Belavin–Drinfeld solutions of the Yang–Baxter equation: Galois cohomology considerations
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Abstract
We relate the Belavin–Drinfeld cohomologies (twisted and untwisted) that have been introduced in the literature to study certain families of quantum groups and Lie bialgebras over a non algebraically closed field \(\mathbb {K}\) of characteristic 0 to the standard nonabelian Galois cohomology \(H^1(\mathbb {K}, \mathbf{H})\) for a suitable algebraic \(\mathbb {K}\)group \(\mathbf{H}.\) The approach presented allows us to establish in full generality certain conjectures that were known to hold for the classical types of the split simple Lie algebras.
Keywords
Belavin–Drinfeld Yang–Baxter Quantum group Lie bialgebra Galois cohomologyMathematics Subject Classification
Primary 20G10 17B37 17B62 17B67 Secondary 17B011 Introduction
The appearance of Galois cohomology in the classification of certain quantum groups is one of the primary goals of this paper. In order to do this we first need to “linearize” quantum groups (in the same spirit that, via the exponential map, complex simply connected simple Lie groups can be studied/classified by looking at their Lie algebras). The linearization problem is an extremely technical construction brought forward as a conjecture in the work of Drinfeld [5] (see also [3] and [4]), and proved in the seminal work of Etingof and Kazhdan (see [6, 7]). An outline of this correspondence can be found in the Introductions of [9, 11], wherein one can also find an explanation of why the description of which Lie bialgebras structures exists on the Lie algebra \({\mathfrak g}\otimes _k k((t)),\) with \({\mathfrak g}\) simple finite dimensional over an algebraically closed field k of characteristic 0, arise naturally in the classification of quantum groups. The approach to the classification of Lie bialgebra structures on \({\mathfrak g}\otimes _k k((t)) \) developed in [9, 10, 11] and [14] is by the introduction of the socalled “Belavin–Drinfeld cohomologies”. The calculation of these cohomologies is mostly done on a casebycase basis in the classical types using realizations of the relevant objects as matrices. The main thrust of the present paper is to realize Belavin–Drinfeld cohomologies as usual Galois cohomologies. This allows for uniform realizationfree proofs in all types of results that were conjectured (and were known to hold on many of the classical types). The methods that we describe also open an avenue for further studies of Lie bialgebra structures over nonalgebraically closed fields.
2 Notation
Throughout this paper \(\mathbb {K}\) will denote a field of characteristic 0. We fix an algebraic closure of \(\mathbb {K}\) which will be denoted by \(\overline{\mathbb {K}}.\) The (absolute) Galois group of the extension \(\overline{\mathbb {K}}/\mathbb {K}\) will be denoted by \(\mathcal {G}.\) ^{1}
If V is a \(\mathbb {K}\)space (resp. Lie algebra), we will denote the \(\overline{\mathbb {K}}\)space (resp. Lie algebra) \(V \otimes _\mathbb {K}\overline{\mathbb {K}}\) by \(\overline{V}.\)
If \(\mathbf{K}\) is a linear algebraic group over \(\mathbb {K}\) the corresponding (nonabelian) Galois cohomology will be denoted by \(H^1(\mathbb {K}, \mathbf{K}).\) (See [13] for details. See also [2, 12, 15] for some of the more technical aspects of this theory that will be used in what follows without further reference). We recall that \(H^1(\mathbb {K}, \mathbf{K}) \) coincides with the usual nonabelian continuous cohomology of the profinite group \(\mathcal {G}\) acting (naturally) on \(\mathbf{K}(\overline{\mathbb {K}}).\)
Let \({\mathfrak g}\) be a split finite dimensional simple Lie algebra over \(\mathbb {K}.\) In what follows \(\mathbf{G}\) will denote a split (connected) reductive algebraic group over \(\mathbb {K}\) with the property that the Lie algebra of the corresponding adjoint group \(\mathbf{G}_\mathrm{ad}\) is isomorphic to \({\mathfrak g}.\) ^{2}
We fix once and for all a Killing couple \((\mathbf{B}, \mathbf{H})\) of \(\mathbf{G}\). The induced Killing couple on \(\mathbf{G}_\mathrm{ad}\), which we denote by \((\mathbf{B}_\mathrm{ad}, \mathbf{H}_\mathrm{ad})\), leads to a Borel subalgebra and split Cartan subalgebras of \({\mathfrak g}\) which will be denoted by \({\mathfrak b}\) and \({\mathfrak h}\) respectively. Our fixed Killing couple leads, both at the level of \(\mathbf{G}_\mathrm{ad}\) and \({\mathfrak g},\) to a root system \(\Delta \) with a fixed set of positive roots \(\Delta _+\) and base \(\Gamma = \{ \alpha _1, \ldots , \alpha _n \}.\) ^{3}
The Lie bialgebra structures that we will be dealing with are defined by rmatrices, which are element of \({\mathfrak g}\otimes _\mathbb {K}{\mathfrak g}\) satisfying \(\mathrm{CYB}(r) = 0\) where CYB is the classical Yang–Baxter equation (see Sect. 3 below and [8] for definitions). For future use we introduce some terminology and notation. Consider the action of \(\mathbf{G}\) on \({\mathfrak g}\otimes _\mathbb {K}{\mathfrak g}\) induced from the adjoint action of \(\mathbf{G}\) on \({\mathfrak g}.\) Let R be a commutative ring extension of \(\mathbb {K}\). If \(X \in \mathbf{G}(R)\) and \( v \in ({\mathfrak g}\otimes _\mathbb {K}{\mathfrak g})_a(R) = ({\mathfrak g}\otimes _\mathbb {K}{\mathfrak g})\otimes _\mathbb {K}R \simeq ({\mathfrak g}\otimes _\mathbb {K}R) \otimes _R ({\mathfrak g}\otimes _\mathbb {K}R)\), then the adjoint action of X in v will be denoted by \(\mathrm{Ad}_X(v).\) ^{4}
Along similar lines if \(\sigma \in \mathcal {G}\) we will write \(\sigma (r)\) instead of \((\sigma \otimes \sigma )(r).\)
3 The Belavin–Drinfeld classification
 1.
Each \(r_\mathrm{BD}\) is an rmatrix (i.e. a solution of the classical Yang–Baxter equation) satisfying \(r + r^{21} = \Omega \) (where \(\Omega \) is the Casimir operator of \({\mathfrak g}.\))
 2.
Any nonskewsymetric rmatrix for \({\mathfrak g}\) is equivalent to a unique \(r_\mathrm{BD}.\)
This leads to two cases, according to whether c is in \(\mathbb {K}\) or not. The first case is treated with the untwisted Belavin–Drinfeld cohomologies, while the second one, in the case when \(\mathbb {K}= k((t))\) with k algebraically closed of characteristic 0, leads to twisted Belavin–Drinfeld cohomologies. These and their relations to Galois cohomology are the contents of the next two sections.
4 Untwisted Belavin–Drinfeld cohomology
Theorem 4.1
Assume that \(\overline{r} = c \, \mathrm {Ad}_{X}(r_\mathrm{BD})\) are as above. Then \(r_\mathrm{BD}\) is rational, i.e. it belongs to \({\mathfrak g}\otimes _\mathbb {K}{\mathfrak g}.\) Furthermore \(X^{1}\gamma (X)\in \mathbf{C}(\mathbf{G}, r_\mathrm{BD})(\overline{\mathbb {K}})\) for all \(\gamma \in \mathcal {G}.\) \(\square \)
We now recall (with our notation) the Belavin–Drinfeld cohomology definitions and results developed in [9]. Let \(r_\mathrm{BD} \in {\mathfrak g}\otimes _\mathbb {K}{\mathfrak g}\) be a Belavin–Drinfeld rmatrix.
Definition 4.2
An element \(X\in \mathbf{G}(\overline{\mathbb {\mathbb {K}}})\) is called a Belavin–Drinfeld cocycle associated to \(\mathbf{G}\) and \(r_\mathrm{BD}\) if \(X^{1}\gamma (X)\in \mathbf{C}(\mathbf{G}, r_\mathrm{BD})(\overline{\mathbb {K}}) \), for any \(\gamma \in \mathcal {G}.\)
The set of Belavin–Drinfeld cocycles associated to \(r_\mathrm{BD}\) will be denoted by \(Z_{BD}(\mathbf{G},r_\mathrm{BD})\). Note that this set contains the identity element of \(\mathbf{G}(\overline{\mathbb {K}})\).
Definition 4.3
Two cocycles \(X_1\) and \(X_{2}\) in \(Z_{BD}(\mathbf{G},r_\mathrm{BD})\) are called equivalent if there exists \(Q\in \mathbf{G}(\mathbb {K})\) and \(C\in \mathbf{C}(\mathbf{G}, r_\mathrm{BD})(\overline{\mathbb {K}})\) such that \(X_{1}=QX_{2}C\).
It is easy to check that the above defines an equivalence relation in the nonempty set \(Z_{BD}(\mathbf{G},r_\mathrm{BD})\)
Definition 4.4
Let \(H_{BD}(\mathbf{G},r_\mathrm{BD})\) denote the set of equivalence classes of cocycles in \(Z_{BD}(\mathbf{G},r_\mathrm{BD})\).
We call this set the Belavin–Drinfeld cohomology associated to \((\mathbf{G}, r_\mathrm{BD}).\) The Belavin–Drinfeld cohomology is said to be trivial if all cocycles are equivalent to the identity, and nontrivial otherwise.
Remark 4.5
The relevance of this concept, as explained in [9], is that there exists a onetoone correspondence between \(H_{BD}(\mathbf{G},r_\mathrm{BD})\) and Lie bialgebra structures \(({\mathfrak g}, \delta )\) on \({\mathfrak g}\) with classical double isomorphic to \({\mathfrak g}\oplus {\mathfrak g}\) and \(\overline{\delta }=\delta _{r_\mathrm{BD}}\) up to equivalence.
Our next goal is to realize \(H_{BD}(\mathbf{G},r_\mathrm{BD})\) in terms of usual Galois cohomology. This will allow us to establish some open conjectures, as well as “interpret” some peculiarities observed with \(H_{BD}(\mathbf{G},r_\mathrm{BD})\) for certain special orthogonal groups.
Proposition 4.6
Proof
The remarkable fact is that the the algebraic \(\mathbb {K}\)group \(\mathbf{C}(\mathbf{G}, r_{BD}))\) is diagonalizable. Indeed since \(r_\mathrm{BD} \in {\mathfrak g}\otimes _\mathbb {K}{\mathfrak g}\) we can reason exactly as in [9] Theorem 1 to conclude that.
Theorem 4.7
\(\mathbf{C}(\mathbf{G}, r_{BD})\) is a closed subgroup of \(\mathbf{H}.\) \(\square \)
Combining this last result with Proposition 4.6 we obtain, with the aid of Hilbert’s theorem 90, that
Corollary 4.8
If the algebraic \(\mathbb {K}\)group \(\mathbf{C}(\mathbf{G}, r_{BD})\) is connected then \(H_{BD} (\mathbf{G},r_\mathrm{BD})\) is trivial. \(\square \)
One of the most important rmatrices is the socalled Drinfeld–Jimbo \(r_\mathrm{DJ}\) given by
Definition 4.9
\(r_\mathrm{DJ} = \sum _{\alpha >0}e_{\alpha }\otimes e_{\alpha } + \frac{1}{2}\, \Omega _0\)
where \(\Omega _0\), as has already been mentioned, stands for the \({\mathfrak h}\otimes _\mathbb {K}{\mathfrak h}\) component of the Casimir operator \(\Omega \) of \({\mathfrak g}\) written with respect to our choice of \(({\mathfrak b},{\mathfrak h}).\)
In [9] it was conjectured that \(H_{BD} (\mathbf{G},r_\mathrm{DJ})\) is trivial under the assumption that \(\mathbf{G}\) be simple and \(\mathbb {K}=\mathbb {C} ((\hbar ))\). The conjecture was established by a casebycase reasoning for most of the classical groups. Further progress on this problem (still for the classical algebras but now with an arbitrary base field of characteristic 0) is given in [11]. The Galois cohomology interpretation we have given provides an affirmative much more general answer to this question.
Theorem 4.10
\(H_{BD} (\mathbf{G},r_\mathrm{DJ})\) is trivial for any split reductive group \(\mathbf{G}\) over a field \(\mathbb {K}\) of characteristic 0.
Proof
We already know that \(\mathbf{C}(\mathbf{G}, r_{DJ})\) is a closed subgroup of our split torus \(\mathbf{H}\). It is also clear from Definition 4.9 that all elements of \(\mathbf{H}(\overline{\mathbb {K}})\) fix \(\overline{r_\mathrm{DJ}}\). This yields \(\mathbf{C}(\mathbf{G}, r_{DJ}) = \mathbf{H}.\) By the last Corollary the Theorem follows.
Remark 4.11
The situation for \(\mathbf{G}=\mathbf{SO}(2n)\) is different. Assume that \(\alpha _n\) and \(\alpha _{n1}\) are the end vertices of the Dynkin diagram of \(\mathfrak {so}(2n)\). Assume also \(\alpha _{n1}=\tau ^k (\alpha _{n})\) for some integer k, where \(\tau : \Gamma _1 \rightarrow \Gamma _2\) defines \(r_\mathrm{BD}\). It was shown in [9] that \(\mathbf{C}(\mathbf{G}, r_\mathrm{BD})=\mathbf{T}\times \mathbb {Z}/2\mathbb {Z}\) in this case and \(\mathbf{C}(\mathbf{G}, r_\mathrm{BD})=\mathbf{T}\) otherwise.
From our results it follows that \(H_{BD} (\mathbf{G},r_\mathrm{BD})\) is trivial in the second case.
We see again that the Galois cohomology point of view “explains” why certain Belavin–Drinfeld cohomolgies are trivial, and why in the case of \(\mathbf{SO}_{2n}\) the appearance of nontrivial classes is natural.
We end this section with a statement, which provides a complete description of nontwisted Belavin–Drinfeld cohomologies in terms of the Galois cohomologies of algebraic groups.
Theorem 4.12
Proof
This is a direct consequence of the various definitions and of Proposition 4.6 (both the statement and the proof). \(\square \)
From Steinberg’s theorem (see [13] Ch III Theorem 3.2.1’) we obtain.
Corollary 4.13
5 Twisted Belavin–Drinfeld cohomologies
In this section we assume that \(\mathbb {K}=k((t))\) where k is algebraically closed of characteristic 0. Fix an element \(j \in \overline{\mathbb {K}}\) such that \(j^2 = t.\) We will denote the quadratic extension \(\mathbb {K}(j)\) of \(\mathbb {K}\) by \(\mathbb {L}\). Twisted Belavin–Drinfeld cohomologies where introduced in [9, 11] to describe a new class of Lie bialgebras structure on \({\mathfrak g}\) whose Drinfeld double (see [8] for the definition and constriction of this object) is isomorphic to \({\mathfrak g}\otimes _\mathbb {K}\mathbb {L}.\)
In this section our reductive group \(\mathbf{G}\) will be assumed to be of adjoint type. Within the general framework described in Sect. 2, our analysis corresponds to the case when in (3.2) the constant c does not belong to \(\mathbb {K}\). As we have seen, then \(c^2 \in \mathbb {K}.\)
Before we recall how these Lie bialgebras appear and what the relevant definitions are, we introduce some notation and give an explicit description of \(\text {Gal}(\mathbb {K})\) and \(\mathrm{Gal}(\mathbb {L})\) that will be used in the proofs.
Fix a compatible set of primitive mth roots of unity \(\xi _m ,\) namely such that \(\xi _{me} ^e = \xi _m \) for all \(e > 0.\) Fix also, with the obvious meaning, a compatible set \(t^\frac{1}{m}\) of mth roots of t in \(\overline{\mathbb {K}}.\) There is no loss of generality in assuming that \(t^\frac{1}{2} = j.\)
Let \(\mathbb {K}_{m} = \mathbb {C}((t^\frac{1}{m})).\) We can then identify \(\mathrm{Gal}(\mathbb {K}_m/\mathbb {K})\) with \(\mathbb {Z}/m\mathbb {Z}\) where for each \(e \in \mathbb {Z}\) the corresponding element \(\overline{e} \in \mathbb {Z}/m\mathbb {Z}\) acts on \(\mathbb {K}_m\) via \( ^{\overline{e}} t^{\frac{1}{m}}_i = \xi ^{e}_{m} t^{\frac{1}{m}}_i.\)
5.1 Definition of the twisted cohomologies
The following result is proved in [11].
Proposition 5.1
 (i)

\( \,\, r=j Ad_X(r_\mathrm{BD})\)
 (iia)

\( \,\, X^{1}\gamma (X)\in \mathbf{C}(\mathbf{G}, r) \,\, \text {for any} \,\, \gamma \in \mathrm{Gal}(\mathbb {L})\)
 (iib)

\( \,\mathrm{Ad}_{X^{1}\gamma _1 (X)}(r_\mathrm{BD})=r_\mathrm{BD}^{21}.\)
To define twisted Belavin–Drinfeld cohomology we will need the following more general result.
Proposition 5.2

\(\gamma (r)=r\) for all \(\gamma \in \mathrm{Gal}(\mathbb {L})\)

\(\gamma _1 (r)=r^{21}\)
Proof
Let \(\gamma \) and \(\gamma '\) satisfy (5.2). Then \(\gamma \gamma ' \in \mathcal {H}\). It follows that \(\mathcal {H}\) is a subgroup of \(\mathcal {G}\) index 2, in fact \(\mathcal {H}= \mathrm{Gal}(\mathbb {L})\). For \(\gamma _1\) we conclude that \(\gamma _1 (r)=rj\Omega =r^{21}\). \(\square \)
Remark 5.3
It is easy to see that if r satisfies the conclusions of the proposition above, then r induces a Lie bialgebra structure on \({\mathfrak g}\).

\( \,\, X^{1}\gamma (X)\in \mathbf{C}(\mathbf{G}, r)(\overline{\mathbb {K}})\,\, \text {for any} \,\, \gamma \in \mathrm{Gal}(\mathbb {L})\)

\( \,\mathrm{Ad}_{X^{1}\gamma _1 (X)}(r_\mathrm{BD})=r_\mathrm{BD}^{21}.\)
Definition 5.4
An element \(X \in \mathbf{G}(\overline{\mathbb {K}})\) is called a twisted Belavin–Drinfeld cocycle for \(\mathbf{G}\) and \(r_\mathrm{BD}\) if \(X^{1}\gamma (X) \in \mathbf{C}(\mathbf{G}, r_\mathrm{BD})\) for any \(\gamma \in \mathrm{Gal}(\mathbb {L})\) and \(\mathrm{Ad}_{X^{1}\gamma _1(X)}(r_\mathrm{BD}) = r_\mathrm{BD}^{21}\).
The definition of equivalent cocycles is just as in the untwisted case.
Definition 5.5
Two twisted Belavin–Drinfeld cocycles X and Y are said to be equivalent if \(Y=QXC\) for some \(C \in \mathbf{C}(\mathbf{G}, r_\mathrm{BD})(\overline{\mathbb {K}})\)and \(Q\in \mathbf{G}(\mathbb {K})\).
It is clear that the above defines an equivalence relation on the set \(\overline{Z}_{BD} (\mathbf{G}, r_\mathrm{BD})\) of twisted Belavin–Drinfeld cocycles.
Definition 5.6
The twisted Belavin–Drinfeld cohomology related to \(\mathbf{G}\) and \(r_\mathrm{BD}\) is the set of equivalence classes of the twisted cocycles. We will denote it by \(\overline{H}_{BD} (\mathbf{G}, r_\mathrm{BD})\).
Note that, unlike the untwisted case, it is not clear that twisted Belavin–Drinfeld cocycles exist.
Remark 5.7
Assume that \(r_\mathrm{BD}\) is rational. Then the twisted Belavin–Drinfeld cohomology \(\overline{H}_{BD} (\mathbf{G}, r_\mathrm{BD})\) gives a onetoone correspondence between equivalence of Lie bialgebra structures on \({\mathfrak g}\) such that over \(\overline{\mathbb {K}}\) they become gauge equivalent to the Lie bialgebra structure defined by \(jr_\mathrm{BD}.\)
5.2 Twisted cohomology for the Drinfeld–Jimbo rmatrix
The only good understanding of twisted Belavin–Drinfeld cohomologies is for the Drinfeld–Jimbo rmatrix \(r_\mathrm{DJ}\) (which is clearly rational). Our main goal is to establish the following.
Theorem 5.8
The set \(\overline{H}_\mathrm{BD}^1 (\mathbf{G}, r_\mathrm{DJ})\) consists of one element.
5.2.1 Construction of S and \(J\in \mathbf{G}(\mathbb {L})\) such that \(\gamma _1(J)=JS\)
Lemma 5.9
Let \(w_0\) be the longest element of the Weyl group W of the pair \((\mathbf{B},\mathbf{H})\). Then there exists an element \(S\in \mathbf{G}(\mathbb {K})\) such that \(S^2=1_{\mathbf{G}(\mathbb {K})}\) and \(S({\mathfrak g}^{\alpha })={\mathfrak g}^{w_0 (\alpha )}\) for all roots \(\alpha \in \Delta \).
Proof
Let \(c\in \mathrm{Aut}({\mathfrak g})\) be the Chevalley involution. Thus \(c^2=\mathrm{Id},\ c({\mathfrak g}^{\alpha })={\mathfrak g}^{\alpha }\) and c restricted to the Cartan subalgebra \({\mathfrak h}\) is scalar multiplication by \(1\). If \(\mathrm{Out} ({\mathfrak g})\) is trivial, then \(w_0({\alpha })=\alpha \) and we take \(S = c\).
In general note that \(w_0\in \mathrm{Out} ({\mathfrak g})\), so we can view this as an element \(d\in \mathrm{Aut}({\mathfrak g})\) of order 2. Clearly, \(cd=dc\) and we set \(S=cd\), which is of order 2.
It remains to be shown that \(S\in \mathbf{G}(\mathbb {K})\). Since both c and d stabilize \({\mathfrak h}\), so does S. From this it follows that \(S({\mathfrak g}^\alpha )={\mathfrak g}^{\theta (\alpha )}\) for some \(\theta \in \mathrm{Aut}(\Delta )\) (the automorphism group of our root system). It is wellknown that \(\mathrm{Aut}(\Delta )\) is a semidirect product of W and \(\mathrm{Out} ({\mathfrak g})\). Moreover, \(S\in \mathbf{G}(\mathbb {K})\) if and only if the restriction of S to \({\mathfrak h}\) is in W. But by construction this restriction is \(\theta =w_0\in W\). \(\square \)
It is clear from Definition 4.9 that \(Ad_S(r_{DJ})=r_{DJ}^{21}\). Since \(\mathbf{C}(\mathbf{G}, r_\mathrm{DJ}) = \mathbf{H}\) we can redefine twisted Belavin–Drinfeld cocycles for \(r_\mathrm{DJ}\) as follows.
Lemma 5.10
 (i)
\(X^{1}\gamma (X) \in \mathbf{H}(\overline{\mathbb {K}})\) for any \(\gamma \in \mathrm{Gal}(\mathbb {L})\), and
 (ii)
\(\mathrm{Ad}_{X^{1}\gamma _1 (X)}(r_\mathrm{BD}) = \mathrm{Ad}_S(r_\mathrm{BD})\).
As we shall see this definition will allow us to compute the corresponding twisted Belavin–Drinfeld cohomology by means of usual Galois cohomology.
Proposition 5.11
Let \(S\in \mathbf{G}(\mathbb {K})\) be as in the previous lemma. Then there exists \(J\in \mathbf{G}(\mathbb {L})\) such that \(\gamma _1 (J)=JS\)
Proof
There exists a unique continuous group homomorphism \(u: \mathcal {G} \rightarrow G(\overline{\mathbb {K}})\) such that \(u(\gamma _1) = S\). Given that \(\gamma _1 (S)=S\) our u is a cocycle in \(Z^1 (\mathbb {K}, \mathbf{G})\).
Note that our element J is a twisted Belavin–Drinfeld cocycle.
5.2.2 Computation of \(\overline{H}_{BD} (\mathbf{G}, r_\mathrm{DJ})\)
The aim of this section to show that \(\overline{H}_{BD} (\mathbf{G}, r_\mathrm{DJ})\) consists of one element generated by the class of the element J constructed above. This will in particular prove Theorem 5.8.
The elements of \(H^1(\mathbb {K}, \tilde{\mathbf{H}})\) mapping to the class of 1 are given by the image of \(H^1(\mathbb {K}, \mathbf{H})\) which is trivial by Hilbert 90. The elements of \(H^1(\mathbb {K}, \tilde{\mathbf{H}})\) mapping to the class of j are given by the image of \(H^1(\mathbb {K}, \mathbf{H}')\) where the \(\mathbb {K}\)group \(\mathbf{H}'\) is a twisted form of \(\mathbf{H}.\) By Steinberg’s theorem \(H^1(\mathbb {K}, \mathbf{H}')\) vanishes. It follows that \(H^1(\mathbb {K}, \tilde{\mathbf{H}})\) has two elements. More precisely.
Theorem 5.12
 1.
The trivial class,
 2.
The class of the cocycle \(u_J\) defined by \(u_J,\ u_J(\gamma )=J^{1} \gamma (J).\) In particular \(u_J(\gamma _1) = S.\)
Theorem 5.13
The map \(X \mapsto \tilde{u}_X\) described above induces an injection \(\overline{H}_{BD} (\mathbf{G}, r_\mathrm{DJ})\rightarrow H^1 (\mathbb {K}, \tilde{\mathbf{H}}) = \{1,j\}.\) More precisely the fiber of the trivial class 1 is empty and that of j consist of the class of the Belavin–Drinfeld cocycle J.
Proof
 1.Suppose that \(\tilde{u}_X\) is in the trivial class \(1\in \{1,j\}\). By definition there exists an element \(h \in \tilde{\mathbf{H}}(\overline{\mathbb {K}})\) such that \(\tilde{u}_X(\gamma ) = h^{1}{^\gamma }h.\) Let \(C \in \mathbf{H}(\overline{\mathbb {K}})\) and \(\epsilon \in \{0,1\}\) be such that \(h =S{^\epsilon }C.\) Since S is fixed by the Galois group \(h^{1}{^\gamma }h = C^{1}{^\gamma }C.\) But this implies, in particular, that \(\tilde{u}_X(\gamma _1) \in \mathbf{H}(\overline{\mathbb {K}})\). This last is false sinceThe fiber of the trivial class 1 under our canonical map is therefore empty.$$\begin{aligned} \tilde{u}_X(\gamma _1)(r_\mathrm{DJ}) = X^{1}\gamma _1(X)(r_\mathrm{DJ}) = r_\mathrm{DJ}^{21}\ne r_\mathrm{DJ}. \end{aligned}$$
 2.Suppose that the class of X is mapped to \(j\in \{1,j\}\). Then \(\tilde{u}_X\) is cohomologous to \(u_J\). By definition there exist \(h =S{^\epsilon }C\) as above such thatfor all \(\gamma \in \mathcal {G}.\) An arbitrary element of our Galois group is of the form \(\gamma _n = n \gamma _1\) Recall that \(J \in \mathbf{G}(\mathbb {L})\) (hence it is fixed by all \(\gamma _n\) with n even), that \(J^{1}\gamma (J) = S \in \mathbf{G}(\mathbb {K})\) and that \(S^2 = 1.\) These easily imply that \(J^{1}\gamma _n(J) = S^n.\) Taking this into account we get from (5.4) that for all \(n \in \mathbb {Z}\)$$\begin{aligned} X^{1}\gamma (X)=C^{1}S^{\epsilon }J^{1}\gamma (J)S^{\epsilon }\gamma (C) \end{aligned}$$(5.4)From these it readily follows that \(Q^{1} := JCX^{1}\) is invariant under the action of \(\mathcal {G}.\) Thus \(Q \in \mathbf{G}(\mathbb {K}).\) Since \(X = QJC\) we have that X and J are equivalent Belavin–Drinfeld cocycles. The fiber of j has therefore exactly one element.$$\begin{aligned} X^{1}\gamma _n (X)=C^{1}J^{1}\gamma _n(J)\gamma _n (C) \quad \mathrm{if} \ \ n \ \ \mathrm{is\, odd} \end{aligned}$$(5.5)
This last result shows that Theorem 5.8 holds. More precisely.
Corollary 5.14
The twisted Belavin–Drinfeld cohomology \(\overline{H}_{BD} (\mathbf{G}, r_\mathrm{DJ})\) consists of one class only, namely the class of the cocycle J. \(\square \)
Footnotes
 1.
For the “untwisted” Belavin–Drinfeld cohomologies \(\mathbb {K}\) will be arbitrary. In the “twisted” case \(\mathbb {K}= k((t))\) where k is algebraically closed.
 2.
The case which is most of interest to us is when \(\mathbf{G}= \mathbf{G}_\mathrm{ad}.\) That said, peculiar phenomena appear when \(\mathbf{G}\) is either \({\mathbf {GL}}_n\) or \({\mathbf {SL}}_n\). Of course \(\mathbf{G}_\mathrm{ad}\) is then \({\mathbf {PGL}}_n\) and \({\mathfrak g}= \mathfrak {sl}_n.\) The case of \(\mathbf{G}= \mathbf{SO}_{2n}\) is also interesting. For all of these reasons we try to maintain our set up as general as possible.
 3.
The elements of \(\Delta \) are to be thought as characters of \(\mathbf{H}_\mathrm{ad}\) or elements of \({\mathfrak h}^*\) depending on whether we are working at the group or Lie algebra level. This will always be clear from the context.
 4.
In contrast to the notation \((\mathrm{Ad}_X \otimes \mathrm{Ad}_X)(v)\) used elsewhere.
 5.
As the reader has probably guessed, it will not necessarily be true that the class of our cocycle will any longer be trivial when viewed as taking values in the smaller group \(\mathbf{C}(\mathbf{G}, r_{BD})\). This subtlety is in fact the reason that allows Galois cohomology to be brought into be picture.
 6.
For example \(\mathbb {K}= \mathbb {C}((t)).\) This is the case most relevant to quantum groups.
 7.
We are in the situation when c in (3.1) is not in \(\mathbb {K}.\) Strictly speaking we should have \(c = aj\) with \(a \in \mathbb {K}^\times \). Since we are working on Lie bialgebras up to equivalence we may assume without loss of generality that \(a = 1\).
Notes
Acknowledgements
We would like to thank Seidon Alsaody for his careful reading of our manuscript.
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