Yangians and Yang-Baxter R-operators for ortho-symplectic superalgebras

Yang-Baxter relations symmetric with respect to the ortho-symplectic superalgebras are studied. We start from the formulation of graded algebras and the linear superspace carrying the vector (fundamental) representation of the ortho-symplectic supergroup. On this basis we study the analogy of the Yang-Baxter operators considered earlier for the cases of orthogonal and symplectic symmetries: the vector (fundamental) R matrix, the L operator defining the Yangian algebra and its first and second order evaluations. We investigate the condition for L(u) in the case of the truncated expansion in inverse powers of u and give examples of Lie algebra representations obeying these conditions. We construct the R operator intertwining two super-spinor representations and study the fusion of L operators involving the tensor product of such representations.


Introduction
The orthogonal and the symplectic groups have similarities, which can be traced back to the existence of an invariant bilinear form in the fundamental representation space. The similarities allow a unified treatment not only of the groups and Lie algebras but also of the Yang-Baxter relations with such symmetries. The orthogonal and symplectic groups are embedded in the ortho-symplectic (OSp) super-group.
In the present paper we study Yang-Baxter relations with ortho-symplectic (osp) supersymmetry. In a recent paper [3] Yang-Baxter relations symmetric with respect to orthogonal or symplectic groups have been considered in a unified formulation, continuing preceeding studies [14], [15], [16], [17], [18] and especially [13,21] where orthogonal and partially symplectic symmetries have been considered. We rely on the similarities, proceed to some extend parallel to [3] and we follow similar motivations. The Yang-Baxter operators, in particular their dependence on the spectral parameter, are generally more involved as compared to the case of general linear symmetry. Compact forms of L operators appear only in cases of particular representations. Such cases are of interest in physical applications, in particular for the theory of quantum integrable systems where the L operators are used as building blocks. The viewpoint of the Yangian algebra and its finite order evaluations is appropriate to understand the cases of explicit solutions for L operators.
The extension to the osp supersymmetric case is of interest in the studies of integrable spin chains of osp-type [26] and exact factorizing S matrices [10] for two-dimensional field theories with ortho-symplectic supersymmetries. It is also relevant to investigations of gauge field theories, in particular because of the conformal transformations form the (pseudo)orthogonal groups. From the mathematical point of view it would be interesting to extend the Drinfeld's classification theorem for the finite-dimensional irreducible representations of the Yangians of the classical type (see [25] and references therein) to the case of the osp-type Yangians.
The generalization of so and sp cases to the supersymmetric osp case is not straightforward. We shall see that it takes much attention and carefully prepared formulations to obtain in a systematic way the correct forms of all the relations starting from the theory of super-matrices of OSp and its super-algebra up to the Yang-Baxter relations and the forms of the L operators related to the first and second order evaluations of the Yangian Y (osp). Interesting and intriguing details are related to sign factors. For deriving a particular relation one could choose the approach of starting from an ansatz up to sign and then invent a test to fix the sign. For our purpose that approach would be not sufficient, not only because of lack of elegance but mostly because it would cost more efforts then the derivations of all formulas from the first principles. The complete formulation of Lie supergroups Osp, their Lie superalgebras and Yangians Y (osp) given in this paper provides a convenient basis for further studies and applications.
We outline the contents emphasizing the main points of the paper. We present a systematic and detailed super-symmetric formulation of the Yang-Baxter relations and the involved operators. To set up the formulation we start in Sect. 2 with the superspace where the ortho-symplectic super-matrices act. Recalling the notion of invariant tensors we prepare the formulation of the fundamental R-matrix of osp-type. The involved invariant operators acting in the twofold tensor product of the superspace are shown to represent the Brauer centralizer algebra [2] (see also [5], [6]). In Section 2 we explain a lot of details and useful notations important in our formulation. In particular we introduce the notion of the sign operator which is useful in many relations. The action of a super-group element U on a higher tensor product of the fundamental superspaces is described by the tensor product of operators U dressed by these sign operators.
Whereas the braid form of the fundamental Yang-Baxter relation coincides for the supersymmetric case with the case without supersymmetry, factors of sign operators make the difference when changing to the R operators including super-permutation. This is explained in Sect. 3. The modifications by factors of sign operators appear in the RLL relations as well as discussed in Sect. 4.
The graded RLL relation involving the osp supersymmetric fundamental R matrix serves as the defining relation of the L operators. It defines the osp Yangian algebra following the known scheme described in Sect. 4: By expanding L(u) in inverse powers of the spectral parameter u one obtains the Yangian algebra generators L (k) ab . Substituting this expansion in the RLL relation one obtains all Yangian algebra relations. The first non-trivial term in the u −1 expansion involves the osp Lie superalgebra generators L (1) ab . We formulate their symmetry relation and the Lie algebra commutation relations as derived from the RLL relation. For the super traceless part G of the matrix L (1) the commutation relations are to be compared with the ones obtained for the infinitesimal OSp supermatrices formulated earlier in Sect. 2. We show that the corresponding basis sets are transformed into each other by the sign operator.
In Sect. 5 we show that in the linear evaluation of Y(osp) the matrix L (1) of generators of osp algebra has to obey the additional constraint in form of a quadratic characteristic identity. We construct the infinite dimensional super spinor representation of osp obeying this constraint as a generalization of the spinor representation of so and the metaplectic representation of sp. We formulate the super spinor representation in terms of an algebra of super oscillators which generalizes the Clifford algebra and the standard multidimensional oscillator algebra.
In Sect. 6 we construct the super spinorial R operator which acts in the tensor product of two super spinor representations. Our systematic supersymmetric formulation allows to extend to the osp case the approach developed for the so case in [13], [21] and for the sp case in [3].
The graded symmetrisation of products of super oscillator operators is treated by the generating function techniques using auxiliary variables obeying a graded multiplication law.
The fusion procedure with superspinor L opertors is considered in Sect. 7. The projection of the tensor product of superspinor representations to the fundamental one applied to the product of such L operator matrices reproduces the fundamental R matrix.
In Sect. 8 we investigate the quadratic evaluation of the Yangian. We formulate the conditions implied by the RLL relation with L depending quadratically on the spectral parameter. The sign operator helps to formulate these conditions in a form similar to the ones in the cases of orthogonal and symplectic symmetry. We construct the solution where the second non-trivial term in L is the quadratic polynomial in the generator matrix appearing in the constraint for the first order evaluation. The conditions imply constraints on the Lie superalgebra representation which are fulfilled by the Jordan-Schwinger ansatz for the generators in terms of graded Heisenberg pairs. In this case the generator matrix obeys the super anticommutator constraint which appeared in Sect. 6 in the superspinorial RLL relation, and as a consequence, the generator matrix obeys a cubic characteristic relation.

.1 The superspace, the super-group OSp and its superalgebra osp
To fix notations we need to formulate the notion of the ortho-symplectic supergroups and their Lie superalgebras (see, e.g., [1] as an introduction to the super analysis and to the theory of Lie supergroups and superalgebras).
We denote by grad(A) the grading of the algebraic object A. Let V (N |M ) be a superspace and z a (a = 1, . . . , N + M ) the graded coordinates in V (N |M ) . We distinguish N even and M odd coordinates z a , denote the grading grad(z a ) of the coordinate z a as [a] = 0, 1 (mod2) and call [a] the degree of the index a. If the coordinate z a is even then [a] = 0 (mod2), and if the coordinate z a is odd then [a] = 1 (mod2). The coordinates z a and w a of two supervectors z, w ∈ V (N |M ) commute as (2.1) Thus the coordinates of the vectors z, w, . . . ∈ V (N |M ) can be considered as generators of a graded algebra. For two homogeneous elements A, B of this algebra we have Further we endow the superspace V (N |M ) with the bilinear form where the super-metric ε ab has the property and the matrix ||ε ab || is inverse to the matrix ||ε ab ||. Here ǫ = ±1. Moreover, we require that the super-metric ε ab is even in the sense that ε ab = 0 iff [a] + [b] = 0 (mod2). This means that and therefore the properties (2.3) can be written as ε ba (the same forε ab ). Taking into account (2.1) we obtain (z · w) = ǫ (w · z), i.e., the bilinear form (2.2) is symmetric for ǫ = +1 and skew-symmetric for ǫ = −1. Further we adopt the following rule for rising and lowering indices, This rule will be applied also for any tensors of higher ranks. According to this rule we have We see that the metric tensor with upper indices ε ab does not coincide with the inverse matrixε ab . Further we shall use only the inverse matrixε ab and never the metric tensor ε ab . So to simplify formulas below we omit the bar in the notationε ab and write simply ε ab keeping in mind that this is the inverse matrix for the metric ε ab . 5 Consider a linear transformation in An alternative rule for lifting and lowering indices (instead of (2.5)) is za = ε ab z b , z a = z b ε ba , where ε ab is the metric tensor (2.6) with upper indices. So, to lower the index we act by the matrix ε ab from the left, while to lift the index we act by the matrix ε ba from the right. In this case one should remember the unusual conditions of inversion: ε ab ε ac = δ c b . Note that is not the rule adopted below We have to mention that the first equation in (2.8) can be also rewritten by using the properties of the supermetric as We represent (2.8) in coordinate-free form as where we have used the concise matrix notations (2.10) Here ⊗ denotes the graded tensor product: where [A] := grad(A) and [B] := grad(B). The sign operator (−) 12 is an extremely useful tool for the R-matrix formulation [12] of quantum supergroups and their particular cases as super-Yangians. This operator was used first in [4]. For more details about the graded tensor product and its consequences for the record of equations, especially the Yang-Baxter equation, see appendix A. 6 We also describe further details of this formalism in the next subsection.
The set of supermatrices U which satisfy equations (2.8) form the supergroup OSp(N |M ) for ǫ = +1 and the supergroup OSp(M |N ) for ǫ = −1 with respect to the usual matrix multiplication. Let us consider the defining relations (2.8) for OSp groups elements close to the unit element U = I + A + . . . . As a result we obtain the conditions for the elements A of the Lie superalgebra osp of the supergroup OSp: 12) or equivalently The coordinate free form of (2.12), (2.13) can be obtained directly from (2.9): Remark. The set of super-matrices A, which satisfy (2.12), (2.14), forms a vector space over C denoted as osp. One can check that for two super-matrices A, B ∈ osp the commutator also obeys (2.12), (2.14) and thus belongs to the vector space osp. It means that osp is an algebra. Any matrix A which satisfies (2.12), (2.14) can be represented as Let us remark shortly that we use the convention that the gradation is carried by the coordinates. There is the opposite convention in which the gradation is carried by the basis vectors. The relation of expressions in both conventions is explained in appendix A. Our convention is of great importance for the formulation of the quantum hyperplane for Lie super-algebras and their quantum deformations.
where ||E a c || is an arbitrary matrix. Let {e f g } be the matrix units, in (2.16) then we obtain the basis { G f g } in the space osp of matrices (2.14): Now any super-matrix A ∈ osp which satisfy (2.12), (2.14) can be expanded in this basis (2.19) By using the explicit representation (2.17) for G a b one can evaluate the supercommutator (2.19) and obtain the defining relations for the Lie superalgebra osp: (2.20) In component free form we write (2.20) as following where we have introduced matrices the K, P ∈ End(V ⊗2 (N |M ) ): The matrix P is called the superpermutation operator since it permutes super-spaces. For example, using this matrix one can write (2.1) as P ab cd w c z d = z a w b . The properties of the operators P and K which will be used below are listed in the appendix B.
Note that the osp conditions (2.14) for the basis matrices ( G f g ) a c are represented in the form Taking into account their property ( G a 3 Moreover one can check the identities (see appendix B)

OSp super-group invariants
Here we give the formulation of invariance of tensors with respect to actions of a supergroup. This formulation will be important for our discussion below of the invariance of the osp-type R-matrices which are solutions of Yang-Baxter equations.
First we recall the case of an ordinary group. Let G be a matrix group which acts in the vector space V. We say that the tensor O a 1 ···a k b 1 ···b j with k upper and j lower indices is This is represented at the level of its Lie algebra G as: For all A ∈ G holds that If the tensor O is an operator in V ⊗k , we have j = k and both conditions can be written in the concise form as where 1, . . . , k are labels of super-vector spaces. Now we extend the above formulation to the case of a supergroup G which acts in a superspace V. Let us start with looking for a compact expression describing an action of a supergroup G on graded tensor products x 1 y 2 ≡ x ⊗ y, where x, y ∈ V. First of all, let us understand the meaning of the action on it by the tensor product of two superoperators Here we introduce the operator (−) 12 acting on the tensor product of two homogeneous vectors as In coordinates the operator (−) 12 has been given above in (2.10). We call it the sign operator. It satisfies The formula (2.30) can be generalized obviously for higher numbers of vector spaces in the tensor product V ⊗k . Let an operator C act the j-th super-space in V ⊗k , then where we define the operator dressed by sign operators as and (−) ℓ,j = (−) ℓj denotes the sign operator which acts nontrivially only in ℓ-th and j-th factors of the tensor product V ⊗k . Then the formula (2.30) is generalized as We specify now the super-operators to the elements of the super-group U ∈ G. Taking into account (2.30) the action of U in V ⊗2 can be expressed using the sign operators (−) 12 (2.35) LetR 12 be an operator acting in V ⊗2 . The invariance ofR 12 w.r.t. the action of G is expressed asR 12R 12 . (2.36) Note that for matrix R 12 = P 12R12 , where P 12 is the superpermutation (2.22), the invariance condition modifies to The generalization of (2.36) for an operator O 1...k acting in V ⊗k for arbitrary k is obtained straightforwardly: Then, for corresponding Lie super-algebra G we obtain (∀A ∈ G): The considerations above were done for operators only. They can be generalized to general tensors. With slight abuse of the notation a general tensor T ∈ V ⊗k ⊗ V ⊗j with k upper and j lower indices can be written as T The infinitesimal form of (2.40) which generalizes (2.39) is We see that the relations of the super-metric invariance (2.9) and (2.14) are just special cases of the general formulas (2.40) and (2.41) for j = 2, k = 0 and for j = 0, k = 2.
3 The fundamental R-matrix and the graded Yang-Baxter equation We have three OSp invariant operators in V ⊗2 (N |M ) : the identity operator 1, the superpermutation operator P and the tensor K. The super-permutation P 12 is a product of the usual permutation P 12 and the sign operator (−) 12 P 12 = (−) 12 P 12 and in coordinates The operator K 12 is defined as In coordinates these operators P, K have been introduced above in (2.22). Their invariance w.r.t. OSp can be proved directly by applying the results of above subsection 2.2.
Using the operators P, K one can construct the set of operators {s i , e i |i = 1, . . . , n − 1} in V ⊗n (N |M ) : which generate the Brauer algebra B n (ω) [2] with the parameter Here N and M are the numbers of even and odd coordinates, respectively. Indeed, one can check directly (see appendix B) that the operators (3.3) satisfy the defining relations for generators of B n (ω) (see, e.g., [5], [6], [7] and references therein) Thus, one can consider eqs. (3.3) as the matrix representation T of the generators of the Brauer algebra B n (ω) in the space V ⊗n (N |M ) . We note that T (3.3) is the special reducible representation of B n (ω). Irreducible representations of Brauer algebra were investigated in many papers (see, e.g. [6], [7], [8] and references therein).
Let us consider the following linear combination of the generators s i , e i ∈ B n (ω) where u is a spectral parameter and Proposition 1. The element (3.7) satisfies the Yang-Baxter equatioň 9) and the unitarity conditionρ Proof. Substitute (3.7) into (3.9), (3.10). One obtains 27 terms in both sides of (3.9). The terms which do not contain the elements e i , e i+1 are cancelled due to the first identity in (3.6) (the remaining terms just cancel each other identically). Other terms which include the elements e i , e i+1 are cancelled due to the identities (3.5), (3.6). In particular to prove (3.9) identities like e i s i±1 e i = e i and s i e i+1 s i = s i+1 e i s i+1 which follow from (3.5), (3.6) are useful. One can consider equation (3.9) as the nontrivial identity in the algebra B n (ω).
The matrix representation T (3.3) of the element (3.7) iš Here we suppress the index i for simplicity. (3.9) implies (see appendix B) thatR(u) satisfies the braid version of the Yang-Baxter equatioň Further we use the R-matrix which includes the super permutation and is the image of the element [7]: The braid version (3.12) of the Yang-Baxter equation has the same form in both the supersymmetric the non supersymmetric cases. However the standard matrix R(u) = PŘ(u) (3.13) satisfies the graded version of the Yang-Baxter equation [22] involving extra sign factors, (3.14) Indeed, after substitutingŘ ij (u) = P ij R ij (u) = (−) ij P ij R ij (u) into (3.12) and moving all standard permutations P ij to the left we write (3.12) in the form is an even matrix, then the following condition holds In particular one can easily check this property for the matrices 1, P, K out of which the operator R(u) is composed. Therefore, Using this property and the identities (2.32) we convert the right hand side of (3.15) to and doing the analogous transformations for the left hand side we represent (3.15) in the form After the exchange of the spectral parameters v → u − v, u → u in (3.19) we obtain the graded version of Yang-Baxter equation (3.14). Finally we stress that (−) 12 in (3.14) can be exchanged with the sign operator (−) 23 by means of similar manipulations as in (3.18). Moreover, if R 12 (u) solves the Yang-Baxter equation (3.14), then the twisted R-matrix is also a solution.
Remark. Eqs. (3.11), (3.13) express uniformly the R-matrices which are invariant under the action of SO, Sp or OSp groups. Recall that for the SO case the R-matrix was found in [10], for the Sp case it was constructed in [11], [17]. For the OSp case such R-matrices were considered in many papers (see, e.g., [23], [26], [4]). Note that the OSp type R-matrix R(u) proposed in [4] coincides with (3.13) in the case ǫ = +1 in the following way The R-matrix (3.13) is related to SO and Sp type R-matrix presented in [3] just by rescaling of the spectral parameter u → −ǫu. The parameter β = 1 − ω 2 is then rescaled to −ǫβ = ǫω 2 − ǫ.

The graded RLL-relation and the Yangian Y(osp)
We start with the graded form of the RLL-relation (see, e.g., [22], [4]) (the different, but equivalent, choice of the RLL-relation will be discussed at the end of this Section, in Remark 3). The R-matrix is of the form (3.13) and the elements of the supermatrix ||L a b (u)|| N +M a,b=1 involve the generators of an associative algebra which we denote by Y(osp). We specify this algebra below. Note that the sign operators in (4.1) are fixed according to the invariance condition (2.37). Consider the product L 1 (−) 12 L 2 (−) 23 (−) 13 L 3 of three L-operators and reorder it with the help of (4.1) as in two different ways in accordance with the arrangement of brackets As a result we obtain (using the properties (2.32), (3.17) of the sign operators) the associativity condition for the algebra (4.1) in the form of the graded Yang-Baxter equation (3.14).
Expand the L-operator in the spectral parameter u as where 1 denotes the unit element in Y(osp). We multiply the R-matrix by and expand the RLL-relation (4.1) in the spectral parameters u −1 and v −1 . The coefficient at u −k v −j in (4.1) gives the defining relations for the infinite dimensional associative algebra Y(osp) which is called the Yangian of the osp-type: The super-commutator is defined as Notice that for elements of two super-matrices ||A a b ||, ||B a b || the super-commutator has the following representation This implies that the relation (4.5) for the Yangian Y(osp) has the following coordinate form: (4.7) Remark 1. We have noticed in section 3 that the relation between the OSp type R-matrix (3.13) and the R-matrix in [3] is given just by rescaling u → −ǫu. Doing the same rescaling for the L-operator we obtain L (k) → −ǫL (k) for k odd and L (k) → L (k) for k even, respectively. After that the Yangian relations in [3] appear just as a special case of the relations (4.7).
Choosing k = 1, j = 3 we obtain the defining relation for the Lie superalgebra osp 12 ] . (4.8) The enveloping algebra of osp is thus a subalgebra in the Yangian Y(osp). Permuting (1 ↔ 2) in (4.8) (we multiply it by super-permutation from both sides) and adding the original and permuted equation we obtain the consistency condition Using this condition and the identities one can simplify defining relation (4.8) of osp as which has the following component form (4.12) Note that (4.11) can be directly obtained from (4.5) with the choice k = 3, j = 1.
Multiplying both sides of (4.9) by K 12 from the left (or from the right) and using the identities we obtain where str(A) ≡ (−1) [a] A a a = ǫε ba A ba is the supertrace of the super-matrix A. One can check that the element str(L (1) ) belongs to the center of the Yangian Y(osp) and therefore to the center of the Lie super-algebra osp.
The Yangian Y(osp) (as well as the gℓ-type Yangian [9] and the so-and sp type Yangians [3]) possess the set of automorphisms which are defined by the assignments . is a scalar function and b k are parameters (in general b k are central elements in Y(osp)). The transformations (4.15) are implied by the form of the defining relations (4.1). As shown in [3] one can use the automorphisms (4.15) to fix L (1) such that str(L (1) ) = 0. In view of this we define the traceless generators These generators satisfy (since we have automorphisms (4.15) and str(L (1) ) is the central element) the same commutation relations (4.8) which we write as: where and we have used (−) 12 P 12 (−) 12 = P 12 . This is to be compared with (2.21), (2.26) if the algebra of elements G a b is represented in the space V (N |M) . Note that the commutation relations (4.17) transform to the commutation relations (2.21) if we redefine the supermetric as ε ab → ǫ(−1) [a] ε ab = ε ba (see also discussion in Remark 3 below). Further, for the traceless generators (4.16) G a b from (4.14) we have (cf. (2.24)) In components this reads as the condition (2.13) for the generators of the Lie superalgebra osp: The operators L 1 (u) and L 2 (v) in (4.1) should be understood as 1 23 , respectively. According to the above consideration of the Yangian Since the L-operator (4.20) satisfies the RLL-elations (4.1) the operator (4.21) should obey commutation relations (4.8) and conditions (4.14). Taking into account (4.21) we represent the traceless part (4.16) as So we have where G 12 and G 12 are defined in ( which yields the equivalent definition of the Yangian Y(osp). In this case the R-matrix (fundamental) representation of the Yangian (4.23) is given by the formula (cf. (4.20)) which leads to the following fundamental representations of the Yangian generators L (1) and their traceless osp generators G (cf. (4.21), (4.22)): We see that the representation of the basis elements of osp in (4.26) coincides with the basis of osp proposed in (2.17) and, thus, the consideration of Subsection 2.1 is relevant to the definition of the Yangian Y(osp) given in (4.23). The next point which we would like to stress here is that RLL relations (4.1) can be rewritten in the form of (4.23): where R 12 (u) is the twisted solution (3.20) of the Yang-Baxter equation (3.14). The twisted matrix R 12 (u) can be obtained from R-matrix (3.13) (compare (4.20) and (4.24)) by the substitution ε ab → ǫ(−1) a ε ab = ε ba . It means that all formulas which we obtain below for the Yangian (4.1) can be easily transformed to the formulas for the Yangian (4.23) by the simple transformation of the supermetric ε ab → ε ba .
5 The linear evaluation of the Yangian Y(osp)

The conditions for linear evaluation
Let us suppose that the L-operator expansion (4.3) terminates after the first term, i.e., Writing the RLL-relation (4.1) with this form of the L-operator and expanding it in u and v we obtain a set of conditions imposed on L (1) . All terms in the RLL-relation (4.1) proportional to u k v ℓ for (k + ℓ) ≥ 3 give trivial conditions which are automatically satisfied. The coefficients at u 2 , v 2 and uv give the defining relations (4.11) for generators L (1) ∈ osp and the condition (4.9). The condition appearing at first powers of u and v is Multiplying it by P 12 from both sides and using (4.10), (4.9) one represents it in the equivalent form These two equations can be obtained directly from (4.5) taking j = 2, k = 3 or j = 3, k = 2.
Finally the condition at zero power of u and v is trivial in view of identities (4.10).
Thus we come to the following statement.
In the case of linear evaluation of L-operator (5.1), in addition to the defining relations (4.11) and the condition (4.9) we obtain only one non-trivial constraint (5.3) on the generators L (1) of osp. Using (4.14) we write (5.3) as the quadratic relation in L (1) : ). Multiplying it by K 12 from one side and using (4.13) we arrive at the quadratic characteristic equation imposed on L (1) For the generators G a b defined in (4.16) with vanishing supertrace, str(G) = 0, this condition simplifies to where β = 1 − ω/2. We arrive at the following statement.
where α is an arbitrary constant, solves the RLL-relation (4.1) if G a b is a traceless matrix of generators of osp, which satisfy eqs. (4.17), (4.18), and in addition obeys the quadratic characteristic identity (5.6).

The super-spinor representation
In this subsection we intend to construct an explicit representation of Y(osp) where the generators of osp ⊂ Y(osp) satisfy the quadratic characteristic equation (5.6) required for the linear evaluation (5.1).
We look for a generalization of the metaplectic or spinor representations of the Sp(n) or SO(n) groups which can be formulated according to [3] based on algebras of bosonic and fermionic oscillators with the defining relation invariant under the group action. We introduce the algebra A of super-oscillators involving both bosonic and fermionic oscillators.
Consider the super-oscillators c a (a = 1, 2, . . . , N + M ) as generators of an associative algebra A with the defining relation where ε ab is the super-metric defined in ( where we have used the condition (2.8) for the elements U ∈ Osp.
With the help of the convention (2.5) for lowering indices one can write the relations (5.7) in the equivalent forms The super-oscillators satisfy the following contraction identity: Further we need the super-symmetrised product of two super-oscillators: and define the operators They satisfy the supercommutation relations (4.12) for generators of osp and obey the quadratic characteristic identity (5.6): Proof. The property (5.13) follows from the definition (5.12) of F ab . The traceless property follows from the identity (5.10). To prove (5.14) we need the following relation: which implies for the supercommutator of two supersymmetrized quadratic operators Then using the definition (5.12) of F ab we obtain Applying the properties of the supermetric ε ab and using the symmetry (5.13) one can show that this relation is equivalent to (5.14). From the contraction identity (5.10) we obtain that which proves (5.15).
Thus the elements F ab ∈ A form a set of traceless generators of osp. Indeed, the elements F ab satisfy the supercommutation relation (4.12) and the symmetry condition (4.19). Moreover they satisfy the quadratic characteristic identity (5.6) for the linear evaluation representation (5.1). It means (see Proposition 3) that the L-operator which solves RLL-equation (4.1) has the form where F a b is defined in (5.12) and α is an arbitrary constant. Note that the appearance of the parameter α in the solution (5.21) is explained by the invariance of the RLL equations (4.1) under the shift of the spectral parameters u → u + α, v → v + α.
Remark. At the end of this subsection we note that for every super-matrix ||A ba || we have where we applied the supercommutation relations between the symmetrized quadratic product c (a c b) and the super-oscillator c d : Equation (5.22) demonstrates that the operators F ab generate any linear transformation of generators c a ∈ A under the adjoint action. Let us consider the graded tensor product A ⊗ A of two algebras of the super-oscillators. It is useful to denote the generators of A ⊗ A as c a ⊗ e = c a 1 and e ⊗ c a = c a 2 where e is the unit element of A. Then formula (5.22) for the adjoint action is generalized to the case of A ⊗ A as follows.
where c ℓ and the dressed supermatrices A {1...k} have been defined in (2.33). Comparing this formula with (2.41) we find that the invariance condition for any function f (c a 1 , c b 2 ) ∈ A ⊗ A is written in the form If grad(f ) = 0, this invariance condition is equivalent to We shall use this condition in Section 6.

The super-spinorial R-operator
We shall construct the R operator intertwining in the RLL relation two super-spinor representations formulated in terms of super-oscillators. We follow here the approach developed for the so-case in [13], [21] and then extended for the sp-case in [3]. We define the L-operator as where G ba are generators of osp and F ab = (−1) b ǫc (a c b) ≡ τ (G ab ) ∈ A are elements G ab in the super-spinor representation τ (see proposition 4). We shall construct the R-operatoř R 12 (u) ∈ A ⊗ A intertwining the L-operators (6.1) via the following RLL-relatioň The operatorŘ 12 (u) acts trivially on the factor Y(osp), whereas e is the unit element of A and as before we denote F ab 1 = F ab ⊗ e, F ab 2 = e ⊗ F ab . We consider the case when grad(Ř 12 (u)) = 0.

The defining conditions
The conditions restricting the R-operator are obtained from the expansion of the RLLrelation (6.2) in the parameter v. The condition at v 2 is trivial. At v 1 we obtain the invariancy condition (5.25) w.r.t. the adjoint action of osp The condition appearing at v 0 is The product of two generators can be rewritten via the supercommutator (4.6) and the superanticommutator The superanticommutator is defined as We introduce the following notation and use the supercommutation relations for osp (4.8) to write (6.4) as This condition is fulfilled only if both sides vanish separately. This becomes the key point of the construction of the R-operatorR(u).

Auxiliary variables
We have to deal with the supersymmetrization of the product of super-oscillators generalizing (5.11), where p(σ) denotes the parity of the permutation σ. Let us explain what we mean byσ.
We denote the basic transposition as σ j ≡ σ j,j+1 permuting the j-th and (j + 1)-st site.

26)
It means that the elements (6.25) are invariant under the action of the Lie superalgebra osp and satisfy the infinitesimal form (5.25) of the invariance condition are generators of osp (see proposition 4).
Proof. According to (6.26) the element U ∈ OSp acts on the product c b k · · · c b 1 as (see (2.34)): where U {k,...,j} = (−) k,k−1 · · · (−) k,j U j (−) k,j · · · (−) k,k−1 . Let the oscillators c a commute as in (5.7), where in the right hand side we put ε ab = 0. Then we have c (b k · · · c b 1 ) = c b k · · · c b 1 and (6.29) gives where the parentheses (. . . ) denote the supersymmetrization. Since the commutation relations of elements U a b and c d are independent of the right hand side of (5.7), the transformation rule (6.30) will be the same for the algebra of super-oscillators (5.7). From (6.26) and in view of the invariance of the bilinear form (2.2) we have the transformation rule for new variablesc a ≡ ε da c d : wherec a = ǫ(−1) [a] c a and j denotes the label of the superspace. Arguing as above we obtain the transformation rule for the supersymmetrized product of the super-oscillators c a :c From eqs. (6.30) and (6.32) we immediately see that the element is invariant under the action (6.26) of the supergroup OSp. Considering now the infinitesimal form of this action U = I + A + . . . and taking into account eqs. (5.24) we deduce the condition (6.27). We present the direct proof of (6.27) in appendix C, giving an alternative of the above proof.

The construction of the R-operator
Having introduced generating functions as an effective formulation of the supersymmetrization of super-oscillators, we are prepared to solve the conditions (6.3) and (6.8) imposed on the R-operatorR 12 (u).
The condition (6.3) says that the R-operator has to be invariant w.r.t. the Lie superalgebra osp. Therefore, it has to be a sum of osp-invariants (6.25) where we use the concise notation Inserting this ansatz into the condition (6.9), we obtain (6.34) We use now the advantage of the generating function formulation developed in the last subsection (in particular we apply relations (6.20), (6.22)) and rewrite the equation as = 0, (6.35) where the notation [±] ab was introduced in (6.21), (6.23). We also see that We want to commute all the partial derivatives ∂ 1 , ∂ 2 to the right and the variables κ 1 , κ 2 to the left and then apply κ 1 = 0, κ 2 = 0. For this purpose we need to know how the operator e λ(∂ 1 ·∂ 2 ) 1 acts on the variables κ 1 , κ 2 First of all Commuting e λ(∂ 1 ·∂ 2 ) 1 through the operator Y (cb)(ad) and imposing κ 1 , κ 2 = 0 we obtain
The bosonic part of osp(N |M ) (in the case ǫ = 1) corresponds to the embedded subalgebra so(N ). Similarly the fermionic part (in the case ǫ = −1) corresponds to the embedded Lie subalgebra so(M ). Hence, restricting ourselves to so ⊂ osp in (6.46) we obtain the recurrence relations for the coefficients r k (u) of the so(d)-symmetric R-operator Moreover, in such a restriction the supersymmetrizers (6.11) appearing in the ansatz (6.33) transfer to the antisymmetrisers. This result coincides with the results obtained in [14], [21]. Indeed, after the rescaling of the spectral parameter u → −u and of the generators c a → √ 2c a one can directly see the coincidence with [21]. The rescaling c a → √ 2c a gives the standard Clifford algebra c a c b + c b c a = 2ε ab for so ⊂ osp which was used in [14], [21] instead of the algebra A (5.7) used in this text. Moreover, the generators F ab of so (5.12) used in our text differ by the factor −ǫ(−1) [b] = −1 from their equivalents in [21]. This is the reason that here and in the left hand side of (6.8) of the spectral parameter is to be rescaled as u → −u.
Similar considerations can be done for the Lie subalgebra sp ⊂ osp. The fermionic part of osp(N |M ) (for ǫ = 1) corresponds to sp(M ) ⊂ osp(N |M ). The bosonic part of osp(N |M ) (for ǫ = −1) corresponds to sp(N ) ⊂ osp(N |M ). Restricting (6.46) to sp ⊂ osp we obtain the recurrence relation for for the sp(d)-symmetric R-operator The supersymmetrizers (6.11) appearing in the ansatz (6.33) transfer to the symmetrizers.

The condition on the generators G
We intend to prove here that from the condition (6.10) follows that {G (bc , G d)a } ∓ = 0..
We study X (cb)(ad) defined in (6.7). It possess obviously the following two symmetries: Using the results of section 6.3 we see that this equation can be rewritten as

Let us investigate all the terms appearing in Z (cb)(ad) + (−1) ([b]+[c])([a]+[d]) Z (ad)(cb)
. It is a third order polynomial in λ: (6.56) The coefficient A (cb)(ad) separates into two parts. The first part contains terms with the structure ∂ 3 1 ∂ 2 whereas the second part contains terms with the structure ∂ 1 ∂ 3 2 . We describe here only the first part, the second one is analysed in the same way. The first part of A (cb)(ad) is: Hence, we see the following symmetry: where (bcd) denotes the supersymmetrization over the indices b, c, d. Let us remark that this supersymmetrization differs from the supersymmetrization of the super-oscillators (5.11). The corresponding symmetrizer is defined like in (6.11) with the replacement σ → σ, where for the elementary permutation of adjacent sites j, j+1 σ j =σ j +[a j ]+[a j+1 ].
We summarize the results of this section in the Proposition 6. The L-operator constructed from the super-oscillator osp generators (5.12) and osp generators G ba , which solve the additional constraint obeys the spinorial RLL-relatioň where super-spinorial R-operatorŘ 12 (u) ∈ A ⊗ A iš Here ω = ǫ(N − M ) (see (3.4)), and A(u), B(u) are arbitrary functions of u.

The fusion of super-spinor L operators
It was shown in [18] (Theorem 3) that the so-type L-operator (i.e. the spinor-vector 7 so-type R-matrix) can be obtained by fusion of two spinor-spinor so-type R-matricesŘ. The main result of Sect. 6 about the generalization of the matrixŘ to the case of the Lie superalgebra osp is given in Proposition 6 (see eq. (6.49) at the end of Subsection 6.3 for the explanation of reducing to the so-case). The vector-vector so-type R-matrix (analog of the osp-type R-matrix (3.13)) was obtained in [18] (Theorem 5) by the fusion of two spinor-vector so-type R-matrices. The standard fusion procedure [19] applied in [18] requires the use of the projector operators V s ⊗ V s → V f which are not simple objects, so that the fusion procedure of [18] is technically non-trivial. In [3], for the cases of the so and sp Lie algebras, we found that the vector-vector R-matrix can be constructed as the fusion of two so-and sp-type L-operators using instead of those projectors the intertwining operators V s ⊗ V s → V f which are realized respectively in terms of gamma-matrices and generators of the oscillator algebra. In this Section we generalize the fusion procedure of the paper [3] to the case of the osp Lie superalgebra.
The RLL relation (4.1) has the following component form, It is convenient to introduce the supertensor product ⊗ s modifying the graded tensor products as It has the important property of associativity, This is checked by the following calculation Using the supertensor product one can represent the graded RLL relation (7.1) in the form 8 : Consider two different L-operators L(v − µ) and L ′ (u − λ) which commute up to the standard sign factor according to grading: This means that for the supertensor products (7.2) we have We assume the RLL relation (7.1), (7.4) to hold for L replaced by L ′ . The property (7.5) allows to apply the "train argument" [24] in the fusion procedure as in the nonsupersymmetric case without extra signs responsible for the grading. Thus the RLL relation (7.1), (7.4) holds for the matrix product where λ, µ are any shifts of the spectral parameter. In components this matrix product reads as where indices a, b, . . . label the coordinates in the vector (fundamental) representation space V f , while indices α, β, . . . are formal indices of the coordinates of the representation space V s in which the super-oscillator algebra A acts. Now the formal matrix (c a ) α β of the super-oscillator generators c a plays the role of the intertwiner: V s ⊗ V s → V f , where V s denotes the space which is dual to V s . The spaces V s and V s are identified with the help of the metric D αβ and inverse metric D αβ which can be used for lowering and rising indices α, β, . . . . Now we define the operator L ′ (u) in (7.6) as following are generators of the osp Lie superalgebra. Indeed one can check directly that they satisfy the graded commutation relations (4.12), (5.13). The elements F ′ define a representation of osp which is contra-gradient to the representation given by elements (5.12). Further, the generators (L (1) ) c d = F c d satisfy the conditions of Proposition 2 by construction. This implies that the generators F ′ c d obey the conditions of Proposition 2 as well. Thus, both operators L ′ and T given in (7.6) and (7.7) satisfy the RLL relations (7.1).
The projection of the elements T b d (u) which are operators in the space V s ⊗ V s to the operators in the space V f gives us the desirable fusion of two L operators to the vectorvector R-matrix. This projection can be done by the invariant contraction of the matrices (7.6) with two intertwiners (c We show now that the fusion expression(7.8) is related to the fundamental R matrix up to a multiplication by certain sign factors which will be fixed at the end of this Section. Traces Tr of products of super-oscillators with definite grading are fixed by the symmetry arguments. In the cases of so, (ǫ = +1) and sp, (ǫ = −1) we had in [3] Tr(c a c b ) = ε ab Tr1 , where Tr1 is a normalization constant which is not important here. In the supersymmetric case of the osp algebra this is modified as follows: To simplify formulas here and below in this Section we write the gradings [a], [b], . . . in sign factors as a, b, . . . . Now we calculate the projection (7.8): In the last line we commute c d 2 with c c 1 and use the identity (5.10) which leads to Then applying formulas for traces (7.9) we write (7.11) as Let arbitrary parameters λ and µ be expressed via one parameter κ as following For this choice of the parameters we finally obtain: where u ′ = u + κ and as usual we denote β = 1 − ω 2 . So we see that the projection (7.8) leads to the result that the fusion of two conjugated super-oscillator L operators decorated by sign factors coincides with the vector-vector (fundamental) osp R-matrix (3.13) and with the twisted R-matrix (−) 12 R(u)(−) 12 (7.14) Recall that the twisted R-matrix (−) 12 R(u)(−) 12 defines the vector (fundamental) representation of L-operator (4.20).
Remark. The Yang-Baxter equation (3.14) for the vector-vector R-matrix follows from the RLL relations (7.1) for the matrices (T b d (u)) defined in (7.6). Indeed, this statement is based on the remarkable identity which generalizes the relations (7.14) and justifies the use of the super-oscillator generators as intertwiners.
8 The quadratic evaluation of the Yangian Y(osp)

The conditions for the quadratic evaluation
We derive the conditions on the terms of a quadratic evaluated L-operator following from the RLL-relation. We investigate a particular solution for the second term. As above we denote the operator A acting non-trivially only in the first space of a tensor product of vector spaces as A 1 and the operator B acting non-trivially only in the second space of the tensor product as B 2 . We introduce a new symbol B 2 for the following object where (−) 12 is the sign operator (2.31) introduced in the first section. It is a particular case of a sign operator dressed operator (2.33). Let us solve the graded RLL-relation (4.1) for a quadratic evaluation of the L-operator with the osp-invariant R-matrix (3.13). Expanding in u, v, we obtain the following set of six equations. The rest of equations is linearly dependent on these six.
. says that at the first level appear the generators of osp. For details, please, see section 5. We discussed there that the generators can be arranged supertraceless and that they satisfy This can be arranged also here as a consequence of equation A. Equation B. is fulfilled if the second level operator N is a linear combination of powers of G, where b j commute with G k for all j, k. Equation C. can be rearranged in the following way: Multiplying this equation from both sides by the super-permutation P we obtain an equivalent equation Adding these two equations, multiplying them by K and using the identities (4.13) we obtain which is obviously solved by One can easily show that

Generators of osp in Jordan-Schwinger form
We introduce a set of graded canonical pairs, variables x a and the corresponding partial derivatives ∂ a , such that (cf. (2.1)) For ǫ = −1 these variables behave like the graded differential forms. (8.11) implies that the invariant bilinear form (x, y) = ε ba x a y b is always symmetric (x, y) = (y, x). One can directly prove that the elements of this graded Heisenberg algebra satisfy the supercommutation relations of osp (4.8) and the symmetry condition (4.19). Therefore, they compose a set of generators of osp. One can easily check that the matrix of generators ( Proof. It is useful to introduce the following operator with the properties Hx a = x a (H + 1), The square of M is then expressed as where we remind ω ≡ ε ef ε ef and introduce concise notation It is useful to use also the following identities After a lengthy calculation we obtain (8.14). The osp representation generated by M ab satisfies the condition (6.60) for the linear Loperators intertwining the super-oscillator with the Jordan-Schwinger type representation.
with F ab (5.12) generating the super-oscillator representation obeys the RLL-relation with the super-spinorial R-operator (6.33 ) constructed in section (6).
Proof. The super-anticommutator condition can be checked by direct calculation using the expression for M ab (8.12). The super-spinorial RLL-relation is fulfilled by the proposition (6). The above cubic characteristic condition of (8.14) follows from the condition (6.60) written in terms of M . This is the consequence of the following: Proposition 9. If the generators G a b of osp obey the condition (6.60) then the matrix G = ||G a b || obeys the cubic characteristic identity (8.14) written in terms of G as Proof. The condition (8.20) can be rewritten as Super-commuting G a 2 c 1 with G a 1 a 2 , this can be further rewritten to The right hand-side is after multiplication from the right by (−1) If we use the properties of the super-metric ε c 1 c 2 and the transposition rule for G 2 the right hand-side can be further rewritten to Comparing the left and right hand-side, we arrive at the statement of the proposition.
The fusion of L(u) = uI − 1 2 F ab M ba and L(u) = uI + 1 2 (F t ) ab M ba with respect to the super-spinor representations generated by F, F t results in an L operator obeying the RLL relation with the vector (fundamental) R matrix (4.1). It is qudratic in u and equivalent to the form (8.2) with N of the form (8.9), shown above to obey the conditions A-C. It obeys also the remaining conditions D-F. The proof can be done by direct calculations using the relations (6.60, 8.14).

Discussion
Yang-Baxter relations with orthosymplectic supersymmetry, in particular the ones involving the fundamental R matrix, can be written in a similar form like the ones with orthogonal or symplectic symmetry. The formulation presented in this paper provides a systematic treatment and displays explicitly the features distinguishing the osp case from the so and sp cases.
We have pointed out that the invariant tensors appearing in the fundamental R matrix represent the Brauer algebra.
L operators can have a simple form in distinguished representations. We have identified the superspinor representation resulting in an L operator linear in the spectral parameter being the generalization of the spinor representation of the so case and the metaplectic representation in the sp case.
The super spinorial R operator which intertwines super-spinor representations (see Proposition 6) has been constructed by the generalizing the methods developed in [13], [21], [3] for the so and sp cases.
The superspinorial RLL relation holds for L operators linear in the spectral parameter and acting in the spinor and the vector (fundamental) representations. It also holds for generalized L operators where the vector (fundamental) representation is replaced by another one obeying a constraint (6.60) represented in the form of a super anticommutator of the elements G ab of the matrix of generators. All these results were summarized in the Proposition 6.
We have investigated the case of the second order Yangian evaluation, in particular the solution for the L operators with all terms expressed as function of the Lie algebra generator matrix G. Its second non-trivial term is proportional to the supertraceless part of G 2 . The Lie algebra representation generated by the matrix elements of G is constraint in such a way that G obeys a condition in terms of a cubic characteristic polynomial. The latter condition is related to the super anticommutator condition appearing in connection with the spinorial Yang Baxter relation. The class of Lie algebra representations constructed by the Jordan-Schwinger ansatz based on graded Heisenberg pairs obeys these constraints.

A The graded tensor product and Yang-Baxter relations
There appear different conventions in the literature regarding the R-matrices and Yang-Baxter equations. In this section we intend to relate the Yang-Baxter equation used, e.g., in [26] to the Yang-Baster equation (3.14).
Let V be a vector superspace V with the basis {|e 1 , . . . , |e N +M }, where {|e 1 , . . . ,|e N } are the basis vectors of the even part of V and {|e N +1 , . . . , |e N +M } are the basis vectors of the odd part of V. The basis of the dual superspace V is { e 1 |, . . . , e N +M |} with the even part { e 1 |, . . . , e N |} and the odd part { e N +1 |, . . . , e N +M |}. We demand that these two bases are dual in the following sense: Unlike the formulation used in the main part of this paper the gradation is now carried by the basis vectors, i.e., grad(|e a ) = grad( e a |) = [a], whereas the coordinates are ordinary numbers from the field F over which the superspace V is constructed (compare with the approach introduced in this article, especially in section 2). The matrix units and the identity operator on V can be expressed as: |e a e a |. and one can immediately check that We introduce the graded tensor product of the superspaces V ⊗ V (2.11). The basis of ] e a 1 | ⊗ e a 2 |} N +M a 1 ,a 2 =1 as can be easily seen, The operator R acting in V ⊗ V has the components w.r.t. the above basis of the form and satisfies The graded permutation is defined as (A.8) In view of the above considerations, it has the following components as expected (compare with (3.1)). It can be expressed using the matrix units as The identity operator on V ⊗ V is This formalism can be obviously extended to V ⊗n for arbitrary n. Due to the Yang-Baxter relation we need to discuss the situation V ⊗3 . Its basis is and the corresponding basis of the dual superspace V ⊗3 is .
Let us remark that the identity operator in V ⊗3 is It is useful to use the shorthand notation e a 1 a 2 a 3 | = e a 1 | ⊗ e a 2 | ⊗ e a 3 |, The YB equation appearing, e.g., in [26] is of the form We show here that if we write it in components, we obtain our form of the Yang-Baxter relation (3.14) provided that the R-matrix is even (see the definition of the even R-matrix (3.16)). The left hand side of (A.14) has the component form and we embed the identity operator (A.12) in V ⊗3 between R 12 and R 13 where we used the properties of the graded tensor product (2.11). We embed the identity operator (A.12) between R 13 and R 23 and obtain (R 12 R 13 R 23 ) a 1 a 2 a 3 b 1 b 2 b 3 = (−1) [ 16) which is exactly the left hand side of (3.14). Similarly, the right hand side of (A.14) has the component form which coincides with the right hand side of (3.14). Thus, the equivalence of (A.14) and (3.14) is established. We recall that this equivalence holds due to the R-matrix being even.

C Direct proof of proposition 5
We shall use the advantage of the generating functions language developed in subsection 6.2. and shall work with two sets of auxiliary variables κ a , κ ′b with the corresponding derivatives ∂ a , ∂ ′b . Then κ b ∂ a ∂ ′b k · · · ∂ ′b 1 + ∂ a 1 · · · ∂ a k ∂ ′b k · · · ∂ ′b 1 ǫ(−1) , where we used the supercommutation relations (6.24). We need two identities: [a] ε aa i ∂ a 1 · · · ∂ a j−1 ∂ a j+1 · · · ∂ a k + We show now that the two underlined terms cancel. We write them here without the factor e (κ·c 1 ) e (κ ′ ·c 2 ) | κ,κ ′ =0 . The first underlined term is The second underlined term is Let us remark that Z 1 , Z 2 are proportional to κ, κ ′ respectively and, therefore, vanish. As we see, the two underlined terms really cancel. The two non-underlined terms cancel, too.