the obtaining a real forms of a semi-simple Lie algebra. The Killing form for the later is indefinite in general (it is negatively- definite for the compact real forms). So one should not be surprised of getting RHF’s with indefinite kinetic energy quadratic form. Of course this is an obstacle for their quantization.

Thus to each involution C satisfying 1)–3) one can relate a RHF of the ATFT. Due to the condition 3) C must preserve the system of admissible roots of g; such involutions can be constructed from theZ2-symmetries of the extended Dynkin diagrams of g studied in [6].

5 Examples

In this Section we provide several examples of RHF of ATFT related to exceptional Kac–Moody algebras with height 1. We show that the involutions C in all these cases are dual to Z2- symmetries of the extended Dynkin diagrams derived in [6].

The examples below illustrate the procedure outlined above and display new types of ATFT.

5.1 E⁽¹⁾₆ Toda f ield theories

The set of admissible roots for this algebra is α1= 1

2(e1−e2−e3−e4−e5−e6−e7+e8), α2 =e1+e2, α₃=e₂−e₁, α₄ =e₃−e₂, α₅ =e₄−e₃, α₆ =e₅−e₄, α0=−1

2(e1+e2+e3+e4+e5−e6−e7+e8),

whereα₁, . . . , α₆form the set of simple roots ofE₆andα₀is the minimal root of the algebra. This is the standard definition of the root system of E6 embedded into the 8-dimensional Euclidean spaceE⁸. The root spaceE6of the algebraE6 is the 6-dimensional subspace ofE⁸ orthogonal to

the vectorse7+e8 ande6+e7+ 2e8. Thus any vector~q belonging toE6 has only 6 independent coordinates and can be written as:

~ q =

5

X

k=1

q_ke_k+q6e⁰₆, e⁰₆ = 1

√3(e6+e7−e8). (8)

Let us fix up the action of the involution C on a generic vector~q inE⁸ by:

C(q_k) =−q_5−k+1 2

4

X

m=1

q_m, for k= 1, . . . ,4,

=q13−k− 1 2

8

X

m=5

q_m, for k= 5, . . . ,8. (9)

This action is compatible with theZ2-symmetryC^#of the extended Dynkin diagram (see Fig.1) and reflects an involution of the Kac–Moody algebraE⁽¹⁾₆ , see [21]. It acts on the root space as follows:

C^#e_k=−e_5−k+1 2

4

X

m=1

em, for k= 1, . . . ,4,

=e13−k−1 2

8

X

m=5

em, for k= 5, . . . ,8,

C^#α1=α6, C^#α3 =α5, C^#αk=αk, k= 0,2,4. (10)

α1

◦ ◦

◦

◦

◦

α5

◦

◦

⇒ ◦

◦

◦

i◦

β3

◦

PP

i 1 PP

i 1

Figure 1. E⁽¹⁾₆ →F⁽¹⁾₄ .

The involution C^# splits the root space E6 into a direct sum of its eigensubspaces: E6 = E+⊕E− with dimE+= 4, dimE−= 2. The vectors:

ee1= 1

2(e5−√

3e⁰₆), ee2 = 1

2(e1+e2+e3+e4), ee3 = 1

2(−e₁−e2+e3+e4), ee4= 1

2(−e₁+e2−e3+e4), ee5 = 1

2(−e₁+e2+e3−e4), ee6 = 1 2(

√

3e5+e⁰₆).

form an orthonormal basis inE6. The first four satisfyC^#ee_k =ee_k,k= 1, . . . ,4, so they spanE+; the last two span E− because C^#ee_j =−ee_j, j = 5,6. In terms of ee_k the admissible root system of F⁽¹⁾₄ takes the standard form:

β0 =−ee2−ee1, β1= 1

2(ee1−ee2−ee3−ee4), β₂ =ee₂−ee₃, β₃ =ee₄, β₄ =ee₃−ee₄. satisfying β0+ 2β1+ 2β2+ 4β3+ 3β4= 0.

Let us take the complex vector ~q(x, t) = ~q⁰(x, t) +i~q¹(x, t) ∈E6 (i.e., of the form (8)) and let p(x, t) =~ ∂~q/∂x. Let us denote their projections onto E± by ~q± and ~p± respectively. Then the densities H^R₁,ω^R₁ for the RHF of AFTF equal:

H^R₁ = 1

2 (~p⁰₊(x, t), ~p⁰₊(x, t))−(~p⁰₋(x, t), ~p⁰₋(x, t))

+ e^{−(~q0+(x,t),β0)} + 2e^{−(~q0+(x,t),β1)}cos((~q¹₋(x, t),ee₅+√

3ee₆)) + 2e^{−(~q0+(x,t),β2)} + 4e^{−(~q0+(x,t),β3)}cos((~q¹₋(x, t),ee5)) + 3e^{−(~q0+(x,t),β4)},

ω₁^R= δ~p₊(x)∧

0 δ~q₊(x)

− δ~p−(x)∧

0 δ~q−(x)

,

If we put ~q−(x, t) = 0 then also ~p−(x, t) = 0 and we get the reduced ATFT related to the Kac–Moody algebraF⁽¹⁾₄ [6].

5.2 E⁽¹⁾₇ Toda f ield theories

The set of admissible roots for this algebra is α1= 1

2(e1−e2−e3−e4−e5−e6−e7+e8), α2 =e1+e2, α₃=e₂−e₁, α₄ =e₃−e₂, α₅ =e₄−e₃, α₆ =e₅−e₄, α₇=e₆−e₅, α₀ =e₇−e₈,

whereα₁, . . . , α₇form the set of simple roots ofE₇andα₀is the minimal root of the algebra. This is the standard definition of the root system of E7 embedded into the 8-dimensional Euclidean space E⁸. The root space E7 of the algebra E₇ is the 7-dimensional subspace of E⁸ orthogonal to the vectore₇+e₈. Thus any vector~q belonging toE7 has 7 independent coordinates and can be written as:

~ q =

6

X

k=1

q_ke_k+q7e⁰₇, e⁰₇ = 1

√

2(e7−e8). (11)

Let us fix up the action of the involution C on a generic vector~q in E⁸ by equation (9). This action is compatible with theZ2-symmetryC^#of the extended Dynkin diagram (see Fig.2) and reflects an involution of the Kac–Moody algebra E⁽¹⁾₇ , see [21]. It acts on the root space as in equation (10) above.

α0

◦

◦

◦

◦

◦

α5

◦ ◦

◦

⇒ ◦

◦

β3

i◦ ◦

◦

1 P

i Pi Pi 1 1

Figure 2. TheE⁽¹⁾₇ → E⁽²⁾₆ RHF of affine TFT.

However its action on the root space E7 is different. It splits E7 into a direct sum of its eigensubspaces: E7=E+⊕E− with dimE+ = 4, dimE−= 3. The vectors:

ee1= 1 2√

2(e1+e2+e3+e4+e5−e6−√

2e⁰₇), ee3 = 1

√

2(−e₁+e4), ee2= 1

2√

2(−e₁−e2−e3−e4+e5−e6−√

2e⁰₇), ee4 = 1

√

2(−e₂+e3),

form an orthonormal basis inE+. Indeed, they satisfy (eej,ee_k) =δ_jk andC^#ee_k=ee_k,k= 1, . . . ,4.

In terms of ee_k the admissible root system ofE⁽²⁾₆ takes the standard form:

β0 =−ee2−ee1, β1=ee2−ee3, β2 =ee3−ee4, β3 = 2ee4, β4 =ee1−ee2−ee3−ee4,

satisfying β₀+ 2β₁+β₂+ 3β₃+ 2β₄ = 0.

The vectors ee5= 1

2(−e₁+e2+e3−e4), ee6 = 1

√

2(e5+e6), ee7= 1

2(−e₅+e6−√ 2e⁰₇), span E− becauseC^#ee_j =−ee_j,j= 5,6,7.

Let us take the complex vector~q(x, t) =~q⁰(x, t) +i~q¹(x, t)∈E7 (i.e., of the form (11)) and let p(x, t) =~ ∂~q/∂x. Let us denote their projections onto E± by ~q± and ~p± respectively. Then the densities H^R₁,ω^R₁ for the RHF of AFTF equal:

H^R₂ = 1

2 (~p⁰₊(x, t), ~p⁰₊(x, t))−(~p⁰₋(x, t), ~p⁰₋(x, t))

+ 2e^{−(~q0+(x,t),β0)}cos((~q¹₋(x, t),ee7)) + 4e^{−(~q0+(x,t),β1)}cos

(~q¹₋(x, t),1

2(ee₅+√

2ee₆−ee₇)

+ 2e^{−(~q0+(x,t),β2)} + 6e^{−(~q0+(x,t),β3)}cos((~q¹₋(x, t),ee₅)) + 4e^{−(~q0+(x,t),β4)},

ω₂^R= δ~p₊(x)∧

0 δ~q₊(x)

− δ~p−(x)∧

0 δ~q−(x)

.

Again, if we put ~q−(x, t) = 0 then also ~p−(x, t) = 0 and we get the reduced ATFT related to the Kac–Moody algebra E⁽²⁾₆ [6].

6 Classical R-matrix method and ATFT

There are several methods to approach the Hamiltonian properties of the ATFT, see e.g. [10, 18,13,11,19]. One of the effective methods is based on the well known classical R-matrix [22]

which is introduced by:

U(x, λ)⊗

0 U(y, µ) = [R(λ, µ), U(x, λ)⊗11 +11⊗U(y, µ)]δ(x−y), (12) where

U(x, λ)⊗

0 U(y, µ) _ij,kl ={U_ij(x, λ), U_kl(y, µ)}.In order to apply this definition effecti- vely we make use of another Lax operator for the ATFT:

ee Le

ψ(x, λ)e ≡ide ψe

dx +U(x, λ)e

ψ(x, λ) = 0,e U(x, λ) =−i

2

r

X

j=1

qj,xHj−λ

r

X

j=0

e^{−(αj,~q)/2}Eαj,

which is gauge equivalent to L (1). Since the Poisson brackets are introduced by (7) the left hand side of (12) takes the form:

U_ij(x, λ)⊗

0 U_kl(y, µ) = i 4

r

X

k=0

e^{−(αk,~q)/2}(µH_αk⊗E_αk −λE_αk⊗H_αk)δ(x−y).

Thus equation (12) becomes an over-determined set of equations for R(λ, µ) which is solved by [22]:

R(λ, µ) = 1 4i

λ^h+µ^h λ^h−µ^h

r

X

k=1

H_k⊗H_k+ 1 2i

X

α∈∆

λ µ

p(α)

µ^hEα⊗E−α

λ^h−µ^h

= 1

4isinh(hη/2) cosh(hη/2)

r

X

k=1

H_k⊗H_k+X

α∈∆

e(p(α)−h/2)η

Eα⊗E−α

! ,

whereh is the Coxeter number ofg,η = ln(λ/µ) andp(α) is the height of the rootαmodulo h.

If α=

r

P

j=1

n_α,jα_j where n_α,j are integers, then p(α) =

r

P

j=1

n_α,j.

The relation (12) allows one to derive the Poisson brackets between the matrix elements of the fundamental solutions T_x⁺₀(x, λ) and T_x⁻₀(x, λ) (or the scattering matrixTx0(λ)) ofL which are defined by:

LT_x^±₀(x, λ) = 0, lim

x→±x0

T_x^±₀(x, λ)e^−iλxJ0 =11, T_x0(λ) = (T_x⁺₀(x, λ))⁻¹T_x⁻₀(x, λ).

Then

T_x^±₀(x, λ)⊗

0 T_x^±₀(y, µ) =

R(λ, µ), T_x^±₀(x, λ)⊗T_x^±₀(y, µ) , and

T_x0(λ)⊗

0 T_x0(µ) = [R(λ, µ), T_x0(λ)⊗T_x0(µ)], (13) These results hold true for potentials on compact support provided we choose x0 large enough so that ~qx = 0 for |x|> x0.

With T_x0(t, λ) we can associate a set of generating functionals τ_k(x₀, λ) of the integrals of motion

Tx0(t, λ) =u⁻¹₀ (x0, t, λ)Dx0(λ)u0(x0, t, λ), Dx0(λ) = exp

r

X

k=1

2τ_k(x0, λ) (α_k, α_k) Hα_k

! . Again we can write the expansion

τ_k(x0, λ) =

∞

X

s=0

I_x^(s)

0,kλ^−s.

An important consequence of (13) is that the functions τ_k(x0, λ) are in involution. Indeed from (13) there follows that:

{trT_x^k₀(λ),trT_x^p₀(µ)}= 0, (14) for any pair of integers k, p. Obviously trT^k(λ) can be expressed in terms of the invari- antsτ_k(λ) (6) of the scattering matrix T(λ)∈ G. Therefore from (14) we find that

{τ_k(x0, λ), τm(x0, µ)}= 0, 1≤k, m≤r, i.e. the integrals of motion I_k^(s) are all in involution:

{I_x^(s)

0,k, I_x⁽ⁿ⁾_0,m}= 0, 0≤k, m≤r, s, n≥0.

In order to derive the corresponding results for the RHF of the ATFT we have to use the fact that the automorphism C induces an automorphism C^∨ on the Lie algebra g and on the Lax operator as follows [23]:

C^∨(Hj) =H_C^#_(ej), C^∨Eα=E_C^#_(α), L(C(~q^∗), x, λ^∗) =C^∨(L(~q, x, λ))^∗.

These relations allow one to prove that the fundamental solution Tx0(x, λ) and the scattering matrixT_x0(λ) for the corresponding RHF model has the properties:

(T_x0(x, λ^∗))^∗=C^∨(T_x0(x, λ)), (T_x0(λ^∗))^∗ =C^∨(T_x0(λ)),

and as a consequence: (τ_k(λ^∗))^∗ =τ¯k(λ)), where ¯k is defined throughC^#(α_k) =α¯k.

7 Conclusions

The RHF of the ATFT models related to the exceptional Kac–Moody algebras E⁽¹⁾₆ and E⁽¹⁾₇ are constructed. These models generalize the ones in [6] since they contain two types of fields

~

q₊(x, t) and~q−(x, t) with different properties with respect to the involutionC. The models in [6]

contain only fields invariant with respect to C.

We outlined the derivation of the Hamiltonian properties through the classical R-matrix approach.

Acknowledgments

One of us (GGG) thanks the organizing committee of the Sixth International Conference “Sym- metry in Nonlinear Mathematical Physics” for the scholarship and for the warm hospitality in Kyiv. The present paper is the written version of the talk delivered by GGG at this conference.

The work of GGG is supported by the Bulgarian National Scientific Foundation Young Scientists Scholarship for the project “Solitons, Differential Geometry and Biophysical Models”. We also acknowledge support by the National Science Foundation of Bulgaria, contract No. F-1410. We also thank an anonymous referee for careful reading of the manuscript and for useful suggestions.

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