Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.


Introduction
A Lie bialgebra is a pair (L, δ) consisting of a Lie algebra L and a linear map δ : L −→ L⊗L, called Lie cobracket, inducing a compatible Lie algebra structure on the dual space L ∨ . This notion originated in [11] as the infinitesimal counterpart of a Poisson Lie group. Shortly after, in [12,13], Lie bialgebras were described as quasi-classical limits of certain quantum groups and received a fundamental role in the quantum group theory.
Having a Lie bialgebra structure δ on a Lie algebra L we can obtain new Lie bialgebra structures using a procedure called twisting. More precisely, let t be a skew-symmetric tensor in L ⊗ L satisfying where CYB(t) := [t 12 , t 13 ] + [t 12 , t 23 ] + [t 13 , t 23 ], and, for example, [(a ⊗ b) 12 , (c ⊗ d) 23 ] := a ⊗ [b, c] ⊗ d. Then the linear map δ t := δ + dt is a Lie bialgebra structure on L. Such a tensor t is called a classical twist.
The most important example of a Lie bialgebra structure is the standard structure δ on a symmetrizable Kac-Moody algebra K := K(A) introduced in [12]. In the case when the Cartan matrix A is of finite type or, equivalently, when K is a finite-dimensional semi-simple Lie algebra, the standard structure δ and all its twisted versions δ t are known to be quasi-triangular, i.e. they are of the form dr for some r ∈ K ⊗ K satisfying the classical Yang-Baxter equation CYB(r) = 0.
When the matrix A is of affine type, the standard structure on K induces a Lie bialgebra structure on [K, K]/Z(K), where Z(K) is the center of K, which we will also call standard. The latter Lie algebra is known (see [23]) to be isomorphic to the loop algebra L σ over a simple finite-dimensional Lie algebra g corresponding to an automorphism σ ∈ Aut C−LieAlg (g) of finite order m. It has the following explicit description L σ = f ∈ g[z, z −1 ] | f (ε σ z) = σ(f (z)) , ε σ := exp(2πi/m).
We denote the induced standard Lie bialgebra structure on L σ by δ σ 0 and call its twists δ σ t := δ σ 0 + dt twisted standard structures. These Lie bialgebra structures are not quasi-triangular, but pseudoquasitriangular as is shown in Theorem 3.3, i.e. they are defined by meromorphic functions r : C 2 −→ g ⊗ g, also known as r-matrices, satisfying the two-parametric classical Yang-Baxter equation ( For example, the trigonometric r-matrix given by a Belavin-Drinfeld (BD) quadruple Q, corresponding to an outer automorphism ν ∈ Aut C−LieAlg (g) (see [3]), gives rise to a twisted standard structure δ σ Q on L σ for any finite order automorophism σ whose coset is conjugate to νInn C−LieAlg (g). It turns out that any r-matrix defining a twisted standard bialgebra structure on L σ is globally holomorphically equivalent to a trigonometric solution in the sense of the Belavin-Drinfeld classification (see Theorem 3.4). We refer to such r-matrices as σ-trigonometric.
We call two twisted standard structures δ σ t and δ σ s (regularly) equivalent if there is a function φ ∈ Aut C[z m ,z −m ]−LieAlg (L σ ) = f : C * −→ Aut C−LieAlg (g) | f is regular and f (ε σ z) = σf (z)σ −1 , called a regular equivalence, such that δ σ t φ = (φ ⊗ φ)δ σ s . The main result of this paper is the classification of twisted standard structures up to regular equivalence. The classification is obtained by reducing our problem to the classification of trigonometric r-matrices up to holomorphic equivalence given in [3]. To deal with the difference between the notions of equivalence we use the geometric formalism of CYBE presented in [7]. More precisely, one of the key results in [7] is that certain coherent sheaves of Lie algebras on Weierstraß cubic curves give rise to so-called geometric r-matrices, satisfying a geometric version of CYBE. In Section 5 we prove the following extension property: Theorem A. A formal equivalence of geometric r-matrices at the smooth point at infinity of the Weierstraß cubic curve gives rise to an isomorphism of the corresponding sheaves of Lie algebras.
It is shown in [1] that all σ-trigonometric r-matrices arise as geometric r-matrices from coherent sheaves of Lie algebras on the nodal Weierstraß cubic with section L σ on the set of smooth points. Since holomorphic equivalences are formal, this result and Theorem A give the desired classification: Theorem B. For any twisted standard structure δ σ t there is a regular equivalence φ of L σ and a BD quadruple Q = (Γ 1 , Γ 2 , γ, t h ) such that is another BD quadruple, then the twisted bialgebra structures δ σ Q and δ σ Q ′ are regularly equivalent if and only if there is an automorphism ϑ of the Dynkin diagram of L σ such that ϑ(Γ i ) = Γ ′ i for i = 1, 2, ϑγϑ −1 = γ ′ and (ϑ ⊗ ϑ)t h = t ′ h . Let Π σ be the set of simple roots of L σ , S Π and p S + ⊆ L σ be the corresponding parabolic subalgebra (see Section 2.2). The standard Lie bialgebra structure δ σ 0 on L σ restricts to a Lie bialgebra structure on the parabolic subalgebra p S + . We refer to this Lie bialgebra structure as the restricted standard structure.
In the special case σ = id and S = Π id \ { α 0 }, where α 0 is the affine root of L id = g[z, z −1 ], the classical twists of the restricted standard structure are in one-to-one correspondence with socalled quasi-trigonometric solutions of CYBE. Such r-matrices were studied and classified in terms of BD quadruples in [26,32]. We use this classification in Section 4.3 to demonstrate the first part of Theorem B in the special case g = sl(n, C). This connection to quasi-trigonometric solutions serves as a motivation for our study of restricted standard structures.
We discover that Theorem B also gives a full classification of classical twists of restricted Lie bialgera structures. More formally, for any classical twist t ∈ p S + ⊗ p S + the structure of L σ guarantees that the regular equivalence between δ σ t and some δ σ Q , given by Theorem B, can be chosen to fix the parabolic subalgebra p S + . The following theorem summarizes this observation.
The theorems stated above provide us with a list of interesting consequences: • Letting σ = id and S = Π id \ { α 0 } in Theorem C we obtain an alternative proof of the classification of all quasi-trigonometric solutions [26,32]; • The necessary and sufficient condition to have an id-trigonometric r-matrix which is not regularly equivalent to a quasi-trigonometric one is the existence of a BD qudruple (Γ 1 , Γ 2 , γ, t h ) such that for any automorphism ϑ of the extended Dynkin diagram of g we have α 0 ∈ ϑ(Γ 1 ). Analyzing Dynkin diagrams, we conclude that any id-trigonometric r-matrix is regularly equivalent to a quasi-trigonometric one if and only if g is of type A n , C n , B 2−4 or D 4−10 ; • We have mentioned that any σ-trigonometric r-matrix is holomorphically equivalent to a trigonometric one in the sense of the Belavin-Drinfeld classification. Combining the structure theory of L σ (Section 2.2) and Theorem B we can improve that result and get more control over that equivalence. More precisely, let ν be an outer automorphism of g, σ be a finite order automorphism of g whose coset is conjugate to νInn C−LieAlg (g) and r σ t be the σ-trigonometric r-matrix defining a twisted standard Lie bialgebra structure δ σ t on L σ . Applying to r σ t the regular equivalence, given by Theorem B, and regrading to the principle grading, i.e. grading corresponding to the Coxeter automorphism σ (1;|ν|) , we obtain a trigonometric r-matrix X depending on the quotient of its parameters: • We answer questions one and two posed at the end of [8] concerning an explicit formula for the quasi-trigonometric solution given by a BD quadruple Q and its connection with the trigonometric solution described by the same quadruple Q (see [3]).
In [29] Montaner, Stolin and Zelmanov classified all Lie bialgebra structures on g[z] by classifying classical twists within each of four possible Drinfeld double algebras. A main point in their argument is the aforementioned classification of quasi-trigonometric solutions [32] or, equivalentely, the classification of classical twists within one of the doubles. From this perspective, our work is a natural step towards the classification of all Lie bialgebra structures on L id = g[z, z −1 ] or, more generally, L σ .
Acknowledgements. The authors are thankful to I. Burban and A. Stolin for introduction to the topic and fruitful discussions as well as to E. Karolinsky for useful comments. R.A. also acknowledges the support from the DFG project Bu-1866/5-1.

Preliminaries
In this section we give a brief review of the theory of Lie bialgebras and loop algebras as well as set up notation and terminology used throughout the paper. Most of the presented results on Lie bialgebras can be found in [10,14] and [27]. A detailed exposition of the theory of loop algebras can be found in [9,23] and [20, Section X.5].

Lie bialgebras, Manin triples and twisting
A Lie coalgebra is a pair (L, δ) consisting of a vector space L over a field k of characteristic zero and a linear map δ : L −→ L ⊗ L, called Lie cobracket, such that for all x ∈ L δ(x) + τ δ(x) = 0 and Alt((δ ⊗ 1)δ(x)) = 0, These conditions guarantee that the restriction of the dual map δ ∨ : (L ⊗ L) ∨ −→ L ∨ to L ∨ ⊗ L ∨ defines a Lie algebra structure. A morphism between two Lie coalgebras (L, δ) and is a Lie algebra, (L, δ) is a Lie coalgebra and the following compatibility condition holds In other words, δ is a 1-cocycle of L with values in L ⊗ L. A linear map between two Lie bialgebras is a Lie bialgebra morphism if it is a morphism of both Lie algebra and Lie coalgebra structures. Lie bialgebras are closely related to Manin triples, i.e. triples (L, L + , L − ), where L is a Lie algebra equipped with an invariant non-degenerate symmetric bilinear form B and L ± are isotropic subalgebras of L with respect to that form, such that L = L + ∔ L − . 2 The definition immediately implies that L ± are Lagrangian subalgebras of L which are paired non-degenerately by B. We say that two Manin triples (L, L + , L − ) and Every Lie bialgebra (L, δ) gives rise to the Manin triple (L ∔ L ∨ , L, L ∨ ) with the canonical bilinear form B given by B(x + f, y + g) := f (y) + g(x) ∀x, y ∈ L, ∀f, g ∈ L ∨ , (2.5) and the Lie algebra structure on L ∔ L ∨ defined by where ad * x := −ad ∨ x is the coadjoint action.
Remark 2.1. The Lie algebra structure (2.6) is the unique Lie algebra structure on L ∔ L ∨ making the canonical form B invariant and L, L ∨ into Lagrangian subalgebras. The space L ∔ L ∨ equipped with this particular Lie algebra structure is called the classical double of (L, δ). ♦ The converse statement is not true, i.e. not every Manin triple (L, L + , L − ) induces a Lie bialgebra structure on L + . However, this is the case when the dual map This condition can be equivalently formulated in the following way: there is a linear map δ : When this condition is satisfied, we say that the Manin triple (L, L + , L − ) defines the Lie bialgebra (L + , δ).
Remark 2.2. Let φ be an isomorphism between two Manin triples M = (L, L + , L − ) and M ′ = L ′ , L ′ + , L ′ − . If M defines a Lie bialgebra structure (L + , δ), then M ′ also defines a Lie bialgebra structure (L ′ + , δ ′ ) and φ| L + : (L + , δ) −→ (L ′ + , δ ′ ) is a Lie bialgebra isomorphism. ♦ Remark 2.3. Generally, there may exist many non-isomorphic Manin triples defining the same Lie bialgebra structure. However, in the finite-dimensional case the condition (2.7) holds automatically and the correspondence between Manin triples and Lie bialgebras described above is one-to-one. ♦ Having a Lie bialgebra structure, we can produce a new bialgebra structure by means of a procedure called twisting. Let (L, δ) be a Lie bialgebra and t ∈ L ⊗ L be a skew-symmetric tensor satisfying the identity for all x ∈ L, defines a new Lie bialgebra structure on L. The skew-symmetric tensor t is called a classical twist of δ.
It was implicitly shown in [26,32,33,34] that the problem of classification of classical twists of some particular Lie bialgebra structures can be reduced to the classification of Lagrangian Lie subalgebras. In the following theorem we summarize and generalize these ideas.
Proof. Let t = x i ⊗ y i ∈ L + ⊗ L + be a classical twist 3 of δ. Define the linear map T : L − −→ L + and the subspace L t ⊆ L by We now show that they meet the requirements of the theorem. The conditions dim(im(T )) < ∞ and L + ∔ L t = L hold by definition. For all w 1 , w 2 ∈ L − we have Therefore, the skew-symmetry of t is equivalent to the skew-symmetry of T and to L t being a Lagrangian subspace. To prove the commensurability of L t and L − we note that ker(T ) = L t ∩ L − and hence dim (L − / (L t ∩ L − )) = dim (im(T )) . (2.11) This shows that L t ∩ L − has finite codimension inside L − . The commensurability now follows from the fact that L − has codimension at most dim(im(T )) inside L t + L − . Finally, the last condition follows from the identity where w 1 , w 2 , w 3 ∈ L − . This identity is obtained by repeating the argument in the proof of [25, Theorem 7] within our framework.
Conversely, given a Lagrangian Lie subalgebra L ′ ⊆ L, satisfying the conditions of the theorem, we define the linear map T : L − −→ L + in the following way: any w ∈ L − can be uniquely written as w + + w ′ , for some w + ∈ L + and w ′ ∈ L ′ ; We let T (w) := w + . Then L ′ = {T w − w | w ∈ L − } and the commensurability of L ′ and L − implies that the rank of T is finite. The other two conditions on T hold because of the relations (2.10) and the Lagrangian property of L ′ . To construct the classical twist t ∈ L + ⊗L + we note that B gives a non-degenerate pairing between the finite-dimensional spaces L − / ker(T ) and im(T ). Let {T w i } n i=1 be a basis for im(T ) and v i + ker(T ) n i=1 be its dual basis for L − / ker(T ). Then for all k ∈ {1, . . . , n}. Since T is completely determined by its action on {w i } n i=1 , we have the equality The identities (2.12) and (2.10) guarantee that t meets the desired requirements and L ′ = L t .
Remark 2.5. It follows that if (L + , δ) is a Lie bialgebra defined by the Manin triple (L, L + , L − ) and t is a classical twist of δ, then the twisted Lie bialgebra (L + , δ + dt) is defined by the Manin triple (L, L + , L t ). Equivalently, for all x ∈ L + and w 1 , w 2 ∈ L − . ♦

Loop algebras
Let g be a fixed finite-dimensional simple Lie algebra over C and σ be an automorphism of g of finite order |σ| ∈ Z + . The eigenvalues of σ are ε k σ := e 2πik/|σ| , k ∈ Z, and we have the following Z/|σ|Z-gradation of g where g σ k is the eigenspace of σ corresponding to the eigenvalue ε k σ . The tensor product equipped with the bracket described by [z i x, z j y] := z i+j [x, y], for all x, y ∈ g and i, j ∈ Z, is a Z-graded Lie algebra over C. The loop algebra L σ over g is the Z-graded Lie subalgebra of L defined by where g σ k+ℓ|σ| = g σ k for all ℓ ∈ Z. It possesses an invariant non-degenerate symmetric bilinear form B, which is given by where κ stands for the Killing form on g.
Remark 2.6. We can extend σ to an automorphism on L by σ(z k x) := (z/ε σ ) k σ(x). Then L σ can be viewed as the Lie subalgebra of L consisting of fixed points of the extended action of σ on L. In particular, we have the identity L = L id . This motivates our choice of notation. ♦

Structure theory (outer automorphism case)
The classification of all finite order automorphisms of g, explained in [22,20,23], gives the following relation at the level of loop algebras: for any finite order automorphism σ there is an automorphism ν of g, induced by an automorphism of the corresponding Dynkin diagram, such that L σ ∼ = L ν .
Therefore, we first describe the structure of L ν and then explain how regrading of L ν carries over the structure theory to L σ . Let g = n ′ − ∔ h ′ ∔ n ′ + be a triangular decomposition of g and ν be an automorphism of the corresponding Dynkin diagram. The induced outer automorphism ν of g is described explicitly by is a fixed set of standard Chevalley generators for g. The order of such an automorphism is necessarily 1, 2 or 3. The subalgebra g ν 0 turns out to be simple with the following triangular decomposition (2.20) Moreover, when |ν| = 2 or 3 the subspace g ν 1 is an irreducible g ν 0 -module. In the case |ν| = 3 it is isomorphic (as a module) to g ν 2 = g ν −1 .
Remark 2.7. For any automorphism ρ of g we have a natural Z-graded Lie algebra isomorphism L σ ∼ = L ρσρ −1 given by z k x −→ z k ρ(x). Since the automorphism ν is defined by its order up to conjugation, this result implies that L ν is also determined by the order of the automorphism ν. ♦ A pair (α, k), where α ∈ h ∨ and k ∈ Z, is called a root if the joint eigenspace is non-zero. Let Φ be the set of all roots and Φ k be the set of roots of the form (α, k). The triangular decomposition (2.20) of g ν 0 gives rise to the polarization For convenience we introduce two more subsets of roots: The elements of Φ + and Φ − are called positive and negative roots respectively. It is clear that we obtain analogues of a triangular decomposition and Borel subalgebras for L ν , namely Let {α 1 , . . . , α n } be a set of simple roots of g ν 0 with respect to (2.20) and α 0 be the corresponding minimal root. For any root α we write α ∨ for the unique element in h such that B(α ∨ , −) = α(−). The set Π := {(α 0 , 1) is called the simple root system of L ν . It satisfies the following properties: 1. Any (α, k) ∈ Φ can be uniquely written in the form (α, k) = n i=0 c i α i , where c i ∈ Z. If the root (α, k) is positive (negative), then the coefficients c i in its decomposition are all non-negative (non-positive); 2. The matrix A := (a ij ), where is a generalized Cartan matrix of affine type. We call it the affine matrix associated to L ν . The Dynkin diagram corresponding to A is called the Dynkin diagram of L ν .
be the set of standard Chevalley generators for g ν 0 with respect to the choice of simple roots we made earlier. Take two elements X ± 0 ∈ L (±α 0 ,±1) such that The subalgebras p S ± := B ± ∔ N S ∓ are the analogues for the parabolic subalgebras in the theory of semi-simple Lie algebras.

Classification of finite order automorphisms and regrading
We now explain the regrading procedure that makes it possible to transfer all the preceding results of this section to L σ for an arbitrary finite order automorphism σ. Let s = (s 0 , s 1 , . . . , s n ) be a sequence of non-negative integers with at least one non-zero element. Using the properties of the simple root system (2.26) we can write for some unique positive integers a i . We define a positive integer m := |ν| n i=0 a i s i . The following results were proven in [20, Theorem X.5.15]: 1. The set {X + j (1)} n j=0 generates the Lie algebra g and the relations define a unique automorphism σ (s;|ν|) of g of order m such that L ν ∼ = L σ (s;|ν|) . In particular, ν = σ ((1,0,...,0);|ν|) ; 2. Up to conjugation any finite order automorphism σ of g arise in this way.
It follows immediately that for any finite order automorphism σ of g there is an automorphism σ (s;|ν|) and an outer automorphism ν of g such that where the second isomorphism, given by conjugation, is described in Remark 2.7. The automorphism σ (s;|ν|) is called the automorphism of type (s; |ν|). Note that the conjugacy class of the coset σ (s;|ν|) Inn C−LieAlg (g) is represented by ν. Now we describe the first isomorphism in the chain (2.31). Define the s-height ht s (α, k) of a root (α, k) ∈ Φ in the following way: decompose (α, k) with respect to the simple root system Π, i.e. (α, k) = n i=0 c i α i and set We introduce a new Z-grading on L ν , called Z-grading of type s, by declaring deg(f ) = 0 for f ∈ h and deg(f ) = ht s (α, k) for f ∈ L ν (α,k) . The isomorphism G s : L ν −→ L σ (s;|ν|) , called regrading, is given by If L ν is equipped with the grading of type s and L σ (s;|ν|) is equipped with the natural grading given by the powers of z, then G s is a graded isomorphism. We write G s ′ s for the resulting regrading Remark 2.8. The grading given by s = 1 = (1, 1, . . . , 1) is called the principle grading and the corresponding automorphism σ (1;|ν|) is the Coxeter automorphism of the pair (g, ν). ♦

Structure theory (general case)
We finish the discussion of loop algebras by pushing the structure theory for L ν to L σ through the chain of isomorphisms (2.31). We do it gradually, starting with the case σ = σ (s;|ν|) , s = (s 0 , s 1 , . . . , s n ).
Let Φ and Π, as before, be the set of all roots and the simple root system of L ν . From the definition of regrading it is clear that G s (h) = h. This allows us to define the joint eigenspaces g σ (α,ℓ) , α ∈ h ∨ , ℓ ∈ Z, using the exact same formula (2.21) and call (α, ℓ) a root of L σ if g σ (α,ℓ) = 0. Using regrading we can describe the root spaces L σ (α,ℓ) := z ℓ g σ (α,ℓ) of L σ in terms of the root spaces of L ν , namely This gives a bijection between roots of L ν and L σ . More precisely, let Φ σ be the set of all roots of L σ , then The subset Π σ := {(α 0 , s 0 ), (α 1 , s 1 ), . . . , (α n , s n )} ⊆ Φ σ is said to be the simple root system of L σ . We again adopt the notation α i for the simple root (α i , s i ). By definition of ht s the root spaces It is evident from (2.34) that the subspaces N ± , B ± ⊆ L ν are fixed under regrading and thus we can unambiguously use the same notations for them considered as subspaces of L σ . Applying regrading to the set of generators Λ of generators of L σ . When S Π σ we use the same notation S S to denote the semi-simple subalgebra of L σ generated by The corresponding parabolic subalgebras of L σ are defined using the same formulas, namely p S ± := B ± ∔ N S ∓ . We also define This gives the triangular decomposition with the same letter ρ. The roots of L σ with respect to the action of the Cartan subalgebra ρ(h) are of the form (αρ −1 , ℓ), where (α, ℓ) is a root of L σ (s;|ν|) , and the root spaces are described by . (2.39) The set of all roots is again denoted by Φ σ , and its subset is called the simple root system of L σ . Applying ρ to the generators (2.36) of L σ (s;|ν|) we get the set of generators of L σ . Later, when there is no ambiguity, the same notations X ± i and H i are used to denote the elements of generating sets (2.41) and (2.36). Combining (2.39) with (2.34) we define where Φ, as before, is the set of all roots of L ν . Note that this notation is in consistence with the one defined earlier.
Remark 2.9. Let σ = ρσ (s;|ν|) ρ −1 and A be the affine matrix associated to L σ , defined in a way similar to (2.27). Then A coincides with the affine Cartan matrix of L ν and so does the Dynkin diagram of L σ . ♦

Connection to Kac-Moody algebras
As the structure theory developed in the preceding subsections suggests, the notion of a loop algebra is closely related to the notion of an affine Kac-Moody algebra. More precisely, let A be an affine matrix of type X (m) N , g be the simple finite-dimensional Lie algebra of type X N and ν be an automorphism of g induced by an automorphism of the corresponding Dynkin diagram with |ν| = m. Then A is the Cartan matrix of L ν and the affine Kac-Moody algebra K(A) is isomorphic to where Cc is the one-dimensional center of K(A), d is the additional derivation element that acts on L ν as z d dz and the Lie bracket is described by 3 The standard Lie bialgebra structure on L σ and its twists Let K(A) be a symmetrizable Kac-Moody algebra with a fixed invariant non-degenerate symmetric bilinear form B. Then it possesses a Lie bialgebra structure δ 0 , called the standard Lie bialgebra structure on K(A), given by . We can immediately see that δ 0 induces a Lie bialgebra structure on where Z(K(A)) is the center of K(A). In particular, when A is an affine matrix and B is the form mentioned in Subsection 2.2.4 we get a Lie bialgebra structure δ ν 0 on L ν . Applying the methods described in Section 2.2 we induce a Lie bialgebra structure δ σ

Pseudoquasitriangular structure
We want to prove that δ σ t is a pseudoquasitriangular Lie bialgebra structure, i.e. it is defined by an r-matrix. We restrict our attention to a special case σ = σ (s;|ν|) . The general result will then follow from the natural isomorphism mentioned in Remark 2.7.
Let C σ k be the projection of the Casimir element C = 3) can be seen as a generalization of well-known r-matrices. P. Kulish introduced r in [28]. More generally r σ (1;|ν|) 0 was introduced in [3] by A. Belavin and V.Drinfeld, which they later, in [5], called the simplest trigonometric solution. M. Jimbo used r ν 0 in [21] and the formula for r id 0 appears in the recent works [26,32] and [8] under the name "quasi-trigonometric r-matrix". ♦ The statement in [3, Lemma 6.22] suggests the following holomorphic relations between functions defined by (3.3).
2. The functions r σ 0 and r σ ′ 0 satisfy the relation Proof. Using the formulas we can easily deduce that the equations α i (µ) = s ′ i /|σ ′ | − s i /|σ| are consistent and define a unique Since L σ is generated by X ± i , identity (3.7) proves the first statement. To verify the second statement we choose a basis {b i (α,k) } for each L σ (α,k) such that Setting n (α,k) := dim(L σ (α,k) ) we can write where Φ + σ stands for the set of positive roots of L σ . Then the Taylor series of r σ 0 in y = 0 for a fixed x is It converges absolutely in |y| < |x| allowing us to perform the following calculation Equality (3.5) now follows by the identity theorem for holomorphic functions of several variables (see [17]).
Having this result at hand we can obtain the desired pseudoquasitriangularity for twisted standard structures δ σ t . Let us call a meromorphic function r : be a finite order automorphism and t ∈ L σ ⊗ L σ . Then r σ t := r σ 0 + t is a skew-symmetric solution of the CYBE if and only if t is a classical twist of δ σ 0 . Moreover, if t is a classical twist of δ σ 0 , then the following relation holds: 4 Proof. First, assume that σ = σ (1;|ν|) and t = 0. In this case [3, Proposition 6.1] implies that r σ 0 is a skew-symmetric solution of the CYBE and (3.11) follows immediately from comparing [3, Equations (6.4) and (6.5)] with the projections of the defining relations (3.1) to L σ . Secondly, applying Lemma 3.2 we get the statement for an arbitrary finite order automorphism σ and t = 0. Finally, since r σ 0 is skewsymmetric, the skew-symmetry of t is equivalent to the skew-symmetry of r σ t and a straightforward computation gives the equality which completes the proof.
We finish this subsection by relating r-matrices of the form r σ t to trigonometric r-matrices in the sense of the Belavin-Drinfeld classification [3].
Applying the function 1 ⊗ L : for h(u) := f (e u/|σ| ). Since h is holomorphic on C, we can find a holomorphic function ϕ : C −→ Aut C−LieAlg (g) such that ϕ ′ (z) = ad(h(z))ϕ(z) and ϕ(0) = id g (see [26, Proof of Theorem 11.3]). The connected component of id g in the group Aut C−LieAlg (g) is exactly the inner automorphisms of g and thus ϕ : C −→ Inn C−LieAlg (g). Finally, the relation (3.17) implies that the r-matrix The set of poles of X is 2πiZ and hence it is a trigonometric r-matrix.
From now on r-matrices of the form r σ t = r σ 0 + t, where σ is a finite order automorphism of g and t is a classical twists of δ σ 0 , are called σ-trigonometric.

Manin triple structure
The standard Lie bialgebra stucture on an affine Kac-Moody algebra (3.1) can be defined using the standard Manin triple (see [13,Example 3.2] and [10, Example 1.3.8]). Restricting that triple to L σ we get a Manin triple defining the standard Lie bialgebra structure δ σ 0 on L σ . More precisely, δ σ 0 is defined by the Manin triple where ∆ is the image of the diagonal embedding of L σ into L σ × L σ and W 0 is defined by The form B on L σ × L σ is given by where B is the form (2.18). From Theorem 2.4 we know that classical twists t of δ σ 0 are in one-to-one correspondence with Lagrangian subalgebras W t ⊆ L σ × L σ complementary to ∆ and commensurable with W 0 . We now describe the construction of such subalgebras using σ-trigonometric r-matrices.
where π (α,k) is the projection of L σ onto L σ (α,k) . For the second statement let us take an arbitrary (w 1 , ). The desired result now follows from the fact that (w 1 , w 2 ) −→ w 2 − w 1 is an isomorphism between W 0 and L σ . and O σ ± := C((z ±m )). 5 The Lie algebra L σ is naturally an O σ -module and hence we can extend it to L σ Equip the product Lie algebra L σ + × L σ − with the following bilinear form where κ( i a i z i , j b j z j ) := i,j κ(a i , b j )z i+j and res z=0 reads off the coefficient of z −1 . The restriction of this form to L σ × L σ is the form (3.21) defined earlier. Consider the subset where N σ ± stands for the completion of N σ ± with respect to the ideal (z ±m ) ⊆ C[z ±m ], B σ ± := h ∔ N σ ± and π ± h : L σ ± −→ h are the canonical projections . Then W 0 is a Lagrangian subalgebra complementary to the diagonal embedding ∆ of L σ into L σ + × L σ − . Since the Lie bracket on W 0 is the restriction of the Lie bracket on W 0 , the Manin triple also defines the standard Lie bialgebra structure δ σ 0 on L σ . The geometric nature of this Manin triple is revealed in [1]: the sheaves used for construction of σ-trigonometric r-matrices can be viewed as formal gluing of twisted versions of W 0 with L σ ∼ = ∆ over the nodal Weierstraß cubic. ♦

Regular equivalence
Let us fix a finite order automorphism σ of g. We now turn to defining the notion of equivalence for twisted standard bialgebra structures on L σ which is compatible with the corresponding pseudoquasitriangular and Manin triple structures. In other words, we want equivalences of Lie bialgebras to induce equivalences of the corresponding Manin triples and trigonometric r-matrices and vice versa. We stress that the notion of holomorphic equivalence used in the Belavin-Drinfeld classification [3] is unsuitable for our purpose, because in general it does not provide isomorphisms of loop algebras.
In the spirit of [33,34] we define a regular equivalence on the loop algebra L σ to be a regular function φ : C * −→ Aut C−LieAlg (g) preserving the quasi-periodicity of L σ , i.e.
The following theorem demonstrates that the notion of a regular equivalence meets all our needs.
We say that two twisted standard bialgebra structures or σ-trigonometric r-matrices are regularly equivalent if one of the equivalent conditions in Theorem 3.7 holds.

The main classification theorem and its consequences
Before stating the main classification theorem we recall the notion of a Belavin-Drinfeld quadruple for an arbitrary finite order automorphism σ, defined in [3], and then associate it with a classical twist of the standard Lie bialgebra structure δ σ 0 . We start with the case σ = σ (s;|ν|) . Let Π σ , Λ σ and Φ σ be as at the end of Section 2.2. A Belavin-Drinfeld (BD) quadruple is a quadruple Q = (Γ 1 , Γ 2 , γ, t h ), where Γ 1 and Γ 2 are proper subsets of the simple root system Π σ , γ : Γ 1 −→ Γ 2 is a bijection and t h ∈ h ∧ h such that The bijection γ induces an isomorphism θ γ : S Γ 1 −→ S Γ 2 , θ γ (z ±s i X ± i (1)) := z ±s γ(i) X ± γ(i) (1), which we extend by 0 to the whole L σ . Let Φ 1 ⊆ Φ σ be the subset of roots that can be written as linear combinations of elements in Γ 1 . For each α ∈ Φ 1 we choose an element b α ∈ L σ α such that B(b α , b − α ) = 1 and construct the following skew-symmetric tensor where Φ + 1 = Φ 1 ∩ Φ + σ and the second sum has only finitely many non-zero terms since θ γ is nilpotent by condition 2.
We write r σ Q , δ σ Q , R Q and W Q instead of r σ , R t σ Q and W t σ Q respectively. In the case s = (1, . . . , 1) the functions r σ Q and R Q as well as the Cayley transform of R Q were studied in details in [3]. Using regrading and Lemma 3.2 we derive the following statements: • r σ t is a skew-symmetric solution of CYBE. Hence Theorem 3.3 implies that t σ Q is a classical twist of δ σ 0 ; • The inhomogeneous system of linear equations constraining t h is consistent. The dimension of its solution space is ℓ(ℓ − 1)/2, where ℓ = |Π σ \ Γ 1 |; • Setting θ ± γ ±1 := θ γ ±1 | N ± , we have and θ Q is the unique gluing of θ γ with the natural isomorphism which coincides with θ γ on the intersection of the domains. The subalgebra W Q is then given by Conjugating σ by ρ ∈ Aut C−LieAlg (g) we extend all statements and constructions given above to an arbitrary finite order automorphism of g.
Theorem 4.1 (The main classification theorem). For any classical twist t of the standard Lie bialgebra structure δ σ 0 on L σ there is a regular equivalence φ of L σ and a BD quadruple is another BD quadruple, the twisted bialgebra structures δ σ Q and δ σ Q ′ are regularly equivalent if and only if there is an automorphism ϑ of the Dynkin diagram of L σ such that ϑ(Γ i ) = Γ ′ i for i = 1, 2, ϑγϑ −1 = γ ′ and (ϑ ⊗ ϑ)t h = t ′ h , which we denote by ϑ(Q) = Q ′ . We put off the proof of the theorem to Section 5. The rest of this section is devoted to various consequences of Theorem 4.1 and to the proof of its first part in the special case g = sl(n, C) and σ = id.

Classification of twists for parabolic subalgebras
To simplify the notation we again assume σ = σ (s;|ν|) . The following results can be stated for an arbitrary finite order automorphism by applying conjugation.
Let S Λ σ be a proper subset of standard generators of L σ . It is easy to see that the standard Lie bialgebra structure δ σ 0 restricts to both S S and p S ± . Such induced Lie bialgebra structures can be defined using modifications of the Manin triple (3.19). For example, the Lie bialgebra structure (p S + , δ σ 0 | p S + ) is defined by the Manin triple where The following theorem gives a classification of classical twists of the restricted Lie bialgebra structure δ σ 0 | p S + or, equivalently, classical twists of δ σ 0 contained in p S + ⊗ p S + .
Theorem 4.2 (The classification theorem for parabolic subalgebras). For any classical twist t ∈ p S + ⊗ p S + of the standard Lie bialgebra structure δ σ 0 on L σ there exists a regular equivalence φ that restricts to an automorphism of p S + and a BD quadruple Q = (Γ 1 , Γ 2 , γ, t h ) such that The proof of Theorem 4.2 is based on the following three structural results for L σ . Proof. 1. : We can write [20,Lemma X.5.6] it follows that α 1 − α 2 = 0 which, in its turn, implies that for any α ′ ∈ h ∨ 1 there exists a unique weight α ∈ h ∨ such that L σ α ′ = L σ α . This observation combined with [h 1 , a] ⊆ a allows us to write (see [23,Proposition 1.5 Take X ∈ L σ −α ∩ p ∩ N − for some α = 0. Let j be the maximal non-negative integer such that the L (−α,−j) -component of X is non-zero. The structure theory of L σ implies dim(L (−α,−j) ) = 1. Assume that (−α, −j + k) is a root for some positive integer k. Decomposing the difference of (−α, −j) and (−α, −j + k) into the sum of simple roots we get a relation of the form n i=0 c i α i = (0, k). Then the identity n i=0 c i α i = 0 and (2.29) imply that k is an integer multiple of n i=0 a i s i . Using [23, Theorem 5.6.b)] we see that (0, k) is a root. Applying [20, Lemma X.5.5'.(iii)] iteratively we see that k≥0 L (−α,−j+k) ⊆ p. (4.12) Following the proof of [24, Lemma 1.5] we now show that p = p S ′ + , where Assume the claim is false. Let (−γ, −ℓ) / ∈ span Z (S ′ ) be a negative root of maximal height such that there exists an element Y ∈ L σ −γ ∩ p ∩ N − with a non-zero L (−γ,−ℓ) -component Y −ℓ . Then there exists (α j , s j ) ∈ Π σ \ S ′ such that [X + j , Y −ℓ ] = 0 and (−γ + α j , −ℓ + s j ) ∈ span Z (S ′ ), where X + j is the standard generator of L σ . Note that equation (4.12) implies γ = α j . By the structure theory of loop algebras we can find Z ∈ L σ (γ−α j ,ℓ−s j ) ⊆ B + ⊆ p such that 14) The invariance of the form B then gives 0 = [Y −ℓ , Z] ∈ L σ (−α j ,−s j ) . Applying formula (4.12) to X = [Y, Z] ∈ p we get (α j , s j ) ∈ S ′ contradicting our choice of (α j , s j ).
3. : Assume that φ(B + ) = B + . Since N + = [B + , B + ], we see that φ also fixes N + and h. By [24, Lemma 1.29] the automorphism φ maps Π σ to a root basis. But since φ fixes N + , this root basis cosists of positive roots and the only root basis in the set of positive roots is Π σ . Hence φ(Π σ ) = Π σ and thus φ induces an automorphism ϑ of the Dynkin diagram of L σ .
Proof of Theorem 4.2. We prove the statement for σ = σ (s;|ν|) . The general result it obtained using conjugation. By Theorem 4.1 there is a regular equivalence φ 1 on L σ and a BD quadruple Since t h is skew-symmetric, this is easily seen to be a coisotropic subspace of h. Then and, in particular, we have the inclusion h 1 ⊆ φ 1 (p S + ). By the first part of Lemma 4.4 we have Since p S + is self-normalizing, φ 1 (p S + ) is self-normalizing as well. Therefore we get h ⊆ φ 1 (p S + ) and consequently B + ⊆ φ 1 (p S + ). Then the second statement of Lemma 4.4 shows that φ 1 (p S + ) = p S ′ + for some S ′ Π σ . The inclusion 4.15 implies that Γ 1 ⊆ S ′ . Define B ′ := φ −1 1 (B + ). The subalgebra B ′ /p S,⊥ + , being the preimage of the Borel subalgebra , is a Borel subalgebra of S S + h = p S + /p S,⊥ + . Therefore, by the conjugacy theorem for Borel subalgebras, there exists an inner automorphism φ 2 of S S + h mapping B ′ /p S,⊥ + to B + /p S,⊥ + . It can be seen from [20, Lemma X.5.5] that ad x is nilpotent on L σ for any x ∈ L σ (α,k) and α = 0. Combining this result with the equality , we can view φ 2 as a regular equivalence on L σ that restricts to an automorphism of p S + and maps B ′ to B + . The composition φ 2 φ −1 1 is then an automorphism of L σ mapping p S ′ + to p S + and fixing the Borel subalgebra B + . The third part of Lemma 4.4 implies that φ 2 φ −1 1 induces an automorphism ϑ of the Dynkin diagram of L σ such that ϑ(S ′ ) = S. Applying the second part of Theorem 4.1 to ϑ we obtain a regular equivalence φ 3 such that (φ 3 × φ 3 )W Q ′ = W Q:=ϑ(Q ′ ) . The composition φ := φ 3 φ 1 and the quadruple Q satisfy all the requirements of the theorem.

Quasi-trigonometric solutions of CYBE
Letting σ = id and S = Π \ {(α 0 , 1)} the corresponding parabolic subalgebra p S + becomes g[z]. The solutions to CYBE of the form r t = r 0 + t, where t ∈ g[z] ⊗2 , are called quasi-trigonometric. Two quasi-trigonometric solutions r t and r s are called polynomially equivalent if there exists a φ ∈ Therefore, a polynomial equivalence is a regular equivalence that restricts to an automorphism of g[z]. Quasi-trigonometric r-matrices were introduced and classified up to polynomial equivalence and choice of a maximal order in [26,32]. More precisely, it was shown that quasi-trigonometric solutions are in one-to-one correspondence with certain Lagrangian subalgebras of g × g((z −1 )). Embedding the Lagrangian subalgebra, corresponding to a quasi-trigonometric solution r, into some maximal order of g × g((z −1 )), the authors of [26,32] obtained a unique quasi-trigonometric solution r Q (given by a BD quadruple Q) polynomially equivalent to r. In this setting we get the following results: • The classification theorem for parabolic subalgebras 4.2 together with Theorem 3.3 gives a new proof of the above-mentioned classification of quasi-trigonometric r-matrices; • In general, a maximal order in which one can embed the Lagrangian subalgebra correspoding to a quasi-trigonometric solution r is not unique. Choosing two different maximal orders we get two different BD quadruples Q and Q ′ and two polynomially equivalent quasi-trigonometric r-matrices r Q and r Q ′ . By Theorem 4.2 this equivalence induces an automorphism ϑ of the Dynking diagram of L that fixes the minimal root, i.e. ϑ( α 0 ) = α 0 . Therefore, any quasitrigonometric solution is polynomially equivalent to exactly one quasi-trigonometric r-matrix r Q , for some BD quadruple Q, if and only if g is of type A 1 , B n , C n , F 4 , G 2 or E 8 .
We note that there exist regularly equivalent quasi-trigonometric solutions r Q and r Q ′ which are not polynomially equivalent (see Figure 1). Therefore regular equivalence is strictly weaker than polynomial one; • It was shown in [26] that for any quasi-trigonometric r-matrix r there exists a holomorphic function φ : C −→ Inn C−LieAlg (g) such that where X is a trigonometric solution in the Belavin-Drinfeld classification [3]. Combining Lemma 3.2 and Theorem 4.2 we get a general version of this statement with more control over the holomorphic equivalence. Precisely, the trigonometric r-matrix r σ (1;|ν|) Q , by definition, always depends on the quotient of its parameters; in order to obtain it from a σ-trigonometric r-matrix r σ t , where the coset of σ is conjugate to νInn C−LieAlg (g), it is enough to apply a regular equivalence composed with the regrading to the principal grading: • Conjecture 1 in [8] is justified: Combining (3.3) with (4.1) we get the explicit formula for a quasi-trigonmetric solution r Q given by a BD quadruple Q, namely • Question 2 in [8] is answered: Let Q be a BD quadruple and h := |σ (1;1) |. The relation between the quasi-trigonmetric solution (4.18) and the trigonometric solution given by the same quadruple Q (see [3]), is described by regrading from id to the Coxeter automorphism σ (1;1) using Lemma 3.2. More precisely, (4.20) Figure 1: Γ 1 , Γ ′ 1 , Γ 2 and Γ ′ 2 leading to regularly but not polynomially equivalent r-matrices r Q and r Q ′ .

Special case g = sl(n, C) and σ = id
The classification of classical twists of the standard Lie bialgebra structure δ 0 := δ id 0 on L = g[z, z −1 ] with g = sl(n, C) can be done without heavy geometric machinery. More precisely, using the theory of maximal orders developed in [33], we can show that the equivalence classes of twisted standard bialgebra structures on L are in one-to-one correspondence with the equivalence classes of quasitrigonometric r-matrices, which were classified in [26] in terms of BD quadruples.
The following lemma explains the way in which orders emerge in our work.
Proof. By Remark 3.6 the standard Lie bialgebra structure δ 0 on L is defined by the Manin triple (g((z)) × g((z −1 )), ∆, W 0 ). (4.21) Therefore, in view of Theorem 2.4 and its proof, it is enough to show that condition 2. corresponds to the commensurability condition on W t and W 0 , or equivalently, to finite dimensionality of the image of the map T = ψ(t) : W 0 −→ ∆. The latter correspondence is justified by the following chain of arguments: the condition dim(im(T )) < ∞ is equivalent to the inclusion for some non-negative integers N and M , which in its turn is equivalent to Since W t is Lagrangian, inclusions (4.23) are equivalent to condition 2. of the theorem.
A subalgebra W ⊆ g((u)) is called an order if there is a non-negative integer N such that Therefore, condition 2. of Lemma 4.5 means that the projections π 1 W t and π 2 W t are orders.
The following two results from [33] play the key role in the classification of classical twists of δ 0 .
Theorem 4.6. For any order W in sl(n, C((u −1 ))) there is a matrix A ∈ GL(n, C((u −1 ))) such that In particular, any maximal order must be of the form A −1 sl(n, C[[u −1 ]])A for some A ∈ GL(n, C((u −1 ))). Let W ⊆ g((z)) × g((z −1 )) be a Lagrangian subalgebra satisfying the conditions of Lemma 4.5. Since the projections π 1 W ⊆ g((z)) and π 2 W ⊆ g((z −1 )) are orders, by Theorem 4.6 there are matrices A ± ∈ GL(n, C((z ±1 ))) such that (4.25) By Sauvage Lemma 4.7 we can find matrices where d ± are diagonal, such that Taking the product and using the fact that P −1 ± sl(n, C[[z ±1 ]])P ± = sl(n, C[[z ±1 ]]) we obtain the inclusion Note that the componentwise conjugation by Q + or d + is a regular equivalence. Applying these conjugations we get By Theorem 3.3 the classification problem of classical twists of the standard Lie bialgebra structure δ 0 is equivalent to the classification problem of id-trigonometric r-matrices. The following lemma reduces the question even further to quasi-trigonometric r-matrices.
Lemma 4.8. Any id-trigonometric r-matrix is regularly equivalent to a quasi-trigonometric one.
Proof. Let r t = r 0 + t be an id-trigonometric r-matrix, where t is a classical twist of δ 0 , and W t be the corresponding Lagrangian subalgebra of g((z)) × g((z −1 )). By the argument preceding the lemma there is a regular equivalence for some classical twist s of δ 0 . We now show that r s is quasi-trigonometric, or equivalently, that be an orthonormal basis for sl(n, C). Then we can write Assume that s α ′ β ′ kℓ = 0 for some α ′ , β ′ ∈ {1, . . . , n} and k, ℓ ∈ Z such that at least one of the indices k or ℓ is strictly negative, i.e. the tensor s contains a negative power of z in one of its components. Since s is skew-symmetric we may assume without loss of generality that k < 0. Then where the sum in the right-hand side contains z k , k < 0. However, by (4.30) the projection π 1 (W s ) is contained in sl(n, C[[z]]) and hence cannot contain negative powers of z. This contradiction shows that both components of s are polynomials in z.
Quasi-trigonometric r-matrices over sl(n, C) were classified (up to regular equivalence) in [26] using BD qudruples we introduced at the beginning of this section. One can show that if we lift the Lagrangian subalgebra W ⊆ g×g((z −1 )), constructed from a BD quadruple Q in [26], to g((z))×g((z −1 )) we get precisely the Lagrangian subalgebra W Q determined by the relation where W Q is given by (4.5). By Lemma 4.5 the Lagrangian subalgebra W Q uniquely determines the classical twist t Q . This gives the classification of classical twists and proves the first part of Theorem 4.1 in the special case g = sl(n, C).
Remark 4.9. The statement of Lemma 4.8 is not surprising. Its general version can be deduced from Theorems 4.1 and 4.2. Precisely, for any finite-dimensional simple Lie algebra g an id-trigonometric solution r Q = r 0 + t Q , given by a BD qudruple Q = (Γ 1 , Γ 2 , γ, t h ), is regularly equivalent to a quasi-trigonometric one if and only if there is an automorphism ϑ of the Dynkin diagram of L such that α 0 ∈ ϑ(Γ 1 ). It is easy to check that this condition is always satisfied for Dynkin diagrams of types A (1) 2−4 and D 4−10 . Therefore, in these cases any id-trigonmetric solution is regularly equivalent to a quasi-trigonometric one. In other cases it is always possible to find a BD quadruple Q, such that r Q is not equivalent to a quasi-trigonometric r-matrix (see Figure 2). ♦

Algebro-geometric proof of the main classification theorem
In this section we give a brief summary of the results in [7], prove the extension property for formal equivalences between geometric r-matrices (see Theorem 5.5) and, finally, combining this property with the results in [1] on geometrization of σ-trigonometric r-matrices we verify Theorem 4.1.

Survey on the geometric theory of the CYBE
Let E be an irreducible projective curve of arithmetic genus 1. Then E is a Weierstraß cubic, i.e. there are parameters g 2 , g 3 ∈ C such that E is the projective closure of E • = V(y 2 −4x 3 +g 2 x+g 3 ) ⊆ P 2 (w:x:y) by a smooth point p at infinity. E is singular if and only if g 3 2 = 27g 2 3 and an elliptic curve otherwise. In the singular case it has a unique singular point s, which is a simple cusp if g 2 = 0 = g 3 and a simple node otherwise. LetȆ be the set of smooth points of E. Fix a non-zero section ω ∈ Γ(E, where Ω E is the dualising sheaf. We view ω as a global regular 1-form in the Rosenlicht sense (see e.g. [2, Section II.6]).
We consider now a coherent sheaf A of Lie algebras on E such that (ii)Ȃ = A|Ȇ is weakly g-locally free, i.e. A| p ∼ = g as Lie algebras for all p ∈Ȇ.
Property (i) gives that A is torsion free and property (ii) ensures that the rational envelope A K of the sheaf A is a simple Lie algebra over the field K of rational functions on E. Together these properties give the existence of a distinguished section, called geometric r-matrix, ρ ∈ Γ(Ȇ ×Ȇ,Ȃ ⊠Ȃ(D)), where D = {(x, x) ∈Ȇ ×Ȇ : x ∈Ȇ} is the diagonal divisor. This section satisfies a geometric version of a generalised CYBE, although, if E is singular, it lacks skew-symmetry in general, which prevents it to solve the CYBE. Thus we demand one more property of A in this case, which ensures skew-symmetry. If E is singular with a singularity s, we can consider the invariant non-degenerate C-bilinear form where the first map is the Killing form of A K over K and res ω s (f ) = res s (f ω) is the residue taken in the Rosenlicht sense.
Now the main statement of the geometric approach to the CYBE is the following. We want to describe ρ as a series, which can be thought of as a Taylor expansion in the second coordinate at the smooth point p at infinity. To do so, let us switch from the sheaf theoretic setting to one localised at the formal neighbourhood of p. There is a unique element u inside the m p -adic completion O p of the local ring (O E,p , m p ), such that u(p) = 0 and ω p = du. We can identify O p with C[ [u]]. Thus the field of fractions Q p can be identified with C((u)). Consequently, we may view O = Γ(E • , O E ) as a subalgebra of Q p = C((u)).
Since g is simple, Whitehead's lemma implies that H 2 (g, g) = 0 and hence all formal deformations of g are trivial (see e.g. [19,Section A.8]). Thus A p , which can be understood as a formal deformation of g by Property (ii) of A, is trivial as a formal deformation, i.e. there exists an , called formal trivialisation, of Lie algebras such that the induced isomorphism is the identity. We obtain an induced Lie algebra isomorphism Q( A p ) = A p ⊗ Qp Q p −→ g((u)) via the C((u))-linear extension of ξ, which we denote by the same symbol. We write the image of Γ(E • , A) ⊆ Q( A p ) under ξ by g(ρ) ⊆ g((u)).
Note that g((u)) is equipped with the invariant non-degenerate C-bilinear form f, g ∈ g((u)). which maps ρ to ∞ k=0 n ℓ=1 f kℓ ⊗y k b ℓ , where {b ℓ } is a basis of g and {f kℓ } is the basis of g(ρ) ⊆ g((u)), uniquely determined by B p (f kℓ , u k ′ b ℓ ′ ) = δ kℓ δ k ′ ℓ ′ . Remark 5.3. Let us clarify what we mean by Taylor expansion in the second coordinate. Let P k = Spec( O p /m k p O p ) and ι k : P k −→ E be the injection, mapping the closed point of P k to p. Then we can consider the pull-back with respect to idȆ \{p} × ι k to obtain the morphism where we have used that (Ȇ \ {p} × P k ) ∩ D) = ∅ and applied the Künneth isomorphism. Mapping Γ(Ȇ \ {p},Ȃ) via ξ to g((u)), using A p /m k p A p ∼ = g[u]/u k g[u] and applying the projective limit with respect to k, yields the desired injection Γ(Ȇ ×Ȇ,Ȃ ⊠Ȃ(D)) −→ (g ⊗ g)((x))[[y]]. ♦ The theorem suggests, that the geometric r-matrix ρ actually determines A completely. Our next goal is to formalise this idea. The construction we present is known in other situations, see e.g. [30]. The algebras O and g(ρ) inherit the ascending filtrations from the natural filtrations of C((u)) and g((u)), namely we have such that O j O k ⊆ O k+j , O j g(ρ) k ⊆ g(ρ) j+k and [g(ρ) j , g(ρ) k ] ⊆ g(ρ) j+k . Therefore, we can consider the associated graded objects 6 gr(O) and gr(g(ρ)), given by O j and gr(g(ρ)) := ∞ j=0 g(ρ) j . (5.7) Note that gr(g(ρ)) is a graded Lie algebra over the graded C-algebra gr(O). Let us denote by gr(g(ρ)) ∼ the associated quasi-coherent sheaf of Lie algebras on Proj(gr(O)) (see e.g. [18, Section II.5]).
Lemma 5.4. We have E = Proj(gr(O)) and the formal trivialisation ξ induces an isomorphism A −→ gr(g(ρ)) ∼ of sheaves of Lie algebras, which we again denote by ξ.
Proof. We can view E

Extension property of formal local equivalences
Now let us consider two coherent sheaves of Lie algebras A 1 and A 2 on E satisfying the conditions (i) -(iii) of Section 5.1 and denote by ρ 1 and ρ 2 the corresponding geometric r-matrices. Fix formal trivialisations ξ i of A i at p and consider the corresponding isomorphisms ξ i : are the Taylor expansions of ρ i described in Theorem 5.2. We are now in a position to show that any formal equivalence of ρ 1 and ρ 2 at p extends to a global isomorphism of the corresponding sheaves.

Proof of the main classification theorem
Fix an automorphism σ ∈ Aut C−LieAlg (g) of finite order m and an outer automorphism ν from the coset σInn C−LieAlg (g). Let t ∈ L σ ⊗ L σ be a classical twist of the standard Lie bialgebra structure δ σ 0 on L σ . In view of Theorem 3.7, to prove the first part of Theorem 4.1 we need to show that there exists a regular equivalence φ ∈ Aut O σ −LieAlg (L σ ), taking values in Inn C−LieAlg (g), and a BD quadruple Q such that (ψ(x) ⊗ ψ(y))r σ t (x, y) = r σ Q (x, y) (5.12) for all x, y ∈ C * , x m = y m . Combining the results of Section 3.1 and [3], we get the following statement. Proof. By Theorem 3.4 and its proof there exists a holomorphic function φ 1 : C −→ Inn C−LieAlg (g) and a trigonometric (in the sense of the Belavin-Drinfeld classification) r-matrix X such that φ 1 (0) = id g and X(u − v) = (φ 1 (u) ⊗ φ 2 (y))r σ t (e u/m , e v/m ). (5.14) Furthermore, it is shown in [3] that there is a holomorphic function φ 2 : C −→ Inn C−LieAlg (g) such that φ 2 (0) = id g and (φ 2 (u) ⊗ φ 2 (v))X(u − v) = r σ (1,ord(ν)) Q (e u/h , e v/h ), (5.15) where h := |σ (1,ord(ν)) |. Combining these results and applying the regrading scheme from Lemma 3.2 we get the desired holomorphic function.
Our next goal is to apply Theorem 5.5 to this holomorphic equivalence to obtain a regular one. Therefore, we need sheaves which give rise to the σ-trigonometric r-matrices from (5.13). These were constructed in [1]. With this choice, we can interpret the series expansion of ρ t , described in Remark 5.3, as the Taylor series of r σ t (e u/m , e v/m ) at v = 0. Therefore, we can apply Theorem 5.5 to the Taylor expansion of (5.13) at v = 0 to obtain an isomorphism ψ : A t −→ A Q , satisfying (ψ ⊠ ψ)ρ t = ρ Q , where A Q := A t Q and ρ Q := ρ t Q .

(5.19)
We finish the proof of the main theorem by explaining when two BD quadruples give rise to equivalent twisted standard structures. Lemma 5.9. Let Q = (Γ 1 , Γ 2 , γ, t h ) and Q ′ = (Γ ′ 1 , Γ ′ 2 , γ ′ , t ′ h ) be two BD quadruples. Then δ σ Q and δ σ Q ′ are regularly equivalent if and only if there exists an automorphism ϑ of the Dynkin diagram of L σ such that ϑ(Q) = Q ′ .