A Drinfeld-type presentation of the orthosymplectic Yangians

We use the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the Yangian for the orthosymplectic Lie superalgebra ${\frak osp}_{N|2m}$ to produce its Drinfeld-type presentation. The results rely on a super-version of the embedding theorem which allows one to identify a subalgebra in the $R$-matrix presentation which is isomorphic to the Yangian associated with ${\frak osp}_{N|2m-2}$.


Introduction
By the original definition of Drinfeld [6], the Yangian Y(a) associated with a simple Lie algebra a is a canonical deformation of the universal enveloping algebra U(a[u]) in the class of Hopf algebras.The finite-dimensional irreducible representations of the algebra Y(a) were classified in his subsequent work [7] with the use of a new presentation which is now often referred to as the Drinfeld presentation.It involves sufficiently many generators which are needed to identify the representations by their highest weights.It is well-known by Levendorskiȋ [27], that this presentation admits a reduced version involving a finite set of generators; see also [18] for its refined form and generalization to symmetrizable Kac-Moody Lie algebras.
The Drinfeld presentation is essential in the theory of Yangian characters or q-characters which were originally introduced by Knight [23] and by Frenkel and Reshetikhin [11] in the quantum affine algebra context.The theory was further developed in [10], [20] and [30], while an extensive review of the applications to integrable systems was given in [26].The isomorphisms between completions of the Yangians and quantum loop algebras constructed by Gautam and Toledano Laredo [12] also rely on the Drinfeld presentations.
The Yangian-type algebra associated with the general linear Lie algebra gl n was considered previously in the work of the Leningrad school on the quantum inverse scattering method, although this name for the algebra was not used; see e.g.review paper by Kulish and Sklyanin [25].In this approach, the defining relations are written in the form of a single RT T -relation involving the Yang R-matrix; see also a brief discussion in [5,Sec. 7.5] explaining connections with integrable lattice models.
An explicit isomorphism between the R-matrix and Drinfeld presentations of the Yangian for gl n can be constructed with the use of the Gauss decomposition of the generator matrix T (u), as was originally outlined in [7]; see [4] for a detailed proof.The same approach was used in [21] to produce such isomorphisms for the remaining classical types B, C and D, while a different method to establish isomorphisms was developed in [19].The construction of [21] was extended in a recent work [8] to the antidominantly shifted Yangians.
The Yangians associated with the general linear and orthosymplectic Lie superalgebras were first introduced in their R-matrix presentations.Nazarov defined the Yangian Y(gl n|m ) in [31] by using a super-version of the Yang R-matrix, while the definition of the orthosymplectic Yangian Y(osp N |2m ) is due to Arnaudon et al. [1], where a super-version of the R-matrix originated in [35] was used.Presentations of the Yangian for sl n|m analogous to [7] and [27] were given by Stukopin [33], while the construction of [4] was extended to the Yangian Y(gl n|m ) by Gow [15], where a Drinfeld-type presentation was devised together with an isomorphism with the R-matrix presentation.More general parabolic presentations of the Yangian Y(gl n|m ), corresponding to arbitrary Borel subalgebras in gl n|m were given by Peng [32]; see also Tsymbaliuk [34].
Our goal in this paper is to give a Drinfeld-type presentation for the orthosymplectic Yangians Y(osp N |2m ) with N 3. To state the Main Theorem, we introduce some notation related to the Lie superalgebras osp N |2m , assuming that m 1.If N = 2n + 1 is odd, they form series B of simple Lie superalgebras, while in the case N = 2n with n 2 they belong to series D. We will consider both cases simultaneously whenever possible.We will assume that the simple roots of osp N |2m are α 1 , . . ., α m+n with α i = ε i − ε i+1 , for i = 1, . . ., m + n − 1, and where ε 1 , . . ., ε m+n is an orthogonal basis of a vector space with the bilinear form such that for i = m + 1, . . ., m + n. (1.1)This choice of simple roots corresponds to the standard Dynkin diagrams given by ♠ ♠ . . .
for osp 2n+1|2m with n 1, and for osp 2n|2m with n 2. In both cases, α m is the only odd simple isotropic root.The associated Cartan matrix C = [c ij ] m+n i,j=1 is defined by c ij = (α i , α j ) for series D, and by for series B. Note that the n × n submatrix [c ij ] m+n i,j=m+1 coincides with the Cartan matrix associated with the simple Lie algebra of type D n or B n , respectively.
Main Theorem.The Yangian Y(osp N |2m ) with N 3 is isomorphic to the superalgebra with generators κ i r , ξ + i r and ξ − i r , where i = 1, . . ., m + n and r = 0, 1, . . . .The generators ξ ± m r are odd, while all the remaining generators are even.The defining relations have the form ) together with the Serre relations for i = j, where we set k = 1 + |c ij |, and the super Serre relations The additional relations obtained by replacing ξ ± m+1 s with ξ ± m+2 s are included for N = 4.
In the formulation of the theorem we used square brackets to denote super-commutator for homogeneous elements a and b of parities p(a) and p(b).The subscripts take all possible admissible values.Note that relations (1.4) and (1.5 Relation (1.6) for i = j = m is implied by (1.7).If m = 1, then the super Serre relations (1.9) are omitted.By omitting relations (1.7) and (1.9), and taking m = 0, we recover the Drinfeld presentation of the Yangian Y(o N ) [7]; cf.[21].
A key role in the proof of the Main Theorem is played by the embedding theorem for the extended Yangians which shows that for any m 1 the extended Yangian X(osp N |2m−2 ) can be regarded as a subalgebra of X(osp N |2m ).Its counterpart for the orthogonal and symplectic Yangians was proved in our work with Jing and Liu [21,Thm 3.1].However, that proof does not fully extend to the super case since the values of the R-matrix R(1) used therein are not defined in general.Instead, we employ R-matrix calculations to produce a different argument.
As a next step, we follow [4], [15] and [21] to use the Gauss decomposition of the generator matrix T (u) in the R-matrix presentation of the extended Yangian X(osp N |2m ) to introduce the Gaussian generators.This leads to a Drinfeld-type presentation of the extended Yangian X(osp N |2m ) (Theorem 6.1) which will then be used to derive the presentation of the Yangian Y(osp N |2m ) ⊂ X(osp N |2m ) given in the Main Theorem.As a part of the proof, we use a description of the center of the extended Yangian in terms of the Gaussian generators (Theorem 5.3).This is a super-counterpart of [21,Thm 5.8], although we use a different argument relying on Jacobi's ratio theorem for quasideterminants; see [13] and [24].
Drinfeld-type presentations of the Yangians Y(osp N |2m ) with N = 1 and N = 2 require some modifications of the relations in the Main Theorem.Such a presentation of the Yangian Y(osp 1|2 ) was given in [2]; it involves additional Serre relations of a different kind.By the embedding theorem, the Drinfeld-type presentation of Y(osp 2|2m ) is largely determined by the case m = 1 which should rely on an isomorphism between the Yangians associated with the Lie superalgebras osp 2|2 and sl 1|2 .
Since the first version of the paper was posted in the arXiv, Drinfeld-type presentations of the super Yangians have been further investigated.Such a presentation for the Yangian Y(osp 1|2m ) was given in [29] and will also appear in the independent work [9] as a part of a more general project involving presentations of the orthosymplectic Yangians associated with arbitrary parity sequences.I am grateful to Alexander Tsymbaliuk for useful discussions and for pointing out a few corrections, including the list of relations for the case N = 4.

Basic properties of the orthosymplectic Yangian
Introduce the involution i → i ′ = N + 2m − i + 1 on the set {1, 2, . . ., N + 2m}.Consider the Z 2 -graded vector space C N |2m over C with the canonical basis e 1 , e 2 , . . ., e N +2m , where the vector e i has the parity ī mod 2 and The endomorphism algebra End C N |2m is then equipped with a Z 2 -gradation with the parity of the matrix unit e ij found by ī +  mod 2. We will identify the algebra of even matrices over a superalgebra A with the tensor product algebra End C N |2m ⊗A, so that a square matrix A = [a ij ] of size N + 2m is regarded as the element where the entries a ij are assumed to be homogeneous of parity ī +  mod 2. The extra signs are necessary to keep the usual rule for the matrix multiplication.The involutive matrix super-transposition t is defined by (A t ) ij = a j ′ i ′ (−1) ī+ θ i θ j , where we set This super-transposition is associated with the bilinear form on the space C N |2m defined by the anti A standard basis of the general linear Lie superalgebra gl N |2m is formed by elements E ij of the parity ī +  mod 2 for 1 i, j N + 2m with the commutation relations We will regard the orthosymplectic Lie superalgebra osp N |2m associated with the bilinear form defined by G as the subalgebra of gl N |2m spanned by the elements Introduce the permutation operator P by and set 2) The R-matrix associated with osp N |2m is the rational function in u given by This is a super-version of the R-matrix originally found in [35].Following [1], we define the extended Yangian X(osp N |2m ) as a Z 2 -graded algebra with generators t (r) ij of parity ī +  mod 2, where 1 i, j N +2m and r = 1, 2, . . ., satisfying the following defining relations.Introduce the formal series and combine them into the matrix Consider the elements of the tensor product algebra End The defining relations for the algebra X(osp N |2m ) take the form of the RT T -relation For m = 0 the algebra X(osp N |0 ) coincides with the extended Yangian X(o N ) associated with the orthogonal Lie algebra o N .As shown in [1], the product T (u − κ) T t (u) is a scalar matrix with where c(u) is a series in u −1 .All its coefficients belong to the center ZX(osp N |2m ) of X(osp N |2m ) and freely generate the center; cf. the Lie algebra case considered in [3].
The Yangian Y(osp N |2m ) is defined as the subalgebra of X(osp N |2m ) which consists of the elements stable under the automorphisms As in the non-super case [3], we have the tensor product decomposition see also [17].The Yangian Y(osp An explicit form of the defining relations (2.4) can be written in terms of the series (2.3) as follows: The mapping defines an automorphism of X(osp N |2m ), while the mapping defines an anti-automorphism.The latter property is understood in the sense that for homogeneous elements a and b of the Yangian.The universal enveloping algebra U(osp N |2m ) can be regarded as a subalgebra of X(osp N |2m ) via the embedding (2.10) This fact relies on the Poincaré-Birkhoff-Witt theorem for the orthosymplectic Yangian which was pointed out in [1] and a detailed proof is given in [17]; cf.[3,Sec. 3].It states that the associated graded algebra for Y(osp ) is generated by the coefficients of the series c(u) and t ij (u) with the conditions i + j N + 2m + 1 for i = 1, . . ., m, m ′ , . . ., 1 ′ and i + j < N + 2m + 1 Moreover, given any total ordering on the set of the generators, the ordered monomials with the powers of odd generators not exceeding 1, form a basis of the algebra.

The embedding theorem
Let A = [a ij ] be a p × p matrix over a ring with 1. Denote by A ij the matrix obtained from A by deleting the i-th row and j-th column.Suppose that the matrix A ij is invertible.The ij-th quasideterminant of A is defined by the formula where r j i is the row matrix obtained from the i-th row of A by deleting the element a ij , and c i j is the column matrix obtained from the j-th column of A by deleting the element a ij ; see [13] If the matrix A is invertible and the (j, i) entry of A −1 is invertible, then the quasideterminant can be found by Suppose that ({i}, L, U) and ({j}, M, V ) are partitions of the set {1, 2, . . ., p} such that |L| = |M| and |U| = |V |.Then, according to Jacobi's ratio theorem for quasideterminants, where B = A −1 and C P Q denotes the submatrix of a matrix C whose rows are labelled by a set P and columns labelled by a set Q; see [13] and [24].Now consider the extended Yangian X(osp N |2m−2 ) with m 1 and let the indices of the generators t ) Proof.The following elements of the algebra End C N |2m will be used throughout the proof: For any A ∈ End C N |2m the product IAI will be understood as an element of End C N |2m−2 , where we identify End C N |2m−2 with the subalgebra of End C N |2m spanned by the basis elements e ij with 2 i, j 2 ′ .Introduce the matrix where T (u) = J T (u)J, and we regard T (u) as the submatrix of T (u) obtained by deleting the last row and column.Write this submatrix in the block form according to the partition {1} ∪ {2, . . ., 2 ′ } of the row and column numbers, so that, in particular, A(u) = t 11 (u).Then using the block multiplication of matrices, we find that ; see e.g.[28,Lemma 1.11.1].Therefore, the (i, j) entry of the matrix G(u) −1 coincides with the series given by the quasideterminant appearing in (3.4).This means that in order to show that the map (3.4) defines a homomorphism, it will be enough to verify that G(u) satisfies the relation where G 1 (u) and G 2 (u) are elements of the tensor product algebra with the copies of the endomorphism algebra respectively labelled by 1 and 2 as in (2.4).Here we use the R-matrix ) so that the operators P and Q are given by the respective formulas (2.1) and (2.2) with the summations restricted to the set i, j ∈ {2, . . ., 2 ′ }.
Introduce elements of the algebra End C N |2m ⊗ End C N |2m by where and Note the relations where the subscripts indicate the element J taken in the respective copy of the endomorphism algebra with the identity component in the other copy.
Multiply both sides of (2.4) by J 1 J 2 from the left and from the right to get the relation where we set e ij ⊗ e ji (−1) .
As a next step, multiply both sides of (2.4) by K + from the left.Since K + P = K − and and hence Performing a similar calculation after multiplying both sides of (2.4) by Ǩ+ from the right, and then rearranging (3.8), we come to the relation Now transform this relation by multiplying both its sides from the left and from the right first by T 2 (v) −1 , then by T 1 (u) −1 , and then by I 1 I 2 .After this transformation, some terms on the right hand side will vanish.Namely, which follows by writing and observing that K − (1 ⊗ e 1 ′ 1 ′ ) = 0 and Similarly, Therefore, taking into account the identities K + = I 1 I 2 K + and Ǩ− = Ǩ− I 1 I 2 , as a result of the transformation, we find that the difference To transform the first product, use the second relation in (3.7) to write Furthermore, due to (3.8), the difference equals the right hand side of (3.8) multiplied from the left and from the right by T 2 (v) −1 and then by T 1 (u) −1 .Hence, replacing R(u − v) T 2 (v) −1 T 1 (u) −1 by the resulting expression, we can bring the first product in (3.10) to the form to see that two products in the expression vanish.Indeed, since K + (e 1 ′ 1 ′ ⊗1) = 0, and K = I 1 K, we get Hence, using (3.11) again to simplify the remaining terms, we find that first product in (3.10) takes the form and (1 ⊗ e 1 ′ 1 ′ ) Ǩ− = 0, both occurrences of Ǩ in (3.12) can be replaced by Ǩ+ , because the components with Ǩ− cancel.Performing a similar calculation for the second product in (3.10), we can conclude that the difference (3.9) equals The expression W can be written as where h 1 ′ (v) denotes the (1 ′ , 1 ′ ) entry of the matrix According to (3.1), the series Hence, (3.2) yields By the defining relations (2.8), we have [t 11 (u), t 11 (v)] = 0, and since the coefficients of the series c(v) belong to the center of the algebra X(osp N |2m ), we get W = 0, thus proving (3.6).
The remaining parts of the theorem are verified by the same argument as for its non-super counterpart [21,Thm. 3.1].Namely, to verify that the homomorphism (3.4) is injective, we pass to the associated graded algebras, where the ascending filtrations on the extended Yangians are defined by setting deg t We point out some consequences of Theorem 3.1 which are verified in the same way as in the non-super case; see [21,Sec. 3].
Suppose that ℓ m + n for type B and ℓ m + n − 1 for type D.
Moreover, the restriction of the map to the Yangian defines injective homomorphisms The embeddings (3.14) possess the following consistency property; cf.[4].We will write ψ for the embedding map ψ ℓ in Corollary 3.2.Then we have the equality of maps

Gaussian generators
Apply the Gauss decomposition to the generator matrix T (u) associated with the extended Yangian X(osp N |2m ), where F (u), H(u) and E(u) are uniquely determined matrices of the form , and H(u) = diag h 1 (u), . . ., h 1 ′ (u) .Recall the well-known formulas for the entries of the matrices F (u), H(u) and E(u) in terms of quasideterminants [14]; see also [28,Sec. 1.11].We have whereas and for 1 i < j 1 ′ .We will need the formulas for the action of the anti-automorphism τ of X(osp N |2m ) defined in (2.9) on the Gaussian generators.
for i < j, and τ : h i (u) → h i (u) for all i.
Proof.We have the following relations for the matrix entries implied by (4.1): for i = 1, . . ., 1 ′ , and for i < j.The required formulas follow by applying τ to both sides of the relations and using the induction on i.
Assuming that ℓ m + n for type B and ℓ m + n − 1 for type D, use the superscript [ℓ] to indicate square submatrices corresponding to rows and columns labelled by ℓ + 1, . . ., (ℓ + 1) ′ .In particular, and H [ℓ] (u) = diag h ℓ+1 (u), . . ., h (ℓ+1) ′ (u) .Furthermore, introduce the product of these matrices by Accordingly, the entries of T [ℓ] (u) will be denoted by t Moreover, the subalgebra X [ℓ] (osp N |2m−2ℓ ) generated by the coefficients of all series t Introduce the coefficients of the series defined in (4.2), (4.3) and (4.4) by Furthermore, set with i = 1, . . ., m + n for type B and with i = 1, . . ., m + n − 1 for type D. In the latter case we also set and e m+n (u) = e m+n−1 m+n+1 (u), We will also use the coefficients of the series defined by The following is a super-version of [21, Lemma 4.3] (the case m = 0 was covered therein); we will use a different argument.Lemma 4.3.Suppose that the indices i, j, k satisfy ℓ + 1 i, j, k (ℓ + 1) ′ and k = j ′ .Then the following relations hold in the extended Yangian X(osp N |2m ), ) Proof.We will assume that m 1 and start by verifying (4.11) for ℓ = 1.The defining relations (2.8) imply that under the given restrictions on the indices, 1k , t 11 (v) = t 1k (v), and hence t On the other hand, Corollary 3.3 implies that t 11 (u), t [1] i j (v) = 0.
By taking the super-commutator of the left hand side with t (1) 1k , we get Therefore, multiplying from the left by t 11 (u) −1 , we arrive at (4.11).Relation (4.12) for ℓ = 1 now follows from (4.11) by applying the anti-automorphism τ and using Lemma 4.1.A similar argument for m = 0 gives another proof of [21,Lemma 4.3].The case of general values of ℓ follows by the application of the homomorphism ψ ℓ and using (3.15) and Proposition 4.2.

Multiplicative formula for c(u)
We will need a formula for the series c(u) defined in (2.5) in terms of the Gaussian generators h i (u) with i = 1, . . ., m + n + 1.To derive it, we first establish some relations between the series h i (u) and show that all their coefficients pairwise commute in X(osp N |2m ).
Proof.Denote the quasideterminant (3.5) by s ij (u) and assume that i and j run over the set P = {2, 3, . . ., 1 ′ }.By applying (3.3), we get Hence, (2.5) and (3.2) imply that s ij (u) coincides with the (i, j) entry of the matrix c(u + κ) T t (u + κ) P P −1 . (5.3) Write the matrix T t (v) P P in the block form according to the partition P = {2, . . ., 2 ′ } ∪ {1 ′ } of the row and column numbers.In particular, we have D(v) = t 11 (v).Using the block multiplication of matrices (see e.g.[28, Lemma 1.11.1]),we find that for any i, j ∈ {2, . . ., 2 ′ } the (i, j) entry of the matrix (T t (v) P P ) −1 is found by Note that this coincides with the (i, j) entry of the matrix S t (v), where As a next step, we verify the identity Indeed, by the defining relations, and so by (2.8).Together with the relation this yields (5.4).Thus, the matrix Since S(u) coincides with the submatrix of (5.3) corresponding to the rows and columns labelled by the set {2, . . ., 2 ′ }, we derive the relation On the other hand, by Theorem 3.1, the matrix S(u) satisfies the RT T relation (2.4) associated with the extended Yangian X(osp N |2m−2 ).Therefore, by (2.5) we have where c ′ (u) is the central series associated with X(osp N |2m−2 ).We thus get the recurrence relation c ′ (u + 1). (5.5) Furthermore, as we pointed out in (3.13), the series c(u) and c ′ (u) are found by where we have taken into account the consistency property of the Gauss decompositions of the matrices T (u) and S(u) as stated in Proposition 4.2.Hence, by relation (5.5) we get The proof of relations (5.1) is completed by the application of the maps ψ ℓ with the use of (3.15) and Proposition 4.2.Relations (5.2) can be deduced by the same argument, starting with the generator matrix T (u) of the extended Yangian X(o N ), or just by applying the non-super version of (5.6) implicitly contained in [21,Sec. 5]; see also [22,Lemma 2.1].

Corollary 5.2. The coefficients of all series h
Proof.Note that the subalgebra of X(osp N |2m ) generated by the coefficients of the series t ij (u) with i, j ∈ {1, . . ., m + n} is isomorphic to the Yangian Y(gl n|m ).The Gauss decomposition of the corresponding generator matrix T (u) was used in [15] to get Drinfeld-type presentations of Y(gl n|m ) and Y(sl n|m ).By changing the parity assumptions of [15] to their opposites (see also [34]), we find that the coefficients of the series h i (u) with i = 1, . . ., m + n generate a commutative subalgebra A of Y(gl n|m ) ⊂ X(osp N |2m ).The relations of Proposition 5.1 imply that the coefficients of the remaining series h i (u) with i = m + n + 1, . . ., 1 ′ can be expressed in terms of the elements of the commutative subalgebra of X(osp N |2m ) generated by A and the coefficients of the central series c(u).
We can now derive a multiplicative formula for the series c(u).
Proof.The relations follow from the recurrence relation (5.5) and the formulas for the respective central series associated with the extended Yangians X(o N ); see [21,Thm. 5.8].

Drinfeld presentation of the extended Yangian
We will rely on the Drinfeld presentations of the Yangian Y(gl n|m ) obtained in [15], and the extended Yangian X(o N ) obtained in [21], to derive the following Drinfeld-type presentation of the algebra X(osp N |2m ).We will use the series introduced in (4.7)-(4.9)along with Theorem 6.1.The extended Yangian X(osp N |2m ) with N 3 and m 1 is generated by the coefficients of the series h i (u) with i = 1, . . ., m + n + 1, and the series e i (u) and f i (u) with i = 1, . . ., m + n, subject only to the following relations, where the indices take all admissible values unless specified otherwise.We have

.2)
For i m + n and all j, and for i = m + n + 1 and j < m + n we have ) where ε m+n+1 = 0.For N = 2n + 1 we have and whereas for N = 2n we have and Moreover, ) and for i < j we have ) We have the Serre relations , and for m 2 the super Serre relations The additional super Serre relations obtained by replacing e m+1 (u 4 ) and f m+1 (u 4 ) by e m+2 (u 4 ) and f m+2 (u 4 ), respectively, are included for N = 4.
Proof.Relations (6.1) follow from Corollary 5.2.To verify the remaining relations, regard the Yangian Y(gl n|m ) as the subalgebra of X(osp N |2m ) generated by the coefficients of the series t ij (u) with 1 i, j m + n.In type D there is another embedding of the Yangian Y(gl n|m ), as the subalgebra generated by the coefficients of the series t ij (u) with i, j running over the set {1, . . ., m + n − 1, (m + n) ′ }.For both types B and D we will also use the embedding of the extended Yangian X(o N ) ֒→ X(osp N |2m ) provided by the homomorphism ψ m ; see Corollary 3.2.
Therefore, some sets of relations between the Gaussian generators follow from [15, Thm.3] (via the change of parity); see also [34].Furthermore, for the values of the indices i m + 1 the relations are implied by the Drinfeld presentation of X(o N ) given in [21] 1 .Note also that most of the relations involving the series f i (u) follow from their counterparts involving e i (u) due to the symmetry provided by the anti-automorphism τ defined in (2.9).
Using these observations, we find that the only cases of (6.2) not covered by the embeddings and the symmetry are i m and j = m + n.In those cases, the relation follows from Corollary 3.3, except for i = m and n = 1 where Lemma 4.3 should be invoked in the same way as in [21,Prop. 5.11].Apart from the Serre and super Serre relations, the remaining relations are verified in the same way as (6.2): if some cases are not covered by [15] and [21], then Corollary 3.3 or Lemma 4.3 should be used.
Turning to the Serre relations, note that the coefficients of the series e i (u) and f i (u) are stable under all automorphisms (2.6) and so they belong to the subalgebra Y(osp N |2m ) of X(osp N |2m ).The relations will be verified in the proof of the Main Theorem in Section 7, where we will see that they are equivalent to the respective relations (1.8) and (1.9).
The above arguments show that there is a homomorphism where X(osp N |2m ) denotes the algebra with generators and relations as in the statement of the theorem and the homomorphism takes the generators h ) to the elements of X(osp N |2m ) with the same name, where we use the expansions for h i (u), e i (u) and f i (u) as in (4.6) and (4.10).We will show that this homomorphism is surjective and injective.
To prove the surjectivity, note the following consequences of (2.8): for 1 i < j m + n for N = 2n + 1, and for  for 1 i j m + n for N = 2n + 1, and for 1 i j m + n − 1 for N = 2n.Moreover, if N = 2n, then we also have m+n−1 (m+n) ′ = t i (m+n) ′ (u) These relations together with the symmetries provided by the automorphism (2.9) and the Poincaré-Birkhoff-Witt theorem for the algebra X(osp N |2m ) imply that it is generated by the coefficients of the series t ij (u) with 1 i, j m + n + 1.Furthermore, it follows from the Gauss decomposition (4.1), that the algebra X(osp N |2m ) is generated by the coefficients of the series h i (u) for i = 1, . . ., m + n + 1 together with e ij (u) and f ji (u) for 1 i < j m + n + 1.
By writing the above relations in terms of the Gaussian generators (cf.[21, Sec.5]), we get e ij (u), e (1) with the same respective conditions on the indices as in (6.14), whereas e (1) for 1 i < j m + n for N = 2n + 1, and for 1 i < j m + n − 1 for N 2n.In both B and D types, we also have for i = 1, . . ., m, while m+n−1 (m+n) ′ = e i (m+n) ′ (u) (6.18) for 1 i m + n − 2 in type D. These relations together with their counterparts for the coefficients of the series f ji (u), which are obtained by applying the anti-automorphism τ and Lemma 4.1, show that the coefficients of the series h i (u) for i = 1, . . ., m + n + 1 and e i (u), f i (u) for i = 1, . . ., m+n generate the algebra X(osp N |2m ) thus proving that the homomorphism (6.13) is surjective.
To prove the injectivity of the homomorphism (6.13), we will apply the argument originally used in [4] and then also in [15] and [21].The first step is to observe that the set of monomials in the generators h ji with r 1 and the conditions i < j i ′ for i = 1, . . ., m and i < j < i ′ for i = m + 1, . . ., m + n, (6.19) taken in some fixed order, with the powers of odd generators not exceeding 1, is linearly independent in the extended Yangian X(osp N |2m ).To see this, introduce an ascending filtration on X(osp N |2m ) by setting deg t and denote by cr the image of c r in the (r − 1)-th component of gr X(osp N |2m ).As with the non-super case considered in [3,Cor. 3.10] (see also [17]), by the decomposition (2.7) and the Poincaré-Birkhoff-Witt theorem, the map where As a next step, working with the algebra X(osp N |2m ), introduce its elements inductively, as the coefficients of the series e ij (u) for i and j satisfying (6.19), by using relations (6.15)- (6.18).Furthermore, the defining relations show that the map and τ : h i (u) → h i (u) for i = 1, . . ., m + n + 1 defines an anti-automorphism of the algebra X(osp N |2m ).Apply this map to the relations defining e ij (u) and use the first relation in (4.5) to get the definition of the coefficients of the series f ji (u) subject to the same conditions (6.19).The injectivity of the homomorphism (6.13) will be proved by showing that the algebra X(osp N |2m ) is spanned by monomials in h To establish the spanning property of the monomials in the e (r) ij in the subalgebra E, it will be enough to verify the relations ī k+ k+ī+ θ i j .(6.22)Note the relations for the elements ē(r) ij implied by (6.15)-(6.18): j j+1 (6.23) for 1 i < j m + n for N = 2n + 1, and for for 1 i < j m + n for N = 2n + 1, and for 1 i < j m + n − 1 for N = 2n.Relation (6.24) also holds for i = j when 1 i m, while m+n−1 (m+n) ′ (6.25) Since the defining relations between the coefficients of the series e i (u) with 1 i m+n−1 are the same as the respective relations in the Yangian Y(gl n|m ), the argument in the proof of [15,Thm. 3] implies (6.22) for the case where all indices i, j, k, l do not exceed m + n.Similarly, if all the indices exceed m, then (6.22) follows from the corresponding relations obtained in the proof of [21,Thm. 5.14].The remaining cases can be verified by the same inductive arguments as in [15] and [21] with the use of relations (6.23)- (6.25).We also give an alternative proof of those cases by following the argument which was already used in [29] for the case N = 1.
Extend the filtration on E to the subalgebra B of X(osp N |2m ) generated by all elements e (r) i and h where h(2) p is the image of h (2)  p in gr B for p = 1, . . ., m + n.As in the Introduction, we will use the orthogonal basis ε 1 , . . ., ε m+n with the bilinear form defined in (1.1).We will take the family of vectors as a system of positive roots for osp 2n+1|2m .The simple roots are α i = α i i+1 for i = 1, . . ., m+n.
For the case of osp 2n|2m , we will take together with α i i ′ = 2ε i for 1 i m, as a system of positive roots.The simple roots are α i = α i i+1 for i = 1, . . ., m + n − 1 and α m+n = α m+n−1 (m+n) ′ .By the already verified cases of (6.22), the elements ē(r) ij with 1 i < j m + n and m + 1 i < j (m + ′ can be written as consecutive commutators of the simple root vectors ē(s) k .Therefore, starting from (6.26) and using induction on the lengths of the positive roots, we derive the relations h(2) p , ē(r) m m+1 , ē(s) m+1 m+2 ] = 0.
Returning to the general values of N = 2n + 1, we will prove by a reverse induction on j, that for 1 i m and i j m + n we have the relation i m+n+1 (6.29) for all r, s 1.For j = m + n, using (6.28) write where the last relation holds due to (6.24).Now suppose that j < m + n and use the induction hypothesis and (6.28) to write , thus proving (6.29).We can now conclude that (6.27) holds for all positive roots α ij , as implied by (6.28) and (6.29).By a similar argument with the use of (6.25), for N = 2n we derive the following respective counterparts of (6.28) and (6.29): for all 1 i < j m + n − 1 and 1 k m + n − 1, and ē(r+s−1) i reproduce the respective part of the Serre-Chevalley presentation of the Lie superalgebra osp N |2m ; see e.g.[16] and [36,Thm 3.4].Then we proceed by induction on r + s, taking commutators with suitable generators h (2) p and applying (6.27).By applying the anti-automorphism (6.21), we deduce from the spanning property of the ordered monomials in the elements e ji in such a way that the elements of F precede the elements of H, and the latter precede the elements of E, we can conclude that the ordered monomials in these elements span X(osp N |2m ).This proves that (6.13) is an isomorphism.
Let E, F and H denote the subalgebras of X(osp N |2m ) respectively generated by all elements of the form e  Relations (1.2)-(1.7) of the Main Theorem are deduced from Theorem 6.1 in the same way as for the Yangians Y(gl n|m ) in [15] and Y(o N ) in [21].Now we prove relations (1.8) and (1.9) and show that they imply the Serre relations and super Serre relations in the algebra X(osp N |2m ).We use the argument originated in the work of Levendorskiȋ [27,Lemma 1.4].Relations (1.4) and (1.5) imply that, as in [27,Cor. 1.5], for a certain polynomial κ i r in the variables κ i p with p r we have κ i r , ξ ± j s = ± (α i , α j ) ξ ± j r+s + linear combination of ξ ± j r+s−2p with 0 < 2p r.The same argument as in [27] shows that the Serre relations both in Y(osp N |2m ) and X(osp N |2m ) are implied by the Serre relations in the Lie superalgebra osp N |2m via the embedding (2.10), which are particular cases of (1.8) with r 1 = • • • = r k = s = 0.
It was already shown in [34,Remark 2.61] how the super Serre relations in Theorem 6.1 follow from relations (1.9) with the use of the polynomials κ i r .The same argument applies to prove that all relations of the form (1.9) are implied by their particular case with r = s = 0 which holds in osp N |2m .
We thus have an epimorphism from the algebra Y(osp N |2m ) defined in the Main Theorem to the Yangian Y(osp N |2m ), which takes the generators κ i r and ξ ± i r of Y(osp N |2m ) to the elements of Y(osp N |2m ) denoted by the same symbols.On the other hand, use the isomorphism X(osp N |2m ) ∼ = X(osp N |2m ) to define the automorphisms of the form (2.6) on the algebra X(osp N |2m ).The injectivity of the epimorphism Y(osp N |2m ) → Y(osp N |2m ) follows from the observation that Y(osp N |2m ) coincides with the subalgebra of X(osp N |2m ) which consists of the elements stable under all these automorphisms.

2 . 4 . 2 .
ij (u) with ℓ + 1 i, j (ℓ + 1) ′ .The following properties of the Gauss decomposition observed in[21, Sec.4]  extend to the super case in the same form.We use the notation of Corollary 3.Proposition The series t [ℓ] ij (u) coincides with the image of the generator series t ij (u) of the extended Yangian X(osp N |2m−2ℓ ) (for ℓ < m) or the extended Yangian X(o N +2m−2ℓ ) (for ℓ m), under the embedding (3.14),

ij = r − 1 for all r 1 .
Denote by t (r)ij the image of t (r) ij in the (r − 1)-th component of the associated graded algebra gr X(osp N |2m ).Introduce the coefficients c r of the series c(u) defined in (2.5) by c(u) = 1 + ∞ r=1 c r u −r

h−F
C[ζ 1 , ζ 2 , . . .] is the algebra polynomials in variables ζ r understood as the images of the respective central elements cr .Under the isomorphism (6.20), the image of e(r) ij in the (r − 1)-th component of gr X(osp N |2m ) corresponds to F ij x r−1 (−1) ī, while the image of f (r) ji corresponds to F ji x r−1 (−1) .Similarly, the image h(r) i of h (r) i in the (r − 1)-th component of gr X(osp N |2m ) corresponds to F ii x r−1 (−1) ī + ζ r /2 for i = 1, . . ., m + n, whereas 2 m+n m+n x r−1 + ζ r /2 for N = 2n,which follows from Theorem 5.3.Hence the claimed linear independence of ordered monomials in X(osp N |2m ) is implied by the Poincaré-Birkhoff-Witt theorem for U(osp N |2m [x]).
ji taken in some fixed order.Denote by E, F and H the subalgebras of X(osp N |2m ) respectively generated by all elements of the form e an ascending filtration on E by setting deg e (r) i = r − 1. Denote by gr E the corresponding graded algebra.Let ē(r) ij denote the image of the element (−1) ī e (r) ij in the (r − 1)-th component of the graded algebra gr E. Extend the range of subscripts of ē(r) ij to all values 1 i < j 1 ′ by using the skew-symmetry conditions ē(r) i j = −ē (r)

i
by setting deg h (r) i = r − 1. Write (6.3) in terms of the coefficients to get the relation h(2) p , ē

1
i m and i j m + n − 1, where r, s 1.They imply that relation(6.27)holds for all positive roots α ij in the case N = 2n as well.The verification of (6.22) is now completed for all values N 3 in the same way as in[29,  Sec.5.2].Namely, relations (6.22) hold in the case r = s = 1 because the defining relations of the theorem restricted to the generators e(1) (r) ij , that the ordered monomials in the elements f (r) ji span the subalgebra F .It is clear that the ordered monomials in h (r) i span H. Furthermore, by the defining relations of X(osp N |2m ), the multiplication mapF ⊗ H ⊗ E → X(osp N |2m )is surjective.Therefore, ordering the elements h

Corollary 6 . 2 .
i .Consider the generators h (r)i with i = 1, . . ., m + n + 1 and r 1, and e (r) ij and f (r) ji with r 1 and conditions(6.19).Suppose that the elements h ji are ordered in such a way that the elements of F precede the elements of H, and the latter precede the elements of E. The following is a version of the Poincaré-Birkhoff-Witt theorem for the orthosymplectic Yangian.The set of all ordered monomials in the elements h ji , where the indices satisfy conditions(6.19)and the powers of odd generators do not exceed 1, forms a basis of X(osp N |2m ).