Series of the solutions to Yang-Baxter equations: Hecke type matrices and descendant R-, L-operators

We have constructed series of the spectral parameter dependent solutions to the Yang-Baxter equations defined on the tensor product of reducible representations with symmetry of quantum algebra. These series are produced as descendant solutions from the $sl_{q}(2)$-invariant Hecke type $R^{r\;r}(u)$-matrices. The analogues of the matrices of Hecke type with the symmetry of the quantum super-algebra $osp_q(1|2)$ are obtained precisely. For the homogeneous solutions $R^{r^2-1\;r^2-1}$ there are constructed Hamiltonian operators of the corresponding one-dimensional quantum integrable models, which describe rather intricate interactions between different kind of spin states. Centralizer operators defined on the products of the composite states are discussed. The inhomogeneous series of the operators $R^{r\mathcal{R}}(u)$, extended Lax operators of Hecke type, also are suggested.


Introduction
The Yang-Baxter equations (YBE), appeared in the early investigations on the problem of the exactly solvability in 2d statistical physics, as well as in (1 + 1)-dimensional scattering theory, and being one of the key relations of the QISM (Quantum Inverse Scattering Method), still remain an actual and attractive subject of the statistical and mathematical physics, with expanding area of applications [1]- [17]. In the study of the solutions to the YBE with the symmetry of quantum algebra as basic constructions the universal R matrices are considered, which can be achieved either by the quantum double principle of Drinfeld or by the Jimbo's compositions involved algebra invariant matrices -projection operators, and different schemes of affinization (or baxterization) are developed for obtaining the spectral parameter dependent solutions [7,9,12,13,16]. It turns out that the range of the spectral parameter dependent solutions to Yang-Baxter equations with the given quantum algebra symmetry is richer, than that which can be constructed via the universal R matrices of the corresponding quantum algebras. For finding the full class of the symmetric YBE solutions it is sufficient to consider the R-matrix in the expansion of the whole basis of invariant operators (projectors), which must be specified for the given set of the representations [12,13,14,28]. In particular, investigating solutions defined on the cyclic and indecomposable representations of the quantum algebra sl q (2) at roots of unity [30,31], we find new solutions and yet a rich variety of the solutions, which are characterised by different structures of decompositions into the projectors and as well by additional (spectral) parameters ( [33]).
As a yet another confirmation of the mentioned observation could be served the existence of a series to the YBE solutions with symmetry of the quantum super-algebra osp q (1|2) defined on the spin-irreps, which differs from the known solutions ( [21,24,19]), and the discussion done in the Section 2 of this work demonstrates the exact derivation of this series. The similar solutions (Hecke type R-operators), as it is known, exist for the quantum algebra sl q (2) [8,10,20,29,28], and this reflects the circumstance that there is an explicit correspondence between the representations of the quantum algebras sl q (2) and osp q (1|2), providing that q → i √ q [25,26,19,27]. Then in Section 3 a descendant series of the mentioned solutions is constructed. The non usual behavior of these R-matrices is the reducible character of the vector spaces on which the operators act for general values of deformation parameter q. The integrable models corresponding to these R-matrices describe interactions between different spins (Section 4), however the Hamiltonian operator derived in accordance to the principles of the Algebraic Bethe Ansatz [4,5,15] has not the conventional form of the superposition of "spin-spin" operators. New formal operators can be proposed for describing these nearest neighborhood rather entangled interactions. In the Section 5 an approach determining the centralizer operators defined on the tensor products of the reducible states is developed, necessary and sufficient relations for them are deduced. By fusion procedure we can find out from the Hecke solutions the descendant inhomogeneous R-matrices also, and it turns out that the matrices R rR defined on the tensor product V r ⊗ U R , with V r being an irrep and U R being a series of the reducible states formed by the truncation of the tensor products of the irreps V r , may constitute "Hecke type" matrices by their structure. In the Section 6 we sketch the scheme of the obtainment of such R rRn (u) operators, defined for each rdimensional irrep and corresponding series of composite representations with definite R n dimensions. These matrices we can refer as the series of the "extended Lax operators", as for the case of the fundamental representation, r = 2, they just coincide with the matrix representations of the ordinary Lax operator. In the Section 7 the summary and some propositions are presented regarded to the "extended" R-, L-operators, and also there are discussed further developments and possible applications of the integrable structures defined on the composite representations. In the next part of the Introduction (Section 1), as well as in the Appendix some preliminary definitions, descriptions and formulas are presented.
Also in the Introduction some questions on the baxterization are analyzed.
Quantum super-algebra osp q (1|2). This graded quantum algebra is constituted by the Here, as usual, [a] q = q a −q −a q−q −1 . Sometimes different definitions for the anti-commutation relation in (1.1) are used, which are equivalent to this one by simple re-scaling of the generators and the deformation parameter [21,24,25,26]. Co-product is defined by the following Here ⊗ denotes the graded tensor product, and I is a unit operator. The quadratic Casimir operator can be written as c = (q  [25,27]. Below we shall use the notation q r = [(−1) r + 1] iπ 4 log q for the factor arising in the case of even dimensional irreps. The odd-dimensional representations are in the full analogy with the non-deformed algebra situation, meanwhile the even-dimensional representations have no well defined limit at q → 1 [25]. The description of the irreducible representations is brought in the Appendix. The decomposition of the tensor products of two irreps is presented by the following linear combination Let us denote j = 2j r − q r and j k = 2j r k − q r k , k = 1, 2. The Clebsh-Gordan q-coefficients (CGC) C j 1 j 2 j i 1 i 2 i are defined by this decomposition, where it is assumed {i 1 + i 2 = i} and also we suppose Here we have presented the formulae of C j 1 j 2 j i 1 i 2 i in such a way to have integer (half-integer) values of the variables j, i for odd (even) dimensional representations as in the case of sl q (2)algebra (for details see Appendix A2). It slightly differs from the notations we have used in [19], [27]. The inverse CG coefficients are defined by Figure 1: YBE, R ij R-matrix and YB equations. As a quasi-triangular Hopf algebra this algebra is equipped with an intertwiner R-matrix, which ensures the operation where a is an arbitrary element of the algebra and σ is the graded permutation operator acting on the elements of the algebra: σ · (a ⊗ c) = (c ⊗ a). R satisfies to the triangle relation (YBE) Here the right and left sides of the equation are acting on the space V 1 ⊗V 2 ⊗V 3 . R ij is defined on the product of the spaces V i and V j , and acts on the remaining state as unity operator. In the so-called "check"-formalism it is usedŘ = P R -matrix, where P is a graded permutation operator acting on the representation spaces as follows, P : . This can be achieved by using the matrix form of the graded tensor products -(a ⊗ c) ij kr = a i k c j r (−1) p k (p j +pr) . The solution to YBE as universal R-matrix in terms of the operators e, f and h, is considered by Drinfeld in the context of the quantum double principle [9]. For the super-algebra osp q (1|2) it is expressed by the following formula [24,25] Here [n] + = (−1) n−1 q n/2 +q −n/2 . The transpose of this matrix with the change q → 1/q also is an intertwiner matrix and is denoted by R − .
The representation of the matrix R r 1 r 2 (u) acting on the tensor product of two irreps V r 1 ⊗ V r 2 is more convenient to write in the "check"-formalism,Ř r 1 r 2 (u). In this case the YBE are written aš The commutation relations in (1.8) and the fusion rules (1.3) give a hint, that theŘ-matrix must be a linear superposition of the invariant matrices -projection operatorsP r r 1 r 2 [12,20,24],Ř r 1 r 2 (u) = r 1 +r 2 −1 the operatorP r r 1 r 2 vanishes on the spaces V r ′ , r ′ = r in the decomposition (1.3), and acts on the space V r , mapping (imaging) it to the space V r of the decomposition V r 2 ⊗ V r 1 . For the homogeneous case the matricesP r r 1 r 1 are the ordinary projection operators P r acting on V r as unity operator, P r V r ′ = δ rr ′ V r ′ . For the quantum super algebra osp q (1|2) the r r (u)-functions, which are the polynomials as for the case of sl q (2) algebra, when both of r 1 and r 2 are odd or even dimensional irreps, can be found e.g. in [12,20,24,19]. The general case of r 1 = r 2 ( mod) 2 is analyzed in detail in [19].
All the solutions with higher spin representations (r 1,2 > 2) in the series (1.12), which satisfy to (1.11) can be obtained by another way, by means of the so called "fusion" technique or "descendant" procedure [8,12] from the fundamental R 2 2 (u) solution. This is the consequence of two factors. One factor is that there is a point u 0 for whichŘ r 1 r 2 (u 0 ) is proportional to the projector with the maximal spinP (r 1 +r 2 −1) r 1 r 2 . The second one is that the left (or right) hand side of the YBE (1.10) itself can be served as a solution of YBE at u = u 0 (or at any point u =ū 0 , for whichŘ(ū 0 ) is a projector or direct sum of the projectors, and thus has the propertyŘ(ū 0 )Ř(ū 0 ) =Ř(ū 0 ) for the homogeneous case; for the general case the permutation must be taken into account in the multiplication of the projectors), in the form ofŘ (r 1 ×r 2 ) r 3 -matrix acting on the product of the vector spaces V r 1 ×r 2 ⊗ V r 3 , where Note, that as a rule, in this paper we are using the upper indexes for R-matrices for denoting the dimensions of the representation spaces on which the R matrices are acting and the down indexes for denoting the positions of the states.
Baxterization: some observations. For the two-dimensional fundamental representations of sl q (2) (definition of this algebra is brought in A3), the spectral parameter dependent R(u)-matrix (1.12) defined on them just can be presented by the following sum, up to multiplication by an arbitrary function (see as example [16,24]) This is true also for the two dimensional representations of the osp q (1|2) super-algebra [19].
From this form it means that the matrices R ± f on the fundamental representations besides of the ordinary constant YBE must satisfy also the other equations, which are For the representations with the higher spins the expansion of R r 1 r 2 (u) (we suppose r 1 ≤ r 2 ) to the series in terms of the parameter q u can contain more terms, i.e. at r 1 > 2 The last two matrices (R + r 1 r 2 , R − r 1 r 2 ) are the braid limits of the corresponding R-matrix and satisfy to the equations (1.14). The matrices R 2 r (u) constitute the matrix representations of the Lax operator L(u) [12,20,19], which keeps the form L(u) = q u L − − q −u L − .
Let us observe what kind of the relations the YBE (1.10) put on the expansion matrices ij (r i ≤ r j ), and supposing r 1 ≤ r 2 ≤ r 3 , we have from the YBE the following set of the equations The solutions to these equations are not unique. As an example we can consider the Rmatrices with the symmetry of the quantum algebra sl q (2) for the case r 1,2 = 3. There are three spectral-parameter dependent solutions R 33 (u) to the homogeneous YBE. Presenting them in the formŘ 33 (u) = q uŘ+ +Ř 0 + q −uŘ− , the first solution can be written aš 20) which is the case n = 3 of the universal solutionŘ nn (u) [12]. The next solution, associating with the Berman-Wenzl-Murakami algebra [29], reads aš This matrix has the same braid limits R ± as the previous one. We see that having the same braid limit constant matrices R ± it is possible to construct different spectral parameter dependent solutions (1.20,1.21). And the third solutioň , is just the solution belonging to the so called Hecke type [16,12] series. All the solutions are brought in the normalized formŘ 33 (0) = I.
The osp q (1|2)-invariant R 33 -matrices, which are equivalent to the solutions (1.20, 1.21) are discussed in details in [19]. And it is known, that in contrast to the case (1.20), the solution (1.21) has no generalization for higher dimensional irreps [19,28].
The osp q (1|2)-invariant analog of the solution (1.22) will be obtained in the next section.
2 A series of Hecke type homogeneous solutions to YBE with osp q (1|2)-symmetry The generalization of the fundamental representation (1.13) for the higher dimensional cases can be tried to do in such a way, that to keep the form R(u) = R (+) q u − R (−) q −u . It is known, that the Hecke type R-matrices, i.e. the matrices, which satisfy the Hecke relation (Ř−q)(Ř −1 +q) = 0, after "baxterization" obtain the mentioned form (e.g. [29] and citations therein). And surely for higher dimensions these R (+/−) -matrices do not coincide with the braid limit matrices R +/− obtained from the universal R-matrix.
Although for the quantum super-algebras osp q (1|2N) the role of the Hecke algebra plays the Birman-Wenzl-Murakami algebra, however, taking into account the equivalence of the quantum algebras sl q (2N) and the quantum super-algebras osp q (1|2N) in respect to their representation spaces [26], one also can expect the existence of the series of Hecke type Rmatrices with the symmetries of these super-algebras. Now let us concentrate our attention on the case of osp q (1|2). For the irreps with the dimensions r = 2 and r = 3 all the solutionš R rr one can obtain by direct matrix calculations and verify that there are the counterparts of the solutions (1.13, 1.22). In the case of the general r let us to look for the solution of (1.11) in a special form, as the solutions at r = 2, 3 admit such expansion (and this is valid in the case of sl q (2) invariant Hecke type operators, too). Here I r is a unit operator in the space V r and f (u) = f (u) − 1, as 2r−1 r ′ =1, △r ′ =2 P r ′ = I r ⊗ I r . Note, that dealing with the homogeneous matrices for simplicity we use the notations P r r i r k ≡ P r ik (r i = r k ), and in some cases just P r , without the indexes i, k denoting the spaces.
It is possible to derive the functions f (u) by various methods exploiting the algebra relations. Here we shall demonstrate an explicit computation in a rather detailed way, using the Clebsh-Gordan coefficients.
The procedure is standard. The right and left sides of the YBE are acting on the space Let us take an arbitrary vector state in that space, suppose v r k ⊗ v r p ⊗ v r t . The projector P 1 ij acts as non-vanishing (unity) operator only on this kind of the productsv r k ⊗ v r −k , which have 0-value of the operator h. Using definitions in (1.4, 1.5) and denoting j 0 = (r − 1)/2, we can write Taking into account these relations let us write down the non-trivial equations which follows from the action of the left and right hands of the YBE with the R-matrices described by is not dependent from the value of t. For defining that factor, let us write explicitly the CGcoefficients for the given case.
Hence, using that ( If to choose the j 0 -state having even norm, then the state {v 1 0 } would have 0 parity, and combining the relations (A.5) and (2.6) we deduce So, we have, using the relation (−1) 2j 0 = 1 for odd dimensional representations, and (2.9) The solutions of the equations (2.4) so have dependence only from the dimension of the representation V r . Equations can easily be solved by passing to the corresponding differential equations, expanding the expressions around a fixed point, e.g. at w = 0. We can write finally the spectral parameter dependent Hecke type solutions aš Here a is an arbitrary number. The "braid"-limits u → ±∞ of (2.10) coincide with the corresponding Hecke type constant solutions, satisfying to (1.14).
The investigation of the case with symmetry of quantum algebra sl q (2) would differ from this consideration only by the gradings of the states and the spin values of the evendimensional irreps. For the algebra sl q (2) the Hecke type solutions have been discussed in the works [11,13,29]. The purpose of this section has been to insist that such kind of series of the solutions exists for the super-algebra osp q (1|2), thus proofing that there is a full correspondence between the YBE solutions with the symmetries of sl q (2) and osp q (1|2).
We can try to construct descendant R-matrices corresponding to the discussed r×r-dimensional solutions. As it is mentioned already, in the standard fusion (descendant) procedure developed for the algebras under consideration [8,12] one leans on the property that there is a . Thus from the matrices R r 1 r 2 and R r 3 r 4 satisfying YBE one can construct the matrix R (r 1 +r 2 −1) (r 3 +r 4 −1) on the product with the maximal spins (Fig. 2).
And such solution is exactly equivalent to the matrix obtained by the Jimbo's constructions.
In the present situation (2.10) there is a point u 0 at whichŘ r r (u 0 ) = I − P 1 = 2r−1 r ′ =3 P r ′ , which can produce (r 2 − 1) × (r 2 − 1)-dimensional R-matrices satisfying YBE defined on the composite spaces. As these solutions can be interesting in the context of the integrable models which describe interactions between different kind of spins, we think it is worthy to obtain the exact form of such matrices. Of course the construction of the intertwiner matrices on reducible spaces has also a mathematical interest. Especially we shall investigate the series of the homogeneous solutions defined on the spaces U r 2 −1 ⊗ U r 2 −1 , as well as the series of the inhomogeneous descendant solutions, which can be referred as the "extended" versions of the ordinary Lax operators.
The discussion hereinafter is proper for both of the symmetries of quantum algebra sl q (2) and super-algebra osp q (1|2), only for the second case the grading of the states must be taken into account. Particularly the tensor product must be replaced by the graded tensor product, and the YBE, when R-matrices are written in non check formulation, would contain additional signs conditioned by the parities (e.g. see [19]). For clarity, in the next sections when concretization will be need we shall use the terminology of the sl q (2)-algebra (basic definitions are brought in the Appendix), but obviously the extension to the case of the quantum superalgebra osp q (1|2) is straightforward.
The descendant solution on the product V r 1 ⊗ V r 2 ⊗ V r 3 ⊗ V r 4 can be presented by the following product of the R-matrices (for this case r 1 = r 2 = r 3 = r 4 = r) Recall that everyŘ ij (u) matrix has the following decompositionŘ ij (u) = I ij + f (u)P 1 ij and the point u 0 is fixed from the equation f (u 0 ) = −1. Expanding the product of the operators in the big parenthesis into the sum of projection operators, and taking into account that the terms which are equivalent to P 1 12 · P , P 1 34 · P or P · P 1 12 , P · P 1 34 are vanishing after multiplying by (I − P 1 12 )(I − P 1 34 ), we come to the expressionŘ 12 34 (u) = (I − P In the course of the calculations done in the previous section we have obtained that the following relations hold (we denote here X ≡ X (j 0 ) = C j 0 j 0 0 we find out that Thus we haveŘ From the operator form of the matrix (3.5) we can check that at u = u 0 the operatorŘ 1234 is (I −P 1 12 )(I −P 1 34 ), which equals to unit operator in the considered (r 2 − 1) × (r 2 − 1)-dimensional space. This follows from the functional dependence of the expressions in (3.5) and and also from the observations, that the first order series expansion of the function f (u) near the point u = 0 equals to f (u) ≈ f 0 u, and that the point It means that at the point u 0 the following expansion is true So we obtain the following expansion near u 0 , .
The expansion of the Hamiltonian operators in terms of the algebra generators at the spaces V 2s and V 2s+1 is obvious just by construction.
The question about the spin structure of H i i+1 we can achieve either by the quantum 6j-symbols or directly by the Clebsh-Gordan coefficients [20,32]. A brief description how to use the quantum 6j-symbols for obtaining the decomposition is brought in the Appendix A3.
From the analysis of the dimensions in the expansions done therein, it follows, that actually, we must clarify whether the following decompositions (I −P

The structure of the Hamiltonian operator
Here we use for the orthogonalized vector states v r k the "ket", "bra" notations, |j, k , r = 2j + 1. The unit operator defined in the space V r can be expressed as I r = k |j,k j,k| j,k|j,k . The projector operator P r 0 r 1 r 2 acting on the tensor product V r 1 ⊗ V r 2 and distinguishing the space V r 0 , r 0 = 2j 0 + 1, we can write in this way, by using the formula (1.4) The unit operator I r 1 ×r 2 = (r 1 +r 2 )−1 r 0 =|r 1 −r 2 |+1 P r 0 r 1 r 2 defined on the space V r 1 ⊗ V r 2 can be written as For the orthosymplectic algebra one must take into account the grading of the vectors, and appropriate signs would appear in the above formulas. The vector states v r k i k can be chosen to be normalized, then the formulas would be more compact. The first term of the Hamiltonian operator corresponding to the cell V 1 ⊗ V 2 ⊗ V 3 ⊗ V 4 can be presented as the following, taking into account that j 1 = j 2 = j 3 = j 4 ≡ j, Correspondingly, the second term of the Hamiltonian operator will be (I −P 1 12 )(I −P 1 34 ) P 1 23 P 1 14 (I −P 1 12 )(I −P 1 34 ) = 2j The obtained operators constitute superpositions of the projectors (P r 0 ) a,b : (V r 0 ) a → (V r 0 ) b , which are algebra invariant operators mapping the different spaces (V r 0 ) a,b with the same spin j 0 = (r 0 − 1)/2 (≡ j 1234 = j ′ 1234 in (4.8)) arising in the fusion of V r ⊗ V r ⊗ V r ⊗ V r one to other. For the expression in Eq. (4.8) r 0 = 1, ..., 2r − 1, and for the case of Eq.(4.9)r 0 = 1). Now let us turn to the question arisen just before this subsection. We can see that the mentioned decomposition in general does not take place. In the case of the H-operator (4.10) here we have used this formal notation -J r ′ r = |v r v r ′ |, where the indexes of the spin projections are omitted. In case of (4.9) r i = r i+1 , r ′ i = r ′ i+1 , as r 1234 = 1 and r 12 = r 34 , r ′ 12 = r ′ 34 . And clearly, the operators J r ′ r in general (r = r ′ ) are not expressed by the algebra generators defined on the states V i .
Detailed expressions and study of such quite large Hamiltonian operators for specific cases are proposed to do in subsequent works.
The actual spaces on which the Hamiltonian operator (4.5) is acting, are the truncated products, and in the next discussions we shall use new notations U R for denoting such Rdimensional composed spaces. Particularly by the action of the projectors the product state i , and will be denoted as U r 2 −1 .

Centralizers and reducible representations.
In fact in (4.10) we deal with the centralizer operator defined on the tensor product of two mixed states U r 2 −1 ⊗ U r 2 −1 . Here we intend by a straightforward construction to reveal the structure of such operators. Let us write down the conditions which the algebra relations put on the centralizers defined on the product U ⊗ U, with composite representations spaces U, and then the extension to general case U ⊗ U ⊗ ... ⊗ U can be done by similar calculations.
Let U consists of some set of irreducible representations: U = r k V r k . The thorough formulation of the operators J r ′ r could be done by means of the ortho-normalised elementary operators If in U there are more than one copies of the irreps with the given same spin-j, one must add an additional index for differentiating them, e.g. -j a . Each linear operator a, evaluating in U can be presented as a superposition The algebra generators on this basis can be presented as (the coefficients β, h, γ below denote the usual matrix elements of the corresponding operators, for the quantum super- The matrix elements of the generators just by definition are the same (up to some elementary transformations, admissible by the algebra relations, see e.g. [19]) for the irreps with the same spin β ja ia ≡ β j i , .... Every element of the center defined on U consists of these elementary projection operators 14) The quadratic Casimir operator is just the sum c = ja c j P ja ja . The centralizer operators c defined on the tensor products of n-copies of the mixed states U, U ⊗ U ⊗ · · · ⊗ U have more rich structure. The sufficient and necessary conditions for them can be obtained straightly from the equations [∆(∆⊗· · · (∆⊗I))[g], c] = 0, presenting the commutation of the operators c with the algebra generators defined on the tensor product of the representations by means of the associative co-product operation. In general one can write any linear operator defined in n U as For the generic centralizer operators c in the simplest case n = 2 (which is enough to consider if we are interested with the nearest-neighbourhood interaction Hamiltonians) the thorough calculations give the following relations on the co- ja,ia , n = 2, with the algebra generators, Here there is taken into account that h j i ≡ i. For the case of the quantum super-algebra osp q (1|2) one must take into account the parities of the states, the graded character of the tensor products and that h j i = i + ı constant for the even dimensional irreps. To obtain the corresponding relations for the general case with arbitrary n is an obvious task, which brings to the evident extension of the Eqs.(5.16), with n summands at the r.h.s. and l.h.s. of the equations.
Particularly the first equation in Eqs.(5.17) ensures the conservation of the spin projection.
The next equations put the relations on the coefficients c {ja,ia} . Note, that for the n-th term in the sum of the l.h.s of the second equation in (5.17) one must take {i ′ bn + 1 → i ′ bn }, and correspondingly for the similar term of the third equation of (5.17) -{i ′ bn − 1 → i ′ bn }.

Extended Lax operators
The Hecke type matrices R rr (u) do not allow generalizations to the inhomogeneous R rr ′ (u) acting on V r ⊗ V r ′ with V r ′ being an irrep, so that R rr R rr ′ R rr ′ = R rr ′ R rr ′ R rr , besides of the case r = 2 with the fundamental irrep V 2 , which gives standard universal Lax operator L, obeying the quantum YBE (see for the references in [19], where the corresponding operator is constructed for the case of osp q (1|2)) However by descendant procedure we can get algebra invariant matrices R rR : V r ⊗ U R satisfying to YBE with U R to be a composite representation. The simplest U R has been discussed in the previous sections. Applying fusion method further one can find descendant R operators defined on the representations larger than V r ⊗ V r − V 1 . And by means of these i i 1  operators one can construct new quantum integrable models on the 1d chains with the action space i U R i . One can expect that the corresponding local quantum Hamiltonian operators would describe interactions between different spins in a rather entangled way, relying to the discussed example of the series of solvable models with homogeneous R r 2 −1 r 2 −1 -matrices.
So, we can construct the series of the "extended Lax operators" L = R rR , satisfying (6.1) with Hecke type R rr , and of course the case of r = 2 would be the usual Lax operator. In the same way, as in the previous sections the descendant series can be constructed by the products of the R rr (u)-operators appropriately fixing the values of the spectral parameters [13]. The matrix deduced from the action of the operators on the space V r ⊗ V r ⊗ V r will beŘ r r 2 −1 (u) = [I ⊗ (I − P 1 )](Ř r r (u + u 0 ) ⊗ I)(I ⊗Ř r r (u))[(I − P 1 ) ⊗ I], (6.2) defined eventually on V r ⊗ V r ⊗ V r − V 1 . Note, that the projection operator (I − P 1 ) at the right of this expression could be omitted due to YBE, which actually ensures it's existence. The extension to the space with n-product V r ⊗ V r ⊗ . . . ⊗ V r we can perform repeatedly using YBE and truncating by the appropriate projectors, achieved by taking step by step u = u 0 for the right-edge R rr (u)-matrices in the Fig.[3]. In the resulting matrix the all spectral parameters are established in accordance to the summation rule of the additive spectral parameters in YBE, and, as it was hinted above, only left hand projections are taking into account. So, the matrix R r×R (u) defined on the truncation of the tensor product V r i ⊗ V r i 1 ⊗ · · · V r in can be written formally as the following expression: (6.4) where the matrices are presented in "non-check" form, and the low indexes of the R-matrices show the spaces on which the operators act in the tensor product. The product of the operators in the second row (6.4) itself is a solution of the YBE defined on V r i ⊗ V r i 1 ⊗ · · · V r in , the expression in (6.3) realizes the projection operation (recall thatŘ rr (u 0 ) = (I − P 1 )).
Step by step acting on the tensor product of the R-matrices these projectors narrow (restrict) the action space from r n+1 -dimensional space to the r × R n -dimensional space V r × U Rn , where U Rn is a reducible space with dimension R n , which can be obtained as for the case of fundamental irrep (r = 2), from the recurrence formulas: R 0 = 1, R 1 = r, R 2 = r 2 − 1, ... R n = r × R n−1 − R n−2 . Correspondingly we can retrieve the structure of the composite space U Rn : repeatedly using the fusion rules (1.3). For the case r = 2, of course we recover R n = n + 1 and in this case U Rn = V n+1 is the (n + 1)-dimensional irreducible representation -n 2 -spin irrep. Generally we can refer to the space U Rn as a truncated product of the irreps ⊗ n V r .
As for the explicit formula for this 'extended' Lax operator, from the discussion above it follows, that R rRn (u) possesses the form P (P R n+1 + P R n−1 + F n (u)P R n−1 ), where F n (u) ≈ n k f (u + (n − k)u 0 ). The proof also can be done by induction, noting, that the operator R r R n+1 (u) differs from R r Rn (u) by the product of (n+1)-operators R rr (u+nu 0 ) n k R rr ((k − 1)u 0 ), and the spectral parameter dependent terms in R r R n+1 (u) proportional to ∼ f (u+nu 0 ) and ∼ F n (u) are eliminating due to relations coming from YBE, and the projectors (I − P 1 ), and the only spectral parameter dependent term which survives is ∼ f (u + nu 0 )F n (u) ⇒ F n+1 (u). So, we can summarize The coefficient f r,n,q is conditioned by the action of the mentioned projection operators, as well by the permutation of the spaces, and can be formulated by means of the appropriate set of the CG-coefficients for each case. It is notable to remark, that it is easy to find out the product n k=1 f (u + (n − k)u 0 ), using the recurrent relations, deduced from the equation .
This gives for the mentioned product a polynomial of this kind (−1) n f (u) 1+an+bnf (u) . Formally these operators keep the form [P a + h(u)P b ] with two projectors, which is typical for the Hecke type operators and ensure the availability of the Hecke relations on the eigen-vectors' space of [P R n+1 + P R n−1 ] which plays the role of the unity operator. Only here one must be careful managing with the multiplication or with the inverse operations, as for the inhomogeneous R-matrices (6.5) we deal with the transposition (permutation) of the vector states V r and U Rn .
For the case of the fundamental irrep, r = 2, fixing a = log q in (2.10) (note, that there for the case of sl q (2) we must take into account that q → ıq 1/2 ) we shall come to , which leads to the usual form of the Lax operator for the algebra sl q (N): L(u) = q u L + − q −u L − [29]. For the general cases with r > 2, the corresponding and similar expansion of the "extended" Lax operator R rR is followed by taking into account the presented above polynomial formulae coming from the recurrent relation (6.6).
All the same results are valid in the non-deformed case also, as the corresponding limit (q → 1) is good defined (recall merely, that for the ortho-symplectic algebra at the classical limit only odd dimensional irreps are existing).
The one-chain Hamiltonian operators, corresponding to the obtained inhomogeneous matrices (6.5), constructed by means of the transfer matrices, where the auxiliary space is the V r -irrep, describe non trivial interactions between the different spins, having the structure expressed by the projection operators, similar to the one, discussed in the previous section.
Strict investigations of such Hamiltonian structures will be carried out in the future.
The set of the composed states, fitting to the truncated tensor products of the spin-irreps, can be built for each case separately. For the simple case n = 2, the normalised (but not orthogonal) states of U r 2 −1 , induced from the initial sublattice, are determined elementary, using the relation (1.5):

Summary and conclusions
Summing up, we can consist, that here new solutions to Yang-Baxter equations defined on the composite states have been investigated, in particular, obtained by the fusion method from the solutions of Hecke type. The Hecke type homogeneous R rr -matrix's series has been constructed for the quantum super-algebra osp q (1|2), defined on the tensor product of two r-dimensional irreducible representations, quite analogous to the corresponding matrix with sl q (2) symmetry. This pattern ascertains that the equivalence of the representation spaces of two algebras implies the equivalence of the solutions to YBE, as here the important role have the basis operators for the R-matrices, i.e. the algebra invariant operators -projectors.
So, the YBE solutions known for the algebra sl q (2n) must be valid for the osp q (1|2n) quantum super algebra after the appropriate changes connected with the gradings and quantum deformation parameter q, as there is full correspondence between their representations [26].
For the Hecke type YBE solutions the corresponding descendant series R r 2 −1 r 2 −1 have been constructed, which are defined on the composite (reducible) r 2 − 1-dimensional states of the sl q (1|2) (or osp q (1|2)) algebra. Also descendant inhomogeneous matrices R rR ("extended" Lax operators), compatible with the mentioned invariant series, have been sug-gested, with definite series of R n -dimensional composite states for each r-dimensional irrep.
Of course, more general "extended" R-operators R RR ′ also could be observed, which would be descendant matrices inherited from the obtained ones. All such type of R-operators produce Hamiltonian operators corresponding to 1d quantum integrable spin models describing non-elementary mixed interactions between different kinds of spins situating on the sites of the chains.
We can summarize the results schematically by the following diagram of the series of YBE solutions with the sl q (2) (osp q (1|2)) symmetry, including the obtained descendant matrices defined on the composite states (and inherited from the Hecke type R rr matrices with r > 2) together with the universal R rr ′ matrices defined on the irreps, which are descendant matrices originated from the matrix on the fundamental irreps R 22  Note, that at the exceptional values of the deformation parameter of the quantum group (i.e, when q is a root of unity), the specter of the irreducible representations is restricted, higher spin irreps are deforming, and new indecomposable representations are arising, and correspondingly, the fusion rules also are deformed, but however in this case also the solutions of YBE defined on the composed states can be found, properly defining the centralisers or the projection operators [30,27].
As it is known, by means of the YBE solutions the braid group representations can be realised, and they can be employed to obtain the link and knot invariants [20,22,23]. Thus one can use the R-matrices defined on the composite spaces for determining link invariants for such extended cases too. And besides of pure mathematical and theoretical interest, the solutions to Yang-Baxter equations on the reducible representations of the quantum algebras also can have practical usage. Particularly, such "extended" R-matrices can be attractive in the context of the recent developments of the mutual interrelations of the quantum entanglement theory (in topological aspects) and the theory of integrable models, or, more precisely, the solutions to YBE, as the essential instruments in the construction of integrable models [17]. As the Hamiltonian operators corresponding to the discussed solutions describe integrable systems having rather large number of degrees of freedom and rich structure of the spin variety (with quite tangled interactions) even for the lattices with few sites, so possible applications may be assumed in different areas of 2d quantum statistical physics, string theories and particle physics.
Combining two relations, (1.4) and (1.5), and using the orthogonality of the v r i -vectors, we obtain that C andC-coefficients are inverse each to other in the following matrix sense (i.e. are forming inverse matrices to each other) From the other hand, as the vectors {v r i } and {v r i } are orthogonal when r and r ′ don't coincide, then it follows that {C j 1 j 2 j i 1 i 2 i } ≈ {C j 1 j 2 j i 1 i 2 i }. The proportionality coefficients we can find from the relation (A.5), where ε j i is the norm of the state v r i . The norm for the graded representations can be defined as follows [24]. Let v r j is an even state, then we can take ε j i = 1 for all i. If v r j is an odd state, then the norm in the irrep V r is indefinite: ε j i = (−1) j−i . For definiteness we can take v r 1 j 1 and v r 2 j 2 as even states, then the irreps V r 1 +r 2 −1−k have positive norms, when k = 0 + 4Z + and have indefinite norms when k = 0 + 2Z + .
In the relations (A.1) the β r i−1 , γ r i -coefficients corresponding to the mentioned normalization can be fixed so, that β r i−1 = γ r i (−1) j−i , and are equal to √ α r i up to a sign.

A.3
Quantum algebra sl q (2). At the end we give also brief definition of the quantum algebra The quadratic Casimir operator is Finite-dimensional irreducible representations V r , dim[V r ] = r, are describing by their Casimir eigenvalues c r = [r/2] 2 q and by "spin" values j = (r − 1)/2, with the analogy of the non-deformed algebra situation.
Quantum 6j-symbols. The associativity of the tensor product of the quantum group is expressed by definition of the quantum 6j-symbols j 1 j 2 j 12 j 3 j j 23 q as follows [20,16]  Here j, j k are the spin values of the corresponding r-dimensional irreps, j = (r − 1)/2. The first diagram corresponds to the tensor product {V j 1 ⊗ V j 2 → V j 12 } ⊗ V j 3 → V j , the second one in the sum corresponds to V j 1 ⊗ {V j 2 ⊗ V j 3 → V j 23 } → V j . Also let us use the notation ρ(j 1 , j 2 ; j) [16] for denoting c− numbers which distinguish the Clebsh-Gordan coefficients corresponding to the projections V j → V j 1 ⊗ V j 2 and V j → V j 2 ⊗ V j 1 . Then for revealing the spin structure of the operator (4.5), one can consider as an elementary cell of the chain lattice (with sublattice structure) the product of the vector spaces V 1 ⊗ V 2 ⊗ V 3 ⊗ V 4 . And taking into account, that both of the terms of the Hamiltonian (4.5, 3.5) contain the projector P 1 23 , we can compare the following relations for the decomposition of the vector products, in order to express the terms containing the projectors P 23 and P 14 acting on the product V 1 ⊗ {V 2 ⊗ V 3 } ⊗ V 4 , by means of the operators acting on the tensor product grouped as As the projector operator P 1 23 acting on the space V 2 ⊗ V 3 maps it into the one dimensional space, then j 23 = 0. And it means j 123 = j 1 , j 1234 = j 14 , so { j 23 j 1 j 123 j 4 j 1234 j 14 } q ≈ δ j 14 ,j 1234 . These equations are valid for each j 1234 which satisfies to |j 1 − j 4 | ≤ j 1234 ≤ (j 1 + j 4 ). The second term in theŘ 1234 -matrix contains the projector P 1 14 , the action of which gives for that term j 14 = 0, and hence j 1234 = 0 and j 12 = j 34 . The external projectors (I − P 1 12 )(I − P 1 34 ) entering into theŘ 1234 -matrix ensure that j 12 = 0 and j 34 = 0. As all the states have the same dimension, i.e. the same spin j = (r − 1)/2, then j 12/34 ∈ [1, ..., 2j].
In the same spirit we can write out the transition operations passing the following