Double Quasi-Poisson Brackets: Fusion and New Examples

We exhibit new examples of double quasi-Poisson brackets, based on some classification results and the method of fusion. This method was introduced by Van den Bergh for a large class of double quasi-Poisson brackets which are said differential, and our main result is that it can be extended to arbitrary double quasi-Poisson brackets. We also provide an alternative construction for the double quasi-Poisson brackets of Van den Bergh associated to quivers, and of Massuyeau–Turaev associated to the fundamental groups of surfaces.


Introduction
We fix a finitely generated associative unital algebra A over a field k of characteristic 0, and we write ⊗ = ⊗ k for brevity. Following Van den Bergh's initial construction [20] Here, the multiplication refers to the outer A-bimodule structure on A ⊗ A, that is a d b = (Here, we define τ (123) : A ⊗3 → A ⊗3 by τ (123) (a 1 ⊗ a 2 ⊗ a 3 ) = a 3 ⊗ a 1 ⊗ a 2 .) This map is an instance of triple bracket : a k-trilinear map, which is also a derivation in its last argument for the outer bimodule structure of A ⊗3 , and which satisfies a generalisation of the cyclic antisymmetry Eq. 1.1 : An important class of double brackets consists of double Poisson brackets. They are such that the associated triple brackets {{−, −, −}} identically vanish. Using Eq. 1.4, this condition can be seen as a version of Jacobi identity with value in A ⊗3 . These structures have also been introduced by Van den Bergh [20], and have been a recent subject of study, see e.g. [4,10,[15][16][17][18][19]22].
Another interesting class of double brackets appears when the unit in A admits a decomposition 1 = s∈I e s in terms of a finite set of orthogonal idempotents, i.e. |I | ∈ N × and e s e t = δ st e s . In that case, we view A as a B-algebra for B = ⊕ s∈I ke s , and we naturally extend the definition of a double bracket to require B-bilinearity, i.e. it vanishes when one of the arguments belongs to B. Then, we say that the double bracket is quasi-Poisson, or that on any a, b, c ∈ A. Condition Eq. 1.6 is an expanded form of the original definition [20, §5.1], and only needs to be checked on generators by the properties of a triple bracket. The main interest of this form is that it is easier to handle in order to classify double quasi-Poisson brackets. Indeed, up to now few cases of double quasi-Poisson brackets are known except associated to quivers [20,21] or fundamental groups of surfaces [14]. To have more examples, we provide a complete classification on the free algebra over one generator, and continue the investigation for two generators (with some restrictions). The reader could then be tempted to say that such examples do not provide particular insights about double quasi-Poisson brackets in general. However, an important result of Van den Bergh is that we can perform fusion [20, §5.3] : we can identify idempotents in an algebra with a double quasi-Poisson bracket, and the resulting algebra also admits a double quasi-Poisson bracket. For example, if we respectively denote by e 1 , e 2 the units of k[t], k s 1 , s 2 viewed as orthogonal idempotents inside k[t] ⊕ k s 1 , s 2 , the fusion algebra obtained by the identification of e 1 and e 2 is nothing else than k t, s 1 , s 2 . Hence, knowing a double quasi-Poisson bracket before fusion gives another one on the free algebra over three generators. Therefore, our classification allows to get double quasi-Poisson brackets over any free algebra in general, though not all of them. Moving to more exotic examples of double quasi-Poisson algebras, there was a major obstruction to use this fusion process up to now, as we needed the double quasi-Poisson bracket to be differential, see § 2.1 for the definition. It was expected by Van den Bergh that this assumption could be removed [20, §5.3], and the main aim of this paper is to prove this result in its most general form. The advantage of our proof of this theorem is to get an explicit form for the double quasi-Poisson bracket in the algebra A obtained by identification of the idempotents e s , e t ∈ A : it is given in terms of the double bracket on A, together with a second double bracket computed in Lemma 2.19 which was uncovered by Van den Bergh [20,Theorem 5.3.1]. Therefore, it becomes easy to see when a double quasi-Poisson bracket has been obtained by fusion. In particular, we can show using our classification of double quasi-Poisson bracket on the free algebra on two generators (with some mild restrictions) provided in § 4.3 that any such double bracket is isomorphic to one obtained by fusion, see Theorem 4.10. This unexpected result suggests that knowing double quasi-Poisson brackets on k[t] and the path algebra of the (double of the) one-arrow quiver t : 1 → 2 may be enough to obtain most examples of double quasi-Poisson algebra structures on free algebras.
A particular subclass of double quasi-Poisson brackets consists in those that admit a distinguished element. To be precise, given a double quasi-Poisson algebra (A, {{−, −}}) as above with a complete set of orthogonal idempotents (e s ) s∈I , a multiplicative moment map is an invertible element = s∈I s with s ∈ e s Ae s such that we have for all a ∈ A and s ∈ I We say that the triple (A, {{−, −}} , ) is a quasi-Hamiltonian algebra. As a continuation of the previous result, Van den Bergh showed that we can also obtain a moment map after fusion inside a quasi-Hamiltonian algebra when the double bracket is differential [20, Theorem 5.3.2]. We also show that this result can be extended to the general case, see Theorem 2.15. As a by-product of our method to prove that we keep a double quasi-Poisson bracket or multiplicative moment map after fusion, we can easily recover the double quasi-Poisson brackets of Van den Bergh [20] and Massuyeau-Turaev [14], see Theorems 3.3 and 3.5.
To finish this introduction, let us recall that double brackets have been introduced by Van den Bergh as a non-commutative version of an antisymmetric biderivation following the non-commutative principle formulated by Kontsevich and Rosenberg [11,12]. More precisely, as explained in §5.1, any double bracket on an algebra A gives rise to an antisymmetric biderivation on the algebra k[Rep(A, n)] for any n ≥ 1, i.e. on the coordinate ring of the representation space Rep(A, n) parametrising n-dimensional representations of A. In the same way, a double (quasi-)Poisson bracket provides a non-commutative notion of a (quasi-)Poisson bracket under this non-commutative principle. Hence, the present study can be understood as giving new examples of quasi-Poisson brackets on representation spaces. This article proceeds as follows. In Section 2, we recall the necessary constructions needed to understand the fusion procedure, and then prove the main result of this paper which is the fusion of quasi-Hamiltonian algebras in the general case. In light of those developments, we give in Section 3 some examples of double quasi-Poisson brackets obtained by fusion. We also give an alternative (though equivalent) construction of Van den Bergh's quasi-Hamiltonian algebras associated to quivers, and those of Massuyeau-Turaev associated to the fundamental group of compact surfaces with boundary. In Section 4, we get some first classification results for double quasi-Poisson brackets. We finish by explaining in Section 5 the notion of quasi-Poisson algebra, which is the structure carried by the coordinate ring of representation spaces of double quasi-Poisson algebras. There are four appendices that contain some computational proofs.

Fusion of quasi-Hamiltonian algebras
We consider finitely generated algebras A, B over a field k of characteristic zero. We assume that A is a B-algebra and, without loss of generality, we identify B with its image in A. Our goal is to prove the main theorems of this paper, which are presented in §2.2. To state and prove these results, we need some preliminary constructions associated to double brackets, which were already introduced by Van den Bergh in [20] for most of them. Since these results easily extend to the case of n-brackets (see below for the definition, noting that double brackets are 2-brackets), we begin by introducing the objects that we will use in full generalities.

Preliminary results
We equip the algebra A ⊗n with the outer A-bimodule structure which is given by b(a 1 ⊗ . . . ⊗ a n )c = ba 1 ⊗ . . . ⊗ a n c. For any s ∈ S n , we introduce the map τ s : A ⊗n → A ⊗n defined by τ s (a 1 ⊗ . . . ⊗ a n ) = a s −1 (1) ⊗ . . . ⊗ a s −1 (n) . Following Van den Bergh [20], we say that a Blinear map {{−, . . . , −}} : A ×n → A ⊗n is a n-bracket if it is a derivation in its last argument for the outer bimodule structure on A ⊗n , and if it is cyclically anti-symmetric : By B-linearity, we mean that the map {{−, . . . , −}} is k-linear in each argument and it vanishes on any subset A ×i−1 × B × A n−i , 1 ≤ i ≤ n. Double and triple brackets as defined in the introduction can be equivalently obtained from the above formulation, for which they correspond to the cases n = 2 and n = 3.

Poly-vector fields and n-brackets
{{a 1 , . . . , a n }} Q = δ n (a n ) δ 1 (a 1 ) ⊗ δ 1 (a 1 ) δ 2 (a 2 ) ⊗ . . . ⊗ δ n−1 (a n−1 ) δ n (a n ) . We say that a n-bracket is differential if it is given by μ(Q) for some Q ∈ (D B A) n . For example, given some δ 1 δ 2 ∈ (D B A) 2 we have a differential double bracket by setting (2.2) for any b, c ∈ A. Any differential double bracket is a linear sum of such double brackets. By [5], we can write is the A-bimodule of non-commutative 1-forms relative to B [9]. The bimodule 1 B A allows us to give conditions for the map μ to be an isomorphism.
We claim that the double bracket is no longer differential on A k . Indeed, any element P ∈ D A k /k is uniquely defined by the image of the generator x, so it can be decomposed as and since we need to satisfy P (x k ) = 0, we obtain that with possible relations between the coefficients (c a,b ). If we consider arbitrary P , Q ∈ D A k /k of that form, we see that the double bracket they define by Eq. 2.1b can be written as for some d a,b ∈ k. Thus, any differential double bracket {{−, −}} on A k is such that {{x, x}} ∈ A k ⊗ A k has homogeneous components of degree ≥ 3, where we define the degree of x a ⊗ x b as a+b. Hence, the double bracket on A k given by {{x, x}} = 1 2 (x 2 ⊗1−1⊗x 2 ) can not be differential.
The algebra D B A is a noncommutative version of the algebra of polyvector fields on a manifold : D B A admits a canonical double Schouten-Nijenhuis bracket, which makes D B A into a double Gerstenhaber algebra [20, §2.7,3.2]. We write this (graded) double where m is the multiplication on the algebra D B A. We note that the following results hold.

Induced brackets and fusion algebras
We now state several ways to get new n-brackets from old ones. Most of these results are straightforward extensions of propositions given in [20, §2.5], which were originally stated in the case n = 2.
Given an algebra A over B and a non-empty subset S ⊂ A, we can consider the universal localisation A S as an algebra over B. The morphism f : A → A S induces a unique map of double derivations f * : Proof Note that a n-bracket on A S needs to satisfy a 1 , . . . , a n−1 , s −1 = −s −1 {{a 1 , . . . , a n−1 , s}} s −1 , for any a 1 , . . . , a n−1 ∈ A S and s ∈ S due to the derivation property. Using the cyclic antisymmetry and the derivation property, we can then always rewrite {{a 1 , . . . , a n }} with a 1 , . . . , a n ∈ A S in terms of sums and products in A S containing only the n-bracket evaluated on elements of A.
We use this result without further mention throughout the text. Next, if e ∈ B is an idempotent, we get a canonical map π e : A → eAe, a → eae, which extends to double derivations as π e * : D A/B → D eAe/eBe , δ → eδe. In the case where B = BeB, we get a non-unique decomposition 1 = i p i eq i , and it yields a trace map Tr : A → eAe given by Tr(a) = i eq i ap i e. It also gives a map Tr : D A/B → D eAe/eBe by setting Tr(δ) = i eq i δp i e, which can be written as Tr(δ)(eae) = eδ (a)p i e ⊗ eq i δ (a)e for any a ∈ A. To extend this to polyvector fields, note that Tr : D B A → eD B Ae : Q → i eq i Qp i e defines a map D B A → D eBe eAe by Proposition 2.5.
Given algebras A, A over B with algebra monomorphisms j : B → A and j : B → A , recall that the free algebra A * B A is given by T k (A ⊕ A )/J , where J is the two-sided ideal generated by the relations a 1 ⊗ a 2 = a 1 a 2 , a 1 ⊗ a 2 = a 1 a 2 , j (b) = j (b) for all a 1 , a 2 ∈ A, a 1 , a 2 ∈ A and b ∈ B. SetĀ = A * B A . The canonical maps i : A →Ā, i : A →Ā yield maps of double derivations i * : D A/B → DĀ /A and i * : D A /B → DĀ /A , which can both be seen to take value in DĀ /B . In particular, they extend to polyvector fields.
In particular, n-brackets are compatible with base changes. We now use these results, and assume that there exist orthogonal idempotents e 1 , e 2 ∈ B. The extension algebraĀ of A along the pair (e 1 , e 2 ) is given bȳ (2.4) where μ = 1 − e 1 − e 2 , and Mat 2 (k) is seen as the k-algebra generated by e 1 = e 11 , e 12 , e 21 , e 2 = e 22 with e st e uv = δ tu e sv . The fusion algebra A f of A along (e 1 , e 2 ) is the algebra obtained fromĀ by discarding elements of e 2Ā +Āe 2 , i.e.
We also say that A f is the fusion algebra obtained by fusing e 2 onto e 1 .
Note  From now on, we denote the compositions Tr •i and Tr •i * simply as Tr. where on the left-hand side we have the associated triple bracket given by Eq. 1.4, while the triple brackets in the right-hand side are defined from Proposition 2.1 with E 3 s ∈ (D B A) 3 . It is then an easy exercise to check that Eq. 2.7 evaluated on a, b, c ∈ A gives (1.6), so that this definition coincides with the one given in the introduction. Note that under the assumption of Proposition 2.2 the double quasi-Poisson bracket {{−, −}} is differential for some Q ∈ (D B A) 2

Main theorems
Hereafter, we assume that A is a B-algebra for B = ke 1 ⊕ . . . ⊕ ke N a semisimple k-algebra. Our aim is to prove the following results.

Image of the gauge elements
We have well-defined double derivations E s ∈ D A/B , 1 ≤ s ≤ N , and we want to know what are their images in the fusion algebra A f , obtained by fusing the idempotent e 2 onto e 1 as in §2. 1 [20, §5.3], but we give a proof for the sake of clarity.) Proof We only need to prove the equality on generators of A f . By Lemma 2.11, we can write any a ∈ A f as a = e + αe − , for a ∈ A and some e + ∈ {e 12   if a = t for t ∈ A , Tr(E 1 )(e 12 u) = (e 12 u)e 1 ⊗ e 1 , Tr(E 2 )(e 12 u) = −e 1 ⊗ (e 12 u) ,

Properties of the double bracket {{−, −}} fus
(2.19) The exact same method works in each case. Note that only ten cases need to be computed as other double brackets can be obtained by cyclic antisymmetry :

Fusion for the double quasi-Poisson bracket
We prove Theorem 2.14. To do so, we need to show that {{−, −, −}} f = for any s = 1, 2 by Lemma 2.17. Similarly, since e 2 = e 21 e 12 , Modulo graded commutators, we can write  20) modulo graded commutators, which finishes the proof.

Fusion for the moment map
Note that f has an inverse If we set The proof consists of checking (2.21) and (2.22) on generators, which is done in Appendix B.

Elementary examples of fusion
Given two double quasi-Poisson algebras (A, {{−, −}}) and (A , {{−, −}} ) over k, we can use Remark 2.13 to get a double quasi-Poisson bracket on A ⊕ A which is B-linear for B = ke 1 ⊕ ke 2 with e 1 = (1, 0) and e 2 = (0, 1). Using Theorem 2.14, we can get a double quasi-Poisson bracket on the fusion algebra (A ⊕ A ) f obtained by fusing e 2 onto e 1 . By iterating this process, we can create new double quasi-Poisson algebras using the different examples given in Section 4. (The same holds for quasi-Hamiltonian algebras if we have moment maps.) Nevertheless, as far as we use differential double brackets, one could argue that this could already be done using Van den Bergh's results [20 which is not differential. The double bracket is in fact quasi-Poisson, e.g. as a consequence of Proposition 4.1.
which is a double quasi-Poisson algebra. Let A be an arbitrary double quasi-Poisson k-algebra. Then we can consider A ⊕ A with idempotents e 1 = (1, 0), e 2 = (0, 1). For B = ke 1 ⊕ ke 2 , A ⊕ A has a B-linear double quasi-Poisson bracket by Remark 2.13. We can form the fusion algebraĀ = (A ⊕ A ) f obtained by fusing e 2 onto e 1 , which we see as an algebra over k by identifying the only remaining non-zero idempotent e 1 with 1. Using Lemma 2.11,Ā is the algebra generated by x and e 12 we 21 for w ∈ A . Thus, we can identifyĀ with A * k A , and see the elements of A as generators of type 1 (2.6a) after fusion, while the elements of A are generators of type 4 (2.6d). Therefore, using Theorem 2.14, we have a double quasi-Poisson bracket onĀ given by if we use Eq. 2.14d in Lemma 2.19, while the double brackets {{x, x}} and w, w for w, w ∈ A are just the ones in A and A respectively.
We can form A s = k[x s ]/(x ks s ) and consider A = ⊕ s A s where we denote each unit by e s so that A is an algebra over B = ⊕ s ke s . Moreover, it has a double quasi-Poisson bracket by Remark 2.13. Fusing e 2 onto e 1 , then e 3 onto e 1 and so on up to e M , we get the fusion algebra which is just a k-algebra. By Theorem 2.14 and Lemma 2.19, A has a double quasi-Poisson bracket given by I have been unable to find a quasi-Hamiltonian algebra whose double bracket is not differential. It is an interesting question to determine if such an example exists, in order to see whether Theorem 2.15 is strictly stronger than [20,Theorem 5.3.2] or not.

Generalities
Let Q be a finite quiver with vertex set denoted I . We define the functions t, h : Q → I that associate to an arrow a either its tail t (a) ∈ I or its head h(a) ∈ I . We form the doubleQ of the quiver Q with the same vertex set I by adding an opposite arrow a * : h(a) → t (a) to each a ∈ Q. We naturally extend h, t toQ, and set (a * ) * = a for each a ∈ Q so that the map a → a * , a ∈Q, defines an involution onQ. We form the path algebra kQ which is the k-algebra generated by the arrows a ∈Q and idempotents (e s ) s∈I labelled by the vertices such that a = e t (a) ae h(a) , e s e t = δ st e s . This implies that we read paths from left to right. We see kQ as a B-algebra with B = ⊕ s∈I ke s .
We define :Q → {±1} as the map which takes value +1 on arrows originally in Q, and −1 on the arrows inQ \ Q. For each a ∈ Q, we also choose γ a ∈ k and set γ a * = γ a . Finally, we associate to kQ the algebra A obtained by universal localisation from the set S = {1 + (γ a − 1)e t (a) + aa * | a ∈Q}. This is equivalent to add local inverses (γ a e t (a) + aa * ) −1 for each a ∈Q (i.e. they are inverses to γ a e t (a) + aa * in e t (a) Ae t (a) ). If γ a = 0, then a −1 := a * (aa * ) −1 satisfies a −1 = (a * a) −1 a * , so that aa −1 = e t (a) and a −1 a = e h(a) ; the same holds for a * .

Furthermore, A is quasi-Hamiltonian for the multiplicative moment map
In Eq. 3.3, we take the product defining s with respect to the ordering on T s . If all γ a = +1, this result explicitly gives the double bracket defined from a poly-vector field P ∈ (D B A) 2 in [20, Theorem 6.7.1], which was written in the above form for particular choices of ordering in [8,Proposition 2.6]. In fact, if all γ a = 0, the result is equivalent to the previous case up to rescaling. If some γ a are equal to zero, our result also encompasses the generalisation proposed in [

Proof of Theorem 3.3
As in the proof of [20,Theorem 6.7.1], we begin with the quiver Q sep which has vertex and arrow sets given by We form the doubleQ sep of Q sep , which amounts to add the arrows We define on it the involution * given by 5) for all b ∈ Q sep and which is zero on every other pair of generators, while the multiplicative moment map is defined as To get a quasi-Hamiltonian structure on A, it remains to fuse all these disjoint quivers ofQ sep according to the ordering that we chose at the vertices ofQ. More precisely, label the vertices in the quiverQ as {1, . . . , |I |}, and label the arrows according to the ordering, that is if the arrow b is the k-th element with respect to the total ordering on T s (going from the minimal to the maximal element in the chain) where s = t (b), we label it a s,k . We use the same names for the arrows inQ sep . To recoverQ, we rename v a 1,1 as 1, then fuse 1 and v a 1,2 which we still name 1, then continue with all vertices labelled v a 1,k for increasing values of k. Next, we do the same for vertices 2, . . . , |I | and recover the quiverQ. In terms of algebras, this means that we consider the fusion algebra obtained after fusing e va 1,2 onto e 1 , then e va 1,3 onto e 1 , and so on. This finally yields the algebra A. Therefore, it suffices to use Theorems 2.14 and 2.15 to get the desired result. We directly find that is given by Eq. 3.3, but understanding the double bracket requires some work. We first show Eqs. 3.1a and 3.1b, where there is nothing to prove if a is not a loop. So assume that a is a loop, and a < t (a) a * . By construction the only new terms arise when we glue w 1 := v a with w 2 := v a * , so to compute these terms we use Theorem 2.14 with the vertices w 1 , w 2 respectively playing the role of 1,2. We have that after fusion a is a generator of third type Eq. 2.6c, so that by Eq. 2.16c the fusion amounts to add a term 1 Similarly, a * is a generator of second type Eq. 2.6b so by Eq. 2.15b we get a term 1 which gives the correct double bracket by adding (3.5). In the case a * < t (a) a, take w 1 := v a * with w 2 := v a and the proof is similar, but now a is of second type and a * is of third type.
Before proving (3.2), we need some preparation. Consider α, β ∈Q and s ∈ I with α < s β, α = β, β * . With the labelling given above, we have that α = a s,k 0 , β = a s,k 1 for some 1 ≤ k 0 < k 1 ≤ |T s |, and v α = v a s,k 0 , v β = v a s,k 1 . WriteQ α for the quiver obtained fromQ sep by fusing all the vertices v a s ,k with either s < s, or s = s with k < s k 1 (i.e. we fuse all vertices up to excluding v β ); set t α and h α for the tail and head maps inQ α . WriteQ β for the quiver obtained fromQ α by additionally fusing the vertex v a s,k 1 (i.e. we fuse all vertices inQ sep up to including v β ). Set again t β and h β for the associated tail and head maps. We let A α and A β respectively denote the algebras obtained from A sep by fusion to arrive at the quiversQ α andQ β . Lemma 3.4 The step of performing fusion from A α to A β amounts to add the following terms in the double quasi-Poisson bracket of A between the elements α, α * and β, β * : Proof We know that h α (β) = t α (β) (otherwise it would contradict the order in which we glue vertices), so we have that α, α * are generators of the first type, β is a generator of the second type and β * is a generator of the third type in the algebra A β obtained after fusing w 1 := v α and w 2 := v β . We have by Eq. 2.14b that the following terms appear in the double quasi-Poisson bracket {{−, −}} β on A β for α, β β : 1 2 (e w 1 ⊗ αβ − α ⊗ β). The first term is non-zero only if h β (α) = t β (β), or t β (α * ) = t β (α), hence we can multiply it by δ t β (α),t β (α * ) . After all fusions are performed, w 1 is just t β (α) and we get Eq. 3.7a.
Using again (2.14b) then twice (2.14c) amounts to add the terms A discussion as in the first case allows to get Eqs. 3.7b-3.7d.
To prove (3.2), we have to show that the equality holds for any kind of ordering when the two arrows meet, as it is trivially zero if they do not. We first show what happens if they meet at exactly one vertex.
c}} by cyclic antisymmetry. This proves (3.2) in this case.
Next, assuming only If b, c meet at two vertices but none of them is a loop, we can conclude by adding together the two corresponding results just derived. Hence, it remains the tedious computation to check the cases when at least b or c is a loop. We now write two illuminating cases where h(b) = t (b) = t (c), and leave to the reader the task to verify all the remaining cases (noting that we only need to check half these cases because of the cyclic antisymmetry) using Eqs. 3.7a-3.7d. Assume , no term contributes to {{b, c}}. Hence, we only need to understand what happens when we glue the vertices corresponding to t (b) = h(b) and t (c), and by Eq.
(Alternatively, we could have used Eq. 3.7c with α = b * , β = c to get the same answer. It is important to remark that we glue vertices not arrows, so that only one of these two cases has to be considered, not both together.) When gluing the vertices ofQ sep corresponding to t (b) and t (c), we get by Eq. 3.7a with α = b, β = c the only term − 1 ⊗ bc contributes to {{b, c}} and we are done.

Double quasi-Poisson brackets for fundamental groups of surfaces
Let denote a compact connected surface with fixed orientation, and such that it has a non-empty boundary ∂ . We denote by g ≥ 0 its genus, and r + 1 ≥ 1 the number of boundary components. Let * ∈ ∂ be a base point, and denote by π 1 ( , * ) the corresponding fundamental group of . The algebra A = kπ 1 ( , * ) can be presented in terms of generators Here, represents the loop around the boundary component containing * (with suitable orientation, see Fig. 1), and we used the multiplicative commutator [α, β] = αβα −1 β −1 .
Note that in the products we write the factors from the left to the right with increasing indices.
Our aim is to give an alternative proof relying only on fusion of the next result due to Massuyeau and Turaev [14], which endows A with a quasi-Hamiltonian algebra structure. (We rescale their double bracket by a factor 1/2.) Hence, this proof is the non-commutative analogue of the fusion process for representation varieties [3].
Theorem 3.5 For the presentation considered above, the algebra A = kπ 1 ( , * ) has a double quasi-Poisson bracket defined for any 1 ≤ i ≤ g by

11)
and for any 1 ≤ k ≤ r and k < l, it is defined by (3.12) Furthermore, for any a = α i , β i , γ k , the double bracket with is given by In particular, is a multiplicative moment map, and A is quasi-Hamiltonian.

Fig. 1
A system of loops on in the cases (g, r) = (1, 0) and (g, r) = (0, 1). They can be used as generators for π 1 ( , * ) after being connected to the base point * ∈ ∂ in a natural way Proof We skip the trivial case g = r = 0 where A = k. If g = 0, r = 1, we have the generators of the boundary components, call them γ, , with corresponding to the component containing * ∈ ∂ . Note that the algebra k[γ ±1 ] has a double quasi-Poisson bracket {{γ, γ }} = 1 2 (γ 2 ⊗1−1⊗γ 2 ) such that γ is a moment map as we show in §4.1. Since it is isomorphic to A 0 = k γ ±1 , ±1 /(γ = ), we have a quasi-Hamiltonian algebra structure on A = A 0 . If g = 1, r = 0, we have two generating cycles α, β and the generator of the boundary component , so that A is just and moment map = [α, β]. By identification, we get a quasi-Hamiltonian algebra structure on A = A 1 . We now prove the general case. We consider g copies of the quasi-Hamiltonian algebra A 1 and r copies of A 0 , and we form A 1 ⊕ . . . ⊕ A 1 ⊕ A 0 ⊕ . . . ⊕ A 0 . By Remark 2.13, this is a quasi-Hamiltonian algebra. We denote the element (0, . . . , 0, 1, 0, . . . , 0) with 1 in i-th position as e i , 1 ≤ i ≤ g + r. By fusing e 2 onto e 1 , then e 3 onto e 1 and so on, we get a quasi-Hamiltonian algebra structure by fusion on where α i , β i , i are the images of α, β, from the i-th copy of A 1 , 1 ≤ i ≤ g, while γ k ,¯ k are the images of γ, in the k-th copy of A 0 . Rewriting the moment map in the algebra obtained by fusion in terms of the i ,¯ k using Theorem 2.15, then removing these unnecessary elements, we can rewrite the latter algebra as This is precisely A. The double quasi-Poisson bracket is then easily obtained from Theorem 2.14, Lemma 2.19, and the ones on A 0 , A 1 . For example, fix 1 < j ≤ g. After the step of fusion of e j onto e 1 , any φ i ∈ {α i , β i } with 1 ≤ i < j is a generator of first type (2.6a) while φ j ∈ {α j , β j } is a generator of fourth type (2.6d), so that φ i , φ j gets a contribution given by Eq. 2.14d. The fusion of e k onto e 1 with k = j does not give any additional term in φ i , φ j , and we obtain (3.10).

Remark 3.6
To see that the double bracket from Theorem 3.5 coincides with the one of Massuyeau-Turaev, note that the double brackets that do not involve the moment map are just those given in [14, §8.3], while for the moment map they are given in [14, §9.2]. In particular, our construction is such that the moment map is the generator of the loop at the boundary component containing * ∈ ∂ . We should also note that our proof applies to the case of a weighted surface discussed in [14, Section 10], i.e. when we fix n k ∈ N × , 1 ≤ k ≤ r, so that the generators α i , β i , γ k (see Eq. 3.15) satisfy the extra constraints γ n k k = 1 for 1 ≤ k ≤ r. Indeed, we can see that the ideal generated by γ n − 1 in A 0 is stable under the double bracket for any n ∈ N × , so that we can start the proof with the algebras k[γ ±1 k ]/(γ n k k − 1) instead of r copies of A 0 . Finally, remark that the way we are gluing components is the algebraic analogue of the boundary connected sum discussed in [14,Appendix B.2].

Remark 3.7
It is an interesting problem to determine whether we can modify the definition of double quasi-Poisson bracket and keep a non-trivial fusion property as in Theorems 2.14 and 2.15. As a motivation, note that for A = kπ 1 ( , * ) the double quasi-Poisson bracket given in Theorem 3.5 was introduced by Massuyeau-Turaev [14] by (cyclically anti-)symmetrizing an operation A ×2 → A ⊗2 denoted by {{−, −}} η . This means that for any a, b ∈ A,  We say that it is an isomorphism of double quasi-Poisson algebras if ψ is an isomorphism of B-algebras, which implies that the inverse ψ −1 : A → A is also an isomorphism of double quasi-Poisson algebras. It seems natural to seek for isomorphisms between the different double quasi-Poisson algebra structures associated to quivers by Van den Bergh [20], or the slight generalisation given by Theorem 3.3. The same problem can be formulated for the double bracket of Massuyeau-Turaev [14] given in Theorem 3.5 if we change the presentation of the fundamental group by swapping factors 1 in Eq. 3.8. In fact, these results easily follow from the next proposition, which is a non-commutative version of [3,Proposition 5.7]. The proof of this statement is quite tedious, so we skip it and we will provide details in further work. Let us simply mention that the isomorphisms between multiplicative preprojective algebras with different orderings, which are given in the proof of [7,Theorem 1.4], are precisely induced by this map.

Elementary classification
All our algebras are over a field k of characteristic 0 for convenience, but the discussion may be adapted to any integral domain (with unit) such that 2 is invertible. One could get rid of the latter localisation by rescaling the defining property (1.6) as in [14].

Polynomial ring in one variable
We begin by classifying all double quasi-Poisson brackets on A = k[t] over B = k. Our argument is similar to the classification of Powell [18,Proposition A.1] in the case of a double Poisson bracket, i.e. when the associated triple bracket (1.4) identically vanishes. We define a degree on A by setting |t| = 1, to get the decomposition A = ⊕ k≥0 kt k in homogeneous components, which can clearly be extended to A ⊗n : an element a 1 ⊗. . .⊗a n is homogeneous of degree k if each a i is homogeneous in A and i |a i | = k.

Proposition 4.1 A has a double bracket which is quasi-Poisson if and only if it is of the form
for λ, μ, ν ∈ k with 4(μ 2 − λν) = 1.
Proof First, we remark that the quasi-Poisson property can be rewritten from Eq. 1.6 as requiring We have thus obtained that {{t, t}} must be of the form (4.1) for some λ, μ, ν ∈ k. The corresponding triple bracket is easily computed (see e.g. [18, Proposition A.1]) and gives  Proof First, remark that when ν = 0, we have by Proposition 4.1 that μ = δ 2 for some δ ∈ {±1}, and = (t − λ) δ is a moment map.
For the converse, we seeĀ as the graded algebra k[t ±1 ], wheret = t − λ has degree +1. We also note that Eq. 4.1 is equivalent to SinceĀ is quasi-Hamiltonian, there exists an (invertible) element that satisfies and which we can decompose as Then, we get by looking at Eq. 4.5 in highest degree that c k 1 t k 1 ,t is of degree at most k 1 + 1. But using the derivation property (1.3), this highest degree is exactly D + k 1 − 1, where D is the maximal degree of t,t given in Eq. 4.4. This implies that D ≤ 2, i.e. there is no component of degree 3 in t,t . We get from Eq. 4.4 that ν = 0.

Algebra with two idempotents
In the previous case, the algebra A was simply a k-algebra with no non-trivial (i.e. distinct from 0,1) idempotent elements. The simplest case where such a decomposition occurs consists in taking the path algebra kQ 1 of the quiver Using Sweedler's notation, this implies that {{t, t}} and {{t, t}} are of the form αt for some α ∈ k. Therefore {{t, t}} = α t ⊗ t, and the cyclic antisymmetry implies α = 0 so that kQ 1 can only be endowed with the zero double bracket. At the same time, it is easy to see that {{t, t, t}} given by Eq. 1.6 vanishes for kQ 1 , so we get the next result.

Lemma 4.3 The zero double bracket is the unique double quasi-Poisson bracket on kQ 1 .
As we have seen in §4.1, the zero double bracket is not quasi-Poisson on k[t], and the fact that it is quasi-Poisson on kQ 1 is only due to the idempotent decomposition which implies t 2 = 0. In fact, if we consider k[t] as the fusion algebra obtained by fusing e 1 and e 2 in kQ 1 , the zero double quasi-Poisson bracket on kQ 1 yields after fusion the case λ = ν = 0 in Proposition 4.1.
To get non-trivial examples of B-linear double brackets, we consider the double quiver Q 1 obtained by adding to Q 1 the arrow s = t * : 2 → 1. If we define a degree on A by setting |s| = |t| = 1 and extend it to A ⊗ A, we can characterise the B-linear double quasi-Poisson brackets on A that have degree at most +4 on generators. By the latter condition, we mean that {{s, s}} , {{t, t}} and {{t, s}} (hence {{s, t}}) are sums of homogeneous terms of degree at most +4.
The proof is given in Appendix C. Thus, the zero double bracket on the path algebra A of the quiver Q 1 Q 1 is also quasi-Poisson by Remark 2.13. We can see A as an algebra over B = ⊕ 4 s=1 ke s , where e s is the elementary path corresponding to the s-th vertex. We can glue the vertices 1 and 3, as well as the vertices 2 and 4. The resulting fusion algebra is just kQ 1 , and we have a double quasi-Poisson bracket by Theorem 2.14 given by Eq. 4.8, where δ = +1 (resp. δ = −1) if we fuse e 3 onto e 1 (resp. e 1 onto e 3 ), and where δ = +1 (resp. δ = −1) if we fuse e 4 onto e 2 (resp. e 2 onto e 4 ).
Example 4.6 Up to localisation, we claim that the algebra A with double quasi-Poisson bracket given by Case 1 with Eq. 4.7b is quasi-Hamiltonian when γ φ = 0. In such a case, we set α = δ 2 for some δ = ±1.
If φ = 0, consider the localisation of A at δγ + st and δγ + ts. This is equivalent to require that the element δγ e 1 + ts is invertible in e 1 Ae 1 , while δγ e 2 + st is invertible in e 2 Ae 2 . We can easily check that 1 = (δγ e 1 + ts) δ and 2 = (δγ e 2 + st) −δ satisfy (1.7). Hence = 1 + 2 is a moment map in the localised algebra.

Free algebra on two generators
Consider A = k s, t with B = k. To obtain new examples of double quasi-Poisson brackets on A, we assume that we have a double bracket such that with coefficients in k that satisfy 4(μ 2 − λν) = 1 and 4(m 2 − ln) = 1. Furthermore, we consider that the double bracket between s and t has the form with all coefficients in k. In other words, if we fix a degree on A by |t| = |s| = 1 and extend it to A ⊗ A, we assume that the double bracket {{t, s}} has degree at most +2. We wish to formulate a classification of the double quasi-Poisson brackets of the above form. To do so, introduce the conditions We say that a double bracket {{−, −}} on A of the form (4.9a)-(4.9b) and (4.10) is reduced if it satisfies either (C1) or (C1'), together with either (C2) or (C2'). It is not difficult to see that, up to an affine change of variables t → t + ρ t , s → s + ρ s , for suitable ρ t , ρ s ∈ k, any double bracket {{−, −}} on A of the form (4.9a)-(4.9b) and (4.10) can be put into reduced form.

Fusion for Proposition 4.8
We can use Theorem 2.14 to obtain the following result.

Theorem 4.10 Up to localisation, any double quasi-Poisson bracket on A of the form (4.9a)-(4.9b) and (4.10) is isomorphic to a reduced double quasi-Poisson bracket obtained by fusion.
The proof follows by combining the different examples that we give now together with Proposition 4.8. Fusing e 1 and e 2 , we get the fusion algebra A f = k t ±1 , s ±1 with double quasi-Poisson bracket given by Eq. 4.11, where μ = + 1 2 (resp. μ = − 1 2 ) if we fuse e 2 onto e 1 (resp. e 1 onto e 2 ) by using Eq. 2.16c (resp. Eq. 2.15b).

Example 4.12 (Fusion for Case 2.)
For any γ ∈ k and δ = ±1, the localisation A of the path algebra kQ 1 at a = δγ + ts and b = δγ + st is a quasi-Hamiltonian B-algebra for B = ke 1 ⊕ ke 2 by Example 4.6 (with φ = 0). The fusion algebra A f obtained by fusing e 2 onto e 1 can be identified with k s, t a,b . It is a quasi-Hamiltonian algebra with double quasi-Poisson bracket

Remark 4.13
After fusion, the case γ = δ = +1 treated in Example 4.12 corresponds to Van den Bergh's quasi-Hamiltonian algebra associated to a one-loop quiver [20] (see Theorem 3.3). The case γ = 0 appears after localisation on A = k s ±1 , t ±1 in [8], and gives the quasi-Hamiltonian structure for the fundamental group of a torus with one marked boundary component [14] (see Theorem 3.5).
Example 4.14 (Fusion for Cases 3,6.) We consider the algebra k s with double quasi-Poisson bracket (4.9b), and kQ 1 for the quiver Q 1 given by t : 1 → 2 endowed with the zero double quasi-Poisson bracket. Consider the direct sum A = kQ 1 ⊕ k s , where we denote the identity of k s as e 3 . This is a double quasi-Poisson algebra by Remark 2.13.
If we fuse e 3 onto e 2 (resp. e 2 onto e 3 ) and call it e 2 , we obtain the fusion algebra A with double quasi-Poisson bracket (4.9b), {{t, t}} = 0 and Then, if we fuse e 2 onto e 1 (resp. e 1 onto e 2 ) which becomes the unit in the fusion algebra A , we have a double quasi-Poisson bracket given by Eq. 4.9b and where μ = 1 2 (resp. μ = − 1 2 ). When α = −μ, we get Eq. 4.13 if n = l = 0, or we get Eq. 4.16 if m = 0.

Generalities on representation spaces
We assume that A is a finitely generated associative algebra over B = ⊕ K s=1 ke s , with e s e t = δ st e s . Following [20,Section 7] (see also [6,Section 4] and [14,Section 3]), let I = {1, . . . , K} and choose a dimension vector α ∈ N I , setting N = s∈I α s . We consider the representation space (relative to B) Rep(A, α). The representation space is the affine scheme whose coordinate ring O (Rep(A, α)) is generated by symbols a ij for a ∈ A, 1 ≤ i, j ≤ N , which satisfy together with the condition that for any 1 ≤ s ≤ K the matrix X (e s ) = ((e s ) ij ) ij is the s-th diagonal identity block of size α s . In other words, we have that (e s ) ij = δ ij if α 1 +. . .+α s−1 +1 ≤ i, j ≤ α 1 +. . .+α s , while it is zero otherwise. Note that this implies 1 ij = δ ij for all 1 ≤ i, j ≤ N . To ease notations, denote by R = O (Rep(A, α)) the coordinate ring, and for any a ∈ A set X (a) to denote the matrix with entries a ij ∈ R.
By definition of Rep(A, α), any element a ∈ A induces functions (a ij ) ij on Rep(A, α), and we would like to extend this definition to derivations. We associate to any δ ∈ D A/B the vector fields δ ij ∈ Der(R), 1 ≤ i, j ≤ N , defined by (5.1) and introduce the vector field-valued matrix X (δ) with (i, j ) entry δ ij . We call the particular disposition of indices in Eq. 5.1 the standard index notation as in [21]. More generally, for an element δ = δ 1 . . . δ n ∈ (D B A) n we define δ ij ∈ n R Der(R) from the matrix identity X (δ) = X (δ 1 ) . . . X (δ n ), and we set tr X (δ) = i δ ii . where, on the left-hand side, J ac : R ×3 → R is defined by while on the right-hand side {{−, −, −}} is the triple bracket (1.4) defined by {{−, −}}, and we write for a = a ⊗ a ⊗ a ∈ A ⊗3 that a ij,kl,uv = a ij a kl a uv .
We will particularly be interested in the case n = 3, which takes the following form.

Quasi-Poisson algebras
Let g be a Lie algebra over k such that g is equipped with a non-degenerate symmetric bilinear form denoted (−|−). Furthermore, assume that the form is g-invariant, i.e.
If we take dual bases (ε i ), (ε i ) under (−|−), then we can define the Cartan trivector φ ∈ 3 g given by Following [14,Section 2] from now on, we assume that g acts on a commutative k-algebra R by derivation, so that the map g → Der(R) is a Lie algebra homomorphism. Denoting by η R the action of η ∈ g on R, the latter means that [η 1 , for any a ∈ R, η 1 , η 2 ∈ g. We say that R is a quasi-Poisson algebra over g if R is equipped with an anti-symmetric biderivation {−, −} such that for any η ∈ g and a, b, c ∈ R  (a, b, c) .
Here, φ R is the image of the Cartan trivector induced by the map g ⊗3 × R ×3 → k given by The operation {−, −} is called a quasi-Poisson bracket. Note that if R g ⊂ R is the subalgebra of g-invariant elements, i.e. R g = {a ∈ R | η R (a) = 0 ∀η ∈ g}, then {−, −} descends to a Poisson bracket on R g since the right-hand side of Eq. 5.8b vanishes. Remark 5.5 In this work, we restrict the definition of quasi-Poisson algebra to the case where φ is the Cartan trivector (5.7), in analogy with [3,20]. Working in greater generalities, Massuyeau and Turaev considered an arbitrary element φ ∈ 3 g, from which we still get a Poisson bracket on R g [14, §2.2]. This notion also encompasses Poisson algebras when we take g = {0}.
Assume that we are also given an arbitrary group G acting on the left on g by Lie algebra automorphisms. (We do not require that g = Lie(G).) For any g ∈ G, we write the action as η → g η, η ∈ g. We say that R is a (G, g)-algebra if R is a g-algebra endowed with a compatible left G-action : We say that R is a quasi-Poisson algebra over the pair (G, g) if R is a (G, g)-algebra and if R is a quasi-Poisson algebra over g such that for any g ∈ G, a, b ∈ R g.{a, b} = {g.a, g.b} , We easily see that if R G ⊂ R is the subalgebra of G-invariant elements, then the quasi-Poisson bracket descends to a Poisson bracket on R G ∩ R g . We now consider R = O (Rep(A, α)) as in §5.1. The algebra R is naturally endowed with an action of GL α = K s=1 GL αs (k), which is given in matrix notation by g.X (a) = g −1 X (a)g for all a ∈ A, g ∈ GL α . We can also consider the Lie algebra g α = K s=1 gl αs (k) of GL α , which acts by derivation on R as η R (X (a)) = [X (a), η], for all a ∈ A, η ∈ g α . We can endow g α with the trace pairing (η 1 |η 2 ) = tr(η 1 η 2 ), and consider the left adjoint action of GL α on g α so that Eq. 5.9 is satisfied. The following result generalises [20,Theorem 7.12.2], see also [14,Lemma 4.4]. (This was already noticed by Van den Bergh without a proof, as mentioned in [20,Remark 7.12.3].)  (Rep(A, α)) is a quasi-Poisson algebra over the pair (GL α , g α ) for the quasi-Poisson bracket defined by Proposition 5.1.
Proof Showing Eqs. 5.8a, 5.10a and 5.10b is easy, so we are left to show Eq. 5.8b on generators of the coordinate ring R . Hence, fix a, b,  If k is algebraically closed, we can use Le Bruyn-Procesi Theorem [13,Theorem 1] to get that A GLα is generated by functions tr X (a), a ∈ A, see e.g. [6,Remark 4.3]. In particular,  N = max(k 1 , . . . , k M ). Combining Example 3.2 and Theorem 5.6, we get that the algebra is a quasi-Poisson algebra over the pair (GL N (k), gl N (k)) with quasi-Poisson bracket When all the (k m ) m are equal, this gives a quasi-Poisson algebra structure on the coordinate ring corresponding to M copies of the space of nilpotent N × N matrices.

Moment maps and Poisson algebra
Consider the quasi-Poisson algebra (R, {−, −}) over the pair (GL α , g α ) obtained from the double quasi-Poisson algebra (A, {{−, −}}) by Theorem 5.6. We now assume that A is a quasi-Hamiltonian algebra, i.e. it is endowed with a moment map ∈ ⊕ s e s Ae s . For any (q s ) ∈ (k × ) K , let q = s q s e s ∈ B × and define the ideal J q generated by the entries of the matrix identity X ( ) − X (q) = 0 N . We can form the algebra R q = R/J q , and denote byr the image of an element r ∈ R under the projection R → R q . We clearly have that J q is GL α -and g α -invariant, so that we can consider the induced actions on R q = R/J q . If we let R t q ⊂ R q denote the subalgebra generated by elements tr(r), r ∈ R, we can see that R t q ⊂ R GLα q ∩ R gα q . The next result follows either from [20,Proposition 6.8.1] and [6,Theorem 4.5], or from [20,Proposition 7.13.2] and quasi-Hamiltonian reduction [3].  Rep(A, α))/(X ( − q))) GLα is a Poisson algebra. Example 5.12 If k is algebraically closed, the double quasi-Poisson bracket of Van den Bergh given in Theorem 3.3 (with γ a = +1 for all a ∈Q) defines a Poisson structure on multiplicative quiver varieties of Crawley-Boevey and Shaw [7], see [20, Theorem 1.1].
of the present paper. The author also thanks A. Alekseev for useful discussions, and the referees for their comments. Part of this work was supported by a University of Leeds 110 Anniversary Research Scholarship.
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Appendix A: Vanishing of the map κ
In this appendix, we prove Lemma 2.20. Note that κ is a linear combination of triple brackets, so it is itself a triple bracket. By definition, it is a derivation in its last argument and is cyclically anti-symmetric. Thus, to show that κ vanishes, it suffices to show that it is equal to zero when applied to generators of A f . Before tackling this task, we use Eq. 1.4 and remark that we can write where 1 A is the identity map. Therefore, evaluated on some elements a, b, c ∈ A f , we can write κ (a, b, c)  so that we will write down the terms A, B, C, A , B , C for the different types of generators.
Using the cyclicity, we only have twenty cases to check. We will only detail the computations in the first few cases, and we will give the final form of the terms A, B, C, A , B , C in the remaining cases so that the reader can check that they sum up to zero. Before beginning with the calculations, we remark that identities involving the double bracket { {−, −} } follow from extension from A to A f which respects the derivation property in each variable. That is, given e + , f + ∈ { , e 12 } and e − , f − ∈ { , e 21 }, we have for any Here, in the left-hand side we have the induced double bracket on A f , while the double bracket in the right-hand side is the original one on A. Recall that we can choose generators a, b ∈ A f that admit such a decomposition by Lemma 2.11.

A.1 All generators of the same type
We drop the idempotent in our computations since this is the unit in A f .
Generators of the second type. Write a = e 12 α, b = e 12 β and c = e 12 γ for α, β, γ ∈ e 2 A . Using Eq. 2.15b, then the derivation property for the outer bimodule structure in the second entry of the double bracket on A f together with Eq. A.2, we get that In the same way, we find so that all terms cancel out together (after using the cyclic antisymmetry, which we will need in each of the remaining cases). With one generator of the third type. Consider c = γ e 21 for some γ ∈ Ae 2 . We get from Eqs. 2.14c and 2.16a that Summing terms together, we get κ = 0.

A.3 Two generators of the second type
Let a = e 12 α, b = e 12 β for α, β ∈ e 2 A . We only collect the final form of the terms A, B, C, A , B , C from now on, and the reader can check that they sum up to zero.
With one generator of the first type. Consider c ∈ A . We get C = 0, while

A.6 Remaining cases
We now take three different types of generators. No generator of the second type. This case and the next one are a bit tedious. We set a ∈ A , b = βe 21 for β ∈ Ae 2 and c = e 12 γ e 21 for γ ∈ e 2 Ae 2 .

Appendix B: Proof of Lemma 2.21
Note that Tr( s ) = s for s = 2, while Tr( 2 ) = e 12 2 e 21 . In particular, using that for s = 2 we have s = e s s e s , we get Tr( s ) = s by understanding that equality in A f .

B.1 Moment map condition for the non-fused idempotents
First, assume that s = 1, 2. Then, using Lemma 2.18, we get  We first look at the case a = t. Using Eq. C.1a, we can find that The first four terms cancel if we take their sum under cyclic permutations, so that we can write Therefore either λ = 0, or the different coefficients vanish i.e. γ = 0, φ 1 = 0 while α 1 = −α 3 and φ 0 = −φ 2 . Doing the computation with s instead of t, we need either l = 0 or the same four conditions. Lemma C.2 If λ = 0 and Eq. C.2b holds, then The same identities are satisfied if l = 0 and Eq. C.2c holds.
Under the conditions from Eq. C.5a, the only terms remaining in { {t, t, s} } are given by st ⊗ t ⊗ e 1 , e 2 ⊗ t ⊗ ts, e 2 ⊗ t ⊗ e 1 and st ⊗ t ⊗ ts with respective coefficients ( Comparing with Eq. C.2b, we get Eq. C.5b.
The method is exactly the same in the case l = 0 assuming that Eq. C.2c holds.
We get by combining Lemmas C.1 and C.2 that if λ = l = 0 as well as α 1 = α 3 , we are in the case 1.a) of Proposition 4.4. If α 1 = α 3 instead, we are in the case 1.b).
We now assume that at least one of the two constants λ, l is nonzero. Hence, if the double bracket (C.1a)-(C.1b) satisfies (C.2a), it must be such that using Lemma C.1.
In the exact same way, we get the next lemma.