Gaussian generators for the Yangian associated with the Lie superalgebra osp (1 | 2 m )

We give a new presentation of the Yangian for the orthosymplectic Lie superalgebra osp 1 | 2 m . It relies on the Gauss decomposition of the generator matrix in the R -matrix presentation. The deﬁning relations between the Gaussian generators are derived from a new version of the Drinfeld-type presentation of the Yangian for osp 1 | 2 and some additional relations in the Yangian for osp 1 | 4 by an application of the embedding theorem for the super-Yangians.


Introduction
The Yangian Y(osp N |2m ) associated with the orthosymplectic Lie superalgebra osp N |2m is a deformation of the universal enveloping algebra U(osp N |2m [u]) in the class of Hopf algebras.The original definition in terms of an R-matrix presentation and basic properties of the Yangian are due to Arnaudon et al. [1].Drinfeld-type presentations of the Yangian Y(osp N |2m ) and extended Yangian X(osp N |2m ) with N 3 were constructed in a recent work [17].Our goal in this paper is to produce similar presentations in the case N = 1 (Theorem 5.4 and Corollary 5.9).
It is well-known that the Yangians associated with simple Lie algebras admit a few presentations which are suitable for different applications in representation theory and mathematical physics.In particular, the Drinfeld presentation originated in [5] was essential for the classification of the finite-dimensional irreducible representations.
Explicit isomorphisms between the R-matrix and Drinfeld presentations of the Yangians associated with the classical Lie algebras were produced in [4] and [12].In the case of the super Yangian for the general linear Lie superalgebra, such an isomorphism between the R-matrix presentation of [18] and a Drinfeld-type presentation of [20] was given in [10]; see also [19] and [21] for generalizations to arbitrary Borel subalgebras.
A key role in the above-mentioned constructions is played by the Gauss decomposition of the generator matrix of the (super) Yangian, which yields a presentation in terms of the Gaussian generators.We use the same approach for the Yangians associated with osp 1|2m in this paper, and our arguments rely on the embedding theorem proved in [17].It allows one to regard the Yangian Y(osp 1|2m−2 ) as a subalgebra of Y(osp 1|2m ), and the same holds for their extended versions.Therefore, a significant part of calculations is reduced to those in the algebras Y(osp 1|2 ) and Y(osp 1|4 ).
A Drinfeld-type presentation of the Yangian Y(osp 1|2 ) was given in [2] with the use of certain Serre-type relations.We give a different version of this presentation involving some additional generators, but avoiding Serre-type relations (Theorem 4.1 and Corollary 4.3).
The finite-dimensional irreducible representations of the algebras X(osp 1|2m ) and Y(osp 1|2m ) were classified in [16].We apply our results to derive the classification theorem in terms of the new presentation of the Yangian Y(osp 1|2m ) (Proposition 5.10).
After we posted the first version of the paper in the arXiv, we were informed by Alexander Tsymbaliuk of the work [7], where similar results will be presented as a part of a more general project involving presentations of the orthosymplectic Yangians associated with arbitrary parity sequences.In particular, some closely related versions of Theorems 4.1 and 5.4 are proved in [7].We are grateful to Alexander for the illuminating discussion of those results and their connection with the work [2]; see Corollary 4.6 below.
The endomorphism algebra End C 1|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 1|2m ⊗ A, so that a square matrix A = [a ij ] of size 2m + 1 is regarded as the element where the entries a ij are assumed to be homogeneous of parity ī +  mod 2. 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 1|2m defined by the anti-diagonal matrix G = [g ij ] with g ij = δ ij ′ θ i .A standard basis of the general linear Lie superalgebra gl 1|2m is formed by elements E ij of the parity ī +  mod 2 for 1 i, j 2m + 1 with the commutation relations We will regard the orthosymplectic Lie superalgebra osp 1|2m associated with the bilinear form defined by G as the subalgebra of gl 1|2m spanned by the elements Introduce the permutation operator P by e ij ⊗ e ji (−1)  ∈ End C 1|2m ⊗ End C 1|2m and set The R-matrix associated with osp 1|2m is the rational function in u given by This is a super-version of the R-matrix originally found in [22].Following [1], we define the extended Yangian X(osp 1|2m ) as a Z 2 -graded algebra with generators t (r) ij of parity ī +  mod 2, where 1 i, j 2m + 1 and r = 1, 2, . . ., satisfying defining relations (2.3) below.Introduce the formal series and combine them into the square matrix T (u) = [t ij (u)]; cf.(2.1).Consider the elements of the tensor product algebra End The defining relations for the algebra X(osp 1|2m ) take the form of the RT T -relation As shown in [1], the products T (u − κ) T t (u) and T t (u) T (u − κ) are scalar matrices with where c(u) is a series in u −1 .All its coefficients belong to the center ZX(osp 1|2m ) of X(osp 1|2m ) and freely generate the center; this can be derived analogously to the Lie algebra case considered in [3].The Yangian Y(osp 1|2m ) is defined as the subalgebra of X(osp 1|2m ) which consists of the elements stable under the automorphisms As in the non-super case [3], we have the tensor product decomposition X(osp 1|2m ) = ZX(osp 1|2m ) ⊗ Y(osp 1|2m ); (2.6) see also [11].The Yangian Y(osp 1|2m ) is isomorphic to the quotient of X(osp 1|2m ) by the relation An explicit form of the defining relations (2.3) can be written in terms of the series (2.2) as follows: In this formula and in what follows, square brackets denote super-commutator for homogeneous elements a and b of parities p(a) and p(b).The assignments define automorphisms of X(osp 1|2m ) [1].We will need their composition (with c = 1) which defines another automorphism The assignment τ : defines an anti-automorphism.The latter property is understood in the sense that for homogeneous elements a and b of the Yangian.Note that the maps σ and τ are not involutive but each of σ 4 and τ 4 is the identity map.The universal enveloping algebra U(osp 1|2m ) can be regarded as a subalgebra of X(osp 1|2m ) via the embedding 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 [11]; cf.[3,Sec. 3].It states that the associated graded algebra for Y(osp 1|2m ) is isomorphic to U(osp 1|2m [u]).The algebra X(osp 1|2m ) is generated by the coefficients of the series c(u) and t ij (u) with the conditions Moreover, given any total ordering on the set of these generators, the ordered monomials with the powers of odd generators not exceeding 1, form a basis of the algebra.The extended Yangian X(osp 1|2m ) is a Hopf algebra with the coproduct defined by For the image of the series c(u) we have ∆ : c(u) → c(u) ⊗ c(u) and so the Yangian Y(osp 1|2m ) inherits the Hopf algebra structure from X(osp 1|2m ).

Gaussian generators
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 [8] Apply the Gauss decomposition to the generator matrix T (u) associated with the extended Yangian X(osp 1|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) .The entries of the matrices F (u), H(u) and E(u) are given by well-known formulas in terms of quasideterminants [9]; see also [15,Sec. 1.11].We have whereas and for 1 i < j 1 ′ .By [17,Lem. 4.1], under the anti-automorphism τ of X(osp 1|2m ) defined in (2.9), for all k and i < j we have Introduce the coefficients of the series defined in (3.2), (3.3) and (3.4) by the expansions Furthermore, set for i = 1, . . ., m.We will also use the coefficients of the series defined by By [17,Prop. 5.1], the Gaussian generators h i (u) satisfy the relations for i = 1, . . ., m. Together with the relation for the central series c(u) defined in (2.4), they imply that the coefficients of all series h i (u) with i = 1, 2, . . ., 1 ′ pairwise commute in X(osp 1|2m ); see [17,Cor. 5.2].We will also recall a formula for c(u) in terms of the Gaussian generators h i (u) with i = 1, . . ., m + 1; see [17,Thm 5.3].We have 4 Drinfeld-type presentations of the Yangians for osp 1|2 We will now suppose that m = 1 and give Drinfeld-type presentations of the algebras X(osp 1|2 ) and Y(osp 1|2 ).Our approach is similar to [2], but we use a different set of generators by adjoining the coefficients of the series e 11 ′ (u) and f 1 ′ 1 (u).This allows us to avoid Serre-type relations used therein.We use notation (3.7) and set e(u) = e 1 (u), f (u) = f 1 (u) and k(u) = k 1 (u).

.2)
Furthermore, ) and ) We also have and Finally, and Proof.As the first step, we will verify that all the above relations hold in the extended Yangian.Relations (4.1) and (4.2) were pointed out in [2] and [16,Sec. 3] along with the identities It is sufficient to verify (4.3), (4.5), (4.7) and (4.9), because the remaining relations will follow by the application of the anti-automorphism τ using (3.5).By (2.7) we have Since h 1 (u) = t 11 (u) and e(v) = t 11 (v) −1 t 12 (v), by multiplying both sides by t 11 (v) −1 from the left we get (4.3).Furthermore, by (3.9) and (3.10) we have There exists a unique power series z(u) in u −1 with coefficients in the center of X(osp 1|2 ) and with the constant term 1, satisfying the relation z(u)z(u + 1/2) = c(u − 1).This implies that h 2 (u) can be expressed by We will use this relation to derive (4.5) from (4.3).By rearranging the latter we get In particular, setting v = u + 1 yields Therefore, we have which implies Since the series z(u) is central, by using (4.13) together with (4.14) and (4.16), we derive the relation which is equivalent to (4.5).Now consider two particular cases of (2.7), and By expanding the super-commutators and eliminating the product t 11 ′ (v) t 11 (u) in the second formula using the first, we come to the relation The right hand side equals Transform it by applying (4.14) to the products e(u)h 1 (v) and e(v)h 1 (u).By taking into account t 11 ′ (u) = h 1 (u)e 11 ′ (u) and multiplying from the left by the inverse of h 1 (u)h 1 (v), we then obtain which is equivalent to (4.7).Finally, to prove (4.9), begin with the following particular case of (2.7), Note its consequence t 12 (u + 1) t 11 ′ (u) = t 11 ′ (u + 1) t 12 (u) which implies Write (4.19) in terms of the Gaussian generators and multiply both sides by h 1 (u Similarly, by multiplying both sides of (4.17) by h 1 (u) −1 h 1 (v) −1 from the left, we obtain Replacing the product e(v)h 1 (u) by (4.14) and rearranging, we come to Substitute this expression into (4.21) and apply (4.14) to the products e(u)h 1 (v) and e(v)h 1 (u).
Multiplying both sides by (u − v)/(u − v + 1), we come to the relation On the other hand, setting v = u + 1 into (4.22),we get Together with (4.15) and (4.20) this yields e(u), e 11 ′ (u) = −2e(u) 3 .(4.23) By using this identity we can simplify the above formula for the super-commutator to ) to get another identity Its use brings the above relation for [e(u), e 11 ′ (v)] to the required form (4.9).Since all relations in the formulation of the theorem hold in the extended Yangian, we have a homomorphism where X(osp 1|2 ) denotes the algebra whose (abstract) generators are the coefficients of series given by the same expansions as in (3.6) and (3.8), with the relations as in the statement of the theorem (omitting the subscripts of e 12 and f 21 ).The homomorphism (4.25) takes the generators to the elements of X(osp 1|2 ) with the same name.We will show that this homomorphism is surjective and injective.The surjectivity is clear from the Gauss decomposition (3.1), formulas (4.11) and the first relation in (4.12).Now we prove the injectivity of the homomorphism (4.25).The same application of the Poincaré-Birkhoff-Witt theorem for the algebra X(osp 1|2 ) as in [17,Sec. 6] shows that the set of monomials in the generators h 2 , e (r) , f (r) , e (r) 11 ′ and f (r) 1 ′ 1 with r 1 taken in some fixed order, with the powers of odd generators not exceeding 1, is linearly independent in the extended Yangian X(osp 1|2 ).Therefore, to complete the proof of the theorem, it is sufficient to verify that the monomials in the generators h 2 , e (r) , f (r) , e (r) 11 ′ and f (r) 1 ′ 1 with r 1 of the algebra X(osp 1|2 ), taken in a certain fixed order, span the algebra.
Define the ascending filtration on the algebra X(osp 1|2 ) by setting the degree of each generator with the superscript r to be equal to r − 1.We will use the bar symbol to denote the image of each generator in the (r − 1)-th component of the graded algebra gr X(osp 1|2 ).The defining relations of X(osp 1|2 ) imply the corresponding relations for these images in the graded algebra, which are easily derived with the use of the expansion formula Namely, relations (4.1) -(4.6) imply while relations (4.7) -(4.10) give This determines all super-commutator relations between the generators of gr X(osp 1|2 ).In particular, we have ).
The spanning property of the ordered monomials now follows from the observation that the super-commutator relations coincide with those in the polynomial current Lie superalgebra a where a is the centrally extended Lie superalgebra osp 1|2 .This completes the proof of the theorem.
The following is a version of the Poincaré-Birkhoff-Witt theorem for the orthosymplectic Yangian which was established in the proof of Theorem 4.1.
By the definition of the Gaussian generators, the coefficients of all series k(u), e(u), f (u), e 11 ′ (u) and f 1 ′ 1 (u) are stable under the action of all automorphisms (2.5) and so they belong to the subalgebra Y(osp 1|2 ) of X(osp 1|2 ).We now derive a Drinfeld-type presentation of the Yangian Y(osp 1|2 ).
Proof.Relation (4.27) follows from (4.1), so we only need to verify (4.28), because (4.29) will then follow by the application of the anti-automorphism τ via (3.5).Since and so Now apply (4.3) and (4.5) to the super-commutators on the right hand side to get The derivation of (4.28) is completed by the application of the following consequence of (4.16), It is clear from the decomposition (2.6) (with m = 1) that the coefficients of the series k(u), e(u), f (u), e 11 ′ (u) and f 1 ′ 1 (u) generate the subalgebra Y(osp 1|2 ) of the extended Yangian X(osp 1|2 ); cf.[12,Prop. 6.1].Therefore, we have an epimorphism from the (abstract) algebra Y(osp 1|2 ) defined by the generators and relations as in the statement of the corollary, to the Yangian Y(osp 1|2 ), which takes the generators to the elements of Y(osp 1|2 ) denoted by the same symbols.Given any series ϕ(u) consider the automorphism of the algebra X(osp 1|2 ) introduced in the proof of Theorem 4.1, defined by and which leaves all the remaining generators fixed; cf.(2.5).Then Y(osp 1|2 ) coincides with the subalgebra of X(osp 1|2 ) which consists of the elements stable under all these automorphisms.Therefore, the epimorphism Y(osp 1|2 ) → Y(osp 1|2 ) can be regarded as the restriction of the isomorphism X(osp 1|2 ) → X(osp 1|2 ), and hence is injective.
By taking the coefficients of v 0 on both sides of (4.18), and applying (3.5), we get e 11 ′ (u) = −e(u) 2 − [e (1) , e(u)] and Therefore, the coefficients of the series e 11 ′ (u) and f 1 ′ 1 (u) can be eliminated from the Yangian defining relations.In other words, we may regard the Yangian Y(osp 1|2 ) as the algebra with generators k (r) , e (r) and f (r) subject to the relations of Corollary 4.3, where all occurrences of e 11 ′ (u) and f 1 ′ 1 (u) are replaced by (4.30).This was the viewpoint taken in [2], where a different presentation of Y(osp 1|2 ) was given with the use of certain Serre-type relations.
To make a more explicit connection with the presentation of the Yangian Y(osp 1|2 ) given in [2, Theorem 3.1], we will use the automorphism σ defined in (2.8).Observe that the subalgebra Y(osp 1|2 ) of X(osp 1|2 ) is stable under σ.We will keep the same notation for the restriction of σ to Y(osp 1|2 ).
Lemma 4.5.The images of the generators of the algebra Y(osp 1|2 ) under the automorphism σ are given by Proof.Since e(u) = t 11 (u) −1 t 12 (u), we find where we used the relation t 11 (u) t 21 (u + 1) = t 21 (u) t 11 (u + 1) implied by (2.7).Similarly, To calculate the image of k(u), first find the image of the series c(u) defined in (2.4).By taking the (1, 1)-entry of the first matrix product in (2.4), we get Hence, the image of c(u) under the map σ equals Therefore, σ : c(u) → c(−u − 5/2) which follows by taking the (1 ′ , 1 ′ )-entry of the second matrix product in (2.4).We can now find the image of the series h 2 (u) by using (4.13).Since c(u) = z(u + 1)z(u + 3/2) and the series z(u) is uniquely determined by this relation, we derive that σ : z(u) → z(−u) and so This implies that σ : k(u) → k(−u).
Corollary 4.6.The Yangian Y(osp 1|2 ) is generated by the coefficients of the series k(u), e(u) and f (u) subject only to relations (4.2), (4.27) and (4.28) together with and the Serre-type relations e(u) 3 = e(u) [e(u), e (1) ] + [e (1) 2 , e(u)], (4.34)We thus have an epimorphism where Y(osp 1|2 ) denotes the algebra whose (abstract) generators are the coefficients of series k(u), e(u) and f (u) with the relations as in the statement of the corollary.The epimorphism (4.36) takes the generators to the elements of Y(osp 1|2 ) with the same name.We only need to show that it is injective.Introduce the series e 11 ′ (u) and f 1 ′ 1 (u) with coefficients in the algebra Y(osp 1|2 ) by formulas (4.30) and proceed as in the proof of Theorem 4.1.It is sufficient to show that the monomials in the generators k (r) , e (r) , f (r) , e (r) 11 ′ and f (r) 1 ′ 1 with r 1 of the algebra Y(osp 1|2 ), taken in a certain fixed order, span the algebra.
Define the ascending filtration on the algebra Y(osp 1|2 ) by setting the degree of each generator with the superscript r to be equal to r − 1 and use the bar symbol to denote the image of each generator in the (r − 1)-th component of the associated graded algebra gr Y(osp 1|2 ).
The generators of Y(osp 1|2 ) used in Corollary 4.6 and in [2, Theorem 3.1] are related as follows.The series e(u) is the same, k(u) corresponds to h(u) in [2] and our f (u) corresponds to −f (u) in [2].The different-looking relations (3.3) and (3.4) in [2] are in fact equivalent to (4.28) and (4.31), respectively.Indeed, to outline the calculation, write (3.3) in our notation and rearrange to get k(u), e (1) .
Take the residue at u − v = 1/2 to derive the relation (1)   and use it to replace the commutator [k(u), e(u)] in the previous formula.Now take the residue at u − v = −1 in the resulting expression to get k(u), e (1) = 2k(u)e(u + 1) − k(u)e(u − 1/2). The (4.37) Coproduct formulas in the Hopf algebra Y(osp 1|2 ) were derived in [2].They can be re-written in terms of the presentation given in Corollary 4.3 as the next proposition shows.Proposition 4.7.For the images of the generator series under the coproduct map we have and Proof.The argument is the same as in [2]: we write the generator series as e(u) = t 11 (u) −1 t 12 (u) and f (u) = t 21 (u) t 11 (u) −1 and apply definition (2.11).To give more details for the first formula, write ∆(e(u)) as which equals As we observed in the proof of Lemma 4.5, t 11 (u) −1 t 21 (u) = f (u + 1) which implies the relation h 1 (u)f (u + 1) = f (u)h 1 (u).Moreover, relation (4.4) implies [h 1 (u), f (1) ] = f (u)h 1 (u).Hence, (4.30) yields Therefore, Furthermore, by Gauss decomposition, Finally, use the Gauss decomposition again to write and recall that f 1 ′ 2 (u) = f (u − 1/2) by (4.11).Now re-arrange (4.4) to bring it to the form The required formula for ∆(e(u)) follows by expressing this image in terms of the generators used in Corollary 4.3.
The image of the series k(u) under the coproduct ∆ can be found by using the relations k(u) = [e (1) , f (u)] = [e(u), f (1) ] implied by (4.2), although its explicit expression has a rather complicated form.
Proof.By inverting the matrices on both sides of (3.1), we get On the other hand, relation (2.4) implies T t (u + κ) = c(u + κ)T (u) −1 .Hence, by calculating the entries of the matrix E(u) −1 and equating the (i, 1 ′ ) entries with i = 2, 3, 4 in this matrix relation, we derive and Observe that relation (4.3) holds in the same form in X(osp 1|4 ), when e(u) is replaced with e 12 (u), e 13 (u) or e 12 ′ (u), thus implying h 1 (u)e(u) = e(u + 1)h 1 (u).Furthermore, Proof.The first relation is immediate from Propositions 5.1 and 5.2, while the second follows from the first by commuting both sides with h 1 (u).Here we rely on [17,Cor. 3.3] implying that h 1 (u) commutes with each of the series e 22 ′ (v) and e 23 (v), and use the commutation relation 12 ] = −h 1 (u)e 12 (u) which follows from (2.7).We point out a consequence of the second relation to be used below.By taking the coefficients of both sides at v −1 , we get e 22 ′ , e 12 (u) = 2e 12 ′ (u). (5.1)
We thus have a homomorphism where X(osp 1|2m ) denotes the (abstract) algebra with generators and relations as in the statement of the theorem and the homomorphism takes the generators to the elements of X(osp 1|2m ) with the same name.We will show that this homomorphism is surjective and injective.
To prove the surjectivity, note that by (2.7), for 1 i < j m, while t (1) for 1 i j m.Relations (5.19), (5.20) and their counterparts obtained by the application of the anti-automorphism (2.9) together with the Poincaré-Birkhoff-Witt theorem for the extended Yangian X(osp 1|2m ) imply that this algebra is generated by the coefficients of the series t ij (u) with 1 i, j m + 1.Hence, due to the Gauss decomposition (3.1), the algebra X(osp 1|2m ) is generated by the coefficients of the series h i (u) for i = 1, . . ., m + 1 together with e ij (u) and f ji (u) for 1 i < j m + 1. Write (5.19) and (5.20) in terms of the Gaussian generators (cf. [12, Sec.5]) to get and e (1) for 1 i < j m, and for i = 1, . . ., m.These relations together with their counterparts for the coefficients of the series f ji (u), which are obtained by applying the anti-automorphism τ via (3.5), show that the coefficients of the series h i (u) for i = 1, . . ., m + 1 and e i (u), f i (u) for i = 1, . . ., m generate the algebra X(osp 1|2m ) thus proving that the homomorphism (5.18) is surjective.Now we turn to proving the injectivity of the homomorphism (5.18).It was shown in the proof of [17,Thm 6.1] that the set of monomials in the generators h taken in some fixed order with the powers of odd generators not exceeding 1, is linearly independent in the extended Yangian X(osp 1|2m ).Furthermore, working now in the algebra X(osp 1|2m ), introduce its elements inductively, as the coefficients of the series e ij (u) for i and j satisfying (5.23) by setting e (r) i i+1 = e (r) i for i = 1, . . ., m and using relations (5.21) and (5.22).The defining relations show that the map and τ : h i (u) → h i (u) for i = 1, . . ., m + 1, defines an anti-automorphism of the algebra X(osp 1|2m ).(We use the same symbol as in (3.5), but this should not cause a confusion since it is used for a differently defined algebra.)Apply this map to the relations defining e ij (u) and use (3.5) to get the definition of the coefficients of the series f ji (u) subject to the same conditions (5.23).Since the images of the elements h ji of the algebra X(osp 1|2m ) under the homomorphism (5.18) coincide with the elements of the extended Yangian X(osp 1|2m ) denoted by the same symbols, the injectivity of the homomorphism (5.18) will be proved by showing that the algebra X(osp 1|2m ) is spanned by monomials in these elements taken in some fixed order.
Extend the filtration on E to the subalgebra B of X(osp 1|2m ) generated by all elements e (r) i and h where h(2) p is the image of h (2)  p in gr B. Lemma 5.5.For all r, s 1 in the algebra gr B we have for 1 i < j m. (5.29) Moreover, for all p = 1, . . ., m we also have for 1 i < j m + 1. (5.30) Proof.Relation (5.12) implies ē(r+1) j−1 j , ē(s) j j+1 = ē(r) j−1 j , ē(s+1) j j+1 (5.31) for all r, s 1.This yields (5.29) for i = j − 1. Continue by induction on j − i (which is the length of the root α i j ) and suppose that j − i 2. Then by (5.26), (5.32) Observe that the commutator [ē (r) ] is zero.Indeed, by the first relation in (5.21), each element e (r) i j−1 ∈ E is a commutator of certain coefficients of the series e i (u), . . ., e j−2 (u).However, the commutator of each of these series with e j (u) is zero by the Serre relations (5.16).Hence, using (5.31), we can write the commutator in (5.32) as Apply the Jacobi identity to this commutator.By the induction hypothesis and (5.26), this equals , as required, completing the proof of (5.29).
To verify (5.30), use induction on j − i with (5.28) as the induction base.For j − i 2 write By the induction hypothesis and (5.29), this equals , where we also used the root relation α i j−1 + α j−1 j = α i j ; see the notation introduced in the beginning of Sec.5.2.
Lemma 5.6.For all r, s 1 and 1 i j m in the algebra gr B we have (5.33) Moreover, for all p = 1, . . ., m we also have (5.34) Proof.We will be proving both relations simultaneously by reverse induction on j starting with j = m (and then an inner induction on i).In this case, relation (5.33) with i = m holds due to (5.10), while using (5.28) with j = m we then derive (5.34).Now take i = m − 1. Relation (5.15) along with (5.33) where we also used (5.26) and (5.27).Hence, by Lemma 5.5 the left hand side of (5.33) can be written as which coincides with ē(r+s−1) m−1 m ′ , as required.Relation (5.34) in the case i = m − 1 and j = m follows by the same calculation as in the proof of Lemma 5.5 with the use of the root relation Continue by reverse induction on i and suppose that i < m − 1. Invoking Lemma 5.5 again and using the induction hypothesis, we get i m ′ by (5.27).This proves (5.33) in the case under consideration; relation (5.34) then also follows.
As a final step, continue by reverse induction on j and suppose that 1 i j < m.By (5.27) we have ē s) j j+1 ] = 0.This relation for r = s = 1 holds as a particular case of (5.25).For arbitrary r, s 1 the relation follows by taking repeated commutators with h(2) p for suitable values of p by using (5.28) and (5.34); it suffices to take p = i and p = j + 1. Hence by Lemma 5.5, where the last equality holds by (5.27), while the second last equality is valid by the induction hypothesis.This proves (5.33), while (5.34) then follows by the same argument as in the proof of Lemma 5.5.
We will now complete the verification of (5.25).Lemmas 5.5 and 5.6 imply the commutation relations for all positive roots α i j .Then the commutator of h(2) p with the left hand side of (5.25) equals k l .
First consider fixed parameters i < j i ′ and k < l k ′ satisfying the following condition: there exist two different values p = a and p = b such that In this case, starting with (5.25) for r = s = 1 and taking repeated commutators of both sides with h(2) b we derive the required relations for the super-commutators by solving the arising system of two linear equations.For instance, starting from [ē with 1 i < j m, we can take p = i and p = j to use the induction step by solving the system of equations ē(r+1) i j ′ = 0.
Consider now the remaining cases, where the above condition on the determinant cannot be satisfied.To verify that [ē (r) i j , ē(s) i j ] = 0 for 1 i < j m, note first that for j = i + 1 this follows from (5.8).Furthermore, if i < k < j for some k, then by the previously verified cases of (5.25), we have ē(r) i j , ē(s) k j ] = 0, as required.For the next case (j = m + 1), observe that by (5.10) Hence, for i < m we have , thus verifying this case.Finally, for 1 i j m we have completing the verification of (5.25).By applying the anti-automorphism (5.24), 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 with the powers of odd generators not exceeding 1, span X(osp 1|2m ).This proves that (5.18) is an isomorphism.
Let E, F and H denote the subalgebras of X(osp 1|2m ) respectively generated by all elements of the form e ji with r 1 with the powers of odd elements not exceeding 1 and satisfying conditions (5.23), forms a basis of the algebra X(osp 1|2m ).
We will now apply Theorem 5.4 to deduce a Drinfeld-type presentation for the Yangian Y(osp 1|2m ).By making use of the series (3.7), introduce the elements κ i r , ξ ± i r and ξ ± r of the algebra X(osp 1|2m ) as the coefficients of the series Since these series are fixed by all automorphisms (2.5), their coefficients belong to the subalgebra Y(osp 1|2m ) of the extended Yangian X(osp 1|2m ).We will use the abbreviation {a, b} = ab + ba.
Proof.The relations are deduced from Theorem 5.4 by the arguments similar to those in [4]; see also [15,Sec. 3.1].In particular, (5.37) and (5.38) are essentially the Yangian relations of type A, while (5.39) -(5.43) follow from Corollary 4.3 via the embedding theorem.To illustrate, we will derive (5.38) with j = i + 1 for ξ − i (u) from the corresponding case of (5.12).We can write the latter in the form to apply (5.47) to the first summand on the right hand side.After expanding the commutators and anti-commutators in the resulting expression, we conclude that it holds due to relation (5.48).The decomposition (2.6) and formula (3.11) imply that the coefficients of the series generate the Yangian Y(osp 1|2m ).The completeness of the relations is verified by using the automorphisms of the form (2.5) on the abstract algebra with the presentation given in the statement of the corollary as with the case m = 1; see the proof of Corollary 4.3.

i
with i = 1, . . ., m + 1 and r 1, and e (r) ij and f (r) ji with r 1 and the conditions i < j i ′ for i = 1, . . ., m, ( i .Define an ascending filtration on E by setting deg e (r) i = r − 1 and denote by gr E the corresponding associated graded algebra.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 .25) where ē(r) ij denotes the image of the element (−1) ī e (r) ij in the (r −1)-th component of gr E and we extend the range of subscripts of ē(r) ij to all values i < j 1 ′ by using the skew-symmetry conditions ē(r) i j = −ē (r) j ′ i ′ (−1) ī+ī θ i θ j .First observe that relations (5.25) hold in the case r = s = 1 because the defining relations of the theorem restricted to the generators e (1) i with i = 1, . . ., m reproduce the respective part of the Serre-Chevalley presentation of the Lie superalgebra osp 1|2m ; see e.g.[6, Sec.2.44].Furthermore, the definitions (5.21) and (5.22) of the elements e (r) ij ∈ E imply the relations in the graded algebra gr E:

i
by setting deg h (r) i = r − 1.Hence, in the associated graded algebra gr B we have h (r) ij , that the ordered monomials in the elements f (r) ji with the powers of odd generators not exceeding 1, 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 1|2m ), the multiplication mapF ⊗ H ⊗ E → X(osp 1|2m )is surjective.Therefore, ordering the elements h

Corollary 5 . 8 .
i .Consider the generators h (r) i with i = 1, . . ., m + 1 and r 1, and e (r) ij and f (r) ji with r 1 and conditions (5.23).Suppose that the elements h (r) i , e (r) ij and f (r) 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 (r) i with i = 1, . . ., m + 1, and the elements e
It is clear from the proof of Corollary 4.6 that its version, where the counterparts of relations (4.28), (4.32) and (4.34) involving the series f (u) are derived by using the anti-automorphism τ instead of the automorphism σ, is also valid.In that version, relation (4.31) is replaced by (4.29), relation (4.33) is unchanged, whereas the Serre-type relation (4.35) is replaced by use of this relation brings[2, (3.3)] to the form (4.28).The equivalence of[2, (3.4)] and (4.31) now follows by applying the automorphism σ.