Orthogonal and symplectic Yangians: oscillator realization

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Orthogonal and symplectic Yangians: oscillator realization To cite this article: D Karakhanyan and R Kirschner 2017 J. Phys.: Conf. Ser Related content - Orthogonal and symplectic Yangians R Kirschner - Scattering in Twisted Yangians Jean Avan, Anastasia Doikou and Nikos Karaiskos - Single OTRA Based Low Frequency Sinusoidal Oscillator Realization Gurumurthy Komanapalli, Neeta Pandey and Rajeshwari Pandey View the article online for updates and enhancements. This content was downloaded from IP address on 18/08/2018 at 08:56

2 International Conference on Recent Trends in Physics 2016 ICRTP2016 Journal of Physics: Conference Series doi: / /755/1/ Orthogonal and symplectic Yangians: oscillator realization D Karakhanyan 1 and R Kirschner 2 1 Yerevan Physics Institute, 2 Alikhanyan br., 0036 Yerevan, Armenia 2 Institut für Theoretische Physik, Universität Leipzig, Leipzig, Germany karakhan@yerphi.am roland.kirschner@itp.uni-leipzig.de Abstract. We solve RLL-relation with fundamental orthogonal/symplectic R-matrix examining polynomial with respect to u expressions for L-operator. We use creation-annihilation operators to construct a realization of Yangian generators. 1. Introduction The simplest and well known mechanism of appearance the projectivity of unitary representations is related to Heisenberg group. Upon passing to central extension in simplest quantization models the Planck constant appears as a measure of the central charge. In infinite-dimensional case the projective representations appear as the groups of automprphysms of canonical anti-commutation relations, i.e. Metaplectic and Metagonal groups. In other words, if the given group of automorphysms of some algebra say, Clifford algebra or group say, Heisenberg group maps the Fock representation fermionic or bosonic to itself, then in that representation space appears, generally speaking, projective representation of that group of automorphysm. In finite-dimensional case the projective representation of the orthogonal group by automorphysms of Clifford algebra leads Cartan in 1913 to discovery of spinor group, which provides the twofold covering of the orthogonal group. Later A. Weil shown that S 1 - covering of the symplectic group is reduced to the Z 2 one [1]. The similarly S 1 -extension of the orthogonal group is Metagonal group. It can be illustrated by the following scheme: MO2n, R Spin2n, R SO2nR, Mpn, R Sp 2 n, R SpnR, here the Metagonal group MO2n, R includes Spin2n, R as S 1 covering, which is in turn is Z 2 covering of SO2nR. Similarly the Metaplectic group contains Spinsymplectic Sp 2 n, R and symplectic SpnR groups [2], [3]. So one can conclude from these considerations that the algebra of fermion and boson oscillators is naturally related to orthogonal and symplectic symmetry. 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 2. Yang-Baxter relations In this work we shall follow to notations of our previous work. The fundamental Yang-Baxter equation: has following solution: R a 1a 2 ur a 3 c 1 b 3 u + vr b 3 c 2 c 3 v = R a 2a 3 b 3 vr a 1b 3 c 3 u + vr c 1 c 2 u R 12 ur 13 u + vr 23 v = R 23 vr 13 u + vr 12 u. 1 R a 1a 2 u = uu + n 2 ɛia 1a 2 + u + n 2 ɛp a 1a 2 ɛ u K a 1a 2, 2 where I a 1a 2 = δ a 1, P a 1a 2 = δ a 1, K a 1a 2 = ε a 1a 2 ε b1, 3 and the choices ɛ = +1 and ɛ = 1 correspond to the SOn and Spn cases respectively. We note that the R-matrix 2 is invariant under the adjoint action of any real form related to the metric ε ab of the complex groups SOn, C and Spn, C. Here ε ab is a non-degenerate invariant metric in V ε ab = ɛ ε ba, ε ab ε bd = δa d, 4 which is symmetric ɛ = +1 for SOn case and skew-symmetric ɛ = 1 for Spn case. We denote by ε bd with upper indices the elements of the inverse matrix ε 1. Namely the existence of the invariant tensors ε ab leads to the above mentioned problems in SOn and Spn cases, e.g., it causes a third term in the corresponding expressions of R-matrices and leads to the dependence on the spectral parameter of second power. Let the index range be a 1, a 2, = 1,..., n for the SOn case and a 1, a 2, = m,..., 1, 1,..., m for the Spn n = 2m case. For the choice ε a 1a 2 = δ a 1a 2 in the SOn case we have R a 1a 2 u = uu + βδ a 1 + u + βδ a 1 uδ a 1a 2 δ b1, β = n/2 1, 5 and for the choice ε a 1a 2 = ɛ a2 δ a 1, a 2 have here ɛ a = signa and ε ab = ε ab in the Sp2m case we R a 1a 2 u = uu + βδ a 1 + u + βδ a 1 uɛ a2 ε b2 δ a 1, a 2 δ,, β = m Linear resolution of gln Yangian The simplest form fundamental R-matrix acquires in the case of gln symmetry: R 12 u = ui 12 + P 12, 7 and choosing linear ansatz for L-operator: Lu = u + G one deduces that RLL-relation: consists of contribution, proportional to u: Rewriting this relation in matrix indices: R 12 ul 1 u + vl 2 v = L 2 vl 1 u + vr 12 u, [G 1 + P 12, G 2 ] = 0 = [G 1, G 2 + P 12 ]. 8 [G a 1 c1, G a 1 c1 ] = δ a 1 c2 G a 2 c1 c1 G a 1 c2, 9 one immediately recognizes gln-algebra. So in the case of gln symmetry the linear ansatz for L-operator solves RLL-relations with fundamental R-matrix upon restriction that generators G realize gln-algebra. 2

4 4. Truncated Yangians of so and sp types Let G be the Lie algebra son or sp2m 2m = n. Then the corresponding Yangian Y G is defined [10] as an associative algebra with the infinite number of generators L k a b with L 0 a b = Iδa b, where I is an unit element in Y G, and Lk a b for k > 0 satisfy the quadratic defining relations. The generators L k a b Y G are considered as coefficients in the expansion of the L-operator L a b u = L k a b u k, L 0 = I, 10 k=0 which satisfies to the Yang-Baxter RLL-relations R a 1a 2 u vl c1 ul c2 v = L a 2 vl a 1 ur c 1 c 2 u v R 12 u v L 1 u L 2 v = L 2 v L 1 u R 12 u v. 11 which play role of the Yangin Y G defining relations. Here R a 1a 2 u v is the Yang-Baxter R-matrix 2. The defining relations 11 are homogeneous in L and allow the redefinition Lu fu Lu + b 0 with any scalar function fu = 1 + /u + /u , where b i are parameters. The Yangian 2, 11 possesses the set of automorphisms Lu u ak u k Lu, k = 1, 2,..., where a is a constant in general a is a central element in Y G. At k = 1 the generators L j are transforming as L 1 L 1 ai n, L 2 L 2 a L 1, L 3 L 3 al 2,..., 12 Taking a = 1 n TrL1 one can fix L 1 to be traceless. Note that TrL 1 is central element in Y G. One has for the fundamental R-matrix 2: 1 Ru v = 1 u 2 v 2 v 1 u 1 v 1 u + β uv 1 1 uv 2 u 2 v + β P ɛ 1 1 K. 13 u 2 v 2 uv 2 u 2 v here β = n 2 ɛ. Then the defining relations for the generators Lk a b of the Yangians Y G: [L k 1, Lj 2 2 ] 2[L k 1 1, L j 1 2 ] + [L k 2 1, L j 2 ] + + P +β[l k 1 1, L j 2 2 ] [L k 2 1, L j 1 2 ] + L k 1 1 L j 2 2 L k 2 1 L j βl k 2 1 L j 2 2 L j 2 2 L k 1 1 L j 1 2 L k βl j 2 2 L k 2 1 P + +ɛ K L k 2 1 L j 1 2 L k 1 1 L j 2 2 L j 1 2 L k 2 1 L j 2 2 L k 1 1 K = 0, 14 where the operators K, P are given in 3, ɛ = +1 for G = son and ɛ = 1 for G = sp2m. The defining relations are obviously completed for infinite set of generators, but it can be truncated at finite values as well, which will impose the additional restrictions on generators L k a b 3

5 At k = 1 we obtain from 14 the set of relations [ ] [L 1 1, Lj 2 2 ] = P 12 ɛk 12, L j 2 2, j, 15 which in particular lead to the statement that TrL 1 is a central element in Y G: [TrL 1, L j a b ] = 0 j. For j = 3 we deduce from 15 the defining relations for the Lie algebra generators G a b L1 a b : [G 1, G 2 ] = [P 12 ɛk 12, G 2 ]. 16 Permutation of indices 1 2 in this equation gives the consistency conditions and the same conditions are obtained from 14 directly. K 12 G 1 + G 2 = G 1 + G 2 K 12, Acting on this equation by K 12 from the left or by K 12 from the right we write it as where we have used K 12 G 1 + G 2 = 2 n TrG K 12 = G 1 + G 2 K 12, 17 K 2 12 = ɛ n K 12, K 12 N 1 K 12 = K 12 N 2 K 12 = ɛ TrN K 12, 18 Here N is any n n matrix. Then, according to the automorphism 12 we redefine the elements G G 1 n TrG in such a way that for the new generators we have TrG = 0 This leads to the antisymmetry conditions for the generators: K 12 G 1 + G 2 = 0 = G 1 + G 2 K 12 G d a ε db + ε ad G d b = The equations 16 and 19 for ɛ = +1 and ɛ = 1 define the Lie algebra G = son and G = sp2m 2m = n, respectively. The defining relations 16 and antisymmetry condition 19 for the generators G ab = ε ad G d b can be written in the familiar form [G ab, G cd ] = ε cb G ad + ε db G ca + ε ca G db + ε da G bc, G ab = ɛ G ba. 20 This means see [10] that an enveloping algebra UG of the Lie algebra G = son, sp2m is always a subalgebra in the Yangian Y G. 5. L operators The Yangian Y G 14 with infinite number of generators can be truncated to some set with finite number of ones restricted to some special representation. We will suppose that the Yangian representation space is realized as a space of matrix polynomials of So Sp-generators G and start with consideration of some general properties of such monomials. Then, thinking generators G and spectral parameter as a dimensionfull variables their ratio and L-operator are dimensionless one will describe the Yangian as Z graded space with finite-dimensional components corresponding to irreducible representations. 1. Linear evaluation of Y G. 4

6 We put equal to zero all generators L k Y G with k > 1 and take the L-operator linear by spectral parameter: L a b u = uδa b + Ga b. 21 Then defining RLL-relation takes the form: uu + βi 12 + u + βp 12 uɛk 12 u + v + G 1 v + G 2 = = v + G 2 u + v + G 1 uu + βi 12 + u + βp 12 uɛk 12. This relation can be rewritten as follows: u + β [G 1, G 2 ] + G 1 G 2 P 12 ɛ[k 12, G 2 ] ɛv[k 12, G 1 + G 2 ] ɛk 12 G 1 βg 2 + ɛg 2 G 1 βk 12 = 0, and has to take place identically by powers of u and v, implying three restrictions on generators G: vc 1,1 = ɛv[k 12, G 1 + G 2 ] = 0, 22 as a coefficient at v, which is takes place upon 22, u + βc 1,2 = u + β [G 1, G 2 ] + G 1 G 2 P 12 ɛ[k 12, G 2 ] = 0, 23 as a coefficient at u and expresses the SO Sp-algebra relations. The remaining terms are: C 1,3 = ɛ K 12 G 1 βg 2 G 2 G 1 βk 12 = One can summarize constraints following from the Yang-Baxter RLL-relation for linear ansatz 21 as: C 1,1 [ɛk 12, G 1 + G 2 ] = 0, 25 Introducing graded oscillators c a = ε ab c b : C 1,2 = [G 1 + P 12 ɛk 12, G 2 ] = 0, 26 C 1,3 = [ɛk 12, G 1 G 2 + G 2 G 1 ] = c a c b + ɛc b c a = ε ab, c a c b + ɛc b c a = δ b a, c a c b + ɛc b c a = ε ba, 28 in orthogonal case ɛ = +1 operators c a will have Grassmann nature and representation is finitedimensional, while in symplectic case one has the ordinary oscillators and an infinite-dimensional representation. Note, that 28 is just half of set of oscillators. Indeed, the algebra of automorphysms, corresponding to Bogolyubov transformation of the set of Fermi Bose creation-annihilation operators is O2n Sp2n. Then one can solve RLL-constraints by simple bilinear ansatz G 1a b = c a c b + ɛ 2 δa b = ɛ 2 c bc a ɛc a c b = ɛc b c a 1 2 δa b, 29 ensuring symmetry properties of G: expressions with lower and upper indices are G 1 ab = c a c b + ɛ 2 ε ab, G 1ab = ɛ c a c b εab. The overall coefficient is chosen according to 26. 5

7 Note that the metric is the unique invariant tensor and the single invariant combination: ε ab c a c b = 1 2 εab c a c b + ɛc b c a = n 2, 30 can be constructed in the case of one set of oscillators. 2. Quadratic evaluation of Y G. Now we put all generators L k Y G with k > 2 equal to zero. In this case the L-operator 10, after a multiplication by u 2, can be written in the form where we introduce Lu = uu a + u G + H. 31 L 1 = G a = G, L 2 = H, 32 where the constant a is chosen to be TrG, so the generator G is traceless. We are going to consider the general quadratic ansatz Lu = uu a + u G + H = u 2 + ug + H, 33 and to solve the restrictions imposed by RLL Yang-Baxter relation. The defining relation 11 has the form: [uu + βi 12 + u + βp 12 ɛuk 12 ]u + v 2 + u + vg 1 + H 1 v 2 + vg 2 + H 2 34 v 2 vg 2 + H 2 u + v 2 u + vg 1 + H 1 [uu + βi 12 + u + βp 12 ɛuk 12 ] = 0. This relation must take place at arbitrary values spectral parameters u and v, i.e. the coefficients at independent monomials u k v r k + r 4 must vanish. The l.h.s. of 34 can be represented as a sum of following eight combinations: C 2,1 = ɛ[k 12, G 1 + G 2 ] = 0, 35 C 2,2 = [G 1 + P 12 ɛk 12, G 2 ] = [G 1, G 2 P 12 ɛk 12 ] = 0, 36 C 2,3 = [K 12, H G2 1 + G H 2 ] = 0, 37 C 2,4 = [G 1 + P 12 ɛk 12, H 2 ] = 0, 38 C 2,5 = [H 1, G 2 P 12 + ɛk 12 ] = 0, 39 C 2,6 = [K 12, H 1 G 2 + H 2 G 1 + G 1 H 2 + G 2 H 1 ] = [K 12, {H 1, G 2 } + {H 2, G 1 }] = C 2,7 = [H 1, H 2 ] + G 2 H 1 H 2 G 1 P 12 ɛ 2 [K 12, {G 1, H 2 ɛ 2 G 2}] = 0, 41 C 2,8 = ɛ K 12 H 1 βg 1 + β 2 H 2 H 2 H 1 βg 1 + β 2 K 12 = Let us now turn to oscillator realization of generators G and H. Consider first the case of one set of oscillators. As we prove above the most general expression for G is given then by 29, for which constraints take place. Then equation 35 allows to determine ε-symmetric traceless part of H, which is zero, because the only ε-symmetric tensor of second rank constructed from oscillators 28 is metric tensor. Then due to identity: G 1 2 ab + βg2 ab = 1 4 nɛ 1ε ab, 43 6

8 one deduces: H 1 ab = αg1 ab + γε ab, where α and γ are arbitrary constants. In other words H 1 ab is an arbitrary tensor, constructed from oscillators 28. Substituting H 1 and G 1 into remaining constraints one obtains identity at γ = α 2 and the quadratic ansatz for L-operator will take the form: Lu = u 2 + u + αg 1 α 2 = u + αu α + G, which is just shifted linear ansatz up to the overall factor and one comes to: Proposition Operator dependence of RLL-relations on single set of oscillators leads to the linear solution. Let us now turn to the case of two sets of oscillators. Oscillators have the natural grading 28. From the other hand in our previous work mentioned that a particular solution corresponding to quadratic ansatz is given by Jordan- Schwinger construction: G ab = x b a ɛx a b, 44 where the variables x a have an opposite grading even for So- and odd for Sp-case: a b ɛ b a = 0, x a x b ɛx b x a = 0, a x b ɛx b a = ε ab. 45 Labeling oscillators by index α = 1, 2: d 1 a = x a, d 2 a = a, 46 one can rewrite commutation relations?? in more compact form: d α a d β b ɛdβ b dα a = ɛ αβ ε ab, 47 where ɛ αβ is two-dimensional Levi-Civita tensor ɛ 12 = 1 = ɛ 21, ɛ 11 = 0 = ɛ 22. The bilinear ansatz 44 is compatible with cyclic identity: {G ab, G cd } + {G ca, G bd } + {G bc, G cd } = 0, 48 upon substitution G ab = G 0 ab relation 48 becomes identity. The identity 48 is necessary to satisfy the constraints following from the RLL-relation. Then it is not hard to check that along with G 0 c 1 c 2 = 1 2 ɛ αβd α c 2 d β c 1 ɛd α c 1 d β c 2 the relation: H = G 2 + βg, 49 solves all constraints and provides the quadratic resolution through oscillators Oscillator realization Let us consider the most general expression for G ab depending on oscillators 28: G ab = k=0 G k ab = k=0 e 1,...,e k A ab,e1...e k c e1... c e k, 50 and try to solve constraints In this expansion the operator G ab and the coefficients A ab,e1...e k is in one-to-one correspondence if the summation in 50 is ordered, say normally, 7

9 when all creation operators c a with a > 0 stand from the left of annihilation ones c a with a < 0. This restriction on summation rule can be lifted if we redefine coefficients A to be ε-antisymmetric: A ab,e1...e i...e j...e k = ɛa ab,e1...e j...e i...e k. 51 Indeed, any tensor A...ei...e j... can be represented as the sum of ε-symmetric and ε-antisymmetric parts: A...ei...e j... = 1 2 A...e i...e j... + ɛa...ej...e i A...e i...e j... ɛa...ej...e i... = A...e i...e j... + A...e i...e j..., and upon contraction with oscillators the symmetric part becomes reducible A...e i...e j...c e i c e j = 1 2 A...e i...e j... + ɛa...ej...e i...c e i c e j = 1 2 A...e i...e j...ε e ie j. So without the loss of generality one can suppose coefficients A ab,e1...e i...e j...e k ε-antisymmetric and traceless by indices e i : A ab,e1...e i...e j...e k ε e ie j = In the first relation: 52 G c1 c 2 + ɛg c2 c 1 = G b bε c1 c 2, 54 generator G can be chosen traceless without loss of generality. Then substituting expansion 50 into 54 one obtains: A c1 c 2,e 1...e k + ɛa c2 c 1,e 1...e k = 0, ε ab A ab,e1...e k = The first two indices a, b are different from the remaining ones e 1... e k. Indeed, generator G a b should rotate tensor quantities, which determine the possible form of corresponding commutator. In particular, the construction 29 can serve as generator, so we start with the simplest quadratic ansatz 29 and examine the most general expression quadratic by oscillators: [c a 1 c c1 ɛ 2 δa 1 c 1, G k a 2 c2 ]. 56 The commutator with A a b zeroth term of expansion 50 vanishes, because these are c-numbers, not operators. Similarly the first term is vanishes too, because we have no any chosen vector. For quadratic term of expansion one has: [c a 1 c c1 ɛ 2 δa 1 c 1, G 1 a 2 c2 ] = [c a 1 c c1 ɛ 2 δa 1 c 1, A a 2 c2,bdc b c d ] = A a 2 c2,c 1 dc a 1 c d A a 2 c2,bdε ba 1 c c1 c d + +A a 2 c2,bc 1 c b c a 1 A a 2 c2,bdε da 1 c b c c1 = 2A a 2 c2,edε ea 1 ε c1 b δ e c 1 δ a 1 b cb c d = = δ a 1 c 2 A a 2 c1,bd c 1 A a 1 c2,bd + ε a 2a 1 A c1 c 2,bd ε c1 c 2 A a 2a 1 bd c b c d. The only chosen tensor we can operate is metric ε ab, so the second term of expansion is given by: G 1 ab = A ab,e 1 e 2 c e 1 c e 2 = 1 2 ε ae 1 ε be2 ɛε be1 ε ae2 c e 1 c e 2 = c a c b ɛ 2 ε ab. 57 Following to these arguments one deduces that the higher term of expansion 50 vanish because commutator 56 contains too many terms and one deduces that the linear by oscillators term of the expansion is unappropriate So the ansatz 29 is the most general in this case. 8

10 Suppose now that we have more than one set of oscillators, which are ε-commutative or ε-anticommutative each to other such that the bilinear combinations 57 are commutative: G i,ab G j,ab = G j,ab G i,ab, i j. 58 One can suppose the following commutation relations between oscillators: c a i c b j + ɛc b jc a i = δ ij ε ab, 1 i, j N. 59 One can construct the quadratic combinations from our oscillators: C ab ij = c a i c b j, 60 Consider first N diagonal ones C ab jj : the ε-antisymmetric by group indices combinations are: G ab j = ɛc a j c b j 1 2 εab, 61 while the ε-symmetric ones are equal to metric tensor. combinations C ab ij i < j can be separated by ε-symmetry: The NN 1/2 non-diagonal S ab ij = 1 2 ca i c b j + ɛc b ic a j = ɛs ba ij = S ab ji, 62 A ab ij = ɛ 2 ca i c b j ɛc b ic a j = ɛa ba ij = A ab ji, 63 here we used commutation rule 59. So we see that the remaining NN 1/2 combinations C ab ij with i > j are expressed by ones with i < j. Using 62 one can construct nn 1/2 scalar combinations s ij = 1 2 ε abs ab ij = 1 2 ε abc a i c b j, 64 contraction A ab ij with metric vanishes. Definitions 62 and 63 can be extended to values i = j: S ab ii = 1 2 ca i c b i + ɛc b ic a i = ε ab, 65 A ab ii = ɛ 2 ca i c b i ɛc b ic a i = G ab i, 66 Then we again suppose the most general ansatz for generators G, depending on n sets of oscillators : G ab = n j=1 γ j s kl G ab j + 1 i<j n and calculate the following algebra between them: α ij s kl A ab ij = [s ij, G ab k ] = δ jk δ ik A ab ij, [s ij, 1 i j n n k=1 α ij s kl A ab ij, 67 G ab k ] = 0, 68 2[s ij, s kl ] = δ jk s il δ ik s jl + δ jl s ki δ il s kj, 69 2[s ij, A ab kl ] = δ jka ab il δ ika ab jl + δ jla ab ki δ ila ab kj, 70 9

11 where and 2[G ab k, Acd ij ] = ɛδ ik + δ jk A bc,ad ij + δ ik δ jk S bc,ad ij. 71 4[A ab ij, A cd kl ] = δ jks bc,ad il +ɛ δ jk A bc,ad il S bc,ad ij A bc,ad ij + δ ik S bc,ad jl + δ ik A bc,ad jl δ jl S bc,ad ki + δ jl A bc,ad ki δ il S bc,ad + δ il A bc,ad kj = ε bc S ad ij ε ca S bd ij + ε ad S bc ij ε db S ac ij, = ε bc A ad ij ε ca A bd ij + ε ad A bc ij ε db A ac ij. kj +, Conclusion The oscillator realization, presented in previous section being the most general way to describe an arbitrary operator dependence, allows to represent generators of the finite resolution of orthogonal or symplectic Yangian. Substituting it into the set of constraints 22-24, 35-42, etc. following from the RLL-relation for the particular truncation. This involving program will allow to construct the complete set of the particular representations compatible with finite resolutions of orthogonal and symplectic Yangians. References [1] Weil A. Sur certains groupes // Acta. Math V P [2] Konstant B. Symplectic spinors // Symposia math. 14, L., AP, 1974, P [3] Vershik A. M. Metagonal and Metaplectic groups // Zap. Nauch. Sem. LOMI 1983 V. 123 P [4] Faddeev L.D., Sklyanin E.K. and Takhtajan L.A., The Quantum Inverse Problem Method// Theor. Math. Phys V. 40 P. 688 [Teor. Mat. Fiz ] [5] Zamolodchikov A.B., Zamolodchikov Al.B. Factorized S Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models,// Annals Phys V. 120 P. 253 [6] Berg B., Karowski M., Weisz P., Kurak V., Factorized Un Symmetric S Matrices in Two Dimensions,// Nucl. Phys V. B 134 P [7] Faddeev L.D., How Algebraic Bethe Ansatz works for integrable model, In: Quantum Symmetries/Symetries Quantiques, Proc.Les-Houches summer school, LXIV. Eds. A.Connes,K.Kawedzki, J.Zinn-Justin. North-Holland, 1998, P , [hep-th/ ] [8] Kulish P.P., Sklyanin E.K., Quantum spectral transform method. Recent developments, Lect. Notes in Physics, 1982 V. 151 P [9] Shankar R., Witten E. The S Matrix of the Kinks of the ψ ψ 2 Model,// Nucl. Phys V. B 141 P. 349 [10] V.G. Drinfeld,Quantum Groups, in Proceedings of the Intern. Congress of Mathematics,Vol. 1 Berkeley, 1986, p [11] Zamolodchikov A B and Zamolodchikov A B 1979 Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models Annals Phys [12] Berg B Karowski M Weisz P and Kurak V 1978 Factorized Un Symmetric s Matrices in Two-Dimensions Nucl. Phys. B [13] Shankar R and Witten E 1978 The S Matrix of the Kinks of the ψ bar ψ**2 Model Nucl. Phys. B [Erratum-ibid. B ] [14] Reshetikhin N Yu 1985 Integrable models of quantum one-dimensional models with On and Sp2k symmetry Theor. Math. Fiz [15] Karakhanyan D and Kirschner R 2010 Jordan-Schwinger representations and factorised Yang-Baxter operators SIGMA [arxiv: [hep-th]] [16] Chicherin D Derkachov S and Isaev A P 2013 Conformal group: R-matrix and star-triangle relation arxiv: [math-ph] and The spinorial R-matrix J. Phys. A arxiv: [mathph] [17] Isaev A P Karakhanyan D and Kirschner R in preparation 10