Yang-Baxter R-operators for osp superalgebras

We study Yang-Baxter equations with orthosymplectic supersymmetry. We extend a new approach of the construction of the spinor and metaplectic $\hat{\cal R}$-operators with orthogonal and symplectic symmetries to the supersymmetric case of orthosymplectic symmetry. In this approach the orthosymplectic $\hat{\cal R}$-operator is given by the ratio of two operator valued Euler Gamma-functions. We illustrate this approach by calculating such $\hat{\cal R}$ operators in explicit form for special cases of the $osp(n|2m)$ algebra, in particular for a few low-rank cases. We also propose a novel, simpler and more elegant, derivation of the Shankar-Witten type formula for the $osp$ invariant $\hat{\cal R}$-operator and demonstrate the equivalence of the previous approach to the new one in the general case of the $\hat{\cal R}$-operator invariant under the action of the $osp(n|2m)$ algebra.


Introduction
The similarities between the orthosymplectic supergroups OSp(N |M ) (here M = 2m is an even number) and their orthogonal SO(N ) and symplectic Sp(M ) bosonic subgroups can be traced back to the existence of invariant metrics in the (super)spaces V (N |M ) , V N and V M of their defining representations. These similarities lead to the consideration of the supergroup OSp and its superalgebra osp in full analogy with the unified treatment (see e.g. [1]) of the groups SO, Sp and their Lie algebras. Moreover these similarities are inherited in the study of solutions of the Yang-Baxter equations that possess such symmetries.
In the present paper, we continue our study [2] of the solutions of the Yang-Baxter equations symmetric with respect to ortho-symplectic groups. We start with the graded RLL-relations with the R-matrix in the defining representation R ∈ End(V (N |M ) ⊗ V (N |M ) ) and find the L-operator, L(u) ∈ End(V (N |M ) ) ⊗ A, where A is a super-oscillator algebra invariant under the action of the OSp(N |M ) group. Then this L operator allows one (via another type of RLL relations) to define a richer and more complicated family of solutions of the Yang-Baxter equations, namely theR-operators, which take values in in the tensor product A ⊗ A and are expressed as an expansion over the invariants in A ⊗ A. The orthogonal and symplectic groups are embedded in the ortho-symplectic super-group OSp, and theR-operators invariant under the so(N ) and sp(M ) algebras can be obtained from the OSp-invariantR-operator as special cases. In the orthogonal case the algebra A is the N -dimensional Clifford algebra and the operatorR is called the spinor R-matrix. In the symplectic case the algebra A is the oscillator algebra andR is called the metaplectic R-operator.
The standard approach to the problem of finding the spinor (so-invariant)R-operator was developed in [3], [4] and is based on the expansion of theR-operator over the invariants I k realized in the spaces A ⊗ A. Here the factors A are the Clifford algebras with the generators (c a ) α β , where α, β and a are respectively spinor and vector indices. Then the invariants I k are given by the contraction of the antisymmetrized products of c 2 ∈ I ⊗ A with the invariant metrics ε a i b i . In that approach we obtain the spinorR-operator as a sum over invariants I k with the coefficients r k which obey recurrence relations. Analogous formulae of the Shankar-Witten (SW) type for theRoperators were deduced for the symplectic case in [1] and then were generalized for the ortho-symplectic case in [2]. Note that we cannot consider these expressions for theRoperators as quite satisfactory, since they do not provide closed formulas for the considered R-operators. For example, in the symplectic and ortho-symplectic cases, the sum over I k is infinite.
On the other hand, it is known that an analogousR-operator invariant under the sℓ(2) algebra can be represented (see [5], [6]) in a compact form of the ratio of two operatorvalued Euler Gamma-functions. Surprisingly, as it was shown in a recent paper [7], the so and sp invariantR-operators (for special Clifford and oscillator representations of so and sp) are also represented in the Faddeev-Tarasov-Takhtajan (FTT) form of the ratio of two operator-valued Euler Gamma-functions.
In the present paper, we generalize the results of [7] to the supersymmetric case and show that the osp invariantR-operator can also be represented in the FTT form. This is the main result of our paper. The natural conjecture is that the osp-invariant SW typeR-operator given as a sum over invariants I k is equal to the osp-invariant FTT typê R-operator given by the ratio of two Gamma-functions. This conjecture is based on the fact that bothR-operators are solutions of the same system of finite-difference equations which arise from the RLL relations.
A complete proof of this conjecture is still missing. In the present paper we propose another simpler and more elegant derivation of the SW type formula for the osp invariant R-operator. This new derivation supports the conjecture of the equivalence of the SW and FTT expressions for theR-operators. Indeed, in the previous derivation, the role of invariant, "colorless", elements in A ⊗ A is played by the operators I k . In the new derivation, we prove that the invariants I k are polynomials of one invariant I 1 ∼ z only and rewrite the RLL relation itself into a "colorless" form from the very beginning in terms of a system of finite-difference equations in the variable z.
We relate this new system of equations to both the SW and the FTT expressions for theR-operator. On one hand, the FTT typeR-operator is its solution. On the other hand we show that the expansion of the SW typeR-operator over I k satisfies this system of finite-difference equations as well.
The paper is organized as follows. In Section 2, we recall some basic facts of the linear algebra on the superspace V (N |M ) with N bosonic and M fermionic coordinates and briefly formulate the theory of supergroups OSp(N |M ) and their Lie superalgebras osp(N |M ). In this section we fix our notation and conventions. In Section 3, we define the osp-invariant solution of the Yang-Baxter equation as an image of a special element of the Brauer algebra in the tensor representation in super-spaces V ⊗r (N |M ) . Section 4 is devoted to the formulation of the graded RLL relations. In this Section, we find a special L-operator that solves the RLL relations in the case of the osp algebra and introduce (see also [2]) the notion of the linear evaluation of the Yangian Y(osp). In Section 5 we define the super-oscillator algebra A and describe the super-oscillator representation for the linear evaluation of the Yangian Y(osp). In particular, we define the set of Osp invariant operators I k in A ⊗ A and their generating function.
In terms of these invariant operators we construct in Section 6 the osp invariantRoperators in the super-oscillator representation. We find two forms for suchR-operator. One of these forms represents theR-operator as a ratio of Euler Gamma-functions. For the sℓ(2) case this type solution was first obtained in [5] (see also [6]) and we call these solutions the FTT typeR-operators. Another form of the osp invariantR-operator in the super-oscillator representation generalizes the SW solution [3] of the spinor-spinor so-invariantR-operator. This solution (see eqs. (6.6) and (6.8)) for the osp-invariant R-matrix in the super-oscillator representation was first obtained in our paper [2] by using the methods developed in [4], [8] and [1]. In [2] we have generalized formulas for the so-type R-matrices (in the Clifford algebra representation) obtained in [3], [8] (see also [9], [10], [11], [12], [1]). In [2] we have also generalized the formulae for sp-type R-matrices (in the oscillator, or metaplectic, representation of the Lie algebra sp), which were deduced in [1]. It has been shown in [2] that all these so-and sp-invariant R-matrices are obtained from (6.6), (6.8) by restriction to the corresponding bosonic Lie subalgebras of osp.
In Section 7 the result for the FTT type R operator is studied in detail in particular cases of osp(N |M ). The arguments of the Gamma-functions involve the invariant operator z ∼ I 1 which decomposes into a bosonic and a fermionic part. The finite spectral decomposition of the fermionic part is considered and used to decompose the R operator with respect to the correspoding projection operators.
In Section 8 we present the new and more direct derivation of the solutions (6.6) and (6.8). Two Appendices are devoted to the proofs of the statements made in the main body of the paper.

The ortho-symplectic supergroup and its Lie superalgebra
Consider (see, e.g., [13], [2]) a superspace V (N |M ) with graded coordinates z a (a = 1, . . . , N + M ). The grading grad(z a ) of the coordinate z a will be denoted as [a] = 0, 1 (mod2). If the coordinate z a is even then [a] = 0 (mod2), and if the coordinate z a is odd then [a] = 1 (mod2). It means that the coordinates z a and w b of two supervectors z, w ∈ V (N |M ) commute as follows Let the superspace V (N |M ) be endowed with a bilinear form which is symmetric for ǫ = +1 and skewsymmetric for ǫ = −1. In eq. (2.2) we define w a ≡ ε ab w b , where, in accordance with the last relation in (2.2), the super-metric ε ab and inverse super-metricε ab have the properties We stress that the super-metric ε ab is an even matrix in the sense that ε ab = 0 iff [a]+[b] = 0 (mod2): In other words the supermatrix ε ab is block-diagonal and its non-diagonal blocks vanish. Using (2.4), the properties (2.3) can be written as According to this rule, we have ε ab =ε acεbd ε cd =ε ba and the metric tensor with the upper indices ε ab does not coincide with the inverse matrixε ab . Further, we use only the inverse matrixε ab and never the metric tensor ε ab .

Consider a linear transformation in
Now we write the relations (2.8) in the coordinate-free form as where the concise matrix notation is used (2.10) Here ⊗ denotes the graded tensor product: . We remark that in our paper we use the convention in which gradation is carried by the coordinates, while there is another convention in which gradation is carried by the basis vectors (see e.g. [14]). The relation of these two formulations is explained in [2]. Consider the elements U ∈ OSp which are close to unity I: U = I + A + . . . . Here dots denote the terms which are much smaller than A. In this case, the defining relations (2.8) give conditions for the supermatrices A which are interpreted as elements of the Lie superalgebra osp of the supergroup OSp: The coordinate free form of relation (2.11) is One can directly deduce these relations from equalities (2.9). The set of super-matrices A, which satisfy (2.11), (2.13), forms a vector space over C which is denoted as osp. One can check that for two super-matrices A, B ∈ osp the commutator [A, B] = AB − BA , (2.14) also obeys (2.11), (2.13) and thus belongs to osp. It means that osp is an algebra. Any matrix A which satisfies (2.11), (2.13) can be represented as where ||E a c || is an arbitrary matrix. Let {e f g } be the matrix units, i.e., matrices with the in (2.15), then we obtain the basis elements { G f g } in the space osp of matrices (2.13): Now any super-matrix A ∈ osp which satisfies (2.11), (2.13) can be expanded over the basis (2.16) where a g f are the components of the super-matrix. Since the elements ( G f g ) a c are even, i.e., ( . It means that the usual commutator (2.14) appears as a super-commutator for the basis elements G f g : where in the component form the super-commutator is . Now we substitute the explicit representation (2.16) in the right-hand side of (2.18) and deduce the defining relations for the basis elements of the superalgebra osp: where we have omitted the matrix indices. Below we use the standard component-free form of notation, where we substitute ( G a i b i ) a k b k → G ik (here i and k are numbers 1, 2, 3 of two super-spaces V (N |M ) in V ⊗3 (N |M ) ). In this notation, taking into account (2.18), the relation (2.19) is written as where we introduce two matrices K, P ∈ End(V ⊗2 (N |M ) ): The matrix P is called superpermutation since it permutes super-spaces, e.g., using this matrix one can write (2.1) as P ab cd w c z d = z a w b . Note that the generators (2.16) of the Lie super-algebra osp can be expressed in terms of P and K as  It means that the defining relations (2.19) can be written in many equivalent forms. At the end of this section we note that the matrix (2.22) is the split Casimir operator for the Lie superalgebra osp in the defining representation.

The OSp-invariant R-matrix and the graded Yang-Baxter equation
Consider the three OSp invariant operators in V ⊗2 (N |M ) : the identity operator 1, the superpermutation operator P and metric operator K. According to definition (2.21), the superpermutation P 12 is a product of the usual permutation P 12 and the sign factor (−) 12 , while the operator K 12 is defined as Their OSp invariance means that (see (2.9)) In particular, it follows from these relations that the comultiplication for the supermatrices U ∈ Osp(N |M ) has the graded form ∆(U ) 12 = U 1 (−) 12 U 2 (−) 12 . In fact this comultiplication follows from the transformation (2.7) applied to the second rank tensor z a 1 · z a 2 .
Using the operators P, K one can construct a set of operators which define the matrix representation T of the Brauer algebra B n (ω) [16], [17] with the parameter Recall that here N and M are the numbers of even and odd coordinates, respectively. Indeed, one can check directly (see Appendix A) that the operators (3.4) satisfy the defining relations for the generators of the Brauer algebra B n (ω) This presentation of the Brauer algebra can be obtained in the special limit q → 1 from the BMW algebra presentation [15] and it is used in many investigations (see, e.g., [18], [19], [20], [21]). We stress that the matrix representation T (3.4) of the generators s i , e i ∈ B n (ω) acts in the space V ⊗n (N |M ) . Let us consider the following linear combination of the unit element 1 ∈ B n (ω) and the generators s i , e i ∈ B n (ω) where u is a spectral parameter and Proposition 1. (see [19], [2]). The element (3.8) satisfies the Yang-Baxter equation

10)
and the unitarity condition The matrix representation T (3.4) of the element (3.8) iŝ Here we suppress index i for simplicity. It follows from (3.10) thatR(u) satisfies the braid version of the Yang-Baxter equation Thus, in the supersymmetric case the braid version (3.12) of the Yang-Baxter equation is the same as in the non supersymmetric case. Further we use the following R-matrix which is the image of the elements [19]:  (3.12) and moving all usual permutations P ij to the left we write (3.12) in the form The matrix R ∈ End(V ⊗2 (N |M ) ) is an even matrix since the following condition holds This follows from the same property for the matrices 1, P, K which compose the operator R(u). Therefore, for arbitrary k we have where the operator (−) ik is defined in (2.10) (i and k are numbers of only two superspaces V (N |M ) in the product V ⊗n (N |M ) where the operator (−) ik acts nontrivially). Using the property (3.17), one can convert (3.15) into the form and after the change of the spectral parameters we obtain the graded version of Yang-Baxter equation (3.14).

Remark 1.
We stress that the sign operators (−) 12 in (3.14) can be substituted by the operators (−) 23 by means of manipulations similar to (3.17). Moreover, if R ij (u) solves the Yang-Baxter equation (3.14), then the twisted R-matrix (−) ij R ij (u)(−) ij is also a solution of (3.14). Remark 2. Eqs. (3.11), (3.13) give unified forms for solutions of the Yang-Baxter equations (3.12), (3.14) which are invariant under the action of all Lie (super)groups SO, Sp and OSp. Recall that for the SO case the R-matrix (3.13) was found in [24] and for the Sp case it was indicated in [25]. For the OSp case such R-matrices were considered in many papers (see, e.g., [23], [14], [26]).

Graded RLL-relation and the linear evaluation of Yangian Y(osp)
We start with the following graded form of the RLL-relation (see, e.g., [26] and references therein) where the R-matrix is given in (3.13). This graded form of the RLL relations is also motivated by the invariance conditions (3.3). It is known (see, e.g., [2], [14] and references therein) that eqs. (4.1) with the R-matrix (3.13) are defining relations for the super-Yangian Y(osp). In [2] we proved the following statement.
where α is an arbitrary constant, solves the RLL-relation (4.1) iff G a b is a traceless matrix of generators of the Lie superalgebra osp, i.e., it satisfies equations (cf. (2.24))
The L-operator (4.2), where the elements G a b satisfy the conditions (4.3), (4.6) and (4.5), is called the linear evaluation of the Yangian Y(osp). Remark 3. The relations (4.4) are written after the exchange 1 ↔ 2 in the form where In particular, it follows from (4.3), (4.7) that the matrix G is traceless Remark 5. The characteristic identity (4.5) is equivalent to the equation provided that the relations (4.3) and (4.6) are satisfied. Remark 6. Comparing the RLL-relations (4.1) and the graded Yang-Baxter equation (3.14), one finds that the latter can be written in the form of the RLL-relation with the L-operator represented as an operator in V ⊗2 Then the operators L 1 (u) and L 2 (v) in (4.1) should be understood as 1 23 , respectively. Taking into account the term proportional to u −1 in (4.8), we represent the traceless generators G a 1 c 1 which satisfy (4.6) as i.e., this formula gives the defining representation T of the generators G a b of osp with the structure relations (4.6) and the conditions (4.3), (4.7). We note that the choice of the basis of osp in (4.9) differs from the choice of the basis of osp in (2.16) by sign factors This is consistent with the equality G 12 = G 21 = (−) 12

Super-oscillator representation for linear evaluation of Y(osp)
In this section we intend to construct an explicit representation of Y(osp) in which the generators of osp ⊂ Y(osp) satisfy the quadratic characteristic equation (4.5). We follow the approach of [2] and introduce a generalized algebra A of super-oscillators that consists of both bosonic and fermionic oscillators simultaneously.
Consider the super-oscillators c a (a = 1, 2, . . . , N + M ) as generators of an associative algebra A with the defining relation where the matrixε ab is defined in (2.3) and (2.5). In view of (2.1), for ǫ = −1, the superoscillators c a with [a] = 0 (mod2) are bosonic and with [a] = 1 (mod2) are fermionic. For ǫ = +1 the statistics of the super-oscillators c a is unusual and we will discuss this in more detail in Remark 8. at the end of this section. Nevertheless, we assume the grading to be standard grad(c a ) = [a] in both cases ǫ = ±1 and therefore the defining relations (5.1) are invariant under the action c a → c ′a = U a c c c of the super-group OSp with the elements U ∈ Osp (see [2]).
With the help of convention (2.6) for lowering indices one can write relations (5.1) in the equivalent forms The super-oscillators c a satisfy the following contraction identities: Further we need the super-symmetrised product of two super-oscillators: and define the operators In [2] we have proved the following statement. : In addition they satisfy the supercommutation relations (4.6) for the generators of osp and obey the quadratic characteristic identity (4.5): Thus, the elements F a b = ǫ ε bd (−1) [b] c (a c d) ∈ A given in (5.5) form a set of traceless generators of osp which satisfy all conditions of Proposition 3 and it means that the following statement holds.
c a c b , obey the RLL equation (4.1) which in the component form is given by 10) and the R-matrix (3.13) is Proof. One can prove this Proposition directly. To simplify the notation, we write (−1) a and (−1) ab instead of (−1) [a] and (−1) [a] [b] . After substituting the L-operator (5.9), the RLL equation (5.10) takes the form Taking into account the representation B a b = ǫ(−1) b c a c b and defining relations (5.1), one checks that the above relation is valid at arbitrary u and v. Remark 7. The Quadratic Casimir operator C 2 of the superalgebra osp(N |M ) in the differential representation (5.5) is equal to the fixed number It means that this realization (5.5) corresponds to a limited class of representations of the superalgebra osp(N |M ). This fact reflects the general statement of [27] that not all representations of simple Lie algebras g of B, C and D types are the representations of the corresponding Yangians Y (g).
14) where the operators x i , ∂ j and b α satisfy (5.12). Note that the super-oscillator algebra which follow from the commonly used properties of the Heisenberg and Clifford algebras: . We shall apply the rules (5.15) below.
Remark 9. Consider the graded tensor product A ⊗ A and denote the generators of the first and second factors in A ⊗ A respectively as c a 1 and c a 2 . Since ⊗ is the graded tensor product, we have (cf. (5.1)) Any element of A ⊗ A can be written as a polynomial f (c a 1 , c b 2 ) and its condition of invariance under the action of the group Osp is written as are the generators of the osp algebras and A ba are the super-parameters (with grad(A ba ) = [a] + [b]). In the case of an even function f , when grad(f ) = 0, this invariance condition is equivalent to Now we introduce the super-symmetrized product c (a 1 · · · c a k ) of any number of superoscillators, which generalizes the super-symmetrized product of two super-oscillators (5.4). The general definition and properties of such super-symmetrized products are given in Appendix B. In [2] we have proved the following statement.

Proposition 6. The elements
∈ A ⊗ A , k = 1, 2, . . . , (5.19) are invariant under the action (2.7) of the supergroup OSp: c a → U a b c b . It means that the elements (5.19) are invariant under the action of the Lie superalgebra osp and satisfy the invariance condition (5.18): where F ab 1 and F ab 2 are the generators (5.17) of the Lie super-algebra osp (see Proposition 4).
It turns out that the invariants (5.19) are not functionally independent. Indeed, we have the following statement.

Proposition 7. The invariants (5.19) satisfy the recurrence relation
where I 0 = 1 and I 1 = ε ab c a 1 c b 2 = c a 1 c 2a . In the representations (5.12), (5.14) and (5.13) Hermitian conjugations of invariant elements (5.19) are Proof. The derivation of the recurrence relation (5.21) is given in Appendix B. To prove (5.22), it is useful to define the invariants where σ 2 = −1, i.e. σ = ±i. Then, the recurrence relation (5.21) for new invariants I k has the form: where I 0 = 1 and we introduce the operator which is Hermitian z † = z in the representations (5.12), (5.14) and (5.13). One can prove the latter statement by making use of the rules (5.15) and commutation relations (5.16).
In view of the recurrence relation (5.24) and initial conditions I 0 = 1 and I 1 = z all invariant operators I k are k-th order polynomials (with real coefficients) of the Hermitian operator z. Therefore all I k are the Hermitian operators I † k = I k , and therefore, taking into account (5.23) and σ * = −σ, we deduce (5.22). Now we introduce a generating function of the Hermitian invariant operators I k : Since the invariants I k are polynomials in z, the generating function (5.26) depends on x and z only. Proof. Using the recurrence relation (5.24) we obtain: Now changing the summation indices and using (5.26) one deduces: where F x (x|z) ≡ ∂ x F (x|z) = ∞ k=0 I k+1 x k k! . The general solution to this ordinary differential equation is given in (5.27) up to an arbitrary constant factor c. The invariants I k are extracted from the generating function (5.26) using the formula from which we fix the constant c = F (0|z) = I 0 = 1.

The construction of the R-operator in the super-oscillator representation
Let T be the defining representation of the Yangian Y (osp). In the previous section we have considered the RLL-relation (4.1) and (5.10) that intertwines L-operators ||L a b (u)|| ∈ T (Y (osp)) ⊗ A (given in (5.9)) by means of the R-matrix (3.13) in the defining representation, i.e., R(u) ∈ T (Y (osp)) ⊗ T (Y (osp)). In other words, the R-matrix in the RLLrelations (4.1) and (5.10) acts in the space V ⊗2 There is another type of RLL-relations which intertwines the L-operators (5.9) by means of the R-matrix in the super-oscillator representation, i.e.,R(u) ∈ A ⊗ A, where ⊗ is the graded tensor product. In components, this type of RLL relations has the form or after substitution of the L-operator (5.9) we havê Here for simplicity we fix α = 1/2 in the definition of the L-operators and associate the first and second factors in A ⊗ A, respectively, with the algebras A 1 and A 2 generated by the elements c a 1 and c b 2 such that [c a 1 , c b 2 ] ǫ = 0 (see (5.16)). The RLL relation (6.2) is quadratic with respect to the parameter v. The terms proportional to v 2 are cancelled, the terms proportional to v givê while the terms independent of v arê where we use the concise notation Inserting this ansatz into the condition (6.4), we obtain (see [2]) the recurrence relation for r k (u) which is solved in terms of the Γ-functions: (6.8) where the parameter ω = ǫ(N − M ) was defined in (3.5) and A(u), B(u) are arbitrary functions of u. Substituting (6.8) in (6.6) gives the expression for the osp-invariant Rmatrix which intertwines two L operators in (6.1).
The methods used in [2] (for derivation of (6.6) and (6.8)) require the introduction of additional auxiliary variables and are technically quite nontrivial and cumbersome. Below in this paper, in Section 8, we give a simpler and more elegant derivation of conditions (6.7). This derivation is based on an application of the generating function (5.27) for the invariants I k , where the explicit form (5.27) is obtained by means of the recurrence relation (5.21).

The Faddeev-Takhtajan-Tarasov type R operator
There is another form of R operators which intertwines the L operators in the RLL equations (6.2) and are expressed as a ratio of Euler Gamma-functions. For the sℓ(2) case this type of solutions for R operator was first obtained in [5] (see also [6] and [28]). The generalization to the sℓ(N ) case (for a wide class of representations of sℓ(N )) was given in [29]. For orthogonal and symplectic algebras (and a very special class of their representations) analogous solutions of (6.2) were recently obtained in [7]. Below we generalize the results of [7] and find the solutions for the super-oscillator Faddeev-Takhtajan-Tarasov type R-operator in the case of osp Lie superalgebras. Proposition 9. The R operator intertwining the super-oscillator L operators in the RLL equations (6.1), (6.2) obeys the finite-difference equation where z = σ c a 1 c 2a and σ 2 = −1. The solution of this functional equation is given by the ratio of the Euler Gamma-functionŝ where r(u, z) is an arbitrary periodic function r(u, z + 2) = r(u, z) which normalizes the solution.
Proof. Taking into account the experience related to the orthogonal and symplectic cases (see [7]), we will look for a solution to the first equation (6.3) aŝ where σ is a numerical constant to be defined. In the last chain of equalities we have used (5.16). In other words, the operatorR 12 (u) acting in V 1 ⊗ V 2 is given by a function of an invariant z bilinear in super-oscillators c a 1 and c a 2 . Note that in the orthogonal and symplectic cases [7] the conventional invariants I k (5.19) are in one-to-one correspondence with polynomials of z of the order k. In the super-symmetric case of the algebras osp we prove this fact in Appendix B (see eq. (B.9) and comment after this equation). To justify the ansatz (6.11), we recall that the super-oscillators belonging to different factors in A ⊗ A and acting in different auxiliary spaces V 1 and V 2 commute according to (5.16) 14) Combining these relations we obtain i.e. z commutes with the sum c a 1 c 1b + c a 2 c 2b , and hence an arbitrary functionR 12 (u|z) depending on z satisfies the invariance conditions (6.3) and (6.5).
Let us introduce and consider a linear combination of (6.13) and (6.14) where the last equation is obtained under the choice Taking into account (6.17), we havê Then multiplying (6.4) by c d ± ε da (or by c d ∓ ε da ) from the left and by c c ± from the right and contracting oscillator vector indices, one obtains four independent scalar relations. Two of them are 3), the definition (6.11) of z and these two relations (6.20) turn to be functional equations onR 12 (u|z): Canceling the common factor ǫ z 2 − ω 2 4 in both sides we obtain a pair of equationŝ which are equivalent for both choices of signs to the one equation (6.9). In a similar fashion the other pair of relations gives identities which are satisfied automatically. Finally, the solution of the functional equations (6.9), (6.21) can be found immediately and is given in (6.10) by the ratio of the Euler Gammafunctions.
We see that the scalar projections (6.20) and (6.22) of the RLL relation are exactly the same as in the non-supersymmetric case [7], i.e. no signs related to grading appear. Moreover, we stress that the functional equation (6.9) is independent of the parameter ǫ, which distinguishes the cases of the algebras osp(N |M ) and osp(M |N ).

(6.23)
In view of the identity Γ(1− x)Γ(x) = π/ sin(πx), these two versions are equivalent to each other up to a special choice of the normalization functions r (±) (u, z). So one can consider only one of the solutions (6.23).
7 TheR operator in special cases of osp(N |2m) In this section, we work out the explicit form of the solution (6.10) in a few particular cases.

The case of osp(M|N) = osp(1|2)
In this case, we have N = 2 and M = 1, and the superalgebra osp(1|2) is described by the bosonic oscillator c 1 ≡ a † , c 2 ≡ a (in the holomorphic representation we have c 1 ≡ x, c 2 ≡ ∂) and by one fermionic variable c 3 ≡ b with the commutation relations (5.1): denotes the anticommutator. To obtain (7.1) from (5.1) and (5.2), we fix there ǫ = −1 and specify the metric matrix as Note that the fermionic variable b can be understood in the matrix representation as a single Pauli matrix (say τ 3 ). To define the operator z in (6.11) we need two copies of superoscillator algebras A 1 and A 2 with the generators c a 1 = (x 1 , ∂ 1 , b 1 ) and c a 2 = (x 2 , ∂ 2 , b 2 ) which act in two different spaces V 1 and V 2 . Then, the invariant operator z in (6.11) looks like 2) where b i satisfy b 2 i = 1 in view of (7.1) and anticommute b 1 b 2 = −b 2 b 1 in order to ensure (6.12). The characteristic equation for the fermionic part of z: (here we took into account (7.1)) allows one to introduce the projection operators: Now any function f of b can be decomposed in these projectors Accordingly, the R-operator (6.10) can also be decomposed as: and finally we havê where r ± (u, x) = r(u, x ± 1 2 ) are periodic functions in x, i.e., the general osp(1|2)-invariant R-operators consist of two independent terms acting on two invariant subspaces, corresponding to eigenvalues ± 1 2 of the fermionic part b ≡ σb 1 b 2 of the invariant operator z. The coefficients in the expansion (7.7) in projectors P ± 1 2 are the functions of the bosonic part x = σ(x 1 ∂ 2 − x 2 ∂ 1 ) of the invariant operator z. These coefficients are nothing but the R-operators for the bosonic subalgebra sℓ(2) ≃ sp(2) ⊂ osp(1|2).

The case of osp(2|2)
In this case, we have two bosonic c 1 = x, c 2 = ∂ and two fermionic c 3 = b 1 , c 4 = b 2 , oscillators which we realize using even and odd variables with the commutation relations (5.1): Here again we fix ǫ = −1 and We introduce two super-oscillator algebras A 1 and A 2 with the generators {c a 1 } and {c a 2 }, respectively. The invariant operator (6.11) is where σ = ±i. The fermionic oscillators b α i ∈ A i with commutation relations (7.8) and (5.16) generate the 4-dimensional Clifford algebra. It is well known (see, e.g., [30]) that the generators of this Clifford algebra can be realized in terms the Pauli matrices τ α : where I 2 is the unit (2 × 2) matrix. The characteristic equation for the fermionic part b = σ The invariant subspaces spanned by the eigenvectors corresponding to eigenvalues 0, ±1 of b are extracted by the projectors: The R-operator is decomposed as follows: 12 (u|x + ℓ)P ℓ . r(u|x + ℓ) Γ 1 2 (x + ℓ + 1 + u) Γ 1 2 (x + ℓ + 1 − u) P ℓ , r(u|z + 2) = r(u|z). (7.13) Note that in view of the periodicity condition r(u|x− 1) = r(u|x+ 1) one can rewrite (7.13) as follows: (7.14) In the pure bosonic case of the orthogonal algebras so(2k), the general solution for theRoperator splits into two independent solutions corresponding to two nonequivalent chiral left and right representations (see [8], [1], [7]). This does not happen here in the supersymmetric case, where the even and odd functions of b are not separated, due to the dependence on the bosonic operator x in the coefficients r(u|x + ℓ) which mixes the chiral representations of so(2k).

The case of osp(n|2)
This case is a generalization of the examples considered in the previous subsections (for n = 1 and n = 2 we respectively reproduce the results for the osp(1|2) and osp(2|2) algebras). Consider the super-oscillator algebra A with two bosonic c 1 = x, c 2 = ∂ and n fermionic generators c 2+α = b α (α = 1, ..., n) with the commutation relations (5.1): where the fermionic elements b α are the generators of the n-dimensional Clifford algebra. This corresponds to the choice of the parameter ǫ = −1 and the metric tensor in the block formε where I n stands for the n × n unit matrix. The invariant operator z ∈ A ⊗ A is One can prove (7.18) by noticing that the operator b is represented as b =z α z α − n/2, where (see (6.16) are respectively the creation and annihilation fermionic operators in the Fock space F which is created from the vacuum |0 : z α |0 = 0 (∀α). Then the operator in the left-hand side of (7.18) is equal to zero since it is zero on all basis vectorsz α 1 · · ·z αm |0 ∈ F (here 1 ≤ α 1 < ... < α m ≤ n and m ≤ n) which are the eigenvectors of b with eigenvalues (m − n 2 ). The projectors P ℓ on invariant subspaces in F spanned by the eigenvectors of b corresponding to eigenvalues (m − n 2 ) ≡ ℓ, where ℓ = − n 2 , − n 2 + 1, ..., n 2 , are immediately obtained from (7.18): The case of even n = 2k We see that eigenvalues of b are integer (or half-integer) when the number n is even (or odd). Thus, for the case osp(n|2) = osp(2k|2), when n = 2k is even, the expansion of the solution (6.10) goes over integer eigenvalueŝ (u|x + ℓ)P ℓ , (7.20) and it implieŝ R osp(2k|2) 12 P ℓ , r(u|z + 2) = r(u|z). (7.21) The case of odd n = (2k + 1) For the case osp(n|2) = osp(2k+1|2), when n = 2k+1 is odd, the expansion of the solution (6.10) goes over half-integer eigenvalues of b: − 2k+1 2 , − 2k−1 2 , . . . , − 1 2 , 1 2 , 3 2 , . . . , 2k+1 2 , and we have the expansion where the periodic function r(u|z + 2) = r(u|z) normalizes the solution.

The case of osp(n|2m)
We consider the osp(n|2m) invariant super-oscillator algebra which is realized in terms of m pairs of the bosonic oscillators c j = x j , c m+j = ∂ j , j = 1, . . . , m and n fermionic oscillators c 2m+α = b α , α = 1, 2, . . . , n, with the commutation relations (5.12) deduced from (5.1) with the choice of the parameter ǫ = −1 and metric tensor (5.13). In this case, the invariant operator z ∈ A ⊗ A defined in (6.11) is Here the operator b is the same as in the previous examples of Section 7.3. Thus, the R operator (6.10) in the case of the algebra osp(n|2m) is expanded over the projection operators P ℓ like in the case of osp(n|2), and the final expression forR osp(2m|n) 12 (u|z) will be given by (7.21) or (7.23): where the projectors P ℓ are defined in (7.19) and

The relation between two approaches
In this section, we give a more direct and elegant derivation of the R matrix solution (6.6), (6.7), that does not require the introduction of additional auxiliary variables (as it was done in [2]) and is based only on using the generating function 4 (5.26), (5.27) of the invariant operators I k . In addition, this derivation partially explains the relationship between the two types of solutions (6.6), (6.7) and (6.10) for the R operator. Now we clarify the relation of the Shankar-Witten form of the R operator (6.6), (6.7) and Faddeev-Takhtajan-Tarasov type R operator given by the ratio of two Euler Gammafunctions in (6.10). First, we write (6.6) in the form where r k (u) = (−σ) k r k (u) and I k = σ k I k . Recall that I k are the Hermitian invariants introduced in the proof of Proposition 7.

2)
(which was used to find the second solution (6.10)) if the coefficients r k (u) satisfy that in terms of r k (u) is written as (6.7).
Proof. One can write (8.2) as We use the relation (5.29) in the form: and obtain Then we use the equations which follow from formula (5.27) for the generating function and write (8.5) in the form Now we apply the identity ∂ k x x = x∂ k x + k∂ k−1 x to move derivatives ∂ x in (8.6) to the right and obtain: In the second term in square brackets we shift the summation parameter k → k + 2 and deduce The resulting expression vanishes due to (8.3). Thus, we prove that the finite-difference equation (8.2) is valid if the coefficients r k (u) satisfy (8.3).
Remark 11. We prove that both R operators (8.1), (8.3) and (6.10) satisfy the same equation (8.2) and indeed obey the RLL relations (6.2). It is worth also to note that the differential operator in the curly brackets in (8.6) coincides (up to change of variable x = σλ) with the differential operator in curly brackets presented in formula (6.43) of our work [2]. This suggests to regard the generating function F (x|z)(1 − x 2 4 ) −1 as a coherent state in the super-oscillator space.
One can check that any element X ∈ B k (ω) of the Brauer algebra