Drinfeld J Presentation of Twisted Yangians

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-lo...

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Hauptverfasser: Belliard, S., Regelskis, V.
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Schriftenreihe:Symmetry, Integrability and Geometry: Methods and Applications
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spelling irk-123456789-1486122019-02-19T01:25:27Z Drinfeld J Presentation of Twisted Yangians Belliard, S. Regelskis, V. 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 sl3 Lie algebra. 2017 Article Drinfeld J Presentation of Twisted Yangians / S. Belliard, V. Regelskis // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 48 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R10; 81R50; 17B37 DOI:10.3842/SIGMA.2017.011 http://dspace.nbuv.gov.ua/handle/123456789/148612 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language English
description 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 sl3 Lie algebra.
format Article
author Belliard, S.
Regelskis, V.
spellingShingle Belliard, S.
Regelskis, V.
Drinfeld J Presentation of Twisted Yangians
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Belliard, S.
Regelskis, V.
author_sort Belliard, S.
title Drinfeld J Presentation of Twisted Yangians
title_short Drinfeld J Presentation of Twisted Yangians
title_full Drinfeld J Presentation of Twisted Yangians
title_fullStr Drinfeld J Presentation of Twisted Yangians
title_full_unstemmed Drinfeld J Presentation of Twisted Yangians
title_sort drinfeld j presentation of twisted yangians
publisher Інститут математики НАН України
publishDate 2017
url http://dspace.nbuv.gov.ua/handle/123456789/148612
citation_txt Drinfeld J Presentation of Twisted Yangians / S. Belliard, V. Regelskis // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 48 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT belliards drinfeldjpresentationoftwistedyangians
AT regelskisv drinfeldjpresentationoftwistedyangians
first_indexed 2025-07-12T18:58:08Z
last_indexed 2025-07-12T18:58:08Z
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 011, 35 pages Drinfeld J Presentation of Twisted Yangians Samuel BELLIARD † and Vidas REGELSKIS ‡§ † Institut de Physique Théorique, Orme des Merisiers batiment 774, CEA/DSM/IPhT, CEA/Saclay, F-91191 Gif-sur-Yvette Cedex, France E-mail: samuel.belliard@cea.fr ‡ Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK E-mail: vidas.regelskis@york.ac.uk § Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK Received May 24, 2016, in final form February 21, 2017; Published online March 01, 2017 https://doi.org/10.3842/SIGMA.2017.011 Abstract. 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 sl3 Lie algebra. Key words: coideal; coisotropic subalgebra; deformation; Manin triple; twisted Yangians 2010 Mathematics Subject Classification: 81R10; 81R50; 17B37 1 Introduction The 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 [17]. The name Yangian is due to V.G. Drinfeld to honour C.N. Yang who found the simplest solution of the Yang–Baxter equation, the rational R matrix [47] (see also [8, 9]). This R matrix and the Yang–Baxter equation were discovered in the studies of exactly solvable two-dimensional statistical models and quantum integrable systems. One of the most important results was the quantization of the inverse scattering method by Leningrad’s school [46] that lead to the formulation of quantum groups in the so-called RTT formalism [44]. These quantum groups are deformations of semisimple Lie algebras and are closely associated to quantum integrable systems. In particular, the representation theory of the Yangian Y(sl2), which is one of the simplest examples of the infinite-dimensional quantum groups, is used to solve the rational 6-vertex statistical model [9], the Heisenberg (XXX) spin chain [23], the principal chiral field model with the SU(2) symmetry group [22, 33]. The mathematical description of quantum groups and of quantization of Lie bi-algebras was presented by Drinfeld in his seminal work [17] (see also [18]). Drinfeld gave a quantization procedure for the universal enveloping algebra U(g̃) for any semisimple Lie algebra g̃.1 The quantization is based on the Lie bi-algebra structure on g̃ given by a skew symmetric map δ : g̃ → g̃ ∧ g̃, the cocommutator. A quantization of (g̃, δ) is a (topological) Hopf algebra (U~(g̃),∆~), such that U~(g̃)/~U~(g̃) ∼= U(g̃) as a Hopf algebra and δ(x) ∼ ( ∆~(X)− σ ◦∆~(X) ) /~ (mod ~), This paper is a contribution to the Special Issue on Recent Advances in Quantum Integrable Systems. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2016.html 1Throughout this manuscript we use g̃ to denote any Lie algebra, while the undecorated g is reserved for finite-dimensional simple complex Lie algebras. mailto:samuel.belliard@cea.fr mailto:vidas.regelskis@york.ac.uk https://doi.org/10.3842/SIGMA.2017.011 http://www.emis.de/journals/SIGMA/RAQIS2016.html 2 S. Belliard and V. Regelskis where σ(a⊗ b) = b⊗ a and X is any lifting of x ∈ g̃ to U~(g̃). The Lie bi-algebra structure on g̃ can be constructed from the Manin triple (g̃, g̃+, g̃−), where g̃± are the isotropic subalgebras of g̃ such that g̃+ ⊕ g̃− = g̃ as a vector space and g̃− ∼= g̃∗+, the dual of g̃+. Then the commutation relations of the quantum group can be obtained by requiring ∆~ to be a homomorphism of algebras, ∆~ : U~(g̃)→ U~(g̃)⊗ U~(g̃). The question of the existence of such a quantization for any Lie bi-algebra was raised by Drinfeld in [19] and was answered by P. Etingof and D. Kazhdan in [20]. They proved that any finite- or infinite-dimensional Lie bi-algebra admits a quantization. Here we will consider only the Yangian case, U~(g̃) = Y(g) with g̃ = L+. We will use the so-called Drinfeld J presentation which is very convenient to approach the quantization problem. In physics, quantum groups are related to quantum integrable models without boundaries and their extensions to models with boundaries. The underlying symmetry of 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 Cherednik [15] and in the context of one-dimensional spin chains with boundaries by Sklyanin [45] in the so-called reflection equation algebra formalism. Mathematical aspects of reflection algebras in the RTT presentation, called twisted Yangians, were first considered by G. Olshanskii in [42] and were further explored in [24, 25, 26, 36, 37, 38, 39] (also see references therein). A similar approach for the q-deformed universal enveloping and quantum loop Lie algebras, Uq(g) and Uq(ĝ), was considered in [6, 14, 40]. A slightly different approach to coideal subalgebras, using Serre–Chevalley presentation of Uq(g) was surveyed by G. Letzter in [30]. Here coideal subalgebras of Uq(g) were constructed using quantum symmetric pairs, see, e.g., [29], and have been classified for all finite-dimensional semisimple Lie algebras. Examples of coideal subalgebras of quantum affine Lie algebras of type A in the Serre–Chevalley presentation were the q-Onsager algebra [3, 4] and the generalized q-Onsager algebras [5]. A generalization of quantum symmetric pairs for infinite-dimensional Lie algebras of type A was proposed in [43] and the general theory of quantum symmetric pairs for Kac–Moody Lie algebras was developed in [28]. A remarkably simple approach to coideal subalgebras based on the Drinfeld’s original construction of Yangians, usually referred to as Drinfeld first or simply as Drinfeld J presentation, was introduced in [16] (see also [32]) and is now conveniently called the MacKay twisted Yangians [14]. Twisted Yangians, which we denote by Y(g, h)tw, are in exact correspondence with the sym- metric pair decomposition of g given by a proper involution θ [1, 27]. The decomposition is given by g = h ⊕ m with θ(x) = x for all x ∈ h, θ(y) = −y for all y ∈ m, the positive and negative eigenspaces of θ. For the half-loop Lie algebra this decomposition is given by L+ = H+ ⊕M+, where the positive eigenspace of θ is the twisted current Lie algebra H+ ∼= (h ⊕ um)[u2]. The reflection equation algebra formalism provides evidence of the existence of a quantization of such a symmetric pair. More generally, P. Etingof and D. Kazhdan have shown in [21] that any homogeneous space G/H admits a local quantization. In [10], the notion of left and right Lie coideal structures2 θ-invariant Lie subalgebras, denoted τ and τ ′, respectively, was introduced by one of the authors. The Lie coideal structure is in one-to-one correspondence with a twisted version of the Manin triple (g̃, g̃+, g̃−) and gives a quantization scheme for the θ-invariant Lie subalgebra leading to a quantum coideal subalgebra. The Lie coideal structure corresponds to the splitting δ = τ + τ ′ of the so-called coisotropic cocommutator δ of a Lie bi-algebra satisfying δ(H+) ∈ M+ ∧H+ [41, 48]. Here we consider the left Lie coideal structure, which allows us to construct a quantization of twisted half-loop Lie algebras leading to quantum coideal subalgebras called twisted Yangians in Drinfeld J presentation. A quantization of the twisted half-loop Lie algebras for rank(g) = 1 case was shown in [10]. In this paper we consider twisted Yangians Y(g, h)tw in the Drinfeld J presentation for any rank(g) ≥ 2. In particular, we present a quantization procedure which holds for all symmetric 2The term “Lie bi-ideal structure” is used in [10] instead. Drinfeld J Presentation of Twisted Yangians 3 pairs of the half-loop Lie algebras of finite-dimensional simple complex Lie algebras of rank g ≥ 2. These symmetric pairs follow from the ones of the simple complex Lie algebras that have been classified by Araki [1] (see also [27]). Such symmetric pairs can be grouped into four classes: h is simple, h = a ⊕ b, h = a ⊕ k and h = a ⊕ b ⊕ k, where a and b are simple Lie subalgebras of h, and k is a one-dimensional Lie algebra. We will further refer to all of these cases as the θ 6= id case. We also consider the trivial involution θ of g. Being trivial at the Lie algebra level, this involution can be extended non-trivially to the half-loop Lie algebra. In this case the θ-fixed subalgebra is the even half-loop Lie algebra g[u2] isomorphic to g[u] as a Lie algebra. We will refer to this setup as the θ = id case. The main results of this paper are Theorems 5.5 and 5.6, which are analogues of the Theorem 2 in [17] for the twisted Yangians for θ 6= id and θ = id cases, respectively. We call the defining relations of the twisted Yangians the horrific relations due to their complex form and similarity to the Drinfeld terrific relations. A proof the horrific relations of Theorem 5.5 is given in Section 7.5. A proof of the horrific relations of Theorem 5.6 is outlined in Section 7.6. The results of this paper provide a uniform way of constructing twisted Yangians for all symmetric pairs (g, h), when g is a simple complex Lie algebra. Twisted Yangians in Drinfeld J presentation have important applications in quantum integrable models, where it is important to know the minimal set of the defining relations of the underlying symmetry algebra. For example, twisted Yangians emerge as the non-abelian symmetries of the principal chiral models defined on a half-line and can be used to find the scalar boundary S-matrices and thus solve the spectral problem [16, 32, 33]. The closure relations are necessary in constructing representation theory of these algebras and allow to classify algebraically all scalar and dynamical boundary conditions and to construct the corresponding dynamical boundary S-matrices; i.e., for the affine Toda models with open boundaries this was done in [5, 11]. Twisted Yangians in Drinfeld J presentation may 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” [7]. We also remark, that twisted Yangians of this type were shown to play an important role in quantum integrable systems for which the RTT presentation of the underlying symmetries is not well understood, for example in the AdS/CFT correspondence [34, 35]. The paper is organized as follows. In Section 2 we recall the basic facts about simple complex Lie algebras and define the symmetric pair decomposition with respect to involution θ. Then, in Section 3, we recall the description of the half-loop Lie algebra L+ of g and 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 coideal structure on H+ that provide the necessary setup for the quantization procedure presented in Section 5. A low rank example is presented Section 6. We give the explicit form of twisted Yangians in Drinfeld J presentation when g = sl3. Section 7 contains the proofs which were omitted in the main part of the paper due to their length and for convenience of the reader. 2 Definitions and preliminaries 2.1 Lie algebra Consider a finite-dimensional simple complex Lie algebra g of dimension dim(g) = n, with a basis {xa} such that [xa, xb] = α c ab xc, α c ab + α c ba = 0, α c ab α e dc + α c daα e bc + α c bd α e ac = 0. (2.1) Here α c ab are the structure constants of g and the Einstein summation rule of the repeated indices is assumed. Let ηab denote the non-degenerate invariant bilinear (Killing) form on g in 4 S. Belliard and V. Regelskis the {xa} basis (xa, xb)g = ηab, that can be used to lower indices {a, b, c, . . .} of the structure constants α d ab ηdc = αabc with αabc + αacb = 0. The inverse of ηab is given by ηab and satisfies ηabη bc = δ ca . Set {x, y} = 1 2(xy + yx). Let Cg = ηab{xa, xb} denote the second order Casimir operator associated to g and let cg be its eigenvalue in the adjoint representation. For a simple Lie algebra it is non-zero and is given by cgδ d c = ηabα e ac α d be = α eb c α d be . (2.2) Here α bc a are the structure constants in the dual basis, namely if {xa} denote the basis of g defined by (xa, x b)g = δ ba , then [xb, xc] = α bc a x a. Constants α bc a satisfy dual co-Jacobi 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 α bc a [xc, xb] = cgxa. (2.3) 2.2 Symmetric pair decomposition Let θ : g → g be an involution, θ2 = id. Then g can be decomposed into positive and negative eigenspaces of θ, i.e., g = h ⊕ m with θ(x) = x for all x ∈ h and θ(y) = −y for all y ∈ m, here dim(h) = h, dim(m) = m satisfying 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 [h, h] ⊆ h, [h,m] = m, [m,m] = h. From the classification of symmetric pairs for simple complex Lie algebras it follows that the invariant subalgebra h is a (semi)simple or reductive Lie algebra which can be decomposed into a direct sum of two simple complex Lie algebras a and b, and a one-dimensional Lie algebra k, at most (see, e.g., [27, Section 5]). We write h = a⊕ b⊕ k. Set dim(a) = a and dim(b) = b. Let the elements Xi ∈ a, Xi′ ∈ b, Xz ∈ k, Yp ∈ m, with i = 1, . . . , a, i′ = 1, . . . , b and p = 1, . . . ,m, (2.4) be a basis of g such that θ(Xα) = Xα for any α ∈ {i, i′, z}, and θ(Yp) = −Yp. Here we use indices i(j, k, . . . ) for elements Xα ∈ a, primed indices i′(j′, k′, . . . ) for elements Xα ∈ b, index α = z for the element Xα ∈ k, and indices p(q, r, . . .) for Yp ∈ m, when needed. Note that Xz is central in h. We will denote the commutators in this basis as follows: [Xα, Xβ] = f γ αβXγ with f γ αβ = 0 if α 6= β or α 6= γ or β 6= γ, [Xα, Yp] = g q αpYq, [Yp, Xα] = g q pαYq, [Yp, Yq] = ∑ α w α pq Xα. (2.5) 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, e.g., ∑ αw α pq Xα = w i pqXi + w i′ pq Xi′ + w z pq Xz for h = a ⊕ b ⊕ k. The Einstein summation rule for the Greek indices will be used in cases when the sum is over a single Drinfeld J Presentation of Twisted Yangians 5 simple subalgabera of h. The notation α 6= 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, which follow from the properties of the Lie bracket, and the homogeneous and mixed Jacobi identities f ν αβ f µ γν + f ν γα f µ βν + f ν βγ f µ αν = 0, f µ αβ g s pµ + g q βp g s αq − g q αpg s βq = 0, w β pq f µ αβ + g r αpw µ qr − g r αqw µ pr = 0, (2.6) 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 pαw β qr − g r qαw β pr = 0 for α 6= β. (2.7) We will further refer to the set {Xα, Yp} = {Xi, Xi′ , Xz, Yp} given by (2.4) and satisfying rela- tions (2.5)–(2.7) as the symmetric space basis for a given Lie algebra g and involution θ. Let κ denote the Killing form on g in the symmetric space basis. It has a block-diagonal form, namely (Xi, Xj)g = (κa)ij , (Xi′ , Xj′)g = (κb)i′j′ , (Xz, Xz)g = (κk)zz, (Yp, Yq)g = (κm)pq, with the remaining entries being trivial. The Casimir element Cg in this basis decomposes as Cg = CX + CY = ∑ α,β (κh) αβ{Xα, Xβ}+ (κm)pq{Yp, Yq}, CX = C + C ′ + Cz = (κa) ij{Xi, Xj}+ (κb) i′j′{Xi′ , Xj′}+ (κk) zz{Xz, Xz}. Here καβ ∈ {(κa)ij , (κb)i ′j′ , (κk) zz}. The block diagonal decomposition of the inverse Killing form can be used to raise the indices of the structure constants. We set f βγ α = κβµf γ αµ , g qνp = κνρg q ρp , w pq ν = (κm)prg 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). Let us now consider the commutation relations. For generators Yp we have [Yp, CX ] = 2 ∑ α g αqp {Yq, Xα}, [Yp, CY ] = 2 ∑ α g qαp {Yq, Xα}. The remaining commutation relations are trivial. Let ca, cb, cz and cm be the eigenvalues of C, C ′, Cz and CY in the adjoint representation, respectively. We have cg = ca+cb+cm+cz. Using (2.3) we find f βν α [Xν , Xβ] = c(α)Xα, w qp γ [Yp, Yq] = c̄(γ)Xγ , (2.8) with α = i(i′), γ = i(i′, z), c̄(α) = cg − c(α), c(i) = ca, c(i′) = cb and c(z) = 0. Using (2.2), (2.5) and the equality above we obtain f µν α f β νµ = c(α)δ β α , w qp α w β pq = c̄(α)δ β α , w qp α w γ pq = 0 for α 6= γ, (2.9) and α = i(i′, z), β = j(j′, z). Finally, for Yq we have∑ α g pαq [Xα, Yp] = 1 2cgYq, ∑ α g rαp g q αr = 1 2cgδ q p . (2.10) 6 S. Belliard and V. Regelskis 3 Symmetric spaces and simple half-loop Lie algebras 3.1 Half-loop Lie algebra Consider the half-loop Lie algebra L+ generated by elements x (k) a with k ≥ 0 and a = 1, . . . , dim(g). It is an infinite-dimensional graded Lie algebra with the grading given by deg ( x (k) a ) = k and the defining relations [x(k)a , x (`) b ] = α c ab x (k+`) c . (3.1) This algebra can be identified with the set of polynomial maps f : C→ g using the Lie algebra isomorphism L+ ∼= g[u] = g⊗ C[u] with x (k) a ∼= xa ⊗ uk. The algebra L+ has another presentation, conveniently called Drinfeld J presentation. Proposition 3.1. The half-loop Lie algebra L+ is isomorphic to the algebra generated by ele- ments xa, J(xb) satisfying [xa, xb] = α c ab xc, J(µxa + νxb) = µJ(xa) + νJ(xb), [xa, J(xb)] = α c ab J(xc), (3.2) [J(xa), J([xb, xc])] + [J(xb), J([xc, xa])] + [J(xc), J([xa, xb])] = 0, (3.3) [[J(xa), J(xb)], J([xc, xd])] + [[J(xc), J(xd)], J([xa, xb])] = 0, (3.4) for any µ, ν ∈ C. The grading is given by deg(xa) = 0 and deg(J(xa)) = 1. The isomorphism is given by the map xa 7→ x (0) a and J(xa) 7→ x (1) a . Relations (3.3) and (3.4) are homogeneous relations of degree 2 and 3. In the rank(g) = 1 case the relation (3.3) is trivial and for the rank(g) ≥ 2 cases the relation (3.4) is implied by (3.3). Since we were unable to locate a complete proof of this proposition in the literature, we have given one in Section 7.1. (An outline of a proof is given in [13, Section 12.1].) 3.2 Twisted half-loop Lie algebra Let θ(xa) = θ ba xb. We cab extend the involution θ of g to the whole of L+ as follows: θ ( x(k)a ) = (−1)kθ ba x (k) b ∼= (−1)kθ(xa)⊗ uk for k ≥ 0, or in the polynomial map point of view, θ(x(u)) = θ(x)(−u). 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+ = span{x ∈ L+ | θ(x) = x}. In the physics literature it is often referred to as the twisted current algebra. Let θ 6= id. Consider the symmetric space basis of g. We write the half-loop Lie algebra L+ in terms of the elements X (k) α and Y (k) q satisfying[ X(k) α , X (`) β ] = f γ αβX (k+`) γ , [ X(k) α , Y (`) p ] = g q αpY (k+`) q ,[ Y (k) p , Y (`) q ] = ∑ α w α pq X (k+`) α , (3.5) for all k, ` ≥ 0. Involution θ acts on these elements by θ ( X (k) α ) = (−1)kX (k) α and θ ( Y (k) p ) = (−1)k+1Y (k) p . Thus the half-loop Lie algebra decomposes as L+ = H+ ⊕M+, where H+ = span { X (2k) α , Y (2k+1) q } is the subalgebra of L+ generated by θ-invariant elements, and M+ = span { X (2k+1) α , Y (2k) q } is the subset of L+ of θ-anti-invariant elements. In the case when θ = id we have g[u]θ ∼= g[u2], that is H+ = span { x (2k) a } and M+ = span { x (2k+1) a } . We now give another presentation of H+, which an analogue of the Drinfeld J presentation of L+. Drinfeld J Presentation of Twisted Yangians 7 Proposition 3.2. Let rank(g) ≥ 2 and θ 6= id. Then the twisted half-loop Lie algebra H+ is isomorphic to the algebra generated by elements Xα, B(Yp) satisfying [Xα, Xβ] = f γ αβ Xγ , [Xα, B(Yp)] = g q αp B(Yq), B(aYp + bYq) = aB(Yp) + bB(Yq), (3.6) [B(Yp), B(Yq)] + ∑ α (c̄(α)) −1w α pq w rs α [B(Yr), B(Ys)] = 0, (3.7) [[B(Yp), B(Yq)], B(Yr)] + 2c−1g ∑ α (κm)tuw α pq g s rα [[B(Ys), B(Yt)], B(Yu)] = 0, (3.8) for all a, b ∈ C. The isomorphism is given by the map Xα 7→ X (0) α and B(Yp) 7→ Y (1) p . The proof is given in Section 7.2. Note that in the contrast to L+, the twisted algebra H+ for rank(g) ≥ 2 has degree-2 and degree-3 defining relations. The rank(g) = 1 case is exceptional. The Drinfeld J presentation in this case has a degree-4 relation instead; see [10, Section 4.2]. Proposition 3.3. Let rank(g) ≥ 2 and θ = id. Then the even half-loop Lie algebra is isomorphic to the algebra generated by elements xi, G(xj) satisfying [xi, xj ] = α k ij xk, G(λxa + µxb) = λG(xa) + µG(xb), [xi, G(xj)] = α k ij G(xk), (3.9) [G(xi), G([xj , xk])] + [G(xj), G([xk, xi])] + [G(xk), G([xi, xj ])] = 0, (3.10) for any µ, ν ∈ C. The isomorphism is given by the map xi 7→ x (0) i and G(xi) 7→ x (2) i . The proof is analogous to that of Proposition 3.1, since g[u2] ∼= g[u] as a Lie algebra. 4 Lie bi-algebras and coideals 4.1 Lie bi-algebra structure of a half-loop Lie algebra A Lie bi-algebra structure on L+ is a skew-symmetric 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 Drinfeld J presentation of L+ by δ(xa) = 0, δ(J(xa)) = [xa ⊗ 1,Ωg], where Ωg = ηabxa ⊗ xb. (4.1) This cocommutator can be constructed from the Manin triple (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[u] the positive half-loop algebra (3.1), and L− = g[[u−1]] the negative half-loop algebra (i.e., n,m < 0; see, e.g., [13, Example 1.3.9]). The triple (L,L+,L−) satisfies axioms of the Manin triple. Definition 4.1 ([18]). A Manin triple is a triple of Lie algebras (g̃, g̃+, g̃−) together with a non- degenerate symmetric bilinear form ( , )g̃ on g̃ invariant under the adjoint action of g̃ such that: • g̃+ and g̃− are Lie subalgebras of g̃; • g̃ = g̃+ ⊕ g̃− as a vector space; • ( , )g̃ is isotropic for g̃± (i.e., (g̃±, g̃±)g̃ = 0); • (g̃+)∗ ∼= g̃−. 8 S. Belliard and V. Regelskis The invariant bilinear form on g̃ = L satisfying the requirements above is given by( x(k), y(l) ) L = −(x, y)gδk+l+1,0, or in the polynomial map point of view (x(u), y(u))L = − res0(x(u), y(u))g where ‘res0’ means taking the coefficient of u−1 in the Laurent series expansion. This data together with the definition of the Manin triple uniquely fixes the cocommutator δ on L+. Remark 4.2. If (g̃, g̃+, g̃−) is a Manin triple for dim(g̃+) = ∞, then in general one must take (g̃+)∗ ∼= g−, where g− is a suitable completion of g̃−. However in our case (L+)∗ ∼= L−, as it is easy to see: (L+)∗ ∼= (⊕ k≥0 g⊗ uk )∗ = ∏ k≥0(g⊗ uk)∗ = ∏ k≥1 g⊗ u−k ∼= L−. Here in the second equality we have used the identity ( ⊕ i≥0 Vi) ∗ = ∏ i≥0 V ∗ i , where Vi 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 (δ(x), y ⊗ z)L⊗L = (x, [y, z])L. (4.2) The cocommutator of the degree zero generators x (0) a = xa is trivial since the degree of [y, z] ∈ L− is strictly less than −1, thus (xa, [y, z])L = 0 for all [y, z] ∈ L−. This implies that (δ(xa), y ⊗ z)L⊗L = 0 for all y, z ∈ L− and thus δ(xa) = 0. The case of degree one generators x (1) a = J(xa) is considered in a similar way. For this case we have the non-trivial pairing ( J(xa), x (−2) b ) L = −ηab. Using x (−2) b = c−1g α ji b [ x (−1) i , x (−1) j ] and the duality relation (4.2) we obtain the constraint( δ(J(xa)), α ji b x (−1) i ⊗ x(−1)j ) L⊗L = −cgηab. Writing δ(J(xa)) = v `ka xk ⊗ x` for some v `ka ∈ C we deduce that v `ka αb`k = −cgηab. By (2.2) we have that α k` a αb`k = cgηab and thus v `ka = −α k` a giving δ(J(xa)) = α `k a xk ⊗ x` = [xa ⊗ 1,Ωg]. 4.2 Lie coideal structure of a twisted half-loop Lie algebra The Lie coideal structure of a twisted half-loop Lie algebra is constructed by employing the anti-invariant Manin triple twist. Here we will consider the left Lie coideal structure. The right Lie coideal is obtained in a similar way. Definition 4.3 ([10]). The anti-invariant Manin triple twist φ of (L,L+,L−) is an automor- phism of L satisfying: • φ is an involution; • φ(L±) = L±; • (φ(x), y)L = −(x, φ(y))L for all x ∈ L+ and y ∈ L−. Drinfeld J Presentation of Twisted Yangians 9 Given an involution θ of g one may uniquely associate the anti-invariant Manin triple twist φ, that is the natural extension of θ to the whole L: φ(x(k)a ) = (−1)kθ ba x (k) b ∼= (−1)kθ(xa)⊗ uk for all k ∈ Z, satisfying the definition above and leading to the symmetric pair decomposition of the Manin triple (L,L+,L−): L± = H± ⊕M± such that φ(x) = x, ∀x ∈ H± and φ(y) = −y, ∀ y ∈M±. (4.3) From the anti-invariance of φ for ( , )L it follows that (H−,H+)L = (M−,M+)L = 0. Thus we must have (H±)∗ ∼= M∓. This is easy to check: (H+)∗ ∼= (⊕ k≥0(h⊕ um)⊗ u2k )∗ = ∏ k≥0 ( (h⊕ um)⊗ u2k )∗ = ∏ k≥1 ( (uh⊕m)⊗ u−2k ) ∼= M−, (H−)∗ ∼= (⊕ k≥1(h⊕ um)⊗ u−2k )∗ = ∏ k≥1 ( (h⊕ um)⊗ u−2k )∗ = ∏ k≥0 ( (uh⊕m)⊗ u2k ) ∼= M+. This decomposition of the Manin triple allows us to construct a Lie coideal structure on H+. Definition 4.4 ([10]). Let φ be an anti-invariant Manin triple twist for (L,L+,L−) which leads to the symmetric space decomposition (4.3). Then the linear map τ : H+ →M+ ⊗H+ is a left Lie coideal structure for (H+,M+) if it is the dual of the following action of H− on M−, τ∗ : H− ⊗M− →M−, x⊗ y 7→ [x, y]L− , (4.4) for all x ∈ H− and y ∈M−. Given as Manin triple (L,L+,L−) and the twist φ the requirement (4.4) uniquely fixes the Left coideal structure τ of (L+,H+). Proposition 4.5. The left Lie coideal structure of (L+,H+), τ : H+ →M+ ⊗H+, is given by θ 6= id: τ(Xα) = 0, τ(B(Yp)) = [Yp ⊗ 1,ΩX ], ΩX = ∑ α,β (κh) αβXα ⊗Xβ, (4.5) θ = id: τ(xa) = 0, τ(G(xa)) = [J(xa)⊗ 1,Ωg]. (4.6) Proof. The construction of the left Lie coideal 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 (τ(x), y ⊗ z)L = (x, [y, z])L with x ∈ H+, y ∈ H− and z ∈M−. Consider the case θ 6= id. For the degree zero generators X (0) α = Xα, we have (Xα, [y, z])L = 0 for all y ∈ H− and z ∈M−. This follows by similar arguments as for the degree zero generators of the half-loop Lie algebra. Hence we have τ(Xα) = 0. 10 S. Belliard and V. Regelskis For elements Y (1) p = B(Yp) we have a non trivial paring ( B(Yp), Y (−2) q ) L = −(κm)pq. Then, using relation Y (−2) q = ∑ α 2c−1g g αpq [ Y (−1) p , X (−1) α ] and the duality relation we obtain∑ α ( τ(B(Yp)), g αr q Y (−1) r ⊗X(−1) α ) L⊗L = −1 2cg(κm)pq. It is clear from properties of the pairing that τ(B(Yp)) = ∑ β v βs p Ys ⊗ Xβ for some v βsp ∈ C. We must have ∑ α v αr p gqαr = −1 2cg(κm)pq. Writing the second identity in (2.10) as ∑ α g rαp gqαr = −1 2cgηpq we find v αrp = −g rαp . Hence τ(B(Yp)) = − ∑ α g rαp Yr ⊗Xα = [Yp ⊗ 1,ΩX ]. The Lie coideal structure for the θ = id case follows from the pairing ( G(xa), x (−3) b ) L = −(κg)ab by similar arguments. � For completeness we recall the remark which was stated by one of the authors in [10]. Remark 4.6. The notion of left (right) Lie coideal 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 δ(h) ⊂ h ∧ g. We have δ(x) = τ(x) + τ ′(x), for x ∈ h with τ ′ = −σ ◦ τ the right ideal structure. 5 Twisted Yangians as quantized Lie coideals 5.1 Quantization of Lie bi-algebras and Lie coideals To obtain a quantization of the Lie coideal we need to introduce some additional notation. 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. These maps are required to satisfy a set of compatibility relations (see, e.g., [13, Section 4]). Since we are interested in left coideal subalgebras, we also need to recall the notion of coideal. Definition 5.1. Let A = (A,∆, ε) be a coalgebra. Then B = (B, , ε) is a left coideal of A if: 1) B is a submodule of A, i.e., there exists an injective homomorphism ϕ : B → A; 2) co-action is a left 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 identi- ties hold: (∆⊗ id) ◦ = (id⊗ ) ◦ , (5.1) (id⊗ϕ) ◦ = ∆ ◦ ϕ; (5.2) 4) ε : B → C is the counit. The relation (5.1) is usually referred to as the coideal co-associativity of B. We will refer to (5.2) as the coideal coinvariance. We call the map ϕ the natural embedding ϕ : B ↪→ A. The definition of a coideal can be naturally extended for a bi-algebra. Namely, let A = (A,µ, η,∆, ε) be a bi-algebra. Then B = (B,m, i, , ε) is a left coideal of A if: the triple (B,m, i), where m Drinfeld J Presentation of Twisted Yangians 11 is the multiplication and i is the unit, is an algebra; B is a subalgebra of A; the triple (B, , ε) is a coideal of (A,∆, ε). However note that ε is not a restriction of ε, that is ε ◦ ϕ 6= ε, since in general B may have a larger center than A. We are now ready to discuss quantization of Lie bi-algebras and their coideals. The next definition, a quantization of a Lie bi-algebra g̃, is due to V.G. Drinfeld. Definition 5.2 ([18]). Let (g̃, δ) be a Lie bi-algebra. We say that a quantized universal en- veloping algebra (U~(g̃),∆~) is a quantization of (g̃, δ), or that (g̃, δ) is the quasi-classical limit of (U~(g̃),∆~), if it is a topologically free C[[~]] module and: 1) U~(g̃)/~U~(g̃) is isomorphic to U(g̃) as a Hopf algebra; 2) for any x ∈ g̃ and any X ∈ U~(g̃) equal to x (mod ~) one has( ∆~(X)− σ ◦∆~(X) ) /~ ∼ δ(x) (mod ~) with σ the permutation map σ(a⊗ b) = b⊗ a. Note that (U~(g̃),∆~) is a topological Hopf algebra over C[[~]] and is a topologically free C[[~]] module. Drinfeld also noted that for a given Lie-bialgebra (g̃, δ), there exists a unique extension of the map δ : g̃ → g̃ ⊗ g̃ to δ : U(g̃) → U(g̃) ⊗ U(g̃) which turns U(g̃) into a co-Poisson–Hopf algebra. The converse is also true. In such a way (U~(g̃),∆~) can be viewed as a quantization of (U(g̃), δ). Let θ be an involution of g̃. We define a quantization of the Lie coideal ( g̃θ, τ ) following [10]. Definition 5.3. Let (g̃, δ) be a Lie bi-algebra and ( g̃θ, τ ) be a left Lie coideal. We say that a left coideal subalgebra ( U~ ( g̃θ ) , ~ ) is a quantization of ( g̃θ, τ ) , or that ( g̃θ, τ ) is the quasi-classical limit of ( U~ ( g̃θ ) , ~ ) , if it is a topologically free C[[~]] module and: 1) (U~(g̃),∆~) is a quantization of (g̃, δ); 2) U~ ( g̃θ ) /~U~ ( g̃θ ) is isomorphic to U ( g̃θ ) as a Lie algebra; 3) ( U~ ( g̃θ ) , ~ ) is a left coideal of (U~(g̃),∆~); 4) for any x ∈ g̃θ and any X ∈ U~ ( g̃θ ) equal to x (mod ~) one has( ~(X)− (ϕ(X)⊗ 1 + 1⊗X) ) /~ ∼ τ(x) (mod ~) with ϕ the natural embedding U~ ( g̃θ ) ↪→ U~(g̃). The Lie coideal structure τ can be extended to τ : U ( g̃θ ) → U(g̃)⊗U ( g̃θ ) such that τ(a1a2) = τ(a1) (a2) + τ(a2) (a1) [10]. However this does not turn ( U ( g̃θ ) , , τ ) into a co-Poisson–Hopf structure. Rather it would be a “one sided coideal-Poisson” extension of the one-sided Lie coideal ( g̃θ, τ ) . In the case of the two-sided coideal structures, the associated Poisson structures are called “Poisson homogeneous spaces”. We refer to [12] and [21] for details on quantization of such structures. 5.2 Yangians and twisted Yangians in Drinfeld J presentation For any set of primitive elements xi1 , xi2 , . . . , xim of any associative bi-algebra over C, set {xi1 , xi2 , . . . , xim} = 1 m! ∑ π xπ(i1)xπ(i2) · · ·xπ(im), where the sum is over all permutations π of the set of indices i1, i2, . . . , im and 〈xi1 , xi2 , . . . , xim〉n = 1 m! ∑ π xπ(i1) · · ·xπ(in) ⊗ xπ(in+1) · · ·xπ(im). (5.3) 12 S. Belliard and V. Regelskis such that ∆({xi1 , xi2 , . . . , xim}) = m∑ n=0 ( m n ) 〈xi1 , xi2 , . . . , xim〉n, (5.4) where ( m n ) denotes the binomial coefficient. Moreover, for any set of indices i1, . . . , im, set a(i1ai2 · · · aim) = ∑ σ aσ(i1)aσ(i2) · · · aσ(im), where the sum is over the cyclic permutations, e.g., α d (ab α e c)d = α d ab α e cd + α d bc α e ad + α d ca α e bd . Next, let us recall the definition of the Yangian as a quantization of the Lie bi-algebra (g[u], δ). Theorem 5.4 ([17]). Let g be a finite-dimensional complex simple Lie algebra. Fix a (non- zero) invariant bilinear form on g and a basis {xa}. There is, up to isomorphism, a unique homogeneous quantization Y(g) := U~(g[u]) of (g[u], δ), with δ given by (4.1). It is topologically generated by elements xa, J(xa) with the defining relations: [xa, xb] = α c ab xc, [xa, J(xb)] = α c ab J(xc), J(λxa + µxb) = λJ(xa) + µJ(xb), (5.5) [J(xa), J([xb, xc])] + [J(xb), J([xc, xa])] + [J(xc), J([xa, xb])] = 1 4~ 2A ijk abc{xi, xj , xk}, (5.6) [[J(xa), J(xb)], J([xc, xd])] + [[J(xc), J(xd)], J([xa, xb])] = 1 4~ 2B ijk abcd{xi, xj , J(xk)}, (5.7) for all xa ∈ g and λ, µ ∈ C. Here A ijk abc = α il a α jm b α kn c αlmn and B ijk abcd = α e cd A ijk abe +α e abA ijk cde. The coproduct is ∆~(xa) = xa ⊗ 1 + 1⊗ xa, ∆~(J(xa)) = J(xa)⊗ 1 + 1⊗ J(xa) + 1 2~[xa ⊗ 1,Ωg]. (5.8) The grading is deg(xa) = 0, deg(~) = 1, deg(J(xa)) = 1. The counit is given by ε~(xa) = ε~(J(xa)) = 0. The antipode is S(xa) = −xa, S(J(xa)) = −J(xa) + 1 4~cgxa. (5.9) An outline of the proof can be found in [13, Section 12.1]. Let us make a remark on the Drinfeld terrific relations (5.6) and (5.7), which are deformations of the relations (3.3) and (3.4), respectively. The coproduct (5.8) is a solution of the quantization condition satisfying the co- associativity property (1⊗∆~) ◦∆~ = (∆~ ⊗ 1) ◦∆~. The right-hand sides (rhs) of the terrific relations are tailored so that ∆~ : Y(g)→ Y(g)⊗Y(g) is a homomorphism of algebras. Indeed, consider the co-action on the left-hand side (lhs) of (5.6). The linear terms in ~ vanish due to the Jacobi identity. What remains are the quadratic terms in ~, that are cubic and totally symmetric in xa. Hence the rhs of (5.6) must be of the form ~2Aijkabc{xi, xj , xk} for some set of coefficients Aijkabc ∈ C. By comparing the terms on the both sides and using the Jacobi identity one finds Aijkabc = A ijk abc. Relation (5.7) is obtained in a similar way.3 We now define twisted Yangians as quantizations of Lie coideals of twisted half-loop algebras for both θ 6= id and θ = id cases. 3Showing (5.7) is more complicated since it requires a heavy usage of (5.6) and Jacobi identity. We were unable to locate the explicit proof of the Drinfeld terrific relations in the mathematical literature available to us. Drinfeld J Presentation of Twisted Yangians 13 Theorem 5.5. Let ( g, gθ ) 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 gθ is the positive eigenspace of θ. Let {Xα, Yp} be a symmetric space basis of g with respect to θ. There is, up to isomorphism, a unique homogeneous quantization Y ( g, gθ )tw := U~ ( g[u]θ ) of ( g[u]θ, τ ) , where τ is given by (4.5). It is topologically generated by elements Xα, B(Yp) with the defining relations: [Xα, Xβ] = f γ αβXγ , [Xα,B(Yp)] = g q αpB(Yq), B(aYp + bYq) = aB(Yp) + bB(Yq), (5.10) [B(Yp),B(Yq)] + ∑ α (c̄(α)) −1w α pq w rs α [B(Yr),B(Ys)] = ~2 ∑ λ,µ,ν Λλµνpq {Xλ, Xµ, Xν}, (5.11) [[B(Yp),B(Yq)],B(Yr)] + 2c−1g ∑ α (κm)tuw α pq g s rα [[B(Ys),B(Yt)],B(Yu)] = ~2 ∑ λ,µ,u Υλµu pqr {Xλ, Xµ,B(Yu)}, (5.12) for all Xα, Yp ∈ g and a, b ∈ C. Here Λλµνpq = 1 3 ( gµtpg λu q + ∑ α (c̄(α)) −1w α pq w rs α gµtrgλus ) w ν tu , (5.13) Υλµu pqr = 1 4 ∑ α ( w α st g λs p g µtq g u αr + ∑ β w α pq f λβ α g µsr g u βs ) + 1 2c −1 g ∑ α,γ (κm)vxw γ pq g y rγ ( w α st g λs y g µtv g u αx + ∑ β w α yv f λβ α g µsx g u βs ) . (5.14) The coideal structure is given by the co-action ~ : Y ( g, gθ )tw → Y(g)⊗ Y ( g, gθ )tw such that ~(Xα) = Xα ⊗ 1 + 1⊗Xα, ~(B(Yp)) = ϕ(B(Yp))⊗ 1 + 1⊗B(Yp) + ~[Yp ⊗ 1,ΩX ], (5.15) where the embedding ϕ : Y ( g, gθ )tw → Y(g) is ϕ(B(Yp)) = J(Yp) + 1 4~[Yp, CX ]. (5.16) The grading is deg(Xα) = 0, deg(~) = 1, deg(B(Yp)) = 1. The counit is ε~(Xα) = ε~(B(Yp)) = 0 for all non-central Xα. In the case when gθ is reductive with its centre k generated by Xz, the counit ε~(Xz) = c with c ∈ C. In the case when gθ is reductive, the one-dimensional representation of gθ is parametrized by the 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 [2]. Similar results can also be deduced for types B, C and D. Theorem 5.6. Let g be a finite-dimensional simple complex Lie algebra of rank(g) ≥ 2. Fix a (non-zero) invariant bilinear form on g and a basis {xi}. There is, up to isomorphism, a unique homogeneous quantization Y(g, g)tw := U~(g[u2]) of (g[u2], τ), where τ is given by (4.6). It is topologically generated by elements xi, G(xi) with the defining relations: [xa, xb] = α c ab xc, [xa,G(xb)] = α c ab G(xc), G(λxa + µxb) = λG(xa) + µG(xb), (5.17) [G(xa),G([xb, xc])] + [G(xb),G([xc, xa])] + [G(xc),G([xa, xb])] = ~2Ψijk abc{xi, xj ,G(xk)}+ ~4 ( Φijk abc{xi, xj , xk}+ Φ ijklm abc {xi, xj , xk, xl, xm} ) , (5.18) 14 S. Belliard and V. Regelskis for all xa ∈ g and λ, µ ∈ C. Here Ψijk abc = α d (ab α k c)r h rij d − α k dr α d (ab h rij c) , (5.19) Φ ijklm abc = 1 5 ( α i rsα d (ab h rjk c) h slm d −Ψjkr abch ilm r ) , (5.20) Φijk abc = 1 9 ( α d (ab W ijk c)d + 1 6Φ (ix(yzj)) abc α r xyα k rz − ( Ψxjy abch kzr y α s zxα i rs + Ψxyz abc h rsk z α i rxα j ys )) , (5.21) with W ijk cd = α i rsh rxy c ( hszkd α j xt α t yz + hsztd α k xt α j yz ) + (( h xyz c hefkd − hxyzd hefkc ) α t yeα i zt α j xf + h jxy c h kzr d α s xr ( α t zyα i st + α t syα i zt ) ) , (5.22) and h bcda = φ bcd a + 2ψ bcd a , h bcd a = φ bcd a − ψ bcd a , ψ bcd a = 1 12 ( α jd a α bc j + α jc a α bd j ) , φ bcd a = 1 24c −1 g ∑ π ( α jk a α π(d)r j α π(b)s k α π(c) sr ) . The coideal structure is given by the co-action ~ : Y(g, g)tw → Y(g)⊗ Y(g, g)tw defined by ~(xa) = xa ⊗ 1 + 1⊗ xa, (5.23) ~(G(xa)) = ϕ(G(xa))⊗ 1 + 1⊗ G(xa) + ~[J(xa)⊗ 1,Ωg] + 1 4~ 2 ( [[xa ⊗ 1,Ωg],Ωg] + c−1g α bc a [[xc ⊗ 1,Ωg], [xb ⊗ 1,Ωg]] ) , (5.24) where the embedding ϕ : Y(g, g)tw → Y(g) is ϕ(G(xa)) = c−1g α bc a [J(xc), J(xb)] + 1 4~[J(xa), Cg]. (5.25) The grading on Y(g, g)tw is deg(xa) = 0, deg(~) = 1, deg(G(xa)) = 2. The co-unit is ε~(xi) = ε~(G(xi)) = 0. Remark 5.7. The co-action (5.24) can be equivalently written as ~(G(xa)) = ϕ(G(xa))⊗ 1 + 1⊗ G(xa) + ~[J(xa)⊗ 1,Ωg] + ~2 ( h bcda xb ⊗ {xc, xd}+ h bcd a {xc, xd} ⊗ xb ) . (5.26) Remark 5.8. The algebras Y(g, gθ)tw and Y(g, g)tw may be considered as flat deformations of the twisted current algebras U(g[u]θ) and U(g[u2]), respectively. It is clear that Y(g, gθ)tw/~Y(g, gθ)tw ∼= U(g[u]θ) and Y(g, g)tw/~Y(g, g)tw ∼= U(g[u2]). The flatness of the deformations then follows from the Poincaré–Birkhoff–Witt (PBW) theorem which is due the fact that both twisted Yan- gians can be embedded to the Yangian Y~(g); the PBW theorem for Y~(g) was demonstrated in [31]. The proof of Theorems 5.5 and 5.6 follows using the same arguments outlined in [13, Sec- tion 21.1]. The uniqueness of co-actions (5.15) and (5.23), (5.24) is demonstrated in Sections 7.3 and 7.4; in particular, the co-action ~ is determined uniquely (up to an isomorphism discussed in the paragraph below) by the coideal compatibility identities (5.1), (5.2) and the property (4) of Definition 5.3. The map (5.16) is the MacKay twisted Yangian formula presented in [16]. The challenging task is to obtain the horrific relations (5.11), (5.12) and (5.18), which are quan- tizations of (3.7), (3.8) and (3.10), respectively. A proof of the first two relations is given in Section 7.5. Proving (5.18) is substantially more difficult. We have given an outline of a proof in Section 7.6. Drinfeld J Presentation of Twisted Yangians 15 Recall that the Yangian Y(g) has a one-parameter group of automorphisms κc, with c ∈ C, given by κc(xa) = xa and κc(J(xa)) = J(xa) + ~cxa, which is compatible with both algebra and Hopf algebra structure. An analogue of this automorphism for the twisted Yangians is a one-parameter family of co-actions (κc ⊗ id) ◦ ~, which for Y(g, gθ)tw is equivalent to one- parameter family of embeddings κc ◦ ϕ; the later does not apply to Y(g, g)tw due to the ~-order term in (5.24). Remark 5.9. We strongly believe that expressions in (5.19)–(5.21) could be further reduced to a more elegant and compact form. We have succeeded to find such a form when g = sl3: Ψijk abc = 1 3A ijk (abc) + α d (ab α k c)l φ lij d − α k dl α d (ab φ lij c) , Φijk abc = −1 6A ijk abc, Φ ijkln abc = 1 36α ir (a α js b A klm c)rs . 6 Coideal subalgebras of the Yangian Y(sl3) In this section we present three examples of twisted Yangians Y ( g, gθ )tw when g = sl3 and gθ = so3 or gθ = gl2 (both θ 6= id cases), or gθ = sl3 (θ = id case). For ease of notation we will denote generators of the former two algebras by symbols h, e, f, k and H, E, F. We start by recalling the Chevalley–Serre presentation of the sl3 Lie algebra and its Yangian. 6.1 The sl3 Lie algebra and the Yangian Y(sl3) Lie algebra sl3 in the Chevalley–Serre presentation is generated by elements ei, fi, hi with i = 1, 2 subject to the defining relations [ei, fj ] = δijhi, [hi, ej ] = aijej , [hi, fj ] = −aijfj , [ei, [ei, ei±1]] = 0, [fi, [fi, fi±1]] = 0, (6.1) where aii = 2 and a12 = −1 = a21 are the matrix elements of the Cartan matrix of sl3. The last two relations are called the Serre relations. Vector space basis of sl3 contains eight elements. In addition to the elements given above there are two more root vectors that correspond to the non-simple roots of sl3, namely e3 = [e1, e2] and f3 = [f2, f1]. The defining relations for the linear basis are obtained by dropping the Serre relations in (6.1) and adding [e1, e2] = e3, [e1, e3] = [e2, e3] = 0, [f1, f2] = −f3, [f1, f3] = [f2, f3] = 0, [e1, f3] = −f2, [e2, f3] = f1, [f1, e3] = e2, [f2, e3] = −e1, [hi, f3] = −f3, [hi, e3] = e3, [e3, f3] = h1 + h2. (6.2) The quadratic Casimir operator is Cg = ∑ 1≤i≤3(eifi + fiei) + 2 3 ∑ 1≤i≤j≤2 hihj and cg = 6. Example 6.1. The Yangian Y(sl3) is the unital associative algebra with sixteen generators ei, fi, hj , J(ei), J(fi), J(hj) with i = 1, 2, 3 and j = 1, 2. The defining relations (5.5), (5.6) are given by (6.1), (6.2) and [J(h1), J(h2)] = 3 4~ 2({e3, f1, f2} − {e1, e2, f3}). (6.3) The remaining relations defined by (5.6) are obtained by the adjoint action of degree-0 generators on (6.3). (The same applies to all horrific relations in the examples given below.) The Hopf algebra structure on Y(sl3) is given by (5.8), (5.9) with Ωg = ∑ 1≤i≤3(ei ⊗ fi + fi ⊗ ei) + 1 3 ∑ 1≤i≤j≤2(hi ⊗ hj + hj ⊗ hi). 16 S. Belliard and V. Regelskis 6.2 Twisted Yangian Y(sl3, so3) tw Let involution θ be defined by θ : e1 7→ −e2, f1 7→ −f2, h1 7→ h2. The action of θ on the remaining elements of g = sl3 is deduced from the constraint θ2 = id. The symmetric space basis is given by gθ(= h) = {h = h1 + h2, e = e1 − e2, f = f1 − f2} and m = {h1 − h2, e1 + e2, f1 + f2, e3, f3}. The positive eigenspace of θ forms the orthogonal subalgebra so3 ⊂ sl3. We denote the generators of this subalgebra by h, e, f. They generate the degree-0 subalgebra of the twisted Yangian Y(sl3, so3) tw. We denote the degree-1 generators by H, E, F, E2, F2. Example 6.2. The twisted Yangian Y(sl3, so3) tw is the unital associative algebra with eight generators h, e, f, H, E, F, E2, F2. The defining relations are the degree-0 Lie relations (of the so3 Lie algebra) [e, f] = h, [h, e] = e, [h, f] = −f, degree-1 Lie relations [h,E] = E, [h,E2] = 2E2, [H, e] = 3E, [e,E] = 2E2, [e,E2] = 0, [h,F] = −F, [h,F2] = −2F2, [H, f] = −3F, [f,F] = −2F2, [f,F2] = 0, [f,E2] = E, [e,F2] = −F, [e,F] = H, [E, f] = H, [H, h] = 0, degree-2 horrific relation [E,F]− [E2,F2] = 1 4~ 2 ( {h, h, h} − 3{e, f, h} ) , degree-3 horrific relation [[E,F],H] = 3 2~ 2 ( {E2, f, f} − {F2, e, e} ) + 15 4 ~ 2 ( {E, f, h} − {F, e, h} ) . The co-action is given by ~(x) = ϕ(x)⊗ 1 + 1⊗ x, ~(Y) = ϕ(Y)⊗ 1 + 1⊗ Y + ~[ϕ0(Y)⊗ 1,Ωh], for all x ∈ {h, e, f} and Y ∈ {E,F,H,E2,F2} such that ϕ(e) = e1 − e2, ϕ(f) = f1 − f2, ϕ(h) = h1 + h2, ϕ0(E) = e1 + e2, ϕ0(F) = f1 + f2, ϕ0(H) = h1 − h2, ϕ0(E2) = e3, ϕ0(F2) = f3, and ϕ(E) = J(e1) + J(e2) + 1 4~[e1 + e2, CX ], ϕ(E2) = J(e3) + 1 4~[e3, CX ], ϕ(F) = J(f1) + J(f2) + 1 4~[f1 + f2, CX ], ϕ(F2) = J(f3) + 1 4~[f3, CX ], ϕ(H) = J(h1)− J(h2) + 1 4~[h1 − h2, CX ]. Here CX = 1 2 ( ef + fe + h2 ) ∈ U(h), Ωh = 1 2(ϕ⊗ id) ◦ ( e⊗ f + f ⊗ e + h⊗ h ) ∈ g⊗ h. The co-unit is ε(x) = ε(Y) = 0. Drinfeld J Presentation of Twisted Yangians 17 6.3 Twisted Yangian Y(sl3, gl2) tw Let involution θ be defined by θ : e1 7→ e1, f1 7→ f1, e2 7→ −e2, f2 7→ −f2, hi 7→ hi, In this case gθ = {h = h1, k = 2h2 +h1, e = e1, f = f1} ∼ gl2 and m = {e2, f2, e3, f3}. We denote the corresponding degree-1 generators by E2, F2, E3, F3. Example 6.3. The twisted Yangian Y(sl3, gl2) tw is the unital associative algebra with eight generators h, e, f, k, E2, F2, E3, F3. The defining relations are the degree-0 Lie relations (of the gl2 Lie algebra)4 [e, f] = h, [h, e] = 2e, [h, f] = −2f, [e, k] = [f, k] = [h, k] = 0, degree-1 Lie relations [h,E2] = −E2, [h,E3] = E3, [k,Ei] = 3Ei, [h,F2] = F2, [h,F3] = −F3, [k,Fi] = −3Fi, [e,E2] = E3, [e,F3] = −F2, [e,F2] = [e,E3] = 0, [f,F2] = F3, [f,E3] = E2, [f,E2] = [f,F3] = 0, degree-2 horrific relations [E2,E3] = 0, [F2,F3] = 0, degree-3 horrific relations [E2, [E2,F3]] = 2~2{E2, f, k}, [F2, [E3,F2]] = −2~2{F2, f, k}. The co-action given by, for x ∈ {h, e, f, k}, ~(x) = ϕ(x)⊗ 1 + 1⊗ x, ϕ(h) = h1, ϕ(k) = 2h2 + h1, ϕ(e) = e1, ϕ(f) = f1 and, for i = 1, 2, ~(Ei) = ϕ(Ei)⊗ 1 + 1⊗ Ei + ~[ei ⊗ 1,Ωh], ϕ(Ei) = J(ei) + 1 4~[ei, CX ], ~(Fi) = ϕ(Fi)⊗ 1 + 1⊗ Fi + ~[fi ⊗ 1,Ωh], ϕ(Fi) = J(fi) + 1 4~[fi, CX ]. Here CX = ef + fe + 1 2h 2 + 1 6k 2 ∈ U(h), Ωh = (ϕ⊗ id) ◦ ( e⊗ f + f ⊗ e + 1 2h⊗ h + 1 6k⊗ k ) ∈ g⊗ h. The co-unit is ε(k) = c ∈ C, ε(x) = ε(Y) = 0 for all x ∈ {e, f, h} and Y ∈ {Ei,Fi} with i = 1, 2. 4The standard gl2 basis {eij}i,j=1,2 with the defining relations [eij , ekl] = δkjeil− δilekj is obtained by setting e11 = −(h + k)/2, e22 = (h− k)/2, e12 = f and e21 = e. 18 S. Belliard and V. Regelskis 6.4 Twisted Yangian Y(sl3, sl3) tw In this case the involution is trivial, θ = id, hence gθ = sl3 and m = ∅. Example 6.4. The twisted Yangian Y(sl3, sl3) tw is the unital associative algebra with sixteen generators ei, fi, hj , G(ei), G(fi), G(hj) with i = 1, 2, 3 and j = 1, 2, obeying the standard sl3 Lie algebra relations of the Cartan–Chevalley presentation and the standard degree-2 Lie relations (5.17) and the following degree-4 horrific relation [G(h1),G(h2)] = ~2 ( {f1, f2,G(e3)} − {e1, e2,G(f3)})− 1 2~ 4({f1, f2, e3} − {e1, e2, f3} ) + ~2 ( {e3, f2,G(f1)}+ {e3, f1,G(f2)} − {f3, e2,G(e1)} − {f3, e1,G(e2)} ) + 1 2~ 2 ( {h1, f2,G(e2)} − {h1, e2,G(f2)} − {h2, f1,G(e1)}+ {h2, e1,G(f1)} ) + 1 4~ 2 ( {h1, e1,G(f1)} − {h1, f1,G(e1)} − {h2, e2,G(f2)}+ {h2, f2,G(e2)} + {h1−h2, f3,G(e3)} − {h1−h2, e3,G(f3)} ) + 1 4~ 4 ( {e1, e1, e2, f1, f3}+ {e1, e2, e2, f2, f3}+ {e1, e2, e3, f3, f3} − {f1, f1, f2, e1, e3} − {f1, f2, f2, e2, e3} − {f1, f2, f3, e3, e3} ) + 1 12~ 4 ( {e1, e2, h1, h1, f3}+ {e1, e2, h1, h2, f3}+ {e1, e2, h2, h2, f3} − {f1, f2, h1, h1, e3} − {f1, f2, h1, h2, e3} − {f1, f2, h2, h2, e3} ) . The co-action given by (5.23), (5.24) with ϕ(G(xi)) = K(xi)+ 1 4~[J(xi), Cg] and with K(xi) = c−1g α kj i [J(xj), J(xk)], where5 K(e1) = 1 3([J(e3), J(f2)] + [J(h1), J(e1)]), K(f1) = 1 3([J(f3), J(e2)]− [J(h1), J(f1)]), K(e2) = 1 3([J(f1), J(e3)] + [J(h2), J(e2)]), K(f2) = 1 3([J(e1), J(f3)]− [J(h2), J(f2)]), K(e3) = 1 3([J(e1), J(e2)] + [J(h1 + h2), J(e3)]), K(f3) = 1 3([J(f1), J(f2)]− [J(h1 + h2), J(f3)]), K(h1) = 1 3 ( [J(e1), J(f1)] + ∑ 1≤i≤3 [J(fi), J(ei)] ) , K(h2) = 1 3 ( [J(e2), J(f2)] + ∑ 1≤i≤3 [J(fi), J(ei)] ) . The co-unit is ε(xi) = ε(G(xi)) = 0. 7 Proofs In the remaining part of the paper we provide proofs omitted in the previous sections. 7.1 A proof of Proposition 3.1 We follow the arguments outlined in [13, Section 12.1] and fill in the gaps. Recall that given a simple Lie algebra g the half-loop Lie algebra L+ is generated by the elements x (k) a with k ≥ 0 and a = 1, . . . ,dim(g), and satisfying (3.1). Let L̃+ denote the algebra generated by the elements xa and J(xa) satisfying (3.2)–(3.4). The map ϕ : L̃+ → L+ given by xa 7→ x (0) a , J(xa) 7→ x (1) a is an algebra homomorphism. It is a direct computation to check that the image of (3.2)–(3.4) holds in L+. 5The non-zero structure constants α kj i for sl3 in the Cartan–Chevalley basis can be read from here. Drinfeld J Presentation of Twisted Yangians 19 To prove that ϕ is surjective we need to show that elements x (0) a and x (1) a generate the whole L+. Indeed, since g is simple, we have [g, g] = g, or equivalently [ L(1),L(1) ] = L(2), where L(k) = span { x (k) a } . By the same arguments we must have, for all k > 0, that [ L(1),L(k) ] = L(k+1). This shows that L+ is generated by L(0) and L(1). Hence it only remains to show that ϕ is injective. We know that L+ = ⊕ k≥0L (k), with dim ( L(k) ) = dim(g), as a vector space. We need to show that an analogous statement holds for L̃+. Denote J (0)(xa) = xa, J (1)(xa) = J(xa) and define recursively J (k)(xa) = c−1g αbca [ J (1)(xc), J (k−1)(xb) ] for all k ≥ 1. (7.1) We need to show that[ J (`)(xa), J (k−`)(xb) ] = α c ab J (k)(xc) for all ` ≤ k, (7.2) so that L̃+ = ⊕ k≥0 L̃ (k) with L̃(k) = span{J (k)(xa)} and dim(L̃(k)) = dim(g). We first demon- strate that[ J (k−`)(xa), J (`)(xb) ] = [ J (k−`′)(xa), J (`′)(xb) ] for all ` < `′ ≤ k. (7.3) We proceed by induction on k, the result being clear when k = 0, 1, by (3.2). For k = 2 there are two inequivalent identities that need to be shown: with ` = 0, `′ = 2 and with ` = 0, `′ = 1. The first identity follows by (7.1) and co-Jacobi identity:[ J (2)(xa), J (0)(xb) ] = c−1g α dc a [[ J (1)(xc), J (1)(xd) ] , J (0)(xb) ] = c−1g α dc a ( α e cb [ J (1)(xe), J (1)(xd) ] + α e db [ J (1)(xc), J (1)(xe) ]) = α c ab J (2)(xc). In a similar way we obtain that [ J (0)(xa), J (2)(xb) ] = α c ab J (2)(xc) thus yielding the required identity. To obtain the second identity we need to contract (3.3) with α cb d giving cg [ J (1)(xa), J (1)(xd) ] + α cb d α e ca [ J (1)(xb), J (1)(xe) ] + α cb d α e ab [ J (1)(xc), J (1)(xe) ] = 0. Now rename the indices b → c, c → e, e → b in the third term and use co-Jacobi identity together with (7.1). This gives [ J (1)(xa), J (1)(xd) ] = α c adJ (2)(xc) = [ J (0)(xa), J (2)(xd) ] . Next, assuming inductively (7.3), we have[ J (s)(xc), [ J (r−`)(xa), J (`)(xa) ]] = α b ca ([ J (r−`+s)(xb), J (`)(xa) ] + [ J (r−`)(xa), J (`+s)(xb) ]) = 0 for all ` ≤ r ≤ k, r − `+ s ≤ k and `+ s ≤ k giving[ J (r−`+s)(xb), J (`)(xa) ] = [ J (`+s)(xb), J (r−`)(xa) ] . Taking all the allowed `, r, s such that r + s ≤ k + 1 we obtain all the necessary identities thus completing the induction. We are now ready to prove (7.2). We use induction on k. The cases with k ≤ 2 are discussed above. Thus, by induction hypothesis, we have[ J (1)(xd), [ J (k−`)(xa), J (`)(xb) ]] = α c ab [ J (1)(xd), J (k)(xc) ] . 20 S. Belliard and V. Regelskis Now contract the lhs with α ad e , use (7.1), (7.3) and co-Jacobi identity: α ad e [ J (1)(xd), [ J (k−`)(xa), J (`)(xb) ]] = cg [ J (k−`+1)(xe), J (`)(xb) ] + α ad e α f db [ J (k−`)(xa), J (`+1)(xf ) ] = cg [ J (k−`+1)(xe), J (`)(xb) ] − 1 2cgα d eb J (k+1)(xd). By doing the same for the rhs we get α ad e α c ab [ J (1)(xd), J (k)(xc) ] = 1 2cgα a eb J (k+1)(xa) yielding [ J (k−`+1)(xe), J (`)(xb) ] = α d eb J (k+1)(xd), which completes the induction. Therefore L̃+ = ⊕ L̃(k). By setting ϕ : J (k)(xa) 7→ x (k) a we obtain a bijection of vector spaces, hence L̃+ ∼= L+. 7.2 A proof of Proposition 3.2 The idea behind the proof is very similar to that of the prove above. Recall that H+ is the subalgebra of L+ generated by elements X (2k) α and Y (2k+1) p with k ≥ 0 and satisfying (3.5). Let H̃+ denote the algebra generated by the elements Xα and B(Yp) satisfying (3.6)–(3.8). The map ψ : H̃+ → H+ given by Xα 7→ X (0) α , B(Yp) 7→ Y (1) p is an algebra homomorphism. It is easy to see that the image of (3.6) holds in H+. To see that the same is true for (3.7) and (3.8) we additionally need to use (2.9) and (2.10). To prove that ψ is surjective we need to show that elements X (0) α and Y (1) p , where Xα runs over a basis for h and Yp runs over a basis for m, generate the whole H+. Indeed, since g = m ⊕ h is simple, we must have [m,m] = h and [m, h] = m, or equivalently [ H(1),H(1) ] = H(2) and[ H(1),H(2) ] = H(3), where H(2k) = span { X (2k) α } and H(2k+1) = span { Y (2k+1) p } . By the same arguments we must have, for all k > 0, that [ H(1),H(k) ] = H(k+1). This shows that H+ is generated by H(0) and H(1). It remains to show that ψ is injective. We know that H+ = ⊕ k≥0H (k), with dim ( H(2k) ) = dim(h) and dim ( H(2k+1) ) = dim(m), as a vector space. We need to show that an analogous statement holds for H̃+. Let B(0)(Xα) = Xα, B(1)(Yp) = B(Yp) and define recursively, for k ≥ 0, B(2k+2)(Xα) = (c̄(α)) −1w qp α [ B(1)(Yp), B (2k+1)(Yq) ] , (7.4) B(2k+1)(Yp) = 2c−1g ∑ α g αqp [ B(1)(Yq), B (2k)(Xα) ] . (7.5) We need to show that, for all ` ≤ k,[ B(2k−2`)(Xα), B(2`)(Xβ) ] = f γ αβB (2k)(Xγ), (7.6)[ B(2k−2`)(Xα), B(2`+1)(Yp) ] = g q αpB (2k+1)(Yq), (7.7)[ B(2k−2`+1)(Yp), B (2`+1)(Yq) ] = ∑ α w α pq B (2k+2)(Xα), (7.8) so that H̃+ = ⊕ k≥0 H̃ (k), with H̃(2k) = span { B(2k)(Xα) } and H̃(2k+1) = span { B(2k+1)(Yp) } , as a vector space. We start by showing (7.6)–(7.8) for small k that follow from (3.6)–(3.8). First, notice that (7.4) implies that (3.7) is equivalent to [ B(1)(Yp), B (1)(Yq) ] = ∑ αw α pq B (2)(Xα). Also, notice that, by (2.8), (2.9) and (7.4),∑ α w α pq [ B(2)(Xα), B(1)(Yr) ] = ∑ α (c̄(α)) −1w α pq w st α [[ B(1)(Yt), B (1)(Ys)], B (1)(Yr) ] = [[ B(1)(Yp), B (1)(Yq) ] , B(1)(Yr) ] . Drinfeld J Presentation of Twisted Yangians 21 Then, by (2.8), (2.10) and (7.4), (7.5),∑ α w α pq g s rαB (3)(Ys) = ∑ α,β 2(cg) −1w α pq g s rαg uβ s [ B(2)(Xβ), B(1)(Yu) ] = ∑ α,β 2(κm)tu(cgc̄(β)) −1w α pq g s rα w β st w vx β [[ B(1)(Yx), B(1)(Yv) ] , B(1)(Yu) ] = 2c−1g (κm)tuw α pq g s rα [[ B(1)(Ys), B (1)(Yt) ] , B(1)(Yu) ] . By combining the expressions above with (3.8) we obtain [ B(2)(Xα), B(1)(Yr) ] = g s αr B (3)(Ys). Consequently, we have that w rp β [ B(1)(Yp), [ B(2)(Xα), B(1)(Yr) ]] = w rp β g s αr [ B(1)(Yp), B (3)(Ys) ] giving w rp β g q pα [ B(3)(Yq), B (1)(Yr) ] + ∑ γ w rp β w γ pr [ B(2)(Xα), B(2)(Xγ) ] = w pr β g q αp [ B(1)(Yr), B (3)(Yq) ] . Using (2.6), (2.9), (7.4), (7.5) and antisymmetry in α and β we obtain c̄(β) [ B(2)(Xα), B(2)(Xβ) ] = ( w pr β g q αp − w pr α g q βp )[ B(1)(Yr), B (3)(Yq) ] = c̄(β)f γ αβB (4)(Xγ), which gives [ B(2)(Xα), B(2)(Xβ) ] = f γ αβB (4)(Xγ). Then, a direct computation using (7.4), (7.5) yields [ B(0)(Xα), B(2)(Xα) ] = f γ αβB (2)(Xγ) and [ B(0)(Xα), B(3)(Yr) ] = g s αr B (3)(Ys) and finally[ B(0)(Xα), B(4)(Xβ) ] = f γ αβB (4)(Xγ). Next we show the following identities that will be necessary in proving (7.7) and (7.8):[ B(2k−2`)(Xα), B(2`+1)(Yp) ] = [ B(2k−2`′)(Xα), B(2`′+1)(Yp) ] , (7.9)[ B(2k−2`+1)(Yp), B (2`+1)(Yq) ] = [ B(2k−2`+1)(Yp), B (2`+1)(Yq) ] . (7.10) We proceed by induction on k, the result being clear when k = 0 and when k = 1 for (7.9). Assuming induction we have[ B(2s)(Xα), [ B(2r−2`+1)(Yp), B (2`+1)(Yp) ]] = g q αp ([ B(2r−2`+2s+1)(Yq), B (2`+1)(Yp) ] + [ B(2r−2`+1)(Yp), B (2`+2s+1)(Yq) ]) = 0 for all ` ≤ r ≤ k, r − `+ s ≤ k and `+ s ≤ k, therefore we must have[ B(2r+2s−2`+1)(Yq), B (2`+1)(Yp) ] = [ B(2s+2`+1)(Yq), B (2r−2`+1)(Yp) ] . Taking all the allowed `, r, s such that r + s ≤ k + 1 we obtain all the necessary identities thus completing the induction. Then, by (7.4), it follows that, for all ` ≤ k, B(2k+2)(Xα) = (c̄(α)) −1w qp α [ B(2`+1)(Yp), B (2k−2`+1)(Yq) ] . (7.11) Relation (7.9) is proved similarly, using[ B(2s+1)(Yq), [ B(2r−2`+1)(Yp), B (2`+1)(Yp) ]] = ∑ α w α qp ([ B(2r−2`+2s+2)(Xα), B(2`+1)(Yp) ] + [ B(2r−2`+1)(Yp), B (2`+2s+2)(Xα) ]) = 0 22 S. Belliard and V. Regelskis and implies that, for all ` ≤ k, B(2k+1)(Yp) = 2c−1g ∑ α g αqp [ B(2`+1)(Yq), B (2k−2`)(Xα) ] . (7.12) We also need to show that, for all ` < `′ ≤ k, B(2k+2)(Xα) = (c(α)) −1f γβ α [ B(2k−2`)(Xβ), B(2`+2)(Xγ) ] . (7.13) Again, we proceed by induction on k. The base of induction was already shown above. Hence, assuming inductively (7.13), we obtain f γβ α [ B(2k−2`)(Xβ), B(2`+2)(Xγ) ] = (c̄(γ)) −1f γβ α w qp γ [ B(2k−2`)(Xβ), [ B(2`′+1)(Yp), B (2`−2`′+1)(Yq) ]] = (c̄(γ)) −1f γβ α w qp γ ( g r βp [ B(2k−2`+2`′+1)(Yr), B (2`−2`′+1)(Yq) ] + g r βq [ B(2`′+1)(Yp), B (2k−2`′+1)(Yr) ]) = (c̄(α)) −1c(α)w rp α [ B(2`+1)(Yp), B (2k−2`+1)(Yr) ] . Here we used (2.6), (2.9) and (7.12). By combining the expression above with (7.11) we complete the induction. We are now in position to prove (7.6)–(7.8) using induction on k. Notice that the base of induction for each of (7.6)–(7.8) was already shown above. The induction step for (7.6) is analogous to the one in Section 7.1, thus we do not repeat the proof. For (7.8), assuming induction, we have[ B(2)(Xβ), [ B(2k−2`+1)(Yp), B (2`+1)(Yq) ]] = ∑ α w α pq [ B(2)(Xβ), B(2k+2)(Xα) ] . By (2.6), (2.10) and (7.12) we obtain∑ β g pβr [ B(2)(Xβ), [ B(2k−2`+1)(Yp), B (2`+1)(Yq) ]] = ∑ β g pβr g s βp [ B(2k−2`+3)(Ys), B (2`+1)(Yq) ] + ∑ β g pβr g s βq [ B(2k−2`+1)(Yp), B (2`+3)(Ys) ] = 1 2cg [ B(2k−2`+3)(Yr), B (2`+1)(Yq) ] − 1 4 ∑ β c̄(β)w β rq B(2k+4)(Xβ). By (2.6) and (7.11) we get∑ α,β g pβr w α pq [ B(2)(Xβ), B(2k+2)(Xα) ] = 1 2 ∑ α,β,γ w γ rq f αβ γ [ B(2)(Xβ), B(2k+2)(Xα) ] = 1 4 ∑ γ c(γ)w γ rq B(2k+4)(Xγ) giving [ B(2k−2`+3)(Yr), B (2`+1)(Yq) ] = ∑ β w β rq B(2k+4)(Xβ), which combined with (7.10) com- pletes the induction. Finally, for (7.7), assuming induction, we have∑ β g βpq [ B(1)(Yp), [ B(2k−2`)(Xα), B(2`)(Xβ) ]] = ∑ β g βpq f γ αβ [ B(1)(Yp), B (2k)(Xγ) ] . (7.14) Using (2.10) and (7.12) we compute∑ β g βpq [ B(1)(Yp), [ B(2k−2`)(Xα), B(2`)(Xβ) ]] Drinfeld J Presentation of Twisted Yangians 23 = ∑ β g βpq g r pα [ B(2k−2`+1)(Yr), B (2`)(Xβ) ] + ∑ β g βpq g r pβ [ B(2k−2`)(Xα), B(2`+1)(Yr) ] = ∑ γ g γrq g p rα [ B(2k−2`+1)(Yp), B (2`)(Xγ) ] + 1 2cg [ B(2k−2`)(Xα), B(2`+1)(Yq) ] . Subtracting the first term in the last equality above from the rhs of (7.14) and using (2.6), (7.12) we get∑ γ (∑ β g βpq f γ αβ − g γr q g p rα )[ B(1)(Yp), B (2k)(Xγ) ] = ∑ γ g r αq g γp r [ B(1)(Yp), B (2k)(Xγ) ] = 1 2cgg r αq B (2k+1)(Yr) yielding [ B(2k−2`)(Xα), B(2`+1)(Yq) ] = g r αq B (2k+1)(Yr), which completes the induction. Thus we have proved that H̃+ = ⊕ H̃(k). By setting ϕ : B(2k)(Xα) 7→ X (2k) α , B(2k+1)(Yp) 7→ Y (2k+1) p we obtain a bijection of vector spaces, hence H̃+ ∼= H+. 7.3 A proof of the uniqueness of the co-action of Y(g, gθ)tw We need to show that the map ~ given by (5.15) together with (5.16) is the unique map satisfying properties given by Definition 5.1 and property (4) of Definition 5.3. The latter property defines the co-action up to the first order in ~, ~(x) = ϕ(x)⊗ 1 + 1⊗ x+ ~τ(x) + O ( ~2 ) , with x ∈ Y(g, h)tw and ϕ the natural embedding Y(g, h)tw ↪→ Y(g). For the degree-0 generators of H+ the Lie coideal structure is trivial and the minimal form of the co-action is given by ~(Xα) = ϕ(Xα)⊗ 1 + 1⊗Xα. (7.15) The coideal compatibility identities (5.1) and (5.2) and the requirement ϕ(Xα)|~→0 = Xα implies that the natural inclusion for Xα is ϕ : Xα ∈ Y(g, h)tw 7→ Xα ∈ Y(g) and that (7.15) is indeed the unique co-action satisfying the required properties. For the degree-1 generators of H+ the Lie coideal structure is non-trivial. The minimal co-action is ~(B(Yp)) = ϕ(B(Yp))⊗ 1 + 1⊗B(Yp) + ~[Yp ⊗ 1,Ωh]. (7.16) As previously, the co-action must satisfy (5.1) and (5.2) and the requirement ϕ(B(Yp))|~→0 = B(Yp). By (5.2) we have ∆~(ϕ(B(Yp))) = ϕ(B(Yp))⊗ 1 + 1⊗ ϕ(B(Yp)) + ~[Yq ⊗ 1,Ωh]. Consider the ansatz ϕ(B(Yp)) = J(Yp) + ~F (0) p with some degree-0 element F (0) p ∈ Y(g). Recall that ∆~(J(Yp)) = J(Yp)⊗ 1 + 1⊗ J(Yp) + 1 2~[Yp ⊗ 1,Ωg]. We rewrite the equality above as ∆~ ( J(Yp) + ~F (0) p ) − (( J(Yp) + ~F (0) p ) ⊗ 1 + 1⊗ ( J(Yp) + ~F (0) p )) = ~[Yp ⊗ 1,Ωh], which equates to ∆~ ( F (0) p ) − ( F (0) p ⊗ 1 + 1⊗ F (0) p ) = 1 2 [Yp ⊗ 1, 2Ωh − Ωg] = 1 2 ∑ α ( g αqp Yq ⊗Xα − g qαp Xα ⊗ Yq ) = 1 2 ∑ α g αqp (Yq ⊗Xα +Xα ⊗ Yq). 24 S. Belliard and V. Regelskis Thus we find that F (0) p = 1 4 ∑ α g αq p (YqXα+XαYq) + cYp = 1 4 [Yq, CX ]− cYp for any c ∈ C, which we can set to c = 0 without loss of generality (this is equivalent to the composition κ−c ◦ϕ with the shift-automorphism κc : J(x) 7→ J(x) + ~cx of Y~(g) for any x ∈ g) giving ϕ(B(Yp)) = J(Yp) + 1 4~[Yp, CX ], or in other words is there is a one-parameter family of embeddings that are equivalent to each other via the shift-automorphism of Y(g). It remains to verify that the coideal compatibility identities (5.1) and (5.2) hold, which imply that (7.16) is the unique co-action, and to show that ϕ is an algebra homomorphism, namely ϕ([bi, bj ]) = [ϕ(bi), ϕ(bj)] and ~([bi, bj ]) = [ ~(bi), ~(bj)] for all bi, bj ∈ Y(g, h)tw, which follow by a direct computation. 7.4 A proof of the uniqueness of the co-action of Y(g, g)tw Similarly as before, we need to show that the map ~ given by (5.23) and (5.24) together with (5.25) is the unique map satisfying properties given by Definition 5.1 and property (4) of Definition 5.3. We first demonstrate an additional lemma that provides us with necessary identities. Lemma 7.1. The following identities hold in U(g) and U(g)⊗ U(g): [[xi ⊗ 1,Ωg],Ωg] = 1 2(α jc i α ab j + α jb i α ac j )(xa ⊗ {xb, xc} − {xb, xc} ⊗ xa), (7.17) α jk i [[xk ⊗ 1,Ωg], [xj ⊗ 1,Ωg]] = α jk i α cr j α bs k α a sr (xa ⊗ {xb, xc}+ {xb, xc} ⊗ xa), (7.18) 6α jk i α cr j α bs k α a sr = α jk i ∑ π α π(c)r j α π(b)s k α π(a) sr + cg ( α jc i α ab j + α jb i α ac j ) . (7.19) Proof. Recall that [a⊗ b, c⊗ d] = [a, c]⊗ {b, d}+ {a, c} ⊗ [b, d]. For the first identity we have [[xi ⊗ 1,Ωg],Ωg] = α bj i ηkc[xj ⊗ xb, xk ⊗ xc] = α bj i ηkc(α a jk xa ⊗ {xb, xc}+ α a bc {xk, xj} ⊗ xa) = α bj i ηkc(α a jk xa ⊗ {xb, xc} − α a jk {xc, xb} ⊗ xa) by ren. k, j ↔ c, b = α bj i α ca j (xa ⊗ {xb, xc} − {xc, xb} ⊗ xa). For the second identity we have α jk i [[xk ⊗ 1,Ωg], [xj ⊗ 1,Ωg]] = α jk i α br k α cs j [xr ⊗ xb, xs ⊗ xc] = α jk i α cs j α br k (α a rs xa ⊗ {xb, xc}+ α a bc {xs, xr} ⊗ xa) = α jk i α cs j α br k α a rs (xa ⊗ {xb, xc}+ {xc, xb} ⊗ xa) by renaming b, c↔ r, s. The third identity is obtained using the following auxiliary identities: α jk i α cr j α bs k α a sr = α jk i α cr j α as k α b sr + 1 2cgα jc i α ab j , α jk i α cr j α bs k α a sr = α jk i α bs k α ar j α c sr + 1 2cgα jb i α ac j . The first auxiliary identity follows by multiple application of the Jacobi identity. In particular, 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 jk i α cr j + α jr i α kc j ) α s rk α ab s = 1 2α jc i α kr j α s rk α ab s = 1 2cgα jc i α ab j . Drinfeld J Presentation of Twisted Yangians 25 The second auxiliary identity follows by the b ↔ c symmetry and renaming j, s ↔ k, r in the first term on the right hand side. We also have that α jk i ∑ π α π(c)r j α π(b)s k α π(a) sr = 2α jk i ( α cr j α bs k α a sr + α cr j α as k α b sr + α ar j α bs k α c sr ) . Hence taking a double sum of the auxiliary identities and adding 2α cr j α bs k α a sr to both sides of the resulting equality reproduces (7.19) as required. � The degree-0 generators have a trivial Lie coideal structure, hence, by the same arguments as before, ~(xi) = ϕ(xi)⊗ 1 + 1⊗ xi, and ϕ : xi ∈ Y(g, g)tw 7→ xi ∈ Y(g) is the natural inclusion. For the degree-2 generators we have ~(G(xi)) = ϕ(G(xi))⊗ 1 + 1⊗ G(xi) + ~[J(xi)⊗ 1,Ωg] + ~2W (0) i , with W (0) i being a degree-0 element in Y(g)⊗ Y(g, g)tw, which can also be viewed as an element in Y(g)⊗ Y(g) via the natural inclusion. Coinvariance (5.2) implies ∆~ ( ϕ(G(xi)) ) = ϕ(G(xi))⊗ 1 + 1⊗ ϕ(G(xi)) + ~[J(xi)⊗ 1,Ωg] + ~2W (0) i . (7.20) We also must have ϕ(G(xi))|~→0 = c−1g α jk i [J(xk), J(xj)]. Let K (2) i = c−1g α jk i [J(xk), J(xj)] and choose the ansatz ϕ(G(xi)) = K (2) i + ~F (1) i with some degree-1 element F (1) i ∈ Y(g). (We do not need to consider degree-0 elements in the ansatz since all xi ∈ Y(g, g)tw.) We then have ∆~ ( K (2) i ) = K (2) i ⊗ 1 + 1⊗K(2) i + 1 2~[J(xi)⊗ 1− 1⊗ J(xi),Ωg] + 1 4c −1 g ~2α jk i [[xk ⊗ 1,Ωg], [xj ⊗ 1,Ωg]], (7.21) which is obtained by making use of the following computation α jk i [J(xk)⊗ 1 + 1⊗ J(xk), [xj ,Ωg]] = α jk i α ab j α c kb (J(xc)⊗ xa − xa ⊗ J(xc)) = 1 2α jk i (α ab j α c kb − α ab k α c jb )(J(xc)⊗ xa − xa ⊗ J(xc)) = 1 2α jk i α b kj α ac b (J(xc)⊗ xa − xa ⊗ J(xc)) = 1 2cgα ac i (J(xc)⊗ xa − xa ⊗ J(xc)) = 1 2cg[J(xi)⊗ 1− 1⊗ J(xi),Ωg]. By comparing the terms linear in ~ in (7.20) and (7.21) we find F (1) i = 1 4 [J(xi), Cg]. Note that ∆~ ( F (1) i ) = F (1) i ⊗ 1 + 1⊗ F (1) i + 1 2 [J(xi)⊗ 1 + 1⊗ J(xi),Ωg] + 1 4~[[xi ⊗ 1,Ωg],Ωg]. Lastly, by comparing the terms quadratic in ~ in (7.20) and using the expressions for K (2) i and F (1) i we find 4W (0) i = [[xi ⊗ 1,Ωg],Ωg] + c−1g α jk i [[xk ⊗ 1,Ωg], [xj ⊗ 1,Ωg]]. as required. The coideal co-associativity identity (5.1) in this case is not that straightforward, thus we demonstrate it explicitly. We need to show that the expression below equates to zero:( (∆~ ⊗ id) ◦ ~ − (id⊗ ~) ◦ ~ ) (G(xi)) 26 S. Belliard and V. Regelskis = ( ∆~(ϕ(G(xi)))− ( ϕ(G(xi))⊗ 1 + 1⊗ ϕ(G(xi)) + ~[J(xi)⊗ 1,Ωg] )) ⊗ 1 + ~2 ( (∆~ ⊗ id) ( W (0) i ) − (id⊗ ~) ( W (0) i ) − 1⊗W (0) i + 1 2α jk i ([xk ⊗ 1,Ωg])⊗ xj ) . Using the fact ϕ(xi) = xi the relation above gives a constraint (∆~ ⊗ id) ( W (0) i ) − (id⊗ ~) ( W (0) i ) +W (0) i ⊗ 1− 1⊗W (0) i + 1 2α jk i ([xk ⊗ 1,Ωg])⊗ xj = 0. (7.22) By Lemma (7.1) the explicit form of W (0) i is equal to W (0) i = 1 8 ( α jc i α ab j + α jb i α ac j ) (xa ⊗ {xb, xc} − {xb, xc} ⊗ xa) + 1 24 ( c−1g α jk i ∑ π α π(c)r j α π(b)s k α π(a) sr + α jc i α ab j + α jb i α ac j ) × (xa ⊗ {xb, xc}+ {xb, xc} ⊗ xa) = 1 12 ( α jc i α ab j + α jb i α ac j ) (2xa ⊗ {xb, xc} − {xb, xc} ⊗ xa) + 1 24c −1 g α jk i ∑ π α π(c)r j α π(b)s k α π(a) sr (xa ⊗ {xb, xc}+ {xb, xc} ⊗ xa). (7.23) Denote W (0) i = h abci xa ⊗ {xb, xc}+ h abc i {xb, xc} ⊗ xa. Then (7.22) is equivalent to 4h abc i xc ⊗ xb ⊗ xa − 4h abci xa ⊗ xb ⊗ xc + α jc i α ab j xa ⊗ xb ⊗ xc = 0, giving 4h cba i − 4h abci + α jc i α ab j = 0. Using (7.23) we find 4h cba i − 4h abci + α jc i α ab j = −1 3 ( α ja i α cb j + α jb i α ca j ) − 2 3 ( α jc i α ab j + α jb i α ac j ) + α jc i α ab j = −1 3 ( α ja i α cb j + α jb i α ac j ) + 1 3α jc i α ab j = 0 by the co-Jacobi identity. By reversing the computations above and using (5.2) one can deduce that co-action ~ is the unique map satisfying the required properties up to the shift (κc⊗id)◦ ~, i.e., there exists a one-parameter family of co-actions that are equivalent to each other via the shift-automorphism κc of Y(g). It remains to verify that the map ϕ is an algebra homomorphism, ϕ([bi, bj ]) = [ϕ(bi), ϕ(bj)] and ~([bi, bj ]) = [ ~(bi), ~(bj)] for all bi, bj ∈ Y(g, g)tw, which follow by a lengthy but direct computation. 7.5 A proof of relations (5.11) and (5.12) Recall the horrific degree-2 and degree-3 relations of the twisted Yangian Y ( g, gθ )tw from the Theorem 5.5: [B(Yp),B(Yq)] + ∑ α (c̄(α)) −1w α pq w rs α [B(Yr),B(Ys)] = ~2 ∑ λ,µ,ν Λλµνpq {Xλ, Xµ, Xν}, [[B(Yp),B(Yq)],B(Yr)] + 2c−1g ∑ α (κm)tuw α pq g s rα [[B(Ys),B(Yt)],B(Yu)] = ~2 ∑ λ,µ,u Υλµu pqr {Xλ, Xµ,B(Yu)}, where, by (5.13) and (5.14), Λλµνpq = 1 3 ( gµtpg λu q + ∑ α (c̄(α)) −1w α pq w rs α gµtrgλus ) w ν tu , Drinfeld J Presentation of Twisted Yangians 27 Υλµu pqr = 1 4 ∑ α ( w α st g λs p g µtq g u αr + ∑ β w α pq f λβ α g µsr g u βs ) + 1 2c −1 g ∑ α,γ (κm)vxw γ pq g y rγ ( w α st g λs y g µtv g u αx + ∑ β w α yv f λβ α g µsx g u βs ) . The coideal structure is given by the co-action ~ : Y ( g, gθ )tw → Y(g) ⊗ Y ( g, gθ )tw defined in (5.15): ~(Xα) = Xα ⊗ 1 + 1⊗Xα, ~(B(Yp)) = ϕ(B(Yp))⊗ 1 + 1⊗B(Yp) + ~[Yp ⊗ 1,ΩX ], where, by (5.16), ϕ(B(Yp)) = J(Yp) + 1 4~[Yp, CX ]. Denote the horrific degree-2 and degree-3 relations as H(2) pq = ~2Λλµνpq {Xλ, Xµ, Xν} and H(3) pqr = ~2Υλµu pqr {Xλ, Xµ,B(Yu)}, respectively. We need to show that co-action ~ extends to a homomorphism of algebras Y ( g, gθ )tw → Y(g) ⊗ Y ( g, gθ )tw . This can be checked by a direct computation. We will demon- strate this in several steps. Degree-2. Set Bpq = [B(Yp),B(Yq)]. We have that ~(Bpq) = ϕ(Bpq)⊗ 1 + 1⊗Bpq + ~[ϕ(B(Yp))⊗ 1 + 1⊗B(Yp), [Yq ⊗ 1,ΩX ]] + ~[[Yp ⊗ 1,ΩX ], ϕ(B(Yq))⊗ 1 + 1⊗B(Yq)] + ~2[[Yp ⊗ 1,ΩX ], [Yq ⊗ 1,ΩX ]]. The ~-order terms in the right hand side of the equality above equate to ~ [ J(Yp) + 1 4~ ∑ α g αrp (YrXα +XαYr)⊗ 1 + 1⊗B(Yp), ∑ α g αrq Yr ⊗Xα ] − (p↔ q) = ~ ∑ α,β g αrq w β pr J(Xβ)⊗Xα + ~ ∑ µ g µtq g s pµYt ⊗B(Ys) + 1 2~ 2∑ α,β g αrp g βsq (∑ γ w γ rs {Xγ , Xα}+ g t αs {Yr, Yt} ) ⊗Xβ − (p↔ q), where (p ↔ q) denote all the terms with indices p and q interchanged. In a similar way the ~2-order term equates to ~2 ∑ α,β g αrp g βsq [Yr ⊗Xα, Ys ⊗Xβ] = 1 2~ 2 ∑ α,β,γ ( g αsp g βtq − g αsq g βtp )( w γ st Xγ ⊗ {Xβ, Xα}+ f γ αβ {Ys, Yt} ⊗Xγ ) . Using mixed Jacobi identities we write∑ α,β ( g αrq w β pr − g αrp w β qr ) J(Xβ)⊗Xα = ∑ α,β,µ w µ pq f αβ µ J(Xβ)⊗Xα,∑ µ ( g µtq g s pµ − g µt p g s qµ ) Yt ⊗B(Ys) = ∑ α,µ w µ pq w st µ Yt ⊗B(Ys) and ∑ α,β ( g t αs ( g αrp g βsq − g αrq g βsp ) {Yr, Yt} ⊗Xβ + ∑ γ f γ αβ ( g αrp g βtq − g αrq g βtp ) {Yr, Yt} ⊗Xγ ) = ∑ α,β ( g αrp ( g t αs g βs q + ∑ γ f β αγ g γt q ) − g αrq ( g t αs g βs p + ∑ γ f β αγ g γt p )) {Yr, Yt} ⊗Xβ = ∑ α,β ( g αrp g tβs g s qα − g αrq g tβs g s pα ) {Yr, Yt} ⊗Xβ = ∑ α,µ w µ pq w rs µ g tαs {Yr, Yt} ⊗Xα. 28 S. Belliard and V. Regelskis Using the identity∑ α,β,γ ( w γ rs ( g αrp g βsq − g αrq g βsp ) {Xγ , Xα} ⊗Xβ + w γ st ( g αsp g βtq − g αsq g βtp ) Xγ ⊗ {Xβ, Xα} ) = ∑ α,β,γ w γ st ( g αsp g βtq + g βsp g αtq ) ({Xγ , Xα} ⊗Xβ +Xγ ⊗ {Xβ, Xα}) and combing all the expressions above we arrive to the following result, ~(Bpq) = ϕ(Bpq)⊗ 1 + 1⊗Bpq + ~ ∑ µ w µ pq ( w st µ Yt ⊗B(Ys) + ∑ α,β f αβ µ J(Xβ)⊗Xα + 1 2~ ∑ α w su µ g tαu {Ys, Yt} ⊗Xα ) + 1 2~ 2 ∑ α,β,γ w γ st ( g αsp g βtq + g βsp g αtq ) ({Xγ , Xα} ⊗Xβ +Xγ ⊗ {Xβ, Xα}) (7.24) and, by similar computations, ϕ(Bpq) = [J(Yp), J(Yq)] + 1 2~ ∑ µ w µ pq ( w st µ {J(Ys), Yt}+ ∑ α,β f αβ µ {J(Xβ), Xα} ) + 1 4~ 2g αsp g βtq [{Ys, Xα}, {Yt, Xβ}]. In a similar way as it was done in the proof of Lemma 7.1, the last line of (7.24) can be split as 1 6~ 2 ∑ α,β,γ,λ,µ w µ pq f λγ µ f βα λ ({Xγ , Xβ} ⊗Xα − 2Xα ⊗ {Xβ, Xγ}) + ~2 ∑ α,β,γ g αsp g βtq w γ st (〈Xα, Xβ, Xγ〉1 + 〈Xα, Xβ, Xγ〉2). (7.25) Here we have used the same notation as in (5.3). The terms in the second line of (7.24) and in the first line of (7.25) do not contribute to ~ ( H (2) pq ) , since w µ pq + ∑ α(c̄(α)) −1w α pq w rs α w µ rs = 0. What remains are the terms in the second line of (7.25) giving ~ ( H(2) pq ) = ϕ ( H(2) pq ) ⊗ 1 + 1⊗H(2) pq + ~2 ∑ α,β,γ ( g αsp g βtq + ∑ λ (c̄(λ)) −1w λ pq w us λ g αsu g βts ) × w γ st (〈Xα, Xβ, Xγ〉1 + 〈Xα, Xβ, Xγ〉2). (7.26) By (5.4) we have ~({Xα, Xβ, Xγ})− ( {Xα, Xβ, Xγ} ⊗ 1 + 1⊗ {Xα, Xβ, Xγ} ) = 3 ( 〈Xα, Xβ, Xγ〉1 + 〈Xα, Xβ, Xγ〉2 ) . (7.27) Comparing (7.26) with (7.27) gives (5.13), as required. Degree-3. Set Bpqr = [Bpq,B(Yr)]. We write ~(Bpqr) = ϕ(Bpqr)⊗ 1 + 1⊗Bpqr + ~B(2) pqr + ~2B(1) pqr + ~3B(0) pqr, (7.28) where B (i) pqr denote elements of degree-i. By consistency, the elements of degree-2 in ~ ( H (3) pqr ) must vanish altogether, since they are of a linear order in ~. This can be shown explicitly. We have B(2) pqr = ∑ α [[J(Yp), J(Yq)]⊗ 1 + 1⊗ [B(Yp),B(Yq)], g αt r Yt ⊗Xα] + ∑ µ w µ pq [ w st µ Yt ⊗B(Ys) + ∑ α,β f αβ µ J(Xβ)⊗Xα, J(Xr)⊗ 1 + 1⊗B(Yr) ] . Drinfeld J Presentation of Twisted Yangians 29 Denote this expression as B (2) pqr = B (2,0) pqr + B (0,2) pqr + B (1,1) pqr , where B (i,j) pqr represents elements of degree-i in the left tensor factor and of degree-j in the right tensor factor. Then B(2,0) pqr = ∑ α,β g αtr ( w β pt [J(Xβ), J(Yq)] + w β qt [J(Yp), J(Xβ)] ) ⊗Xα + ∑ α,β,µ w µ pq f αβ µ [J(Xβ), J(Yr)]⊗Xα = 2 ∑ α,β,γ ( g αtr (w β pt g s βq − w β qt g s βp ) + ∑ µ w µ pq f αβ µ g s βr ) c−1g g γus [J(Yu), J(Xγ)]⊗Xα + 1 2~ 2c−1g ∑ α,β,γ ( g αtr (w β pt g γu q − w β qt g γu p ) + ∑ µ w µ pq f αβ µ gγur ) × ∑ ijk A ijk βuγ{xi, xj , xk} ⊗Xα = 2 ∑ α,β,γ w β pq g t βr g αs t g γus c−1g [J(Yu), J(Xγ)]⊗Xα + O1 ( ~2 ) , (7.29) where O1(~2) is a short–hand notation for the terms in the third line, and we have used the mixed Jacobi identities and the Drinfeld terrific relation (5.6) in the form [J(Xβ), J(Xq)] + 2c−1g g s βq g γu s [J(Xγ), J(Yu)] = 1 2~ 2c−1g ∑ γ ∑ ijk g γuq βijkβuγ{xi, xj , xk}. Here the sum ∑ i xi spans all of the symmetric space basis, xi = {Yp, Xα}; the same applies to xj and xk. In a similar way we find B(0,2) pqr = ∑ α g αtr Yt ⊗ ( g u pα [B(Yu),B(Yq)] + g u qα [B(Yp),B(Yu)] ) + ∑ α w α pq w st α Yt ⊗ [B(Ys),B(Xr)] = ∑ α,β ( g αtr (g u pαw β uq − g u qα w β up ) + w α pq w st α w β sr ) (c̄(β)) −1w vs β Yt ⊗ [B(Ys),B(Yv)] + ~2 ∑ α,λ,µ,ν ( g αtr (g u pα Λλµνuq − g u qα Λλµνup ) + w α pq w st α Λλµνsr ) Yt ⊗ {Xλ, Xµ, Xν} = ∑ α,β w α pq g t αr g βt t w vs β (c̄(β)) −1Yt ⊗ [B(Ys),B(Yv)] + O2 ( ~2 ) , (7.30) where we have used the mixed Jacobi identities and (5.11). The notation for O2(~2) is the same as before. The remaining elements give B(1,1) pqr = ∑ µ,β w µ pq ( w st µ w β tr + ∑ α f αβ µ g s αr ) J(Xβ)⊗B(Ys) = ∑ α,β w α pq g t αr g sβ t J(Xβ)⊗B(Ys). (7.31) Now it is easy to check that leading terms in (7.29)–(7.31) do not contribute to ~ ( H (3) pqr ) , since (by (2.10))∑ β w β pq g t βr + 2c−1g ∑ α,β (κm)uvw α pq g s rα w β su g t βv = 0. (7.32) Next step is to consider the degree-1 term in (7.28). We write B (1) pqr = B (1,0) pqr + B (0,1) pqr . Using (7.24), (7.25) and (5.16) we find B(1,0) pqr = 1 2 ∑ α,µ w µ pq (∑ β,γ f αβ µ g γsr [J(Xβ)⊗Xα, 2Ys ⊗Xγ + {Ys, Xγ} ⊗ 1] 30 S. Belliard and V. Regelskis + w su µ g tαu [{Ys, Yt}, J(Yr)]⊗Xα ) + 1 2 ∑ γ,µ w µ pq g γu r ( w st µ [{J(Ys), Yt}, Yu] + ∑ α,β f αβ µ [{J(Xβ), Xα}, Yu] ) ⊗Xγ + 1 2 ∑ α,β,γ w γ st ( g αsp g βtq + g βsp g αtq ) [{Xγ , Xα} ⊗Xβ +Xγ ⊗ {Xβ, Xα}, J(Yr)⊗ 1] = 1 2 ∑ γ,λ,µ w µ pq (∑ α,β g βsr ( 2f αλ µ f γ αβ + f γα µ f λ αβ ) + w λ tr ( w tu µ g sγu + w su µ g tγu ) + g γur ( w ts µ w λ tu + ∑ α f αλ µ g s αu )) {Ys, J(Xλ)} ⊗Xγ + 1 2 ∑ α,β,γ,µ ( w µ pq f αβ µ g γsr g u βs + w β st g αs p g γtq g u βr ) × (2J(Yu)⊗ {Xα, Xγ}+ {Xα, J(Yu)} ⊗Xγ + {Xγ , J(Yu)} ⊗Xα) + 1 2 ∑ α,β,γ w γ ts ( w β pq w ut β g αsr + g u rβ ( g βsp g αtq + g αsp g βtq )) {Xγ , J(Yu)} ⊗Xα. (7.33) The first sum in the last equality can be simplified using the mixed Jacobi identities, giving 1 2 ∑ γ,λ,µ w µ pq g t µr ( g uλt gγsu + ∑ α g αst fγλα ) {Ys, J(Xλ)} ⊗Xγ . This component does not contribute to ~ ( H (3) pqr ) due to (7.32). The second degree-1 element, B (0,1) pqr , is B(0,1) pqr = 1 2 ∑ α,β w β pq ( w st β g αu r [Yt ⊗B(Ys), 2Yu ⊗Xα + {Yu, Xα} ⊗ 1] + w su β g tαu [{Ys, Yt} ⊗ [Xα,B(Yr)] ) + 1 2 ∑ α,β,γ w γ st ( g αsp g βtq + g βsp g αtq ) [{Xγ , Xα} ⊗Xβ +Xγ ⊗ {Xβ, Xα}, 1⊗B(Yr)] = 1 2 ∑ α,β w β pq ( 2w st β g αu r g v sα + w vs β g αur g t sα + w ts β g uα s g v αr ) {Yt, Yu} ⊗B(Yv) + ∑ α,β,γ w γ ts ( w β pq w ut β g αsr + g u rβ ( g αsp g βtq + g βsp g αtq )) × ( Xγ ⊗ {Xα,B(Yu)}+ 1 2{Xα, Xγ} ⊗B(Yu) ) . (7.34) The first sum in the last equality above can be simplified to 1 2 ∑ α,β w β pq g s rβ g αt s w uv α {Yt, Yu} ⊗ B(Yv), and does contribute to ~ ( H (3) pqr ) due to (7.32). Set Aαγupqr = 1 2 ∑ β,µ ( w µ pq f αβ µ g γsr g u βs + w β st g αs p g γtq g u βr + (α↔ γ) ) , Bαγu pqr = 1 2 ∑ β w γ ts ( w β pq w ut β g αsr + g u rβ ( g αsp g βtq + g βsp g αtq )) . Using the mixed Jacobi identities (2.6) we find that Υαγu pqr = ( Aαγupqr + 2c−1g ∑ µ (κm)tuxw µ pq g s rµA αγu stu = Bαγu pqr + 2c−1g ∑ µ (κm)tuw µ pq g s rµB αγu stu ) satisfying Υαγu pqr = Υγαu pqr . The contribution of the remaining sums in (7.33) and (7.34) to ~ ( H (3) pqr ) is∑ α,γ Υαγu pqr ( J(Yu)⊗ {Xα, Xγ}+ 2{Xα, J(Yu)} ⊗Xγ + 2Xγ ⊗ {Xα,B(Yu)}+ {Xα, Xγ} ⊗B(Yu) ) . (7.35) Drinfeld J Presentation of Twisted Yangians 31 Using (5.4) and (5.3) we find that ~({Xα, Xγ ,B(Yu)})− (ϕ({Xα, Xγ ,B(Yu)})⊗ 1 + 1⊗ {Xα, Xγ ,B(Yu)}+ O3(~)) = J(Yu)⊗ {Xα, Xγ}+ {Xα, Xγ} ⊗B(Yu) + ({J(Yu), Xα} ⊗Xγ +Xγ ⊗ {Xα,B(Yu)}+ (α↔ γ)), where O3(~) denotes linear in ~ terms that appear due to (5.16). Comparing the relation above with (7.35) and using symmetry Υαγu pqr = Υγαu pqr gives (5.14), as required. The remaining elements in ~ ( H (3) pqr ) , that we have denoted by ~3B(0) pqr, are cubic in ~ and, by summing with ~O1(~2) of (7.29) and ~O2(~2) of (7.30), and repeating similar steps as we did above, give precisely the element ~2O3(~), thus the relations (5.11) and (5.12) are compatible with the co-action (5.15). 7.6 An outline of a proof of relations (5.18)–(5.22) The following lemma, which follows by a direct computation, will be necessary in what follows below. Lemma 7.2. In any associative algebra generated by elements xa(b,c,...) over C the following identities hold: {xi, {xj , xk}} = {xi, xj , xk}+ 1 12 [x(j , [xk), xi]], {xj , xk, xl, xm} = 1 3{xj , x(k, {xl, xm)}} − 1 36 [[xj , x(k], {xl, xm)}], {1⊗ xi, 1⊗ xj , xa ⊗ {xb, xc}} = xa ⊗ {xi, xj , {xb, xc}}, {xi ⊗ 1, 1⊗ xj , xa ⊗ {xb, xc}} = {xi, xa} ⊗ {xj , {xb, xc}} − 1 12 [xa, xi]⊗ [xj , {xb, xc}], {xi ⊗ 1, 1⊗ xj , {xb, xc} ⊗ xa} = {xi, {xb, xc}} ⊗ {xj , xa} − 1 12 [{xb, xc}, xi]⊗ [xj , xa]. Recall the horrific degree-4 relation (5.18) of the twisted Yangian Y(g, g)tw in Theorem 5.6: [G(xa),G([xb, xc])] + [G(xb),G([xc, xa])] + [G(xc),G([xa, xb])] = ~2Ψijk abc{xi, xj ,G(xk)} + ~4(Φijk abc{xi, xj , xk}+ Φ ijklm abc {xi, xj , xk, xl, xm}), where, by (5.19)–(5.21), Ψijk abc = α d (ab α k c)r h rij d − α k dr α d (ab h rij c) , Φ ijklm abc = 1 5 ( α i rsα d (ab h rjk c) h slm d −Ψjkr abch ilm r ) , Φijk abc = 1 9 ( α d (ab W ijk c)d + 1 6Φ (ix(yzj)) abc α r xyα k rz − ( Ψxjy abch kzr y α s zxα i rs + Ψxyz abc h rsk z α i rxα j ys )) . with W ijk cd = α i rsh rxy c ( hszkd α j xt α t yz + hsztd α k xt α j yz ) + (( h xyz c hefkd − hxyzd hefkc ) α t yeα i zt α j xf + h jxy c h kzr d α s xr ( α t zyα i st + α t syα i zt ) ) and h bcda = φ bcd a + 2ψ bcd a , h bcd a = φ bcd a − ψ bcd a , ψ bcd a = 1 12 ( α jd a α bc j + α jc a α bd j ) , φ bcd a = 1 24c −1 g ∑ π ( α jk a α π(d)r j α π(b)s k α π(c) sr ) . Denote the degree-4 relation as H (4) abc = ~2Ψijk abc{xi, xj ,G(xk)}+ ~4 ( Φijk abc{xi, xj , xk}+ Φ ijklm abc {xi, xj , xk, xl, xm} ) . (7.36) 32 S. Belliard and V. Regelskis The coefficients Ψijk abc, Φijk abc, Φ ijklm abc ∈ C must be designed to make ~ extend to a homomorphism of algebras Y(g, g)tw → Y(g) ⊗ Y(g, g)tw. We will find these coefficients by acting with ~ on (7.36) and equating the numeric factors of some elements on both sides of the resulting expression. These elements are {xi, xj} ⊗ G(xk), xi ⊗ {xj , xk}, xi ⊗ {xj , xk, {xl, xm}}. All the remaining elements will be referred as the unwanted terms (UWT). Denote Gad = [G(xa),G(xd)]. Then H (4) abc = α(ab dGc)d. Using (5.24) we write ~(Gcd) = ϕ(Gcd)⊗ 1 + 1⊗Gcd + ~G(3) cd + ~2G(2) cd + ~3G(1) cd + ~4G(0) cd , where G (i) cd denote elements of degree-i and can be computed explicitly using (5.25) and (5.26). For our purpose we need to consider certain elements of G (2) cd and G (0) cd only. For G (2) cd we have G (2) cd = (α k cr h rij d − α k dr h rij c ){xi, xj} ⊗ G(xk) + UWT, (7.37) and for G (0) cd we find G (0) cd = α i rsh rjk c hslmd xi ⊗ {{xj , xk}, {xl, xm}}+ h jln c h krs d [{xl, xn}, {xr, xs}]⊗ {xj , xk} + ( h trs c hlmnd − htrsd hlmnc ) {xl, {xr, xs}} ⊗ [xt, {xm, xn}] + UWT. Using Lemma 7.2 and symmetries hrjkc = hrkjc , h rjk c = h rkj c we reduce the expression above to G (0) cd = α i rsh rjk c hslmd xi ⊗ {xj , xk, {xl, xm}}+ 1 3W ijk cd xi ⊗ {xj , xk}+ UWT, (7.38) where W ijk cd = α i rsh rxy c ( hszkd α j xt α t yz + hsztd α k xt α j yz ) + (( h xyz c hefkd − hxyzd hefkc ) α t yeα i zt α j xf + h jxy c h kzr d α s xr ( α t zyα i st + α t syα i zt ) ) . Next step is to find the corresponding elements in the rhs of ~ ( H (4) abc ) . Using (5.26), (5.4) and Lemma 7.2 we obtain ~({xi, xj ,G(xk)}) = {xi, xj} ⊗ G(xk) + ~2habck xa ⊗ {xi, xj , {xb, xc}} + 1 6~ 2h abc k ( α r bi α s cr xs ⊗ {xj , xa}+ α r bj α s cr xs ⊗ {xi, xa} ) − 1 6~ 2habck (α r ai α s jb + α r aj α s ib )xr ⊗ {xs, xc}+ UWT, ~({xi, xj , xk}) = x(i ⊗ {xj , xk)}+ UWT, ~({xi, xj , xk, xl, xm}) = 1 3x(i ⊗ {xj , x(k, {xl, xm))}} − 1 36x(i ⊗ [[xj , x(k], {xl, xm))}] + UWT. This gives ~ ( H (4) abc ) = ~2Ψijk abc{xi, xj} ⊗ G(xk) + ~4 ( 1 3Φ (ij(klm)) abc + Ψjkr abch ilm r ) xi ⊗ {xj , xk, {xl, xm}} + ~4 ( Φ (ijk) abc − 1 36Φ (ix(y(zj))) abc α r xyα k rz + 1 6 ( Ψ (xj)y abc h kzr y α s zxα i rs + Ψ (xy)z abc hrskz α i rxα j ys )) xi ⊗ {xj , xk}+ UWT. Drinfeld J Presentation of Twisted Yangians 33 Then, by substituting (7.37) and (7.38) into ~(α(ab dGc)d) = ~ ( H (4) abc ) and comparing the elements on the both sides of the resulting equality we find Ψijk abc = α d (ab α k c)r h rij d − α k dr α d (ab h rij c) , Φ (ij(klm)) abc = 3 ( α i rsα d (ab h rjk c) h slm d −Ψjkr abch ilm r ) , Φ (ijk) abc = 1 3α d (ab W ijk c)d + 1 18Φ (ix(yzj)) abc α r xyα k rz − 1 3 ( Ψxjy abch kzr y α s zxα i rs + Ψxyz abc h rsk z α i rxα j ys ) , where in the last expression we used the symmetries Ψijk abc = Ψjik abc and Φ (ij(klm)) abc = Φ (ij(kml)) abc . Finally, using 1 3Φ (ijk) abc {xi, xj , xk} = Φijk abc{xi, xj , xk}, 1 15Φ (ij(klm)) abc {xi, xj , xk, xl, xm} = Φijklm abc {xi, xj , xk, xl, xm}, we obtain (5.19), (5.20) and (5.21). To complete the proof it would require to compute all elements in ~(α(ab dGc)d) = ~ ( H (4) abc ) as we did in Section 7.5 above. Here we will not attempt to reach this goal. 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Algebra 215 (2011), 411–419, arXiv:0810.5160. https://doi.org/10.1070/RM1996v051n02ABEH002772 http://arxiv.org/abs/hep-th/9409025 https://doi.org/10.1142/S0129055X02001156 http://arxiv.org/abs/math.QA/0107213 https://doi.org/10.1142/S0129055X03001813 http://arxiv.org/abs/math.QA/0208140 http://arxiv.org/abs/1005.1371 https://doi.org/10.1007/BFb0101183 http://arxiv.org/abs/1206.6498 https://doi.org/10.1088/0305-4470/21/10/015 https://doi.org/10.1007/BF01018718 https://doi.org/10.1007/BF01018718 https://doi.org/10.1103/PhysRevLett.19.1312 https://doi.org/10.1016/j.jpaa.2010.04.026 http://arxiv.org/abs/0810.5160 1 Introduction 2 Definitions and preliminaries 2.1 Lie algebra 2.2 Symmetric pair decomposition 3 Symmetric spaces and simple half-loop Lie algebras 3.1 Half-loop Lie algebra 3.2 Twisted half-loop Lie algebra 4 Lie bi-algebras and coideals 4.1 Lie bi-algebra structure of a half-loop Lie algebra 4.2 Lie coideal structure of a twisted half-loop Lie algebra 5 Twisted Yangians as quantized Lie coideals 5.1 Quantization of Lie bi-algebras and Lie coideals 5.2 Yangians and twisted Yangians in Drinfeld J presentation 6 Coideal subalgebras of the Yangian Y(sl3) 6.1 The sl3 Lie algebra and the Yangian Y(sl3) 6.2 Twisted Yangian Y(sl3,so3)tw 6.3 Twisted Yangian Y(sl3,gl2)tw 6.4 Twisted Yangian Y(sl3,sl3)tw 7 Proofs 7.1 A proof of Proposition 3.1 7.2 A proof of Proposition 3.2 7.3 A proof of the uniqueness of the co-action of Y(g,g)tw 7.4 A proof of the uniqueness of the co-action of Y(g,g)tw 7.5 A proof of relations (5.11) and (5.12) 7.6 An outline of a proof of relations (5.18)–(5.22) References

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