Sine Gordon Model in the Homogeneous Higher Grading

Transcription

1 Journal of Physics: Conference Series PAPER Sine Gordon Model in the Homogeneous Higher Grading To cite this article: Alexander Zuevsky 2017 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - Ground State Properties of a Local Sine- Gordon Model with TwoScatters Chen Yu-Guang, Chen Hong and Zhang Yu-Mei - The complex sine-gordon model on a half line Peter Bowcock and Georgios Tzamtzis - Quantum complex sine-gordon model on a half line Peter Bowcock and Georgios Tzamtzis This content was downloaded from IP address on 13/02/2018 at 17:58

2 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ International Conference on Recent Trends in Physics 2016 ICRTP2016 Journal of Physics: Conference Series doi: / /755/1/ Sine Gordon Model in the Homogeneous Higher Grading Alexander Zuevsky Institute of Mathematics, Czech Academy of Sciences, Prague Abstract. A construction of equations and solutions for the sine Gordon model in the homogeneous grading as an example of higher grading affine Toda models are considered. 1. Introduction and preliminaries The problem of construction of exactly solvable models and their solutions is a very important problem in the theory of integrable models in general and in application to real dynamical systems in Physics. One of the ways to proceed is to us the the Lie-algebraic method to construct non-linear exactly solvable models in classical regions. This method is very well known and elaborated [10]. Applying the zero-curvature conditions on elements of connection containing Lie algebra generators in appropriate grading subspaces, we obtain systems of equations of motion associated to a specific Lie algebra. The main motivation to use the theory of Lie algebras in exactly solvable models is its effectiveness. We are able not only to re-construct the equations of motion but also to find exact solutions starting from internal algebraic symmetries based on deep algebraic symmetries of systems under consideration. In [4] the higher grading generalization to the conformal affine Toda models was considered. Elements of the higher then number one grading subspaces are taking into account while connection elements are constructed. The main example was the principal grading case. In this paper we conside an alternative, new case, which is corresponds to the homogeneous grading of the Lie algebra. We derive the systems of equations generalizing the case of the sine-gordon equation and provide quantum group solutions Affine Kac Moody Lie algebras In this subsection we recall some facts about affine Kac Moody algebras [7], [4]. Consider an untwisted affine Kac-Moody algebra Ĝ endowed with an integral grading Ĝ = n Z Ĝn, and denote Ĝ± = n>0 Ĝ±n. By an affine Lie algebra we mean a loop algebra corresponding to a finite dimensional simple Lie algebra G of rank r, extended by the center C and the derivation D. According to [7], integral gradings of Ĝ are labelled by a set of co-prime integers s = s 0, s 1,... s r, and the grading operators are given by Q s H s + N s D 1 2N s T rh 2 s C. 1 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ Here H s r a=1 s ala v H 0, N s r i=0 s im ψ i, ψ = r a=1 mψ a α a, m ψ 0 = 1. H0 is an element of Cartan subalgebra of G; α a, a = 1, 2,... r, are its simple roots; ψ is its maximal root; m ψ a the integers in expansion ψ = r a=1 mψ a α a ; and la v are the fundamental co weights satisfying the relation α a lb v = δ ab. The principal grading operator Q ppal is given by 1 where N s = h is Coxeter number. Therefore Ĝ0 = {Ha 0, a = 1, 2,... r ; C; Q ppal }, Ĝ m = {E 0, E 1 }, α m α h m Ĝ m = {E 0, E 1 } where 0 < m < h, and α m are positive roots of height m, and C is α m α h m the center. The element B is parameterized as B = e ϕ H 0 e ν C e ηq ppal = e ϕ H0 e ν C e ηq ppal, where H 0 was defined in [4] as H 0 a = H 0 a 1 N s T r H s H 0 a C = H 0 a 2 α 2 a N s C, and ν = ν 2 δ h ϕ,, and l a being the fundamental weights of G. Let us denote by H n, E±, n D, C with δ = r a=1 la α 2 a the Chevalley basis generators of ŝl 2. The commutation relations are [H m, H n ] = 2 m C δ m+n,0, [H m, E n ±] = ±2 E m+n ±, [E m +, E n ] = H m+n + m C δ m+n,0, [D, T m ] = m T m, T m H m, E m ±, where C is the center. The grading operator for the principal grading s = 1, 1 is Q 1 2 H0 + 2D. Then the eigensubspaces are Ĝ0 = {H 0, C, Q}, Ĝ2n+1 = {E+, n E n+1 }, n Z, Ĝ 2n = {H n }, n { Z 0} Quantized universal enveloping algebra U qŝl 2 In the spirit of [2], [5], the quantised enveloping algebra U q sl 2 is an associative algebra generated by X +, X, H with q-deformed commutation relations X + X X X + = q H q H q q 1 1, HX ± X ± H = ±2X ±. It possesses a Hopf algebra structure with the deformed adjoint action ad X ± q a = X ± aq H/2 q 1 q H/2 ax ±, ad H q a = Ha ah, for all a U q sl 2. Let us recall the second Drinfeld realization of the quantized universal enveloping algebra U qŝl 2, i.e., U q ŝl 2 without grading operator [2], [6], which is a natural quantum analogue of the algebra ŝl 2 in the loop realizations. U qŝl 2 is an associative algebra generated by {x ± k, k Z; a n, n { Z 0}; γ ± 1 2, K}, where γ ± 1 2 belong to the center of the algebra, satisfying [2k] γ the commutation relations [K, a k ] = 0, [a k, a l ] = δ k γ k k, l k, Kx ± q q 1 k K 1 = q ±2 x ± k, [ a n, x ± ] k = ± [2n] n n γ 2 x ± n+k, [ x + k, ] x γ n = k n/2 ψ k+n γ n k/2 φ k+n, x ± q q 1 k+l x± l q ±2 x ± l x± k = q±2 x ± k x± l+1 x± l+1 x± k. The generators φ k and ψ k, k Z + are related to a k and a k by means of the expressions ψ m z m = Kexp q q 1 a k z, k φ m z m = K 1 exp q q 1 a k z, k i.e., k=0 k=0 ψ m = 0, m < 0; φ m = 0, m > 0. Here [k] = qk q k. q q 1 It is easy to define the grading operators corresponding to the principal and homogeneous grading of U qŝl 2 by analogy with the grading of U qg where G is a simple Lie algebra. The principal grading can be realized with the help of the operator D p x = 1 d 2 qk 1 dq KxK 1 K + 2λ d dλ x, where x Uqŝl 2 and λ is an affinization parameter. The power of λ is denoted by the subscript of U qŝl 2 generators. Then the grading subspaces are q Ĝ 0 = {K, γ}, qĝ2n+1 = {x + n, x n+1, n Z}, qĝ2n = {a n, n { Z 0}}. The grading operator for the homogeneous grading is D h x = 2λ d dλ x, so that the grading subspaces are qĝ0 = {K, γ, x + 0, x 0 }, qĝn = {x + n, x n, a n, n { Z 0}}. The level one irreducible integrable highest weight representation of U qŝl 2 can be constructed in the following way [6]. Let P = Z α 2, Q = Zα be the weight/root lattice of sl 2. Consider the group algebras F [P ], F [Q] of P and Q. The multiplicative basis of F [P ] s a 2

4 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ is formed by e α 2 n, n Z. The F [Q]-module is split into F [P ] = F [P ] 0 F [P ] 1 where F [P ] n = F [Q]e α 2 n. The sl 2 -module structure on the space W = F [a 1, a 2,...] F [P ] is given by the action of the a k, k { Z 0} and e α, α = a 0 generators in accordance with the rules a k f e β = a k f e β, k < 0, a k f e β = [a k, f] e β, k > 0, e α f e β = f e α+β, α f e β = α, βf e β, K = 1 q α, γ = q id. Then W is a U qŝl 2-module. Its submodules are isomorphic to irreducible highest weight modules V Λ n with the highest vectors Λ n = 1 e αn 2, n = 0, Higher grading affine Toda system In this and the next sections we recall [4] the affine Toda system consrtuction. Consider a two dimensional manifold M with local coordinates z ±. Up to a gauge transformation, 1, 0- component lying in see subsection 1.1 l n=0 Ĝ+n and 0, 1-component in l n=0 Ĝ n of a flat connection A in the trivial holomorphic principal fibre bundle M Ĝ M l > 0 is fixed integer satisfy the zero curvature condition [ + + A +, + A ] = 0. 2 The components A ± are the following we keep notations of [4] for k = ±: A k = δ k, B F + δ k,+ B 1 + δ k, F. 3 Here B is a mapping M Ĝ0 Ĝ0 is a group with the Lie algebra Ĝ0 and F ± 1 m l 1 are mappings to l Ĝ±n F ± = E ±l + where E ±l are some fixed elements of Ĝ±l and F ± m Ĝ±m, 1 m l 1. Substituting 3 into 2 one arrives at the equations of motion l 1 m=1 l 1 + B B 1 = [E l, B E l B 1 ] + [Fn, B F n + B 1 ], 4 F ± m, l m 1 ± Fm = [E l, B ±1 F ± l m B 1 ] [Fn+m, B±1 F n ± B 1 ]. 5 Since Q s, C Ĝ0 then B can be parameterized as B = b e η Qs e ν C where b is a mapping to G 0, the subgroup of Ĝ0 generated by all elements of Ĝ0 other than Q s and C. Substituting B into the equations of motion 4 5 one has l 1 + bb ν C = e lη [E l, b E l b 1 ] + e nη [Fn, b F n + b 1 ], 6 l m 1 F m + = e l mη [E l, b 1 F l m b] + e nη [F m+n +, b 1 Fn b], 7 l m 1 + Fm = e l mη [E l, b F + l m b 1 ] e nη [Fm+n, b F n + b 1 ], 8 + η Q s =

5 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ The case l = 1 Now consider the case l = 1. Let us parameterize the element B in the homogeneous grading of Ĝ, [4]. From the equations 6 9 for an infinite dimensional Lie algebra Ĝ in the principal grading we obtain the affine Toda field theory systems of equations Let define + φ + 4µ β r i=1 α m i i expβα α 2 i φ ψ 2 exp β φ = i Jg, ξ = ξ i g ξ i ξ 0 g ξ 0 m j, for representation vectors ξ i see below and a group-like element g. The formal general solution to the above equation was introduced in [12]: e βλ i φ = e βλ i φ 0 Jγ µ 1 + z +µ z γ 0, λ i 1 = e βλ i φ 0 JB, λ i 1 11 The general solutions to the matter fields F i ± may be written in the following form. For m = 1 in 6 9 one has [4] for g µ = µ 1 + µ, i F + 1 i; i = f + i r k il φ φ le = el=0 ν 0 + i gµ i; i J 1 g µ, i. Here i; i denotes an element of the Verma module which is result of the action of the lowering generator on the highest state vector. The fact that 11 is indeed a solution to 10 may be checked by using the representation theory of Ĝ. A map g : M G appearing in the gradient form of the flat connection A ± = g 1 ± g, may be factorized according to the Lie algebra decomposition G = G G 0 G + by the modified Gauss decomposition g = µ ν + γ 0 or g = µ + ν γ 0+ with maps γ 0± : M G 0, µ ±, ν ± : M G ±. The grading condition provides the holomorphic property of µ ±, i.e., they satisfy the initial value problem Ẽ ± z ± = ± µ ± z ± = µ ± z ± Ẽ±z ±, 12 M Ẽ mφ ± ±, m=1 Ẽ ± mφ ± = α + m Φ ±m α z ± X ±α, 13 with arbitrary functions Φ ±m α z ± determining the general solution to the system. Note that the summations in 13 are performed over the set of positive roots + m of G = m Z G m in the subspace G m. 3. Soliton solution for the sine Gordon in homogeneous grading Another way to construct soliton solutions [13] to the sine Gordon equation is to consider the formal general solution 10 in the homogeneous grading and to use vertex operators [7] which are related to the homogeneous Heisenberg subalgebra of ŝl 2. Take the general solution 11 to the affine Toda system 10. In the homogeneous grading the mappings γ ± can be parameterized as γ ± = e dφ de cφc e φ± 0 x± 0, where d is the grading operator, c is the center of ŝl 2 and x ± k are generators of the subspaces Ĝk corresponding to the homogeneous grading. The mappings µ ± satisfy 12 where κ ± z ± = a ±1 + φ ± x ± 1. In order to obtain a soliton solution we put φ± = 0, φ ± 0 = 0. Then the general solution reduces to e βφz+,z = Jg a,µ, Λ

6 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ where g a,µ = e a +1z + µ0e a 1z The following group element µ0 in 14 µ0 = e α 2 N N [ exp 1 α+1 iq n Φζ n ] e α 2 ζ 1 2 α n, generates an N soliton solution. Here the action of the operators 1 2 α and e α 2 on the highest vectors Λ n = 1 e α 2 n, n = 0, 1 is the same as in the case of U qŝl 2 [7] when q = 1. The operator Φζ is given by a Φζ = exp n n ζn exp a +n n ζ n, and diagonalises the action of a ±k, k IN, i.e., [a ±k, Φζ] = ζ ±k Φζ. The product of two vertex operators can be normal ordered as Φζ 1 Φζ 2 = Xx : Φζ 1 Φζ 2 :, where Xx = exp x 2n /n = exp log1 x 2. When x = 1, Xx vanishes which results in Φζ Φζ = 0. Therefore the exponential of Φζ operator terminates after the first order. In the limit q 1 soliton soliton, antisoliton antisoliton and soliton -antisoliton scattering reduce to the classical case, i.e., F a ζ 1 F b ζ 2 = 1 Xx x Xx 1 F b ζ 2 F a ζ 1, where x 2 = ζ 2 /ζ 1, a, b denote soliton antisoliton, and the factor 1/x comes from the commutation of e α 2 ζ 1 2 α 1 and e α 2 ζ 1 2 α 2 operators. Therefore the vertex operator generating a classical soliton solution is F ζ = Q Φζ e α 2 ζ 1 2 α 2. Taking into account the properties of the operator F ζ we rewrite the solution 14 for g ζ = α+1 iqφζ e α 2 ζ 1 2 α e βφz+,z = J g ζ, Λ 1 = 1 + iqe ζz+ ζ 1 z 1 iqe ζz+ ζ 1 z ζ. The antisoliton solution can be associated with the vertex operator F ζ = Q Φζ e α 2 ζ 1 2 α. 4. Homogeneous higher grading generalization of the affine Toda model The Lie-algebraic way to construct non-linear exactly solvable models in classical regions is very well known and elaborated [10]. Applying the zero-curvature conditions on elements of connection containing Lie algebra generators in appropriate grading subspaces, we obtain systems of equations of motion associated to a specific Lie algebra. In [4] of which we keep notations the higher grading generalization to the conformal affine Toda models was considered. Elements of the higher then number one grading subspaces are taking into account while connection elements are constructed. The main example of [4] is the principal grading case. In this paper we consider an alternative, the homogeneous grading case. We derive the systems of equations generalizing the case of the sine Gordon equation and provide quantum group solutions. We start with the equations 6 9 of [4] see subsection 2. Consider the case l = 1. In the principal grading we obtain from 6 the sine Gordon equation. Recall that In the homogeneous grading of G the grading subspaces are G n = { H n, E±} n. We take E 1 = E E 1, E 1 = E E1. 15 Consider a particular case when we parameterize the group element b as b = e φh0. 16 Then, substituting 15 and 16 into 6 8 we get the following system of equations ± φ = e η e 2φ e 2φ, ± ν = e η e 2φ + e 2φ, ± η = 0, 5

7 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ i.e., in the first equation is again the sine Gordon equation. The solution to the field φ is then the standard classical solution 11, [12] see subsection 2. Now consider the case l = 2. The equations corresponding to the principal grading can be found in [4]. Here again we take b = e φh0 though it this is not the most general choice of the group element parameterization since it does not contain dependence on the E± 0 elements from Ĝ, i.e., we have send corresponding fields near those generators to zero. Let us also put F 1 + = κ+ H 1 + f + + E1 + + f + E1, F1 = κ H 1 + f+ E f E 1. Then the system 6 8 gives the following system of equations: ± φ = e η e 2φ e 2φ + e η f + f + e 2φ f f + + e2φ, ± ν = 2e η e 2φ e 2φ + e η 2κ + κ + f f + + e2φ f + f + e 2φ, ± η = 0 + κ = e η f + e 2φ f + + e2φ, κ + = e η f e 2φ f + e2φ, + f + = 2eη κ +, f + + = 2eη κ, + f = 2eη κ +, f + = 2eη κ, e 2φ f + + κ = κ + f +, e 2φ f + κ = κ + f. 19 The formal general solution to age given in [4]: e φ = e φ+ 0 φ 0 Jg µ Λ 1, 20 for the φ field and for the F 1 ± + elements i F i i; i = ek i1φ 0i φ 0i + i γ 0 b γ µ 1 + µ i; i. In the homogeneous grading case, taking into account the parameterization of b element, we get 1 F i + 1; 1 = e2φ 0 φ e φ 0 H0 e φh0 e φ+ 0 H0 g µ 1; 1. Here we have made use of the properties of the i-th fundamental representation corresponding to the homogeneous grading of the Lie algebra ŝl 2. Thus, using 20 we get 1 F + i 1; 1 = e2φ 0 φ gµ 1; 1 J 1 g µ, Λ Dirac equations Let s switch notations similarly to Dirac field components, i.e., ψ R = f + +, ψ L = f+, ψ R = f +, ψ L = f. Now use the extra conditions 19, substituting them into 17. Then we see that the second summands in the first two formulae in 17 vanish, i.e., the final equations are + φ = e η e 2φ e 2φ, + ν = 2e η e 2φ e 2φ 2e η κ + κ, + η = 0, 21 i.e., equations 21 do not differ much from the corresponding equations with l = 1. Now suppose that η = η 0 = const. Then substitute the last four equations of 18 on f = ψ fields into the first two on κ ± fields. Then we get + ψ R = 2e 2η 0 + ψ L = 2e 2η 0 that can be rewritten as ψr e 2φ ψ R e 2φ, + ψr = 2e 2η 0 ψr e 2φ ψ R e 2φ, 22 ψl e 2φ ψ L e 2φ, + ψl = 2e 2η 0 ψl e 2φ ψ L e 2φ, 23 + ω R = 0, + τ R = 2e 2η 0 ω R e 2φ e 2φ, + ω L = 0, + τ L = 2e 2η 0 ω L e 2φ e 2φ, 24 where ω R,L = ψ R,L + ψ R,L, τ R,L = ψ R,L ψ R,L. The upshot is that using such a parametrization b = e φh0 we arrive at three systems of sine Gordon like systems when η is a constant. 6

8 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ l = 2. The general case Let s consider such b G 0 that involves all generators of the G 0 in the homogenous gradation. Take for instance b = e φ +E 0 + e φh 0 e φ E 0, Then for l = 2 the equations are + bb νc = e 2η [ E 2, be 2 b 1] + [ F 1, bf + 1 b 1], 25 F + 1 = eη [ E 2, b 1 F 1 b], + F 1 = eη [ E 2, bf + 1 b 1], + η = 0, 26 where F 1 + = κ+ H 1 + f + + E1 + + f + E1, F1 = κ H 1 + f+ E f E 1. Let us take κ+ = κ = 0. Then we have + φ + φ + φ = e 2η [ e 2φ 1 φ 2 e 2φ 2φ + φ ] +e η [ e 2φ f + f + + φ2 f+ + f φ2 + 2f + + f φ φ + f + + f e2φ], + ν = 2e 2η [ e 2φ 1 φ 2 e 2φ 2φ + φ ] +e η [ e 2φ f + f + + φ2 f+ + f φ2 + 2f + + f φ φ + f + + f e2φ], + φ + 2 φφ + φ 2 + φ e 2φ = e 2η [ 2φ + e 2φ 1 φ 2 ] 2φ +e η [ 2f+ e 2φ f + f + + φ2 φ + φ f + + ], + φ e 2φ = e 2η [ 2φ + e 2φ 1 φ 2 ] φ + e η [ 2f + + e 2φ f + f + + φ2 φ + φ f + + ], + η = 0, 27 + f+ = [ eη 2 f + + φ + e 2φ f + + φ2 + f + φ +], + f = [ eη 2 f + + φ + e 2φ f + + φ2 + f + φ ] + e 2φ f + + φ2 + f φ2 + 2f + + φ φ + f + + e2φ = 0 28 f + + = [ eη 2 φ e 2φ f+ φ2 +f φ +f ], f + = [ 2 φ e 2φ f+ φ2 +f φ +f ], e 2φ f+ φ2 +f 1 + φ2 e 2φ f + φ φ + f = When κ + 0 and κ 0 the system of equations is more complicated. 6. Solitonic solutions from general solutions In [12] it was shown how to extract solitonic solutions from the formal general solutions of the affine Toda field equations. Let s take γ 0 ± = 1 in 11 to be a constant function. Then the mappings µ ± are µ ± = µ 0 ±e z ±E ± with µ 0 ± being some fixed mappings independent of z ±. Next take Ẽ± in 13 as E ± E ±l + l 1 N=1 c± N E ±N where E ± are elements of a Heisenberg subalgebra of Ĝ, namely [E +, E ] = ΩC. One can consider principal of homogeneous Heisenberg subalgebras for that purpose. In this paper we only deel with the principal case while the homogeneous case will be discussed elsewhere. Thus, we arrive at a special solution to 10 e βλ i φ = e βλ i φ 0 J g E,µ, λ i 1, 30 for g E,µ = e x ±E ± µ 0 e x ±E ±. In order to compute these solutions explicitly we have to remove E ± -dependence from 30 moving E + to the right and E to the left. Then we should find such a µ 0 = N i=1 ev i so that V i would be eigenvectors with respect to the adjoint action of E ±, i.e., [E ±, V i ] = ω i ± V i. Then it turns out [12] that resulting expressions provide us with solitonic solutions to the equations under considerations while parameters ω i ± characterize solitons. 7. Quantum group soliton solution for sine Gordon in homogeneous grading As in [11] one can show that the affine Toda models are co-invariant with respect to the lightcone quantization. Namely, the equation of motion are preserved in form though a standard normal ordering has to be introduced as well as some infinite constant comming from quantum versions of Lax pair to generate equations using Lie algebra elements in quantum case. At the 7

9 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ same time infinite constants do not appear in final formal solutions to the light-cone quantized versions of equations. In order to find quantum solutions, one has to replace [3], [8], [9] group elements as well as state vectors formal general solutions by their quantum group counterparts. In this subsection we write examples of quantum group solutions to the quantized affine Toda model in the specific case of the higher grading sine Gordon equation the cases l = 1, 2, 3. Recall [13], that the homogeneous grading subspaces of U q ŝl 2 are q Ĝ 0 = {K, γ, x + 0, x 0 }, qĝn = {x + n, x n, a n, n {Z 0}} The case l = 1 From the commutation relations for x ± m and a m see subsection 1.2 it follows that in this realization of the quantum group U q ŝl 2, the generators x ± m, a m G m, x ± 0 G 0. The solution e βλ j φ = e βλ j φ 0 Je a 1z + e Qφ e a 1z, Λ j, where Λ 0 = 1 1, Λ 1 = 1 e α 2 and the homogeneous grading quantum vertex operator is a φ = exp k [2k] q 7k 2 ζ k exp a k [2k] q 5k 2 ζ k e α 2 q 3 ζ α+i 2. Using the fact that [5] [a k, φ ] = q 7k [k] 2 k ζk, φ, k > 0, [a k, φ ] = q 5k [k] 2 k ζ k, φ, k > 0, we commute exp a 1 z + with expqφ to the right and expqφ with expa 1 z to the left. The commutation of exp a 1 z + withexpa 1 z gives exp z + z [2]. Thus it follows that Λ j e a 1z + e Qφ e a 1z Λ j = Λ j exp Qe q 2 7 z + ζ q 2 5 z ζ 1 φ exp z + z [2] Λ j, recall that exp a 1 z + and expa 1 z act on Λ j and Λ j as identities. Then we expand exp Qe q 7 2 z + ζ q 5 2 z ζ 1 φ as a series and apply the action of powers of operators a exp k [2k] q 7k 2 ζ k and exp a k [2k] q 5k 2 ζ, k to the left and to the right. Powers of operators φ act on the second part of tensor product as follows: φ n 1 e α 2 = q 3 ζ i+n αn e 2, φ n 1 1 = q 3 n 1 αn i+ ζ 2 1 e 2. Thus we have for g Q = e z +z [2] exp Qe q 2 7 z + ζ q 2 5 z ζ 1 q 3 ζ i 2 q 3 ζ i 2 exp e α 2, In the limit q 1 we obtain ordinary soliton solutions. e βλ j φ = e βλ j φ 0 Jg Q, Λ j The case l = 2 As in [13], if we put φ ± x,1 = 0, then E ± = a ±2 + a ±1, and one can integrate the equations for q µ ± to obtain q µ ± z ± = q µ ± 0e a ±2+a ±1 z ±. Then the quantum soliton solution to the quantized 17 is : e β φz +,z :=: e β φ 0 z +,z : Jg a, Λ 1 q, where g a = e a +1+a +2 z + qµ0e a 1+a 2 z, and q µ0 should be chosen the same as in [13]. Then we have : e β φz +,z :=: e β φ 0 z +,z : Jg α, Λ 1 q =: e β φ 0 z +,z : 1+iW 2Q 1 iw 2 Q ζ 1 2, 8

10 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ where g α = e α 2 exp i 1 α+1 Q W 2 2 qφζ e α 2 ζ 1 2 α, W 2 = exp Similarly, Thus, Finally, 7.3. Case l=3 The states Λ 0 m = m+1 q 1 F + 1 1; 1 q = e 2 φ 0 φ + q 1 g a 1; 1 q Jg a, Λ 1 q. 7k 2 [k] k ζk z + 2 q 1 F + 1 1; 1 q = e 2 φ 0 φ + q 1 e a +1+a +2 z + g α e a 1+a 2 z 1; 1 q Jg α, Λ 1 q. q 1 F 1 + 1; 1 q = e 2 φ 0 φ iw 2 Q[2] 1 iw 2Q 1+iW 2 Q ζ 1 2. a m k 1, Λ 1 m = m+1 are annihilated by the action of G n, n m. Therefore for F + m we have 3 1 Λ 1 e 3 a k z +e a k z Λ1 Qφ e m = 1 Λ 1 exp Qe q 7 2 z + ζ q 5 2 z ζ 1 φ exp z + z 3 a m k e α 2, Λ 0 1 = Λ 0, Λ 1 1 = Λ 1, 3 [2k] k [k] Action by the operators e a k z + on Λ1 m, m = 1, 2, 3 we get for instance, e 3 e 3 5k 2 a k z + Λ1 m. a k z + Λ1 3 = Λ 1 3 z + C 2 + a 2 Λ z 2 +C 1 C 2 Λ 1 1, [k] k ζ k z. where C k = [2k] k [k]. Then we expand eφ again and act on the states. Therefore we get an infinite series over Λ 1 3, Λ 1 2, Λ 1 1 which contain C k, k = 1, 2, 3, z + and tensor -part due to powers of e α 2 q 3 ζ α+i Conclusions In this paper we considered the alternative case of the homogeneous grading of the symmetry algebra of the affine Toda systems and, in particular, the sine-gordon equations. The Liealgebraic method helps us to understand the use pure algebraic nature of exact integrability of the dynamical system under consideration. We show explicitly that even in the new case of homogeneous grading it is possible to derive the equations of motion and provider explicit solutions both in classical and quantum regions. The knowledge of the quantum group structures underlying the homogeneous higher grading case leads to specific form of the quantum group soliton solutions for the sine-gordon equation. As possible way to develop, we could mention the search for the explicit solutions associated to all higher grading subspaces of the affine Lie algebra. 9

11 International Conference on Strongly Correlated Electron Systems SCES 2016 IOP Conf. Series: Journal of Physics: Conf. Series doi: / /807/11/ Acknowledgments We would like to thank the Orgnizers of the International Conference of Strongly Correlated Electron Systems, Alexander Zuevsky Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic References [1] Delius G W and Hueffmann A 1996 J. Phys.A29 [2] Drinfeld V G 1985 Sov. Math. Dokl. 32 [3] Fedoseev I A and Leznov A N 1984 Phys. Lett. 141B, 1-2, [4] Ferreira L A, Gervais J-L, Guillen, J S and Saveliev M V 1996 Nuclear Phys. B [5] Jimbo M 1985 Lett. Math. Phys. 10, 63 [6] Jimbo M, Miki K, Miwa T and Nakayashiki A 1992 Phys. Let. A [7] Kac V G 1990 Infinite dimensional Lie algebras. Third edition Cambridge university Press, Cambridge [8] Leznov A N and Fedoseev I A 1982 Theor. Math. Phys. 53 3, 1175 [9] Leznov A N and Mukhtarov M A 1987 Theor. Math. Phys [10] Leznov A N and Saveliev M V 1992 Group-Theoretical Methods for Integration of Non Linear Dynamical Systems. Progress in Physics Series, v. 15, Birkhauser Verlag, Basel [11] Mansfield P 1983 Nucl. Phys. B 222 [12] Olive D I Turok N and Underwood J W R 1993 Nucl. Phys. B401 [13] Saveliev M V and Zuevsky A B 2000 Internat. J. Modern Phys. A