Goldman-Turaev formality implies Kashiwara-Vergne
aa r X i v : . [ m a t h . G T ] D ec Goldman-Turaev formality implies Kashiwara-Vergne
Anton Alekseev ∗ , Nariya Kawazumi † , Yusuke Kuno ‡ and Florian Naef § Abstract
Let Σ be a compact connected oriented 2-dimensional manifold with non-emptyboundary. In our previous work [3], we have shown that the solution of generalized(higher genus) Kashiwara-Vergne equations for an automorphism F ∈ Aut( L ) of afree Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra g (Σ) and its associated graded gr g (Σ). In this paper, we prove the converse: if F induces an isomorphism g (Σ) ∼ = gr g (Σ), then it satisfies the Kashiwara-Vergneequations up to conjugation. As an application of our results, we compute the degreeone non-commutative Poisson cohomology of the Kirillov-Kostant-Souriau doublebracket. The main technical tool used in the paper is a novel characterization ofconjugacy classes in the free Lie algebra in terms of cyclic words. Let Σ be a compact connected oriented surface with non-empty boundary and K afield of characteristic zero. The K -linear space g = g (Σ) spanned by free homotopyclasses of loops in Σ has an interesting Lie bialgebra structure, the Lie bracket beingthe Goldman bracket [10] and the Lie cobracket being the Turaev cobracket [26]. (Tobe more precise, one needs to fix a framing on Σ in order to define the Lie cobracketon g and it actually depends on the choice of framing.) As was shown in [3], one cannaturally define the graded version gr g of the Goldman-Turaev Lie bialgebra, and itturns out to be isomorphic to the necklace Lie bialgebra structure [6, 9, 25] associatedto a certain quiver determined by the topological type of Σ.The Lie bialgebras g and gr g admit natural completions which we denote by b g andgr b g , respectively. They are isomorphic as filtered K -vector spaces, but not canonically.The formality question for the Goldman-Turaev Lie bialgebras is whether there existsa filtered Lie bialgebra isomorphism from b g to gr b g such that the associated graded ∗ Department of Mathematics, University of Geneva, 2-4 rue du Lievre, 1211 Geneva, Switzerland e-mail:[email protected] † Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan e-mail:[email protected] ‡ Department of Mathematics, Tsuda University, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo 187-8577,Japan e-mail:[email protected] § Department of Mathematics, Massachusetts Institute of Technology, 182 Memorial Dr, Cambridge,MA 02142, USA e-mail:[email protected] b g . This question has been studied during the last severalyears by various approaches: the study first began with formality for Goldman brackets[16, 17, 22, 23, 24, 12] and then has been deepened to formality for Turaev cobrackets[21, 2, 4, 3, 13]. One motivation for considering this question comes from the study ofthe Johnson homomorphisms of mapping class groups [18].In this paper, we impose a restriction on a map b g → gr b g by assuming that it isinduced by a group-like expansion [20], which is a notion related to 1-formality of a freegroup of finite rank. In order to explain this notion, for the moment we assume that theboundary of Σ is connected. (The general case needs a more careful treatment and willbe explained in Section 2.) The group algebra K π of the fundamental group π = π (Σ)has a decreasing filtration defined by powers of the augmentation ideal. This defines acompletion c K π , and the associated graded gr c K π . Since the fundamental group π is afree group of finite rank, gr c K π is canonically isomorphic to the completed tensor algebra b T ( H ) generated by the first homology H = H ( π, K ) ∼ = H (Σ , K ). Furthermore, we canidentify c K π with gr c K π in a non-canonical way. In this context, a group-like expansionis a complete Hopf algebra isomorphism θ : c K π → b T ( H ) = gr c K π such that gr θ = id. Any group-like expansion induces a filtered K -linear isomorphism θ : b g → gr b g . This follows from the fact that there is a natural identification g = K ( π/ conj) ∼ = | K π | := K π/ [ K π, K π ] , where π/ conj is the set of conjugacy classes in π and [ K π, K π ] is the K -linear span ofelements of the form ab − ba with a, b ∈ K π . Our goal is to characterize group-likeexpansions which induce Lie bialgebra isomorphisms θ : b g → gr b g .As was shown by Kawazumi-Kuno [16][18] and Massuyeau-Turaev [22] [23] indepen-dently, if θ satisfies the boundary condition θ ( ζ ) = e ω , where ζ ∈ π is the boundary loop of Σ and ω ∈ ∧ H ⊂ b T ( H ) is the 2-tensor corre-sponding to the intersection pairing on Σ, then the induced map θ : b g → gr b g is a Liealgebra isomorphism. Group-like expansions satisfying θ ( ζ ) = e ω are called symplecticexpansions (in the case where the boundary of Σ is connected). In this paper, converselywe prove the following theorem: Theorem 1.1 (For the general case, see Theorem 2.5) . Assume that the boundary of Σ is connected. If a group-like expansion θ induces a Lie algebra isomorphism θ : b g → gr b g ,then θ is conjugate to a symplectic expansion, i.e., there exists a group-like element g such that θ ( ζ ) = ge ω g − . In our previous work [2, 3], the formality question for the Lie bialgebra g has beenstudied in connection with the Kashiwara-Vergne problem from Lie theory [14, 5] and2ts generalization to surfaces of positive genus. In this approach, one fixes generators ofthe group π and decomposes any group-like expansion as follows: θ F = F − ◦ θ exp . Here, θ exp is the group-like expansion determined by the choice of generators of π and F is a complete Hopf algebra automorphism of b T ( H ); in other words, F ∈ Aut( b L )where b L ⊂ b T ( H ) is the completed free Lie algebra generated by H . In this way, theproperties of θ are encoded in the properties of F . In the formulation of the generalizedKashiwara-Vergne problem in [3], there are two equations (KV I) and (KV II) for theautomorphism F . The equation (KV I) is equivalent to θ F ( ζ ) = e ω . The second equation(KV II) depends on the choice of framing on Σ and is related to the formality of theTuraev cobracket. Recall the following result from [3]: Theorem 1.2 ([3], Theorem 5.12) . Suppose that F ∈ Aut( b L ) satisfies (KV I) . Then, θ F = F − ◦ θ exp induces a Lie bialgebra isomorphism θ F : b g → gr b g if and only if F satisfies (KV II) . Since the generalized (higher genus) Kashiwara-Vergne problem admits solutions(with the exception of certain framings on genus one surfaces) [3, § b g → gr b g has been settled.In this paper, we introduce a modification of equations (KV I) and (KV II), whichwe denote by (KV I’) and (KV II’). For an explicit form of these two equations, seeTheorem 2.9. As an application of Theorem 1.1, we prove the following result: Theorem 1.3 (=Theorem 2.9) . Let θ F = F − ◦ θ exp be a group-like expansion. Then, θ F induces a Lie bialgebra isomorphism θ F : b g → gr b g if and only if F satisfies equations (KV I’) and (KV II’) . This result is an improvement on Theorem 1.2, and it gives a complete algebraic char-acterization of group-like expansions which induce Lie bialgebra isomorphisms θ : b g → gr b g .The results described above are based on a novel characterization of conjugacy classesin the free Lie algebra. In more detail, let H be a symplectic vector space, b L the (degreecompleted) free Lie algebra generated by H , ω ∈ ∧ H ⊂ b L the element representing thesymplectic form, and | b T ( H ) | = b T ( H ) / [ b T ( H ) , b T ( H )]the degree completed space of cyclic words with alphabet defined by H . We denotethe natural projection b T ( H ) → | b T ( H ) | by x
7→ | x | . We say that an element a ∈ b L isconjugate to b ∈ b L if there is a group-like element g ∈ exp( b L ) such that a = gbg − . Thefollowing result plays a key role in the paper: Theorem 1.4 (= Theorem 3.5) . The element a ∈ b L is conjugate to ω if and only if | exp( a ) | = | exp( ω ) | . Acknowledgements.
We are grateful to the Simons Center for Geometry in Physicsat the Stony Brook University where part of this work was conducted. Research of AAwas supported in part by the project MODFLAT of the European Research Council(ERC), by the grants number 178794 and 178828 and by the NCCR SwissMAP of theSwiss National Science Foundation (SNSF). Research of NK was supported in part bythe grants JSPS KAKENHI 15H03617, 26287006 and 18K03283. Research of YK wassupported in part by the grant JSPS KAKENHI 26800044 and 18K03308. Research ofFN was supported by the grant of Early Postdoc Mobility grant 175033 of the SwissNational Science Foundation.
Contents Setup and statement of results
Let Σ = Σ g,n +1 be a compact connected oriented surface of genus g with n + 1 boundarycomponents, where g and n are non-negative integers. Label the boundary components ofΣ by integers 0 , , . . . , n , and choose a basepoint ∗ on the 0th boundary component. Thenthe fundamental group π = π (Σ , ∗ ) has a set of free generators α i , β i , γ j , i = 1 , . . . , g , j = 1 , . . . , n , such that γ j is freely homotopic to the j th boundary component withpositive orientation and g Y i =1 α i β i α i − β i − n Y j =1 γ j = γ , where γ is the based loop around the 0th boundary component with negative orientation.The first homology group H = H (Σ , K ) of the surface Σ has a 2-step decreasingfiltration defined by H (1) = H and H (2) = { x ∈ H | h x, y i = 0 for all y ∈ H } , where h· , ·i : H × H → K is the intersection pairing. Letgr H := H/H (2) ⊕ H (2) be the associated graded vector space. The homology classes of α i and β i give rise to abasis of H/H (2) , we denote the corresponding basis elements by x i and y i . The homologyclasses of γ j , denoted by z j , give rise to a basis of H (2) .Let A = b T (gr H ) be the completed tensor algebra over gr H . In other words, A is thecompleted free associative algebra generated by variables x i , y i , z j . We assign weightsto the generators as follows:wt( x i ) = wt( y i ) = 1 , wt( z j ) = 2 . (1)Then, the algebra A becomes graded and thus filtered. Besides, A naturally carries thestructure of a complete Hopf algebra. We denote by b L the set of primitive elements in A . It is identified with the completed free Lie algebra generated by gr H .As was shown in [3, § { K π ( m ) } m ≥ oftwo-sided ideals of the group algebra K π such that K π (0) = K π , α i − , β i − ∈ K π (1),and γ j − ∈ K π (2). Furthermore, the (completion of the) associated graded of thisfiltration is canonically isomorphic to A [3, Proposition 3.12].Let c K π = lim ←− m K π/ K π ( m ) be the completion of K π with respect to the filtrationdescribed above. We have gr c K π = gr K π = A . Definition 2.1 ([3], Definition 3.19) . A group-like expansion of π is an isomorphism θ : c K π → A of complete filtered Hopf algebras such that the associated graded map isthe identity: gr θ = id. 5or example, the map θ exp defined by the following values on generators θ exp ( α i ) = e x i , θ exp ( β i ) = e y i , θ exp ( γ j ) = e z j is a group-like expansion. Any group-like expansion θ can be written as θ F = F − ◦ θ exp for some F ∈ Aut( b L ) with gr F = id. For a (topological) associative K -algebra A , we denote | A | := A / [ A , A ] , where [ A , A ] is the (closure of the) K -span of elements of the form ab − ba with a, b ∈ A .If A is filtered, then | A | is naturally filtered. Let | · | : A → | A | be the natural projection.The space g = g (Σ) := | K π | is canonically isomorphic to the K -span of homotopyclasses of free loops in Σ. As was shown by Goldman [10], the space g has a Lie bracket[ · , · ] defined in terms of intersections of free loops. By using self-intersections of freeloops, Turaev [26] introduced a Lie cobracket on the space g / K , where denotes theclass of a constant loop. By fixing a framing f on Σ (that is, a choice of trivializationof the tangent bundle of Σ), one can lift it to a Lie cobracket on the space g , which wedenote by δ f . The triple ( g , [ · , · ] , δ f ) becomes a Lie bialgebra [3, § · , · ] and the framed Turaev cobracket δ f extend naturally tothe Lie bialgebra structure on the completion b g = | c K π | . Moreover, they induce a Lie bial-gebra struture on the associated graded space gr b g , which we denote by (gr b g , [ · , · ] gr , δ f gr ).One can also view the space gr b g ∼ = | A | as the space spanned by cyclic words in x i , y i , z j . The Lie bracket [ · , · ] gr and Lie cobracket δ f gr coincide with the necklace Lie bialgebrastructure associated to the quiver with g circles and n edges emanating from a distin-guished vertex, where the Lie bracket was introduced by Bocklandt-Le Bruyn [6] andGinzburg [9] and the Lie cobracket by Schedler [25]. Any group-like expansion θ inducesan isomorphism θ : b g = | c K π | ∼ = → | A | = gr b g of complete filtered K -vector spaces. Definition 2.2 ([3], Definition 3.21) . A group-like expansion θ is called tangential iffor any j = 1 , . . . , n , there is a group-like element g j ∈ A such that θ ( γ j ) = g j e z j g j − .Furthermore, a tangential group-like expansion θ is called special if θ ( γ ) = e ω , where ω = P i [ x i , y i ] + P j z j . 6ote that the elements ω = P i [ x i , y i ] and P j z j (once we choose the 0th boundarycomponent of Σ) are independent of the choice of generators α i , β i , γ j , and, hence, so isthe element ω . Remark 2.3.
For the defining conditions for special expansions, see also [18, § n = 0, the boundary condition θ ( γ ) = e ω was first turned into a definition byMassuyeau [20]. In this case, special expansions are called symplectic expansions. Specialexpansions exist for any g and n . For examples for g = 0 or n = 0, see [11, 1, 15, 19]. Forthe general case, there is a gluing argument which proves existence of special expansions,see [3, § Theorem 2.4 (Kawazumi-Kuno [16, 18], Massuyeau-Turaev [22, 23]) . Every special ex-pansion θ induces a Lie algebra isomorphism between the completed Goldman Lie algebra ( b g , [ · , · ]) and its associated graded, (gr b g , [ · , · ] gr ) . The main result of this paper is the converse of Theorem 2.4 (up to conjugacy):
Theorem 2.5.
Let θ : c K π → A be a group-like expansion and assume that θ induces aLie algebra isomorphism between ( b g , [ · , · ]) and (gr b g , [ · , · ] gr ) . Then, θ is conjugate to aspecial expansion. Namely, θ is tangential and there exists a group-like element g ∈ A such that θ ( γ ) = g e ω g − . Remark 2.6.
Denote by ι j : Z = π ( S ) → π/ conj the maps ι j : t n
7→ | γ nj | inducedby inclusions of the boundary components into Σ. These maps are independent of theconcrete choice of generators γ j ∈ π and they are compatible with filtrations if oneassigns a filtration degree 2 to the generator t ∈ Z . Note that the complete Hopf algebra d KZ admits a unique group-like expansion θ exp : d KZ → K [[ τ ]] given by θ exp ( t ) = exp( τ ),where τ is the primitive generator of degree 2. The map ι j induces a map of associatedgraded gr ι j : K [[ τ ]] → | A | . With this notation, Theorem 2.5 is equivalent to the followingstatement:Let θ : c K π → A be a group-like expansion. Then, θ induces a Lie algebraisomorphism between ( b g , [ · , · ]) and (gr b g , [ · , · ] gr ) if and only if it preserves allthe boundary components in the sense that for j = 0 , . . . , n the followingdiagram commutes: b g θ / / gr b g d KZ ι j O O θ exp / / K [[ τ ]] gr ι j O O Let Aut( b L ) be the group of filtration preserving automorphisms of b L . We say that F ∈ Aut( b L ) is tangential if for any j = 1 , . . . , n there is a group-like element f j ∈ A suchthat F ( z j ) = f j − z j f j . Note that any F ∈ Aut( b L ) extends to a filtration preserving7utomorphism of A = U ( b L ), and thus induces a filtration preserving automorphism of | A | = gr b g . Also, since F is filtration preserving, gr F is defined as an automorphism ofgr b L = b L .It turns out that Theorem 2.5 can be restated as a property of the automorphismgroup of the Lie algebra (gr b g , [ · , · ] gr ). Theorem 2.7.
Let F ∈ Aut( b L ) such that gr F = id and assume that it induces anautomorphism of the Lie algebra (gr b g , [ · , · ] gr ) . Then, F is tangential and there exists agroup-like element f ∈ A such that F ( ω ) = f − ωf .Proof. Let θ : c K π → A be a special expansion. Then, by Theorem 2.4, it inducesa Lie algebra isomorphism between ( b g , [ · , · ]) and (gr b g , [ · , · ] gr ). If F ∈ Aut( b L ) satisfiesassumptions of Theorem 2.7, then the map F ◦ θ a group-like expansion and it alsoinduces a Lie algebra isomorphism between ( b g , [ · , · ]) and (gr b g , [ · , · ] gr ). By Theorem 2.5,we conclude that F ◦ θ is conjugate to a special expansion. Therefore, F = ( F ◦ θ ) ◦ θ − maps z j ’s and ω to their conjugates, as required. In this section, we combine Theorem 2.5 with our previous result [2, 3] and derivea necessary and sufficient condition for group-like expansions which induce Goldman-Turaev formality.In [3, § g with n + 1 bound-ary components and a choice of framing f on it, we introduced the Kashiwara-Vergneproblem KV ( g,n +1) f , which asks to find a tangential automorphism of b L satisfying twoequations (KV I) and (KV II). The original Kashiwara-Vergne problem [14, 5] corre-sponds to the case of ( g, n + 1) = (0 , F ( ω ) = log g Y i =1 ( e x i e y i e − x i e − y i ) n Y j =1 e z j =: ξ. (KV I)Since θ exp ( γ ) = e ξ , F satisfies (KV I) if and only if θ F = F − ◦ θ exp is a special expansion.To write down the second equation, we need more material from [3]. Lettder + = (tder ( g,n +1) ) + = { ( u, u , . . . , u n ) ∈ Der + ( b L ) × b L ⊕ n | u ( z j ) = [ z j , u j ] } be the Lie algebra of tangential derivations on b L and letTAut = TAut ( g,n +1) = { ( F, f , . . . , f n ) ∈ Aut + ( b L ) × b L ⊕ n | F ( z j ) = e − f j z j e f j } be the group of tangential automorphisms on b L . The divergence cocycle div : tder + → | A | is defined as follows:div( u ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g X i =1 ( ∂ x i u ( x i ) + ∂ y i u ( y i )) + n X j =1 ∂ z j u ( z j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∈ | A | . a ∈ A without constant term, we denote a = g X i =1 (( ∂ x i a ) x i + ( ∂ y i a ) y i ) + n X j =1 ( ∂ z j a ) z j . (2)Now, choose a framing f on Σ. For any immersed closed curve γ in Σ, one can defineits rotation number rot f ( γ ) ∈ Z with respect to f . Put p := g X i =1 (rot f ( | β i | ) x i − rot f ( | α i | ) y i ) ∈ H/H (2) ⊂ gr H ⊂ A and define the cocycle c f : tder + → | A | by c f ( u ) = X j rot f ( | γ j | ) | u j | . The cocycles div and c f integrate to group 1-cocycles j : TAut → | A | and C f : TAut → | A | . More explicitly, if F = exp( u ) ∈ TAut with u ∈ tder + then j ( F ) = e u − u · div( u ). Set j f := j − C f . Finally, introduce the element r ∈ | A | by r = g X i =1 (cid:12)(cid:12)(cid:12)(cid:12) log (cid:0) e x i − x i (cid:1) + log (cid:0) e y i − y i (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) . With the notation as above, the second Kashiwara-Vergne equation is of the form j f ( F ) = r + | p | + n X j =1 | h j ( z j ) | − | h ( ξ ) | for some h j , h ∈ K [[ s ]]. (KV II)We recall the following result: Theorem 2.8 ([3]) . Let f be a framing on Σ and F ∈ TAut be a solution of theequation (KV I) . Then, θ F = F − ◦ θ exp is a Lie bialgebra isomorphism from ( b g , [ · , · ] , δ f ) to (gr b g , [ · , · ] gr , δ f gr ) if and only if F satisfies (KV II) . Combining Theorem 2.5 and Theorem 2.8, we obtain the following result:
Theorem 2.9.
Let θ be a group-like expansion. Then, θ induces a Lie bialgebra iso-morphism between ( b g , [ · , · ] , δ f ) and (gr b g , [ · , · ] gr , δ f gr ) if and only if there exists a tangentialautomorphism F ∈ TAut such that θ = θ F = F − ◦ θ exp , F ( ω ) = e − ℓ ξe ℓ for some ℓ ∈ b L, (KV I’) and j f ( F ) + rot f ( | γ | ) | ℓ | = r + | p | + n X j =1 | h j ( z j ) | − | h ( ξ ) | for some h j , h ∈ K [[ s ]] . (KV II’)9 emark 2.10. Since | · | vanishes on commutators, the expression | ℓ | is of degree 1.We can ignore the terms in ℓ proportional to z j ’s because they can be absorbed in thelinear terms of functions h j . Proof.
First, we compute the cocycle j f on inner automorphisms. For ℓ ∈ b L let u ℓ ∈ tder + be the inner derivation with generator ℓ : u ℓ ( a ) = [ a, ℓ ] for any a ∈ b L . Theelement F ℓ := exp( u ℓ ) ∈ TAut is an inner automorphism given by conjugation by e ℓ : F ℓ ( a ) = e − ℓ ae ℓ .We computediv( u ℓ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g X i =1 ( ∂ x i [ x i , ℓ ] + ∂ y i [ y i , ℓ ]) + n X j =1 ( ∂ z j [ z j , ℓ ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g X i =1 ( x i ( ∂ x i ℓ ) − ℓ + y i ( ∂ y i ℓ ) − ℓ ) + n X j =1 ( z j ( ∂ z j ℓ ) − ℓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (1 − g − n ) | ℓ | . In the third line we used the cyclic invariance | x i ( ∂ x i ℓ ) | = | ( ∂ x i ℓ ) x i | and formula (2)for ℓ . Also, we have c f ( u ℓ ) = ( P j rot( | γ j | )) | ℓ | . Since u ℓ acts trivially on the space | A | ,integration yields j f ( F ℓ ) = (div − c f )( u ℓ ) = (1 − g − n − n X j =1 rot( | γ j | )) | ℓ | = − rot f ( | γ | ) | ℓ | . (3)In the last equality we have used the Poincar´e-Hopf theorem n X j =1 rot f ( | γ j | ) − rot f ( | γ | ) = χ (Σ) = 1 − g − n. Now let θ be a group-like expansion and assume that it induces a Lie bialgebraisomorphism θ : b g → gr b g . By Theorem 2.5, there exists an element F = ( F, f , . . . , f n ) ∈ TAut and a group-like element g ∈ A such that θ = θ F = F − ◦ θ exp and θ ( γ ) = g e ω g − . By setting ℓ := log F ( g ) ∈ b L we obtain F ( ω ) = e − ℓ ξe ℓ , which implies (KVI’). By construction, the automorphism F ′ := F − ℓ ◦ F satisfies (KV I). Since the actionof F − ℓ on | A | is trivial, θ F ′ = θ on b g . By Theorem 2.8, F ′ satisfies (KV II). This impliesthat F satisfies (KV II’), since j f ( F ′ ) = j f ( F − ℓ ) + F − ℓ · j f ( F ) = rot f ( | γ | ) | ℓ | + j f ( F )by (3) and the fact that F − ℓ acts trivially on | A | . This completes the proof of “only if”part. The other direction can be proved by the same method, so we omit it.10 Free Lie algebras and cyclic words
In this section, we will prove several statements about conjugacy classes in free Liealgebras and their characterization in terms of cyclic words. These statements are themain technical content of the paper.
Let g be a Lie algebra over a field K of characteristic zero. Later we will take g to bethe free Lie algebra over a finite dimensional K -vector space. By the PBW theorem, wehave a natural decomposition U ( g ) = ∞ M m =0 Sym m g , (4)where we denote by Sym m g the m th symmetric power of the vector space g . Theisomorphism (4) is given by the maps Sym m g → U ( g ), x x · · · x m m ! X σ ∈ S m x σ (1) x σ (2) · · · x σ ( m ) , for m ≥
0. Here x x · · · x m stands for the symmetric tensor product of x , x , . . . , x m ∈ g (see, e.g. [7] Ch. 1, §
2, no. 7). It should be remarked that the vector space Sym m g isspanned by the set { x m ; x ∈ g } . Now we consider the Lie algebra abelianization of theassociative algebra U ( g ), | U ( g ) | = U ( g ) / [ U ( g ) , U ( g )] . We denote the quotient map by | · | : U ( g ) → | U ( g ) | , u
7→ | u | . The decomposition (4)descends to abelianizations: Theorem 3.1.
We have the direct sum decomposition | U ( g ) | = ∞ M m =0 | Sym m g | . (5) Proof.
Since g generates U ( g ), we have [ U ( g ) , U ( g )] = [ g , U ( g )]. Recall that the decom-position (4) is a g -module decomposition. Hence, we have[ g , U ( g )] = ∞ X m =0 [ g , Sym m g ] ⊂ ∞ X m =0 ([ g , U ( g )] ∩ Sym m g ) . This means that the subspace [ g , U ( g )] = [ U ( g ) , U ( g )] is homogeneous with respect tothe decomposition (4), and this implies the direct sum decomposition in the theorem.11et V be a finite dimensional K -vector space, T = T ( V ) = L ∞ m =0 V ⊗ m the tensoralgebra over V , and L = L ( V ) the free Lie algebra over V . If we denote by ∆ : T → T ⊗ T the (standard) coproduct of the Hopf algebra structure on T , then L is identified with theset of primitive elements, i.e., L = { a ∈ T ; ∆ a = a ⊗ ⊗ a } , and we have T = U ( L ).For our purpose we need completions of T and L ; we denote by b T = b T ( V ) = Q ∞ m =0 V ⊗ m the completed tensor algebra over V and by b L = b L ( V ) the completed free Lie algebraover V .The Lie algebra L admits a grading with finite dimensional graded components givenby tensor powers of V : L = L ∞ q =0 ( L ∩ V ⊗ q ). This implies that decompositions (4) and(5) extend to b L (with direct sums replaced by direct products): b T = U ( b L ) = ∞ Y m =0 Sym m b L, | b T | = ∞ Y m =0 | Sym m b L | . This observation has the following interesting corollaries:
Theorem 3.2.
Let u and v ∈ b L such that | exp( u ) | = | exp( v ) | ∈ | b T | . Then, | u m | = | v m | ∈ | b T | for all m ≥ .Proof. We have, | exp( u ) | = ∞ X m =0 m ! | u m | = ∞ X m =0 m ! | v m | = | exp( v ) | . By decomposition (5) for the Lie algebra b L , this implies that the series in the middleare equal term by term: | u m | = | v m | , as required.Similarly, one can prove the following statement: Theorem 3.3.
Let u and v , . . . , v n ∈ b L satisfy | exp( u ) | ∈ (cid:12)(cid:12)(cid:12)P nj =0 K [[ v j ]] (cid:12)(cid:12)(cid:12) . Then, wehave | u m | ∈ P nj =0 K | v j m | for all m ≥ .Proof. Observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =0 K [[ v j ]] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∩ (cid:12)(cid:12)(cid:12) Sym m b L (cid:12)(cid:12)(cid:12) = n X j =0 K | v j m | . The component of | exp( u ) | in (cid:12)(cid:12)(cid:12) Sym m b L (cid:12)(cid:12)(cid:12) is | u m | /m !. Hence, we conclude | u m | ∈ n X j =0 K | v jm | , as required. 12 .2 Conjugacy theorems This subsection is devoted to two conjugacy theorems. As in the preceding sections, fora K -vector space V , we denote by b T = b T ( V ) the completed tensor algebra over V , andby b L = b L ( V ) the completed free Lie algebra over V .We are now ready to formulate the main technical results of the paper: Theorem 3.4.
Let V be a finite dimensional K -vector space. Suppose that an element z ∈ b L has non-trivial linear term, z ∈ b L \ Q l> ( b L ∩ V ⊗ l ) , and that another element a ∈ b L satisfies | exp( a ) | = | exp( z ) | ∈ | b T | . Then we have a = gzg − for some group-likeelement g ∈ exp( b L ) . Theorem 3.5.
Let V be a finite dimensional K -symplectic vector space, whose symplec-tic form we denote by ω ∈ ∧ V . Suppose that an element a ∈ b L satisfies | exp( a ) | = | exp( ω ) | ∈ | b T | . Then we have a = gω g − for some group-like element g ∈ exp( b L ) . To prove these theorems we need some preliminary lemmas. Let V be a finite di-mensional K -vector space. Lemma 3.6.
Let z be an element of the sets V \ { } or ∧ V \ { } . Then, we have { u ∈ b T ⊗ b T ; [∆ z, u ] = 0 } = K [[ z ]] ⊗ K [[ z ]] . Proof.
It suffices to show that the LHS is included in the RHS. As was proved in Propo-sition 5.6, [3], we have { a ∈ b T ; [ z, a ] ∈ K [[ z ]] } = K [[ z ]] . (6)In fact, the element z is reduced in the sense of § { x , . . . , x n } of V (with no relation to z ). Let u be an element of theLHS. We may assume that under the grading defined by powers of V it is homogeneousof some degree m ≥
0. One can write uniquely u = u ⊗ m X k =1 u i ...i k ⊗ x i . . . x i k , u ∈ V ⊗ m , u i ...i k ∈ V ⊗ ( m − k ) . Then, since z is primitive, we have0 = [∆ z, u ] = [ z, u ] ⊗ m X k =1 ([ z, u i ...i k ] ⊗ x i . . . x i k + u i ...i k ⊗ [ z, x i . . . x i k ]) . (7)We claim that u i ...i k ∈ K [[ z ]] for all k and i . . . i k . First consider the case z ∈ V \{ } .Then, equation (7) is equivalent to the following family of equations0 = [ z, u ] ⊗ , X i [ z, u i ] ⊗ x i and0 = X i ,...,i k [ z, u i ...i k ] ⊗ x i . . . x i k + X i ,...,i k − u i ...i k − ⊗ [ z, x i . . . x i k − ]13or k ≥
2. By (6), we have u ∈ K [[ z ]] and u i ∈ K [[ z ]] for any i . Suppose k ≥ u i ...i k − ∈ K [[ z ]] for all i . . . i k − . Then, the third equation above im-plies [ z, u i ...i k ] ∈ K [[ z ]]. Again, by (6), we obtain u i ...i k ∈ K [[ z ]]. This completes theinduction.In the case z ∈ ∧ V \ { } , equation (7) is equivalent to the following family ofequations: 0 = [ z, u ] ⊗ , X i [ z, u i ] ⊗ x i , X i ,i [ z, u i ,i ] ⊗ x i x i and0 = X i ,...,i k [ z, u i ...i k ] ⊗ x i . . . x i k + X i ,...,i k − u i ...i k − ⊗ [ z, x i . . . x i k − ]for k ≥
3. The same argument as above applies to give u i ...i k ∈ K [[ z ]].In both cases, we have u i ...i k = λ i ...i k z m − k for some λ i ...i k ∈ K . Hence, if we write v k = P i ,...,i k λ i ...i k x i . . . x i k ∈ V ⊗ k , we have u = P mk =0 z m − k ⊗ v k , which implies0 = [∆ z, u ] = m X k =0 z m − k ⊗ [ z, v k ] . Again, by (6), v k ∈ K [[ z ]]. This completes the proof of the lemma. Lemma 3.7.
Let z be an element of the sets V \ { } or ∧ V \ { } . Then, we have { a ∈ b T ; [ z, a ] ∈ b L } = b L + K [[ z ]] . Proof.
It suffices to show that the LHS is included in the RHS.By the PBW decomposition (4), we can consider the projection ̟ : T = U ( L ) = L ∞ m =0 Sym m L → Sym L = L , and a linear endomorphism E : T = L ∞ m =0 Sym m L → T defined by E | Sym m L = m Sym m L , if m ≥ , , if m ≤ . If we denote the multiplication by µ : T ⊗ T → T and use the symbolsym( ℓ , . . . , ℓ m ) := X σ ∈ S m ℓ σ (1) · · · ℓ σ ( m ) for ℓ i ∈ L , 1 ≤ i ≤ m , then one deduces that µ (1 T ⊗ ̟ )(∆ a − a ⊗ − ⊗ a ) | a =sym( ℓ ,...,ℓ m ) equals m (sym( ℓ , . . . , ℓ m )) if m ≥
2, and 0 if m ≤
1. Hence, we have Eµ (1 T ⊗ ̟ )(∆ a − a ⊗ − ⊗ a ) = ( a for a ∈ L ∞ m =2 Sym m L , and0 for a ∈ Sym L ⊕ Sym L. (8)14ssume that a ∈ b T satisfies [ z, a ] ∈ b L . We may assume a is homogeneous, in partic-ular, that a is an element of T = U ( L ). Moreover we may assume a ∈ L ∞ m =2 Sym m L .If we write u = ∆ a − a ⊗ − ⊗ a , then0 = ∆([ z, a ]) − [ z, a ] ⊗ − ⊗ [ z, a ] = [∆ z, ∆ a ] − [ z ⊗ ⊗ z, a ⊗ ⊗ a ] = [∆ z, u ] . Hence, Lemma 3.6 implies that u = P ∞ i,j =0 λ ij z i ⊗ z j for some λ ij ∈ K , and so∆ a − a ⊗ − ⊗ a = ∞ X i,j =0 λ ij z i ⊗ z j . (9)Applying (8) to (9), we obtain a = Eµ (1 T ⊗ ̟ )( P ∞ i,j =0 λ ij z i ⊗ z j ) = Eµ ( P ∞ i =0 λ i z i ⊗ z ) = E ( P ∞ i =0 λ i z i +1 ) ∈ K [[ z ]] . This completes the proof.Now we can begin the proof of Theorem 3.4. One of the keys to the proof is thefollowing.
Proposition 3.8 (Proposition A.2 in [2]) . Let x ∈ V \ { } and a ∈ b T . If | ax l | = 0 forall l ≥ , then a ∈ [ x, b T ] .Proof of Theorem 3.4. It suffices to prove the theorem in the case z ∈ V \ { } . Indeed,write z = z + z ′ with z ∈ V \ { } and z ′ ∈ b L ∩ Q l> V ⊗ l . Then, by the universalmapping property of b L , there is a continuous Lie algebra endomorphism ϕ of b L suchthat ϕ ( z ) = z , ϕ ( Q l>p V ⊗ l ) ⊂ Q l>p V ⊗ l for any p ≥
0, and the associated graded of ϕ with respect to the filtration { Q l>p V ⊗ l } p ≥ is the identity. From these properties onecan deduce that ϕ is a topological automorphism of b L . Thus the theorem for z impliesthat for z .For the rest of the proof, we suppose z ∈ V \ { } . We denote the Baker-Campbell-Hausdorff series by ∗ : b L × b L → b L , ( u, v ) u ∗ v = bch( u, v ). Namely we haveexp( u ∗ v ) = exp( u ) exp( v ) ∈ b T .From Theorem 3.2 follows | a m | = | z m | (10)for any m ≥
1. By induction on k ≥
1, we will prove that there exist elements u k ∈ b L ∩ V ⊗ k such thatexp(ad( u k )) exp(ad( u k − )) · · · exp(ad( u ))( a )(= exp( u k ∗ u k − ∗ · · · ∗ u )( a ) exp( u k ∗ u k − ∗ · · · ∗ u ) − ) ≡ z (mod ∞ Y l>k +1 V ⊗ l ) . Since | a | = | z | , we have a ≡ z + b (mod Q ∞ l> V ⊗ l ) for some b ∈ b L ∩ V ⊗ . Thecomponent of degree ( m + 1) in equation (10) reads m | b z m − | = 0 for any m ≥ b = [ z, u ] for some u ∈ V = b L ∩ V ⊗ andexp(ad( u ))( a ) ≡ z + b + [ u , z ] = z (mod ∞ Y l> V ⊗ l ) . k ≥
2. By the inductive assumption we haveexp(ad( u k − )) · · · exp(ad( u ))( a ) ≡ z + b k +1 (mod ∞ Y l>k +1 V ⊗ l )for some b k +1 ∈ b L ∩ V ⊗ ( k +1) and u , . . . , u k − ∈ b L . The degree ( m + k ) part of equation(10) reads m | b k +1 z m − | = 0 for any m ≥
2. Hence, by Proposition 3.8, we have b k +1 =[ z, u ′ k ] for some u ′ k ∈ V ⊗ k . Applying Lemma 3.7 to equation [ z, u ′ k ] = b k +1 ∈ b L , weobtain u ′ k = u k + λ k z k , where u k ∈ b L ∩ V ⊗ k and λ k ∈ K . Therefore, b k +1 = [ z, u k ] andexp(ad( u k )) exp(ad( u k − )) · · · exp(ad( u ))( a ) ≡ z + b k +1 + [ u k , z ] = z (mod ∞ Y l>k +1 V ⊗ l ) , as required.The sequence { v k = u k ∗ u k − ∗ · · · ∗ u } ∞ k =1 converges to an element v ∞ ∈ b L by degreecounting. Taking g = exp( − v ∞ ) ∈ exp( b L ), we obtain g − ag = z ∈ b L . This completesthe proof.The proof of Theorem 3.5 is quite similar to that of Theorem 3.4, so we omit itexcept for a symplectic analogue of Proposition 3.8. Proposition 3.9.
Let V be a finite dimensional K -symplectic space with symplectic form ω ∈ ∧ V . If an element a ∈ b T satisfies | aω l | = 0 for all l ≥ , then there is an element b ∈ b T such that a = [ ω , b ] . In order to prove this statement, we may assume that a is homogeneous. Thus, it issufficient to prove the following proposition. Proposition 3.10.
Let a ∈ V ⊗ m for some m ≥ . Assume that for some p ≥ we have | aω l | = 0 for all l ≥ p . Then, there is an element b ∈ V ⊗ m − such that a = [ ω , b ] . The proof of this proposition is postponed to Section 5.
In this section, we prove Theorem 2.5 and explain some applications to non-commutativePoisson geometry.
We consider the situation of Section 2 and use the notation introduced there. We applyresults of the preceding section to V = gr H and A = b T ( V ) = b T (gr H ). Note that if n >
0, the expression ω = ω + P nj =1 z j has a non-trivial linear term, and if n = 0, then ω = ω = P i [ x i , y i ] is a symplectic form on V .Recall that as a vector space the associated graded of the Goldman Lie algebra gr b g is isomorphic to | A | = | b T (gr H ) | . The following theorem gives a description of its center:16 heorem 4.1 (Theorem 5.4 in [3]) . Z (gr b g , [ · , · ] gr ) = | K [[ ω ]] | ⊕ n M j =1 | K [[ z j ]] ≥ | . This result has been proved in [8] by using Poisson geometry of quiver varieties. Analternative elementary proof is given in [3, § ≤ j ≤ n we denote by γ j the loop along the j th boundary component(with positive orientation), and that γ is the loop along the 0th boundary component(with negative orientation). Proposition 4.2.
Let θ : c K π → A be a group-like expansion and assume | θ ( γ j ) | ∈ Z (gr b g , [ · , · ] gr ) . Then, we have | θ ( γ j ) | = ( | exp( z j ) | , for j ≥ , | exp( ω ) | , for j = 0 . Proof.
Recall the grading on A in which x i and y i have degree 1 and z j has degree 2.Under this grading, z j ’s and the expression ω = ω + P nj =1 z j are homogeneous and havedegree 2. By Theorem 4.1, we have | θ ( γ j ) | ∈ | K [[ ω ]] | ⊕ n M j =1 | K [[ z j ]] ≥ | . By Theorem 3.3, for any m ≥ | (log θ ( γ j )) m | ∈ K | ω m | ⊕ n M j =1 K | z jm | . Note that all the terms on the right hand side have degree exactly 2 m . Furthermore,note that log θ ( γ j ) ≡ ( z j , if j ≥ ,ω, if j = 0 , (mod terms of degree ≥ . Therefore, | (log θ ( γ j )) m | ≡ ( | z j m | , if j ≥ , | ω m | , if j = 0 , (mod terms of degree ≥ m + 1) . But | (log θ ( γ j )) m | contain no terms of degree higher than 2 m . Hence, the equalities aboveare verified without error terms of higher degree which proves the proposition. Proof of Theorem 2.5.
Let θ : c K π → A be a group-like expansion which induces a Liealgebra isomorphism ( b g , [ · , · ]) ∼ = → (gr b g , [ · , · ] gr ). Since for each boundary loop γ j the ex-pression | γ j | , 0 ≤ j ≤ n , is in the center for the Goldman bracket, we have | θ ( γ j ) | ∈ (gr b g , [ · , · ] gr ). Hence, by Proposition 4.2, | θ ( γ j ) | = | exp(log θ ( γ j )) | equals | exp( z j ) | if j ≥
1, and | exp( ω ) | if j = 0.Suppose j ≥ n ≥
1. Then, by Theorem 3.4, we have some group-like element g j such that log θ ( γ j ) equals g j z j g j − for j ≥
1, and g ωg − for j = 0. In the caseof n = 0, one has ω = ω and Theorem 3.5 implies that log θ ( γ ) = g ωg − for somegroup-like element g . Therefore, the expansion θ is conjugate to a special expansionwhich proves the theorem. Recall the context of non-commutative differential calculus. Let A = K hh u , . . . , u s ii bea free associative algebra with s even generators, and D • A = K hh u , . . . , u s , ∂ , . . . , ∂ s ii be the free associative algebra with s even generators u , . . . , u s and s odd generators ∂ , . . . , ∂ s . The algebra D • A carries a double bracket in the sense of van den Bergh definedby formula { ∂ i , u j } = δ i,j ⊗ , { ∂ i , ∂ j } = { u i , u j } = 0 . The space of cyclic words | D • A | = D • A / [ D • A , D • A ] = ∞ M k =0 | D kA | carries the induced graded Lie bracket (the non-commutative analogue of the Schoutenbracket on polyvector fields) [27]. Note that | D A | = | A | , D A = Der( A, A ⊗ A ) is thespace of double derivations of A , | D A | = Der( A, A ) is the Lie algebra of derivations of A , and | D A | is the space of double brackets on A . A double bracket Π ∈ | D A | is Poissonif and only if the non-commutative Schouten bracket vanishes: [Π , Π] = 0. A Poissondouble bracket Π induces a differential d Π = [Π , · ] on | D • A | .There is a natural map ∂ : | D kA | → Hom K ( | A | ⊗ k , | A | )defined by differentiating k elements of | A | by k double derivations ∂ i contained in anelement of | D kA | . Note that the right hand side also carries a Schouten bracket and themap ∂ is a Lie homomorphism.Let E be the double derivation defined by formula E ( a ) = a ⊗ − ⊗ a for every a ∈ A . Define the graded quotient space D • A as follows D • A = coker( D •− A α αE | −→ | D • A | ) . That is, D kA = | D kA | / | D k − A E | . Proposition 4.3.
The map ∂ vanishes on | D • A E | . roof. Since E ( a ) = a ⊗ − ⊗ a , we have E ( | a | ) = a − a = 0. Hence, | D k − A E | acts byzero on | A | ⊗ k .Proposition 4.3 implies that the map ∂ descends to a map (which we denote by thesame letter) ∂ : D • A −→ Hom K ( | A | • , | A | ) . (11) Proposition 4.4.
The subspace | D • A E | ⊂ | D • A | is a Lie ideal under the Schouten bracket.Proof. We compute,[ | a | , | Eb | ] = |{| a | , E } b | + | E {| a | , b }| = | E {| a | , b }| ∈ | D • A E | , where we have used {| a | , E } = − E ( | a | ) = 0. This shows that | D • A E | is indeed a Lie idealin | D • A | .As a simple example, consider | D A E | . This space is spanned by elements of the form | aE | which induce inner derivations on A : {| aE | , b } = ab − ba . The Lie algebra | D A | isisomorphic to the Lie algebra derivations of A and, hence, D A is isomorphic to the Liealgebra of outer derivations of A .Proposition 4.4 implies that the Schouten bracket descends to D • A and makes themap ∂ above into a Lie homomorphism. If Π ∈ | D A | is a Poisson double bracket,the differential d = [Π , · ] also descends to D • A and defines a non-commutative Poissoncohomology theory.We now choose A = b T (gr H ), where gr H is the associated graded of the first homologyof the surface Σ with generators x i , y i of degree 1 and generators z j of degree 2. Weconsider the double bracket (see Section 5.1 in [3])Π = g X i =1 | ∂ x i ∂ y i | + n X j =1 | z j ∂ z j ∂ z j | . The first term on the right hand side is the symplectic double bracket induced by theintersection pairing on the first homology, and the second term is the Kirillov-Kostant-Souriau (KKS) linear bracket corresponding to boundary components. It induces theassociated graded of the Goldman Lie bracket.
Remark 4.5.
In this paper, we assume double brackets to be skew-symmetric. Thisexplains the difference in the form of the bivector Π above and the bivector Π gr in [3](where no skew-symmetry assumption was made).The main result of this section is the following theorem: Theorem 4.6. H ( D • A , d = [Π , · ]) = Z ( | A | ) ,H ( D • A , d = [Π , · ]) = M i K | ∂ z i | . emark 4.7. Let Rep(
A, N ) = Hom( A, End( K N )) be the space of N -dimensional rep-resentations of A . In [27], van den Bergh constructs a Lie homomorphism | D • A | → T poly (Rep( A, N )) GL N . Under this correspondence, elements of the form | αE | are in theimage of the map ( gl N ⊗ T •− → T • poly ) GL N given by the gl N action by conjugation.Hence, D • A maps to polyvector fields on the quotient space Rep( A, N ) /GL N and thenon-commutative Poisson cohomology H ( D • A , d = [Π , · ]) maps to the Poisson cohomol-ogy of Rep( A, N ) /GL N . Proof of Theorem 4.6.
In degree zero, the cohomology of d are elements | a | ∈ | A | = D A such that {| a | , ·} is an inner derivation of A . By Theorem A.1 in [2], this is equivalentto | a | being in the center of the associated graded of the Goldman bracket, as required.In degree one, let u ∈ | D A | and [ u ] ∈ D A = Der( A, A ) / inn( A, A ) such that d ([ u ]) = 0.That is, the class of [Π , u ] in D A vanishes and therefore ∂ ([Π , u ]) = [ ∂ (Π) , ∂ ( u )] = 0.Then, u induces the derivation ∂ ( u ) of the Lie bracket ∂ (Π) : | A | ⊗ | A | → | A | .Assume that u ∈ Der ≥− ( A ). Then, by Lemma 4.8, u is tangential and u ( ω ) = [ ω, l ]for some l . Hence, u is special up to an inner derivation. As shown in [3, § | a | ∈ | A | such that u = {| a | , ·} which implies [ u ] = d | a | .If u is of degree ( − P j λ j | ∂ z j | . That is, u ( x i ) = u ( y i ) = 0 , u ( z j ) = λ j . Note that [ | z∂ z ∂ z | , | ∂ z | ] = | ∂ z ∂ z | = 0 , where we have used the fact that ∂ z is odd. Note also that the image of d : D A → D A is agraded vector space with degree bounded from below by ( − −
2) define nontrivial cocycles, as required.Assume ( n, g ) = (1 , φ ∈ Der( A ) is called tangential if φ ( z i ) = [ a i , z i ] for all i = 1 , . . . , n for some a i ∈ A . Set z = − ω = − ω − P nj =1 z j . Wesay that φ ∈ Der( A ) is fully tangential if it is tangential and in addition φ ( z ) = [ a , z ]for some a ∈ A . Lemma 4.8.
Assume that φ ∈ Der ≥− ( A ) induces a derivation of the graded Goldmanbracket on | A | . Then, φ is a fully tangential derivation.Proof. Let φ ∈ Der( A ) be such that it induces a derivation of the graded Goldmanbracket. Then, it preserves the center of the graded Goldman Lie algebra. Recall thatthe center is spanned by elements | z j k | for j = 0 , . . . , n and k ≥
0. In particular, for all l and N we have | φ ( z Nl ) | = N | z N − l φ ( z l ) | = n X j =0 | f j ( z j ) | , for some f j ∈ K [[ s ]] of degree at least N −
1. Setting z l = 0 in the equation above, weobtain for N ≥ X j = l | f j ( z j ) | = 0 . | z j k | are linearly independent for k ≥
2, we obtain that for N ≥ N | z N − l φ ( z l ) | = | f l ( z l ) | . Using Propositions 3.8 and 3.9 we conclude that φ ( z l ) is of the form φ ( z l ) = [ a l , z l ] + g l ( z l )for some a l ∈ A and g l ∈ K [[ s ]]. Using the relation | P nj =0 z j | = 0 we obtain that0 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ ( n X j =0 z j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =0 ([ a j , z j ] + g j ( z j )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =0 g j ( z j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . This equality implies that functions g j ( z j ) are at most linear. Since derivations of weight( −
2) were excluded by assumptions, we have g j ( z j ) = 2 λz j for some λ ∈ K . Then, definea derivation ψ of A by ψ ( x i ) = λx i , i = 1 , . . . , gψ ( y i ) = λy i , i = 1 , . . . , gψ ( z j ) = 2 λz j , j = 1 , . . . , n. The difference φ − ψ is now fully tangential, hence preserves the graded Goldmanbracket, and thus ψ preserves the graded Goldman bracket. This implies that the gradedGoldman bracket is of weight zero. Since it is actually of weight ( − n, g ) = (1 , In the proof of Proposition 3.10, it will be convenient to identify | b T | with a vectorsubspace of cyclically invariant elements of b T through the embedding | b T | ֒ → b T definedby | x x · · · x l | 7→ l − X k =0 ν k ( x x · · · x l ) . Here x k ∈ V for 1 ≤ k ≤ l and ν is the cyclic permutation: ν : x x · · · x l x · · · x l x . Recall that for l = 0 , | V ⊗ l | = V ⊗ l .Let dim V = 2 g be the dimension of the symplectic vector space V and C : V ⊗ → K the non-degenerate pairing defined by the symplectic form ω . Denote Q = Ker( C ) andlet π : V ⊗ → Q ⊂ V ⊗ be the projection corresponding to the direct sum decomposition V ⊗ = Q ⊕ K ω . 21n what follows, we use the following simple facts: C ( ω ) = 2 g , π ( ω ) = 0, and( C ⊗ l ⊗ V )( xω l ) = (1 V ⊗ C ⊗ l )( ω l x ) = ( − l x (12)for any x ∈ V and l ≥ m = 0 , , , The case m = 0 . Since | ω l | = l ( ω l + ν ( ω l )), C ⊗ l | ω l | = l ((2 g ) l + ( − l g ) . If l ≥
2, the right hand side of this equation is nonzero. Hence, | aω l | = 0 implies a = 0,as required. The case m = 1 . If deg a = 1, then | aω l | = P lj =0 ω j aω l − j + P l − j =0 ν ( ω j aω l − j ).Since 1 V ⊗ C ⊗ l : ω j aω l − j ( − j (2 g ) l − j a, ν ( ω j aω l − j ) ( − l − j (2 g ) j a, we have (1 V ⊗ C ⊗ l ) | aω l | = l X j =0 ( − j (2 g ) l − j a + l − X j =0 ( − l − j (2 g ) j a = (2 g ) l + 2( − l l − X j =0 ( − g ) j a. If l ≥
2, the coefficient of a is not zero. Hence, | aω l | = 0 (for l sufficiently large) implies a = 0. The case m = 2 . If deg a = 2, then | aω l | = P lj =0 ω j aω l − j + P lj =0 ν ( ω j aω l − j ).We have π ⊗ C ⊗ l : aω l (2 g ) l π ( a ) ω j aω l − j ≤ j ≤ lν ( ω j aω l − j ) ( − l π ( ν ( a )) for any 0 ≤ j ≤ l Here, the second case follows from π ( ω ) = 0 and the third case from (12). Therefore,( π ⊗ C ⊗ l ) | aω l | = (2 g ) l π ( a ) + ( l + 1)( − l π ( ν ( a )) . Now assume that | aω l | = 0 for any l ≥ p . Then, the right hand side of the aboveformula vanishes for any l ≥ p , and this shows that π ( a ) = 0 and π ( ν ( a )) = 0. Theequation π ( a ) = 0 implies that a is a multiple of ω . From the case m = 0, we deducethat a = 0.To consider the case of m = 3, we need the following lemma. Lemma 5.1. V ⊗ = [ ω , V ] ⊕ ( V ⊗ Q ) . roof. Let a ∈ [ ω , V ] ∩ ( V ⊗ Q ). Since a ∈ V ⊗ Q , (1 V ⊗ C )( a ) = 0. On the otherhand, there is an element b ∈ V such that a = [ ω , b ] = ω b − bω , and (1 V ⊗ C )( a ) = − b − (2 g ) b = − (2 g + 1) b . Therefore, b = 0 and a = 0. Thus [ ω , V ] ∩ ( V ⊗ Q ) = { } . Bycounting dimensions, the assertion follows. The case m = 3 . In view of Lemma 5.1, it is sufficient to prove the following assertion:let a ∈ V ⊗ Q and assume that there exists some p ≥ | aω l | = 0 for any l ≥ p . Then a = 0.Assume a ∈ V ⊗ Q and introduce the notation a = a a a . Let us apply 1 V ⊗ π ⊗ C ⊗ l to | aω l | = l X j =0 ω j aω l − j + l X j =0 ν ( ω j aω l − j ) + a a ω l a . Since 1 V ⊗ π ⊗ C ⊗ l : aω l (2 g ) l aω j aω l − j ≤ j ≤ lν ( aω l ) ( − l (1 V ⊗ π )( a a a ) ν ( ω j aω l − j ) ≤ j ≤ la a ω l a ( − l (1 V ⊗ π )( a a a ) , we have 0 = (1 V ⊗ π ⊗ C ⊗ l ) | aω l | = (2 g ) l a + ( − l (1 V ⊗ π )( a a a + a a a ) . Since this equality holds true for any l ≥ p , we deduce that a = 0.The case m ≥ Proposition 5.2.
Let m ≥ , a ∈ Q ⊗ V ⊗ m − ⊗ Q , and b ∈ V ⊗ m − . Assume that thereexists some p ≥ such that | aω l + bω l +1 | = 0 for all l ≥ p . Then, a = 0 .The case m ≥ . Let a ∈ V ⊗ m and assume that | aω l | = 0 for any l ≥ p . By thedirect sum decomposition V ⊗ = Q ⊕ K ω , we can uniquely write a = ω b ′ + b ′′ ω + c, where b ′ ∈ V ⊗ m − , b ′′ ∈ Q ⊗ V ⊗ m − , and c ∈ Q ⊗ V ⊗ m − ⊗ Q . For any l ≥ p ,0 = | aω l | = | ( ω b ′ + b ′′ ω + c ) ω l | = | cω l + ( b ′ + b ′′ ) ω l +1 | . Applying Proposition 5.2, we obtain c = 0, a = ω b ′ + b ′′ ω , and | ( b ′ + b ′′ ) ω l | = 0 forany l ≥ p + 1. By the inductive assumption, there exists some d ∈ V ⊗ m − such that b ′ + b ′′ = [ ω , d ]. Then a = ω b ′ + ([ ω , d ] − b ′ ) ω = [ ω , b ′ ] + [ ω , d ] ω = [ ω , b ′ + dω ] , as required. This completes the proof of Proposition 3.10.Finally, let us prove Proposition 5.2. We use the following two lemmas, which canbe proved by straightforward computations.23 emma 5.3. Let m be an odd integer ≥ and l ≥ ( m +1) / . For any u , u , . . . , u m ∈ V ,we have ( π ⊗ V ⊗ m − ⊗ π )(1 V ⊗ m ⊗ C ⊗ l ) | u u · · · u m ω l | =( π ⊗ V ⊗ m − ⊗ π )((2 g ) l u u · · · u m + ( − l Φ( u )) , where Φ( u ) = X ≤ k ≤ m − k odd ( − k − (cid:16) Cont( u m − k +1 · · · u m − ) u m ω k − u u · · · u m − k +Cont( u · · · u k ) u k +1 · · · u m ω k − u (cid:17) . Here,
Cont( u · · · u k ) = C ( u , u ) · · · C ( u k − , u k ) ∈ K . Lemma 5.4.
Let m be an even integer ≥ and l ≥ m/ . For any u , u , . . . , u m ∈ V ,we have ( π ⊗ V ⊗ m − ⊗ π )(1 V ⊗ m ⊗ C ⊗ l ) | u u · · · u m ω l | =( π ⊗ V ⊗ m − ⊗ π )((2 g ) l u u · · · u m + (cid:16) l − m (cid:17) ( − l Φ ( u ) + ( − l Φ ( u )) , where Φ ( u ) = ( − m − Cont( u · · · u m − ) u m ω m − u , Φ ( u ) = X ≤ k ≤ m − k odd ( − ( k − (cid:16) Cont( u m − k +1 · · · u m − ) u m ω k − u u · · · u m − k +Cont( u · · · u k ) u k +1 · · · u m ω k − u (cid:17) . Proof of Proposition 5.2.
First assume that m is odd and ≥
5. We apply Lemma 5.3 to u = a + bω . Since a ∈ Q ⊗ V ⊗ m − ⊗ Q , ( π ⊗ V ⊗ m − ⊗ π )( a ) = a . Since π ( ω ) = 0,( π ⊗ V ⊗ m − ⊗ π )( bω ) = 0. Hence0 = (2 g ) l a + ( − l ( π ⊗ V ⊗ m − ⊗ π )Φ( a + bω )for any l ≫
0. Therefore, a = 0.Next, assume that m is even and ≥
4. We apply Lemma 5.4 to u = a + bω . Then,we obtain 0 = (2 g ) l a + (cid:16) l − m (cid:17) ( − l Φ ( a + bω ) + ( − l Φ ( a + bω )for any l ≥ p . We can find sufficiently large integers l , l , l such thatdet (2 g ) l ( l − m )( − l ( − l (2 g ) l ( l − m )( − l ( − l (2 g ) l ( l − m )( − l ( − l = 0 . Therefore, we can conclude that a = 0. 24 eferences [1] A. Alekseev, B. Enriquez and C. Torossian, Drinfeld associators, braid groups andexplicit solutions of the Kashiwara-Vergne equations, Publ. Math. Inst. Hautes´Etudes Sci. , 143–189 (2010)[2] A. Alekseev, N. Kawazumi, Y. Kuno and F. Naef, The Goldman-Turaev Lie bial-gebra in genus zero and the Kashiwara-Vergne problem, Adv. Math. , 1–53(2018)[3] A. Alekseev, N. Kawazumi, Y. Kuno and F. Naef, The Goldman-Turaev Lie bialge-bra and the Kashiwara-Vergne problem in higher genera, arXiv:1804.09566 (2018)[4] A. Alekseev and F. Naef, Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection, C. R. Math. Acad. Sci. Paris (2017), no. 11, 1138–1147.[5] A. Alekseev and C. Torossian, The Kashiwara-Vergne conjecture and Drinfeld’sassociators, Ann. of Math. , 415–463 (2012)[6] R. Bocklandt and L. Le Bruyn, Necklace Lie algebras and noncommutative sym-plectic geometry, Mah. Z. , no. 1, 141–167 (2002)[7] N. Bourbaki, Groupes et alg`ebres de Lie , Hermann, Paris, 1971-72.[8] W. Crawley-Boevey, P. Etingof and V. Ginzburg, Noncommutative geometry andquiver algebras, Adv. Math. , 274–336 (2007)[9] V. Ginzburg, Non-commutative symplectic geometry, quiver varieties, and operads,Math. Res. Lett. , no.3, 377–400 (2001)[10] W. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surfacegroup representations, Invent. Math. , 263–302 (1986)[11] N. Habegger and G. Masbaum, The Kontsevich integral and Milnor’s invariants,Topology , 1253–1289 (2000)[12] R. Hain, Hodge theory of the Goldman bracket, arXiv:1710.06053[13] R. Hain, Hodge Theory of the Turaev Cobracket and the Kashiwara–Vergne Prob-lem, arXiv:1807.09209[14] M. Kashiwara and M. Vergne, The Campbell-Hausdorff formula and invariant hy-perfunctions, Invent. Math. , 249–272 (1978)[15] N. Kawazumi, Harmonic Magnus expansion on the universal family of Riemannsurfaces, preprint, arXiv: math.GT/0603158 (2006)2516] N. Kawazumi and Y. Kuno, The logarithms of Dehn twists, Quantum Topol. ,347–423 (2014)[17] N. Kawazumi and Y. Kuno, Intersections of curves on surfaces and their applicationsto mapping class groups, Annales de l’institut Fourier , 2711–2762 (2015)[18] N. Kawazumi and Y. Kuno, The Goldman-Turaev Lie bialgebra and the Johnsonhomomorphisms, Handbook of Teichmuller theory, ed. A. Papadopoulos, VolumeV, EMS Publishing House, Zurich, 97–165 (2016)[19] Y. Kuno, A combinatorial construction of symplectic expansions, Proc. Amer. Math.Soc. , 1075–1083 (2012)[20] G. Massuyeau, Infinitesimal Morita homomorphisms and the tree-level of the LMOinvariant, Bull. Soc. Math. France , 101–161 (2012)[21] G. Massuyeau, Formal descriptions of Turaev’s loop operations, Quantum Topol. , 39–117 (2018)[22] G. Massuyeau and V. G. Turaev, Fox pairings and generalized Dehn twists, Annalesde l’institut Fourier , 2403–2456 (2013)[23] G. Massuyeau and V. G. Turaev, Tensorial description of double brackets on surfacegroups and related operations, draft (2012)[24] F. Naef, Poisson brackets in Kontsevich’s “Lie world”, preprint, arXiv:1608.08886(2016)[25] T. Schedler, A Hopf algebra quantizing a necklace Lie algebra canonically associatedto a quiver, Int. Math. Res. Notices 2005 no. 12, 725–760[26] V. G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci.´Ecole Norm. Sup.24