Drinfeld J Presentation of Twisted Yangians

We present a quantization of a Lie coideal structure for twisted half-loop algebras of finite-dimensional simple complex Lie algebras. We obtain algebra closure relations of twisted Yangians in Drinfeld J presentation for all symmetric pairs of simple Lie algebras and for simple twisted even half-loop Lie algebras. We provide the explicit form of the closure relations for twisted Yangians in Drinfeld J presentation for the ${\mathfrak{sl}}_3$ Lie algebra.


Introduction
Yangian Y(g) is a flat quantization of the half-loop Lie algebra L + ∼ = g[u] of a finite dimensional simple complex Lie algebra g [Dri 85]. The name Yangian is due to V. G. Drinfel'd to honour C. N. Yang who found the simplest solution of the Yang-Baxter equation, the rational R matrix [Yan 67] (see also [Bax 72,Bax 82]). This R matrix and the Yang-Baxter equation were discovered in the studies of the exactly solvable two dimensional statistical models and quantum integrable systems. One of the most important result was the quantization of the inverse scattering method by Leningrad's school [FST 79] that lead to the formulation of quantum groups in the so-called RTT formalism [FRT 89]. These quantum groups are deformations of semi-simple Lie algebras and are closely associated to quantum integrable systems. In particular, the representation theory of the Yangian Y(sl 2 ), which is one of the simplest examples of an infinite dimensional quantum groups, is used to solve the rational 6-vertex statistical model [Bax 82], the XXX Heisenberg spin chain [FadTak 81], the principal chiral field model with the SU (2) symmetry group [FadRes 86, Mac 04].
The mathematical formalism for quantum groups and for quantization of Lie bi-algebras was presented by Drinfel'd in his seminal work [Dri 85] (see also [Dri 87]). Drinfel'd gave a quantization procedure for the universal enveloping algebra U(g) for any semi-simple Lie algebrag. 1 The quantization is based on the Lie bi-algebra structure ong given by a skew symmetric map δ :g →g ∧g, the cocommutator. A quantization of (g, δ) is a (topological) Hopf algebra (U (g), ∆ ), such that U (g)/ U (g) ∼ = U(g) as a Hopf algebra and where σ • (a ⊗ b) = b ⊗ a and X is any lifting of x ∈g to U (g). The Lie bi-algebra structure ong can be constructed from the Manin triple (g,g + ,g − ), whereg ± are isotropic subalgebras ofg such thatg + ⊕g − =g as a vector space andg − ∼ =g * + , the dual ofg + . Then the commutation relations of the quantum group can be obtained by requiring ∆ to be a homomorphism of algebras U (g) → U (g) ⊗ U (g). The question of the existence of such quantization for any Lie bi-algebra was raised by Drinfeld in [Dri 92] and was answered by P. Etingof and D. Kazhdan in [EtiKaz 96]. They proved that any finite or infinite dimensional Lie bi-algebra admits a quantization. Here we will consider only the Yangian case, U (g) = Y(g) withg = L + . We will use the so-called Drinfel'd basis approach which is very convenient to approach the quantization problem.
In physics, quantum groups are related to unbounded quantum integrable models and their extensions to models with boundaries. The underlying symmetry of the models with boundaries is given by coideal subalgebras of quantum groups that were introduced in the context of 1+1 dimensional quantum field theories on a half-line by Cherednick [Che 84] and in the context of one dimensional spin chains with boundaries by Sklyanin [Skl 88] in the so-called reflection algebra formalism. Mathematical aspects of reflection algebras in the RT T formalism, called twisted Yangians, were first considered by G. Olshasnkii in [Ols 90] and were models with boundaries in [BasBel 11, BelFom 12]. Twisted Yangians in Drinfel'd basis can also be used to solve the spectral problem of a semi-infinite XXX spin chain for an arbitrary simple Lie algebra using the 'Onsager method' [BasBel 10]. We also remark, that twisted Yangians of this type were shown to play an important role in quantum integrable systems for which the RT T presentation of the underlying symmetries is not known, for example in the AdS/CFT correspondence [MacReg 10, MacReg 11].
The paper is organized as follows: in section 2 we recall basic definitions of simple complex Lie algebras and define the symmetric pair decomposition with respect to involution θ. Then, in section 3, we recall definitions of a half-loop Lie algebra L + of g given in the Drinfel'd basis, and introduce the Drinfel'd basis of a twisted half-loop Lie algebra H + with respect to the symmetric pair decomposition of L + . In section 4 we construct the Lie bi-algebra structure on L + and Lie bi-ideal structure on H + that provide the necessary data to achieve the quantification presented in section 5. The special case g = sl 3 is fully considered in section 6, where we have presented all the corresponding twisted Yangians. Section 7 contains the proofs which were omitted in the main part of the paper due to their length and for the convenience of the reader.
Acknowledgements: The authors would like to thank P. Baseilhac, N. Crampé, N. Guay, N. Mackay and J. Ohayon for discussions and their interest for this work. V.R. acknowledges the UK EPSRC for funding under grant EP/K031805/1.

Definitions and preliminaries
2.1. Lie algebra. Consider a finite dimensional complex simple Lie algebra g of dimension dim(g) = n, with a basis {x a } given by (2.1) [x a , x b ] = α c ab x c , α c ab + α c ba = 0, α c ab α e dc + α c da α e bc + α c bd α e ac = 0. Here α c ab are the structure constants of g and the Einstein summation rule of the dummy indices is assumed. We will further always assume g to be simple. Let η ab denote the non-degenerate invariant bilinear (Cartan-Killing) form of g in the {x a } basis (2.2) (x a , x b ) g = η ab = α d ac α c bd , that can be used to lower indices {a, b, c, . . .} of the structure constants α d ab η dc = α abc with α abc + α acb = 0.
The inverse of η ab is given by η ab and satisfies η ab η bc = δ c a . Set {a, b} = 1 2 (ab + ba). Let C g = η ab {x a , x b } denote the second order Casimir operator of g and let c g be its eigenvalue in the adjoint representation. For a simple Lie algebra it is non-zero and is given by The elements α bc a satisfy the co-Jacobbi identity, which is obtained by raising one of the lower indices of the Jacobi identity in (2.1). Moreover, contracting α bc a with the Lie commutator in (2.1) gives 2.2. Symmetric pair decomposition. Let θ be an involution of g. Then g can be decomposed into the positive and negative eigenspaces of θ, i.e. g = h ⊕ m with θ(h) = h and θ(m) = −m, here dim(h) = h, dim(m) = m satifying h + m = n. Numbers h and m correspond to the number of positive and negative eigenvalues of θ. This decomposition leads to the symmetric pair relations From the classification of the symmetric pairs for simple complex Lie algebras it follows that the invariant subalgebra h is a (semi)simple Lie algebra which can be decomposed into a direct sum of two simple complex Lie algebras a and b, and a one dimensional centre k at most. We write h = a ⊕ b ⊕ k (see e.g. section 5 in [Hel 78]). Set dim(a) = a and dim(b) = b. Let the elements with i = 1 . . . a, i ′ = 1 . . . b and p = 1 . . . m, (2.5) be a basis of g such that θ(X α ) = X α for any α ∈ {i, i ′ , z}, and θ(Y p ) = −Y p . We will further use indices i(j, k, ...) for elements X α ∈ a, primed indices i ′ (j ′ , k ′ , ...) for elements X α ∈ b, index α = z for the central element X α ∈ k, and indices p(q, r, . . .) for Y p ∈ m, when needed. We will denote the commutators in this basis as follows: The structure constants above are obtained from the ones of g by restricting to the appropriate elements.
Here and further we will use the sum symbol α to denote the summation over all simple subalgebras of h. The Einstein summation rule for the Greek indices will be used in cases when the sum is over a single simple subalgabera of h. The notation α = γ means that indices α and γ correspond to different subalgebras of h. The structure constants given above satisfy the (anti-)symmetry relations f γ αβ + f γ βα = 0, g q µp + g q pµ = 0, w µ pq + w µ qp = 0, and the homogeneous and mixed Jacobi identities , w β pq f µ αβ + g r αp w µ qr − g r αq w µ pr = 0, with α(β, γ, ...) = i(j, k, ...) ∈ a or α(β, γ, ...) = i ′ (j ′ , k ′ , ...) ∈ b and α (w α pq g s rα + w α qr g s pα + w α rp g s qα ) = 0, g r pα w β qr − g r qα w β pr = 0 for α = β. (2.8) We will further refer to the set {X α , Y p } = {X i , X i ′ , X z , Y p } given by (2.5) and satisfying relations (2.6-2.8) as to the symmetric space basis for a given Lie algebra g and involution θ.
The Killing form of g has a block diagonal decomposition with respect to the symmetric space basis, namely with the remaining entries being trivial. The Casimir element C g in this basis decomposes as The decomposition of the inverse Killing form can be used to raise the indices of the structure constants. We set f βγ α = κ βµ f γ αµ , g qν p = κ νρ g q ρp , w pq ν = (κ m ) pr g q νr . with α(β, γ, µ...) = i(j, k, ...) or i ′ (j ′ , k ′ , ...) and ν(ρ) = i(j) or i ′ (j ′ ) or z(z). Consider the commutation relations. For the generator Y p we have The remaining commutation relations are trivial. Let c a , c b , c z and c m be the eigenvalues of C, C ′ , C z and C Y respectively. We have c g = c a + c b + c m + c z . Using (2.4) we find 3. Symmetric spaces and simple half-loop Lie algebras 3.1. Half-loop Lie algebra. Consider a half-loop Lie algebra L + generated by elements {x This algebra can be identified with the set of polynomial maps f : C → g using the Lie algebra isomorphism The half-loop Lie algebra has another basis conveniently called the Drinfel'd basis: Proposition 3.1. The half-loop Lie algebra L + admits a Drinfel'd basis generated by elements {x a , J( for any µ, ν ∈ C. In the rank(g) = 1 case the level-2 loop terrific relation 3.3 becomes trivial and for the rank(g) ≥ 2 case the level-3 loop terrific relation 3.4 follows from level-2 loop terrific relation. The isomorphism with the standard loop basis is given by the map A proof is given in section 7.1.

3.2.
Twisted half-loop Lie algebra. Let us extend the involution θ of g to the whole of L + as follows: The twisted half-loop Lie algebra H + ∼ = g[u] θ is a fixed-point subalgebra of L + generated by the elements stable under the action of the (extended) involution θ, namely H + = {x ∈ L + | θ(x) = x}. In physics literature it is ofted referred to as the twisted current algebra.
Consider the symmetric space basis of g. We write the half-loop Lie algebra L + in terms of the elements {X for all k, l ≥ 0 and α = λ. The action of θ on this basis is given by θ(X  The twisted half-loop Lie algebra can be defined in terms of the Drinfel'd basis: Proposition 3.2. Let rank(g) ≥ 2. Then the twisted half-loop Lie algebra admits a Drinfel'd basis generated by elements {X α , B(Y p )} satisfying The isomorphism with the standard twisted half-loop basis is given by the map A proof is given in section 7.1. Note that in the contrast to L + , the twisted algebra H + for rank(g) ≥ 2 has level-2 and level-3 higher-order defining relations, which we call horrific relations. This is due to the fact that even and odd levels of H + are not equivalent. The rank(g) = 1 case is exceptional. The Drinfel'd presentation in this case has level-4 relation instead (see [BelCra 12], section 4.2).
Proposition 3.3. Let rank(g) ≥ 2. Then the even half-loop Lie algebra admits a Drinfel'd basis generated by elements for any µ, ν ∈ C. The isomorphism with the standard half-loop basis is given by the map i . The proof follows directly by Proposition 3.1, since g[u 2 ] ∼ = g[u] as Lie algebra.
4. Lie bi-algebras and bi-ideals 4.1. Lie bi-algebra structure of a half-loop Lie algebra. A Lie bi-algebra structure on L + is a skewsymmetric linear map δ : L + → L + ⊗ L + , the cocommutator, such that δ * is a Lie bracket and δ is a 1-cocycle, δ([x, y]) = x.δ(y) − y.δ(x), where dot denotes the adjoint action on L + ⊗ L + . The cocommutator is given for the elements in the Drinfel'd basis of L + by This cocommutator can be constructed from the Manin triple Lie algebra (L, L + , L − ), with L = g((u −1 )) the loop algebra generated by elements {x (n) } with x ∈ g, n ∈ Z and defining relations (3.1) (but with n, m ∈ Z), L + = g  . A Manin triple is a triple of Lie bi-algebras (g,g + ,g − ) together with a nondegenerate symmetric bilinear form ( , )g ong invariant under the adjoint action ofg: •g + andg − are Lie subalgebras ofg; •g =g + ⊕g − as a vector space; • ( , )g is isotopic forg ± (i.e. (g ± ,g ± ) L = 0); Here the bilinear form is given by (x (k) , y (l) ) L = (x, y) g δ k+l+1,0 .
Remark 4.1. If (g,g + ,g − ) is a Manin triple for dim(g + ) = ∞, then (g + ) * ∼ =ḡ−, whereḡ − is the completion ofg − . However in our case (L + ) * ∼ = L − , as it is easy to see: Here in the second equality we have used the identity where V i denotes a finite dimensional vector space; an equivalent identity is used in the last equality.
The cocomutator is obtained using the duality between L + and L − . Recall that δ * : L − ⊗ L − → L − is the Lie bracket of L − . We can deduce the cocommutator δ of L + from the duality relation The cocommutator of the level zero generators x Let us consider an ansatz δ(J(x a )) = v lk a x k ⊗ x l for some v lk a . Then we must have v lk a α blk = c g η ab . It follows from (2.2) that δ(J(x a )) = α lk a x k ⊗ x l = [x a ⊗ 1, Ω g ]. 4.2. Lie bi-ideal structure of twisted half-loop algebras. The Lie bi-ideal structure of twisted halfloop algebras is constructed by employing the anti-invariant Manin triple twist. Here we will consider the left Lie bi-ideal structure. The right Lie bi-ideal is obtained in a similar way.
Definition 4.2 ([BelCra 12]). The anti-invariant Manin triple twist φ of (L, L + , L − ) is an automorphism of L satisfying: • φ is an involution; for all k ∈ Z). The twist φ gives the symmetric pair decomposition of the Manin triple (L, L + , L − ): Then we must have (H ± ) * ∼ = M ∓ . This is easy to check: This decomposition of the Manin triple allows us to construct the Lie bi-ideal structure on H + .
for all x ∈ H − and y ∈ M − .
We are now ready to define the Lie bi-ideal structure for (L + , H + ).
Proposition 4.1. The Lie bi-ideal structure of (L + , H + ), τ : H + → M + ⊗ H + is given by Proof. The construction of the Lie bi-ideal structure τ from the anti-invariant Manin triple twist is similar to the one of the Lie bi-algebra structure from the Manin triple. We have to consider the duality relation Consider the case θ = id. For the level zero generators X This follows by similar arguments as for the level zero generators of the half-loop Lie algebra. Hence we have τ (X α ) = 0.
For the level one generators Y ] and the duality relation we obtain

Let us consider an ansatz
The Lie bi-ideal structure for the θ = id case follows from the pairing (G(x a ), x For completeness we give a remark which was stated by one of the authors in [BelCra 12].
Remark 4.2. The notion of left (right) Lie bi-ideal is related to the notion of co-isotropic subalgebra h of a Lie bi-algebra (g, δ). It is a Lie subalgebra which is also a Lie coideal, meaning that

Quantization
To obtain a quantization of Lie bi-ideal we need to introduce some additional algebraic structures. Recall the definition of a bi-algebra and of a Hopf algebra. A bi-algebra is a quintuple (A, µ, ı, ∆, ε) such that (A, µ, ı) is an algebra and (A, ∆, ε) is a coalgebra; here A is a C-module, µ : A ⊗ A → A is the multiplication, ∆ : A → A ⊗ A is the comultiplication (coproduct), ı : C → A is the unit and ε : A → C is the counit. A Hopf algebra is a bi-algebra with an antipode S : A → A, an antiautomorphism of algebra. (1) B is a sumbodule of A, i.e. there exists an injective homomorphism ϕ : B → A; (2) coaction is a coideal map : B → A ⊗ B and is a homomorphism of modules; (3) coalgebra and coideal structures are compatible with each other, i.e. the following identities hold: (4) ǫ : B → C is the counit.
where m is the multiplication and i is the unit, is an algebra; (2) B is a subalgebra of A, i.e. there exists an injective homomorphism ϕ : B → A; (3) the triple (B, , ǫ) is a coideal of (A, ∆, ε).
The relation (5.1) is usually referred to as the coideal coassociativity of B. We will refer to (5.2) as the coideal coinvariance. We will refer to the map ϕ as to the natural embedding ϕ : B ֒→ A.
The next definition, a quantization of a Lie bi-algebrag, is due to Drinfel'd [Dri 87]: Definition 5.3. Let (g, δ) be a Lie bi-algebra. We say that a quantized universal enveloping algebra (U (g), ∆ ) is a quantization of (g, δ), or that (g, δ) is the quasi-classical limit of (U (g), ∆ ), if it is a topologically free C[[ ]] module and: (1) U (g) / U (g) is isomorphic to U(g) as a Hopf algebra; (2) for any x ∈g and any X ∈ U (g) equal to x (mod ) one has Note that (U (g), ∆ ) is a topological Hopf algebra over C[[ ]] and is a topologically free C[[ ]] module. Drinfel'd also noted that for a given Lie-bialgebra (g, δ), there exists a unique extension of the map δ :g → g ⊗g to δ : U(g) → U(g) ⊗ U(g) which turns U(g) into a co-Poisson-Hopf algebra. The converse is also true. In such a way (U (g), ∆ ) can be viewed as a quantization of (U(g), δ).
In the remaining part of this section we will consider a quantization of symmetric pairs of half-loop Lie algebras (g,g θ ) = (L + , H + ). We will recall the coproduct of the Yangian Y(g) = U (L + ) which follows from the Lie bi-algebra structure on L + . Then we will construct the coaction of the twisted Yangian Y(g, g θ ) tw = U (L + , H + ) which will follow from the Lie bi-ideal structure on H + and the coideal compatibility relations (5.1) and (5.2). And finally, we will recall the defining relations of the Drinfel'd Yangian and give the main results of this paper, the defining relations of the twisted Yangians in Drinfel'd basis.
5.1. Coproduct on Y(g). The coproduct is given by This is the simplest solution of the quantization condition satisfying the coassociativity property The grading on Y(g) is given by As in the previous section, we need to consider the cases θ = id and θ = id separately.
Proposition 5.2. Let θ = id. Then the coideal subalgebra structure is given by The grading on Y(g, g) tw is given by deg(x a ) = 0, deg( ) = 1 and deg(G(x a )) = 2.
The proofs of Propositions 5.1 and 5.2 are stated in section 7.2. The map (5.9) is the MacKay twisted Yangian formula [DMS 01]. The next remark is due to Lemma 7.1: Remark 5.1. It will be convenient to write the coaction (5.11) in the following way

Yangians and twisted Yangians in Drinfel'd basis.
For any elements x i1 , x i2 , . . . , x im of any associative algebra over C, set where the sum is over all permutations π of {i 1 , i 2 , . . . , i m } and Theorem 5.1. Let g be a finite dimensional complex simple Lie algebra. Fix a (non-zero) invariant bilinear form on g and a basis {x a }. There is, up to isomorphism, a unique homogeneous quantization Y(g) := U (g[u]) of (g[u], δ). It is topologically generated by elements x a , J (x a ) with the defining relations: where β ijk abc = α il a α jm b α kn c α lmn , γ ijk abcd = α e cd β ijk abe + α e ab β ijk cde , (5.24) for all x a ∈ g and λ, µ ∈ C. The coproduct and grading is given by (5.3), (5.4) and (5.6), the antipode is The counit is given by ε (x a ) = ε (J (x a )) = 0.
An outline of a proof can be found in chapter 12 of [ChaPre 94]. Let us make a remark on the Drinfel'd terrific relations (5.22) and (5.23), which are deformations of the relations (3.3) and (3.4), respectively. The right-hand sides (rhs) of the terrific relations are such that ∆ : Y(g) → Y(g) ⊗ Y(g) is a homomorphism of algebras. Choose any total ordering on the elements x a , J (x b ). Then it is easy to see that the basis of Y(g) is spanned by the totally symmetric polynomials {x a1 , . . . , x am , J (x b1 ), . . . , J (x bn )} with m + n ≥ 1, m, n ≥ 0, and ordering a i · · · a m , b i · · · b n . Moreover, the defining relations must be even in . Indeed, consider the coaction on the left-hand side (lhs) of (5.22). The linear terms in vanish due to the Jacobi identity. What remains are the 2 -order terms cubic and totally symmetric in x a . Hence the rhs of the terrific relation must be of the form 2 A ijk abc {x i , x j , x k } for some set of coefficients A ijk abc ∈ C. By comparing the terms on the both sides of the equation and using the Jacobi identity one finds A ijk abc = β ijk abc . The level three terrific relation (5.23) is obtained in a similar way. 2 Theorem 5.2. Let (g, g θ ) be a symmetric pair decomposition of a finite dimensional simple complex Lie algebra g of rank(g) ≥ 2 with respect to the involution θ, such that g θ is the positive eigenspace of θ. Let {X α , Y p } be the symmetric space basis of g with respect to θ. There is, up to isomorphism, a unique homogeneous quantization Y(g, g θ ) tw := U (g [u], g[u] θ ) of (g [u], g[u] θ , τ ). It is topologically generated by elements X α , B(Y p ) with the defining relations: for all X α , Y p ∈ g and a, b ∈ C. The coaction and grading is as in Proposition 5.1. The counit is ǫ (X α ) = ǫ (B(Y p )) = 0 for all non-central X α . In the case when g θ has a non-trivial centre k generated by X z , then ǫ (X z ) = c with c ∈ C.
In the case when h has a central element, the one dimensional representation of h has a free parameter c ∈ C. This parameter corresponds to the free boundary parameter of a quantum integrable model with a twisted Yangian as the underlying symmetry algebra. For Lie algebras of type A, this parameter also appears in the solutions of the boundary intertwining equation leading to a one-parameter family of the boundary S-matrices satisfying the reflection equation [AACDFR 04].
Theorem 5.3. Let g be a finite dimensional simple complex Lie algebra of rank(g) ≥ 2. Let {x i } be a basis of g. Fix a (non-zero) invariant bilinear form on g and a basis {x i }. There is, up to isomorphism, a unique homogeneous quantization Y(g, g) tw := U (g[u], g[u 2 ]) of (g[u], g[u 2 ], τ ). It is topologically generated by elements x i , G(x i ) with the defining relations: for all x a ∈ g and λ, µ ∈ C. The coaction and grading is as in Proposition 5.2. The co-unit is ǫ (x i ) = ǫ (G(x i )) = 0.
Theorems 5.2 and 5.3 can be proven using essentially the same strategy outlined in chapter 12 of [ChaPre 94]. The complicated part is to obtain the horrific relations (5.27), (5.28) and (5.33), which are quantizations of (3.8), (3.9) and (3.11), respectively. A proof of the first two horrific relations is given in the first part of section 7.3. Proving the third horrific relation is substantially more difficult. We have given an outline of a proof in the second part of section 7.3. We believe that coefficients of the horrific relation (5.33) could be further reduced to a more elegant and compact form. We have succeeded to find such a form for the sl 3 Lie algebra: Remark 5.2. For g = sl 3 the coefficients of the horrific relation (5.33) get simplified to The Yangian Y(g) has a one-parameter group of automorphisms τ c , c ∈ C, given by τ c (x a ) = x a and τ c (J (x a )) = J (x a ) + c x a , which is compatible with both algebra and Hopf algebra structure. An analogue of this automorphism for the twisted Yangian Y(g, g θ ) tw is a one-parameter group of automorphism of embeddings (5.9) given by τ c (ϕ(X α )) = X α and τ c (ϕ There is no analogue of such an automorphism for the twisted Yangian Y(g, g) tw , since it is not compatible with the relation (5.33).

Coideal subalgebras of the Yangian Y(sl 3 )
In this section we present three coideal subalgebras Y(g, h) tw of Y(sl 3 ), with h = so 3 , h = gl 2 and h = sl 3 (θ = id case). We will denote generators of the first two algebras by symbols h, e, f, k and H, E, F. We will start by recalling the definition of the Lie algebra sl 3 in the Serre-Chevalley basis and in the Cartan-Chevalley basis.
6.1. The sl 3 Lie algebra. Lie algebra sl 3 in the Serre-Chevalley basis is generated by {e i , f i , h i | i = 1, 2} subject to the defining relations The quadratic Casimir operator of U(sl 3 ) is given by (c g = 6) Definition 6.1. Let Y(sl 3 ) denote the associative unital algebra with sixteen generators e i , f i , h j , J (e i ), J (f i ), J (h j ) with i = 1, 2, 3, j = 1, 2 and the defining relations (6.2) and Definition 6.2. The Hopf algebra structure on Y(sl 3 ) is given by 6.3. Orthogonal twisted Yangian Y(sl 3 , so 3 ) tw . Let the involution θ be given by The action of θ on the rest of the algebra elements is deduced by the constraint θ 2 = id. The symmetric space basis for sl 3 is given by g θ = {h = h 1 + h 2 , e = e 1 − e 2 , f = f 1 − f 2 } and m = {h 1 − h 2 , e 1 + e 2 , f 1 + f 2 , e 3 , f 3 }.
Definition 6.6. The algebra Y(sl 3 , gl 2 ) tw admits a unique left co-action given by for all x ∈ {h, e, f, k} and The co-unit is given by for all x ∈ {e, f, h} and Y ∈ {E i , F i } (i = 1, 2).
Definition 6.7. Let Y(sl 3 , sl 3 ) tw denote the associative unital algebra with sixteen generators e i , f i , h j , G(e i ), G(f i ), G(h j ) with i = 1, 2, 3, j = 1, 2, obeying the standard sl 3 Lie algebra relations of the Cartan-Chevalley basis and the standard level-2 Lie relations for any x, y ∈ {e i , f i , h j }, a, b ∈ C, and the following level-4 horrific relation Definition 6.8. The algebra Y(sl 3 , sl 3 ) tw admits a left co-action given by a . The cases n + m = 0 and n + m = 1 are given by (3.2). The case n + m = 2 follows from the level-2 Drinfel'd terrific relation (3.3). Indeed, we can rewrite (7.2) as ]) = 0. For n = m = 1 it is equivalent to the level-2 Drinfel'd terrific relation and for n = m this equality follows from definition (7.1) and the Jacobi identity (2.1). Let us recall that for the rank(g) = 1 case, this level-2 terrific relation is trivial and one has to consider the level-3 terrific relation (5.23) instead, which can be constructed in a similar way.
Define level-n generators by α are the level-0 and level-1 Drinfel'd generators, respectively. Then (7.2) is equivalent to for some N αβ ∈ C × satisfying N αβ = −N βα . We will prove (7.3) by induction. The base of induction is given by the cases with 0 ≤ m + n ≤ 2. Now suppose (7.3) is true for some n + m = k ≥ 3. The action of adh

Acting with adh
(1) α and using (7.6) we get To obtain the remaining relations we act with adh (1) α on the third relation in (7.3). This gives α+β .
The level-2 generators are defined by . They are required to satisfy the following identities: The first identity is the Lie algebra relation for the level-2 generators. It follows by a direct calculation: ] by (7.14) (2) (X γ ), by the mixed Jacobi identity w pr γ f γ αβ + w qp β g r αq + w qr β g p qα = 0 and by (7.14). The second identity ensures that there are exactly h = dim(g θ ) level-2 generators. It gives the level-2 horrific relation (3.8): Now consider the level-3 generator defined by We require B (3) (Y p ) to satisfy the following identities: The first identity is the Lie algebra relation for the level-3 generators. The second identity ensures that there are exactly m = dim(m) level-3 generators. Let us show the first identity. We will need the following mixed Jacobi relation (7.18) g βq p f µ αβ − g µr p g q rα − g µq r g r αp = 0.
For the second identity in (7.17) we have which combined with (7.19) gives the level-3 horrific relation (3.9). For completeness, let us recall that for the rank(g) = 1 case, the level-2 and level-3 horrific relation are trivial and one has to consider a level-4 horrific relation instead. This can be shown in a similar way.
We need to show that (7.11) and (7.13) hold for all n ≥ 1, and that (7.12) holds for all n ≥ 2 provided they hold for n = 0 and n = 1, respectively. We will use the symmetric space basis of the Cartan decomposition of g and an induction hypothesis to complete the proof. For simplicity, we will restrict to a special case of a symmetric pair with the Cartan decomposition given by h = l h ⊕ (⊕ α h α ) and m = (⊕ µ m µ ) with α ∈ ∆ h and µ ∈ ∆ m , the roots of h and m, respectively. This case corresponds to the symmetric pair of type AIII.
We will use the lower case letters to denote the generators of even levels and the upper case letters for the odd levels. We define level-(2k + 2) generators {h (1) µ = E µ are the level-0 and level-1 Drinfel'd basis generators. Let α, β ∈ ∆ h , µ, ν, λ ∈ ∆ m , and γ, δ ∈ ∆ denote any root. The relations (7.11), (7.12) and (7.13) in this basis are equivalent to Suppose that the relations above hold for all levels up to 2k + 1 ≥ 3, the marginal case being the base of induction consisting of (7.23), (7.24) and (7.25) with m = n = 0, which give level-0, level-1 and level-2 relations, respectively, and (7.24) with m = 0 and n = 1 giving level-3 relations.
We have demonstrated that all level-(2k + 2) relations hold provided all the defining relation of level no greater than (2k + 1) hold. It remains to show that level-(2k + 3) relations in (7.24) also hold. Consider (7.23) with 2n + 2m + 2 = 2k + 2. Acting with −adE (1) µ on the first relation we find Set δ = γ and use the induction hypothesis and definition (7.22). Then Next, act with −adE (1) µ on the second relation in (7.23). We obtain µ+β , which combined with the initial expression gives the required relation for arbitrary 2n + 2m = 2k + 3, namely µ+β . To end this section we remark that we would welcome a proof of the Drinfel'd basis for the half-loop and twisted half-loop Lie algebras, which would not be based on the Cartan decomposition of the underlying Lie algebra.

Proofs of coactions.
7.2.1. Proof of Proposition 5.1. The Lie bi-ideal structure on H + defines the coaction up to the first order in , For the level zero generators of H + the Lie bi-ideal structure is trivial and the minimal form of the coaction is given by The coideal compatibility relations (5.1) and (5.2), and the classical limit requirement ϕ(X α )| →0 = X α implies that ϕ(X α ) = X α is the natural inclusion ϕ : X α ∈ Y(g, h) tw → X α ∈ Y(g).
For the level one generators of H + the Lie bi-ideal structure is non trivial. The simplest coaction is given by As previously, the coaction must satisfy relations (5.1) and (5.2), and in the classical limit we must obtain ϕ(B(Y p ))| →0 = B(Y p ). By (5.2) it follows Consider an ansatz ϕ(B(Y p )) = J (Y p ) + F (0) p with some level zero element F We choose F . It remains to check the coideal coassociativity (5.1) and the homomorphism property ϕ( In what follows we will need the following auxiliary Lemma: Lemma 7.1. In a simple Lie algebra g the following identities hold: i α cs j α br k (α a rs x a ⊗ {x b , x c } + α a bc {x s , x r } ⊗ x a ) = α jk i α cs j α br k α a rs (x a ⊗ {x b , x c } + {x c , x b } ⊗ x a ) by ren. b, c ↔ r, s. The third identity is obtained using the following auxiliary identities 2 α jk i α cr j α bs k α a sr = α jk i α cr j (α bs k α a sr + α as k α b sr ) + 1 2 c g α jc i α ab j , (7.41) 2 α jk i α cr j α bs k α a sr = α jk i α bs k (α cr j α a sr + α ar j α c sr ) + 1 2 c g α jb i α ac j . (7.42) The first auxiliary identity follows by multiple application of the Jacoby identity: α jk i α cr j (α bs k α a sr − α as k α b sr ) = α jk i α cr j α s rk α ab s = 1 2 α jc i α kr j α s rk α ab s = 1 2 c g α jc i α ab j . The second auxiliary identity follows by the b ↔ c symmetry and renaming j, s ↔ k, r.
To complete the proof we need to calculate all elements in (H (4) abc ) as we did in the proof of Theorem 5.2 above. Here we will not attempt to reach this goal. To give all the details of the proof would take more pages than the present paper contains itself, and thus we refrain from making such an attempt. An important question is whether the coefficients Ψ ijk abc , Φ ijk abc andΦ ijklm abc can be written in an elegant and compact form. We have not succeeded in accomplishing this, so we will leave it as an open question for further study.