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Poisson-Gerstenhaber brackets in representation algebras

We introduce cyclic bilinear forms on coalgebras and use them to generalize Van den Bergh's Poisson brackets in representation algebras.


Introduction
Van den Bergh [8] introduced double Poisson algebras as a non-commutative version of Poisson geometry, see also Crawley-Boevey [2] for a related approach. To give a more precise formulation, note that the pair (A, N ) determines a commutative algebra A N and a homomorphism t N from A to the algebra Mat N (A N ) of N × N matrices over A N characterized by the following universal property: for any commutative algebra B and any homomorphism s : A → Mat N (B), there is a unique homomorphism r : A N → B such that s = Mat N (r)t N (see [7,4]). One calls A N the coordinate algebra of Rep(A, N ). Van den Bergh showed that a double Poisson bracket in A induces a Poisson bracket in A N for all N ≥ 1.
The aim of this paper is to generalize Van den Bergh's Poisson bracket in A N to a wider class of commutative algebras associated with A. Fix a commutative ring K which will be the ground ring of all modules, algebras, and coalgebras. For an algebra A and a coalgebra M (possibly, non-counital), we define a commutative representation algebra A M . It is characterized by the universal property as above with Mat N (B) replaced by the convolution algebra Hom K (M, B). The algebra A M is described here via generators {a α | a ∈ A, α ∈ M } and certain relations. For the coalgebra M = (Mat N (K)) * dual to Mat N (K), we recover A N because Theorem 1.1 derives a Poisson bracket in A M from a double Poisson bracket in A and a cyclic bilinear form on M . For constructions of double brackets, see [8,9,5,6]. We give here two constructions of cyclic bilinear forms. One construction starts from a K-valued conjugation-invariant function on a finite group G and produces a cyclic form on the coalgebra (K[G]) * dual to the group algebra K[G]. In particular, this derives cyclic forms on (K[G]) * from the traces of linear representations of G of finite rank. The second construction starts from a symmetric Frobenius algebra and produces a cyclic form on the dual coalgebra. For example, the algebra of matrices Mat N (K) with N ≥ 1 is a symmetric Frobenius algebra with Frobenius pairing being the trace of the product of matrices. This determines a cyclic bilinear form v on (Mat N (K)) * . The bracket {−, −} v in A N is the original Van den Bergh's Poisson bracket. Note that symmetric Frobenius algebras naturally arise in Topological Quantum Field Theory, see [3].
We We also introduce an equivariant generalization of the bracket {−, −} v and discuss connections with the topology of surfaces.
The main body of the paper is written in the more general setting of graded algebras. Instead of (double) Poisson brackets we consider their graded versions, the (double) Gerstenhaber brackets.
This work was partially supported by the NSF grant DMS-1202335.

Representation algebras
We begin with generalities on graded algebras and coalgebras and then introduce the representation algebras.
2.1. Graded algebras. By a module we mean a module over the fixed commutative ring K. By a graded module we mean a Z-graded module A = ⊕ p∈Z A p . An element a of A is homogeneous if a ∈ A p for some p; we call p the degree of a and write |a| = p. For any integer n, the n-degree |a| n of a homogeneous element a ∈ A is defined by |a| n = |a| + n. Note that the n-degree of 0 ∈ A is an arbitrary integer.
An algebra is a module endowed with an associative bilinear multiplication. A graded algebra is a graded module A endowed with an associative bilinear multiplication such that A p A q ⊂ A p+q for all p, q ∈ Z. Thus, |ab| = |a|+|b| for homogeneous a, b ∈ A. A graded algebra A is unital if it has a two-sided unit 1 A ∈ A 0 . Unless explicitly stated to the contrary, we do not require algebras to be unital. Given graded algebras A and B, a graded algebra homomorphism A → B is an algebra homomorphism A → B carrying A p to B p for all p ∈ Z. The graded algebras and graded algebra homomorphisms form a category denoted GA.
Non-graded algebras will be identified with graded algebras concentrated in degree 0.

Coalgebras.
A coalgebra is a module M endowed with a coassociative linear map µ : M → M ⊗ M called the comultiplication (here and below ⊗ = ⊗ K ). We do not suppose M to be graded. The image of any α ∈ M under µ expands (nonuniquely) as a sum i α 1 i ⊗ α 2 i where α 1 i , α 2 i ∈ M and the index i runs over a finite set. To shorten the formulas, we will drop the index i and the summation sign and write simply µ(α) = α 1 ⊗α 2 . The coassociativity of µ reads then ( If a counit exists, then it is unique. A coalgebra having a counit is counital. Unless explicitly stated to the contrary, we do not require coalgebras to be counital.
2.3. The algebra A M . From now on, the symbol A denotes a graded algebra and the symbol M denotes a coalgebra with comultiplication µ. Consider the algebra A M with generators {a α }, where a runs over A and α runs over M , subject to the following two sets of relations: (i) (the bilinearity relations) for all k ∈ K, a, b ∈ A, and α, β ∈ M , k a α = (ka) α = a kα , (a + b) α = a α + b α , a α+β = a α + a β ; (ii) (the multiplicativity relations) for all a, b ∈ A and α ∈ M , A fully developed version of the latter relation is (ab The bilinearity relations ensure that a α 1 b α 2 = i a α 1 i b α 2 i does not depend on the choice of the expansion. We turn A M into a graded algebra by declaring that the generator a α is homogeneous of degree |a| for all homogeneous a ∈ A and all α ∈ M . A typical element of A M is represented by a non-commutative polynomial in the generators with zero free term. One can alternatively define A M as the tensor algebra The construction of A M is functorial: a graded algebra homomorphism f : A → B induces a graded algebra homomorphism f M : A M → B M by f M (a α ) = (f (a)) α for all a ∈ A, α ∈ M . For a fixed M , the formulas A → A M , f → f M define an endofunctor of the category of graded algebras GA.
2.4. The universal property. We now formulate the universal property of A M . Note that the linear maps from M to a graded algebra B form a graded module The map inverse to (2.4.1) carries a graded algebra homomorphism s : A → H M (B) to the algebra homomorphism r = r s : A M → B defined on the generators by r(a α ) = s(a)(α). That this rule is compatible with the bilinearity relations between the generators of A M follows from the linearity of s. The compatibility with the multiplicativity relations: Clearly, r preserves the grading, and the maps s → r s and r → s r are mutually inverse.
Lemma 2.1 implies the following universal property of A M : for any graded algebra B and any graded algebra homomorphism s : A → H M (B), there is a unique graded algebra homomorphism r : A M → B such that the diagram commutes. Here the horizontal arrow is the graded algebra homomorphism carrying a ∈ A to the map M → A M , α → a α . This homomorphism corresponds to the identity automorphism of A M via (2.4.1). The algebra A M has the same universal property as A M with B running over commutative graded algebras.

The algebra
2.6. Examples. 1. Consider the coalgebra M = (Mat N (K)) * dual to the matrix algebra Mat N (K) with N ≥ 1. For i, j ∈ {1, ..., N }, let τ i,j ∈ M be the linear functional on Mat N (K) carrying each matrix to its (i, j)-term. These functionals form a basis of M . The comultiplication in M carries τ i,j to r τ i,r ⊗ τ r,j for all i, j. Given a graded algebra A, we write a i,j for the generator a (τi,j ) of A M . The defining relations of A M in these generators are as follows: 2. Consider the group algebra A = K[G] of a finite group G and the dual coalgebra M = A * . The underlying module of M is free with basis {δ g } g∈G dual to the basis {g} g∈G of A. The comultiplication in M is given by Given a graded algebra A, we write a g for the generator a (δg) of A M . The defining relations of A M in these generators are as follows: 3. If, in the previous example, A = K[π] is the (non-graded) group ring of a group π, then the algebra A M is generated by the set {a g } a∈π,g∈G subject only to the third relation in (2.6.1) for all a, b ∈ π, g ∈ G.

Brackets and double brackets
We recall the notions of Gerstenhaber algebras and double brackets. By an n-graded biderivation in a graded algebra A we mean an n-graded bracket {−, −} in A such that for any homogeneous a, b, c ∈ A,   An n-graded biderivation in a graded algebra is Gerstenhaber if its Jacobi form is equal to zero (this is the n-graded Jacobi identity). An n-graded Gerstenhaber biderivation is also called an n-graded Gerstenhaber bracket. A graded algebra equipped with an n-graded Gerstenhaber bracket is called an n-graded Gerstenhaber algebra. For non-graded algebras (i.e., for graded algebras concentrated in degree zero) and n = 0, we say "Poisson" instead of "Gerstenhaber".
An example of a 0-graded Gerstenhaber algebra in provided by an arbitrary graded algebra with bracket {a, b} = ab − (−1) |a||b| ba for homogeneous a, b.
3.2. Double brackets. Double brackets in algebras were introduced by Van den Bergh [8]. The exposition here follows [6]. We begin with notation.
The tensor product of m ≥ 2 copies of a graded algebra A is denoted A ⊗m . For a permutation (i 1 , . . . , i m ) of (1, . . . , m) and n ∈ Z, the symbol P i1···im,n denotes the n-graded permutation A ⊗m → A ⊗m carrying a 1 ⊗ · · · ⊗ a m with homogeneous a 1 , . . . , a m ∈ A to (−1) t a i1 ⊗ · · · ⊗ a im where t ∈ Z is the sum of the products |a i k | n |a i l | n over all pairs of indices k < l such that i k > i l . Set P i1···im = P i1···im,0 .
We define two A-bimodule structures on A ⊗2 by a(x ⊗ y)b = ax ⊗ yb and a * (x ⊗ y) * b = (−1) |a||xb|+|b||y| xb ⊗ ay for any homogeneous a, b, x, y ∈ A. An n-graded double bracket in A with n ∈ Z is a linear map { {−, −} } ∈ End(A ⊗2 ) satisfying the following conditions: (i) (the grading condition) for any integers p, q, (ii) (the n-graded antisymmetry) { {−, −} } P 21,n = −P 21 { {−, −} }; (iii) (the n-graded Leibniz rules) for all homogeneous a, b, c ∈ A, Note that (3.2.2) is a consequence of (3.2.1) and the antisymmetry. Observe also that each x ∈ A ⊗ A expands (non-uniquely) as a sum i , x ′′ i ∈ A are homogeneous and the index i runs over a finite set. In the sequel, we will drop the index and the summation sign and write simply x = x ′ ⊗ x ′′ . In this notation, for any a, b ∈ A. The n-graded antisymmetry condition may be rewritten as For non-graded algebras and n = 0, we say "Poisson" instead of "Gerstenhaber".

Cyclic bilinear forms and induced brackets
We define cyclic bilinear forms on coalgebras and introduce the associated brackets in representation algebras.
Given α, β ∈ M , we can expand v(α ⊗ β) as a finite sum i x i ⊗ y i with x i , y i ∈ M . To shorten the formulas, we will suppress the index and the summation sign and write α β for x i and β α for y i . Thus, we write v(α ⊗ β) = α β ⊗ β α . In this notation, Lemma 4.1 says that for any α, β ∈ M , In the notation above, this can be rewritten as The set of cyclic bilinear forms on M is a submodule of (M ⊗ M ) * . In particular, v = 0 is cyclic. More interesting examples will be given in Sections 5 and 6.
The following lemma is our main technical result. It derives a bracket in the representation algebra A M from a double bracket in A and a cyclic form on M .
Proof. The uniqueness of {−, −} v is obvious because the symbols a α generate A M .
To prove the existence, consider the graded algebra A generated by the symbols  Pick homogeneous a, b, c ∈ A and set Therefore, the bracket {−, −} : A ⊗ A → A M annihilates the multiplicativity relations. It also annihilates the commutativity relations: It remains only to prove (4.2.3). In the rest of the argument, we write Using the commutativity of A M , we deduce that for any ϕ, ψ, ρ ∈ M , Permuting b and c in (4.2.6), we similarly obtain that We rewrite the last summand as follows. Note that These formulas and the commutativity of A M imply that As a result, we obtain that Cyclically permuting a α , b β , c γ , we obtain Below we prove that The equalities w 1 = u 1 and z 1 = t 1 are tautological. The formula w 2 = u 2 would follow from the identity Similarly, the formulas w 3 = u 3 , z 2 = t 3 , z 3 = t 2 would follow from the identities Formula (  which we now prove. For α, β, γ ∈ M , Hence, We must show that To this end, we take (4.2.1) and replace α with γ. This gives Applying id M ⊗µ 3 to both sides, we obtain that We apply to both sides the linear map This gives (4.2.13) and completes the proof of (4.2.7).
The cyclicity of v gives σX = Xσ and τ Y = Y τ . Then We can now formulate the main result of this paper.

Cyclic structures on algebras and coalgebras
We reformulate cyclic bilinear forms on coalgebras in terms of so-called cyclic structures on coalgebras and algebras.

Cyclic structures on coalgebras.
A cyclic structure on a coalgebra M is a linear map M → M * , α → α such that for any α, β ∈ M , Proof. For α, β ∈ M , Clearly, vp 21 = p 21 v if and only if ad v is a cyclic structure on M . This proves (a). Claim (b) is obvious; the inverse bijection carries a cyclic structure M → M * , α → α to the bilinear form α ⊗ β → α(β) on M .

5.2.
Cyclic structures on algebras. Dualizing cyclic structures on coalgebras we obtain the following notion. A cyclic structure on an algebra A is a linear map A * → A, α → α such that α(aβb) = β(bαa) for any α, β ∈ A * and a, b ∈ A. Here and below A * = Hom K (A, K).
Given a coalgebra M , a cyclic structure M * * → M * , x → x on the dual algebra M * induces (under additional assumptions) a cyclic structure on M . Recall that multiplication in M * is defined by (ab)(α) = a(α 1 )b(α 2 ) for a, b ∈ M * and any α ∈ M . Let e : M → M * * be the evaluation map carrying α ∈ M to the functional In other words, the evaluations of a ⊗ b on the vectors The cyclic identity is easily verified on the basis vectors: for any a, b, g, h ∈ G, The associated cyclic bilinear form v on M is computed by v(δ g ⊗ δ h ) = F (gh) for g, h ∈ G. Given a graded algebra A with double bracket Note that a linear representation ρ : G → GL N (K) with N ≥ 1 determines a conjugation-invariant function G → K, g → tr(ρ(g)). The corresponding cyclic bilinear form on M carries δ g ⊗ δ h to tr(ρ(gh)) for any g, h ∈ G.

Cyclic bilinear forms from Frobenius algebras
We show how to derive cyclic bilinear forms from symmetric Frobenius algebras. We begin by recalling basic definitions concerning Frobenius algebras.

6.2.
From Frobenius algebras to cyclic forms. The following theorem derives from every symmetric Frobenius algebra a cyclic bilinear form on the dual coalgebra. Theorem 6.1. Let A be a symmetric Frobenius algebra with Frobenius pairing (−, −). For any α ∈ A * , there is a unique α ∈ A such that α(a) = (a, α) for all a ∈ A. The map A * → A, α → α is a cyclic structure on A and on A * . The associated cyclic bilinear form v on A * is given by v(α ⊗ β) = β(α) = (α, β) for any α, β ∈ A * .
The computation of v follows from the remark at the end of Section 5.2.
We will combine Theorem 6.1 with the following construction of symmetric Frobenius algebras. Let A be an algebra whose underlying module is free of finite rank. We say that θ ∈ A * is trace-like if the bilinear form (−, −) θ on A defined by (a, b) θ = θ(ab) for a, b ∈ A is symmetric and non-degenerate. Then the pair (A, (−, −) θ ) is a symmetric Frobenius algebra. The associated cyclic bilinear form on the coalgebra A * will be denoted v θ . 6.3. Examples. 1. The group algebra A = K[G] of a finite group G has a tracelike θ ∈ A * which carries the neutral element of G to 1 and all other elements of G to 0. The cyclic form v θ on A * is the bilinear form of Section 5.3 determined by the function θ| G : G → K.
2. In analogy with quartenions we define a unital algebra A = K1 A ⊕Ki⊕Kj⊕Kk with unique (associative) multiplication such that i 2 = j 2 = k 2 = ijk = −1 A . The linear map θ : A → K carrying 1 A to 1 K and i, j, k to 0 is trace-like. This turns A into a (non-commutative) symmetric Frobenius algebra and yields a cyclic bilinear form v θ on the dual coalgebra A * .
3. For any n ≥ 1, the truncated polynomial algebra A = K[x]/x n+1 has a tracelike θ ∈ M = A * defined by θ(x n ) = 1 and θ(x i ) = 0 for i = 0, 1, ..., n − 1. Let (u i ) n i=0 be the basis of M dual to the basis (x i ) n i=0 of A. The comultiplication µ in M , the cyclic structure M → A, α → α, and the form v θ on M are computed by It is understood that if i + j − n < 0, then the right-hand side is equal to zero.
This bracket was first introduced by Van den Bergh [8]. 5. In generalization of the previous example, consider a symmetric Frobenius algebra (A, (−, −)) with cyclic structure A * → A, α → α and with cyclic bilinear form v on A * as in Theorem 6.1. For any integer N ≥ 1, the matrix algebra Mat N (A) acquires a symmetric Frobenius pairing . This determines a cyclic structure α → α and a cylic bilinear form, v N , on the coalgebra M = (Mat N (A)) * . We identify elements of M with matrices α = (α ij ) i,j where i, j ∈ {1, ..., N }, α ij ∈ A * for all i, j, so that α(a) = i,j α ij (a ij ) for any a = (a ij ) i,j ∈ Mat N (A). It is easy to see that α = (α ji ) i,j and v N (α ⊗ β) = i,j v(α ij ⊗ β ji ) for any α, β ∈ M . 6.4. Remark. Each 2-dimensional Topological Quantum Field Theory (TQFT) gives rise to a unital commutative Frobenius algebra, see [3]. Combining with Theorem 6.1, we conclude that every 2-dimensional TQFT gives rise to a cyclic bilinear form on a coalgebra.

A group action on A M
We show that coalgebra automorphisms of M act on A M by algebra automorphisms and study the behavior of the bracket (4.2.2) under this action. For counital M , we exhibit a group of automorphisms of A M preserving the bracket.  Proof. The first claim is obvious. It is easy to check that for any If v is cyclic, then p 21 commutes with both v and ω ⊗ ω and hence commutes with v ω . Hence, v ω is cyclic.
For any graded algebra A, there is a natural left action of Ω = Aut(M ) on A M . Any ω ∈ Ω acts on the generators by ωa α = a ω(α) . This extends uniquely to a graded algebra automorphism of A M . The compatibility with the bilinearity relations is obvious. The compatibility with the multiplicativity relations: where the third equality holds because µω = (ω ⊗ ω)µ. The action of Ω on A M induces an action of Ω on A M = Com( A M ) by graded algebra automorphisms. Proof. It is easy to see that if the identity {ωx, ωy} v = ω {x, y} v ω holds for generators of A M , then it holds for any x, y ∈ A M . Given a, b ∈ A and α, β ∈ M , Note for the record that the action of any u ∈ U on M is dual to the conjugation by u in M * . Indeed, for any a ∈ M * and α ∈ M , Proof. Let α, β ∈ M . Applying ε ⊗ ε : M ⊗ M → K to both sides of (4.2.1), we obtain that v(α ⊗ β) = v(β ⊗ α). This equality may be rewritten as α(β) = β(α) where the overbar denotes the cyclic structure M → M * determined by v.
By Lemma 7.2, in order to prove (ii), it is enough to show that v u = v for all u ∈ U . We first verify that u α = uαu −1 for any α ∈ M . We have Evaluating on any β ∈ M , we obtain where the penultimate equality is obtained by evaluating u ⊗ u −1 on both sides of (5.1.1). We conclude that u α = uαu −1 . Next, identifying u ∈ U with its image in Aut(M ), we obtain for any α, β ∈ M , We conclude that v u = v. This proves (ii). Clearly, (ii) implies (iii).

Coderivations.
A coderivation in a coalgebra M is a linear map δ : M → M such that This formula may be rewritten as the identity for Proof. The uniqueness is obvious, and we need only to construct the action in question. We derive from each δ ∈ coder(M ) a derivation δ in A M such that δ(a α ) = a δ(α) for all generators a α of A M . We check the compatibility with the defining relations of A M . The compatibility with the bilinearity relations is obvious. The compatibility with the multiplicativity relations: The map coder(M ) → der( A M ), δ → δ is linear and preserves the Lie bracket: for δ, d ∈ coder(M ), a ∈ A, α ∈ M ,

Proof. We verify first that the bracket {−, −} v is invariant under the action on
A simple computation using the Leibniz rules shows that it is enough to prove this identity for the generators x = a α , On the other hand, We only need to prove that for all α, β ∈ M , or, equivalently, Since δ is a coderivation, we easily compute that for any β ∈ M , Hence, evaluating the left-hand side of (8 The first and third terms cancel out by the assumption v(δ ⊗ id M ) = −v(id M ⊗δ).
The remaining two terms give This proves (8.2.3).
To accomplish the proof, it remains to show that v(δ ϕ ⊗ id M ) = −v(id M ⊗δ ϕ ) for all ϕ ∈ M * . We evaluate both sides on any α ⊗ β ∈ M ⊗ M . Recall the cyclic structure M → M * , γ → γ determined by v. Applying id M ⊗ε M and ε M ⊗ id M to both sides of (5.1.1) we obtain Using these formulas and the identity γ(β) = β(γ) for all γ ∈ M , we obtain

Compatibility.
To relate the actions of Aut(M ) and coder(M ) constructed above, we use the language of Lie pairs introduced in [5]. A Lie pair is a pair (G, g) where G is a group and g is a Lie algebra equipped with a (left) action of G by Lie algebra automorphisms w → g w where w runs over g and g runs over G. A morphism of Lie pairs (G ′ , g ′ ) → (G, g) is a pair (a group homomorphism F : G ′ → G, a Lie algebra homomorphism f : g ′ → g) such that f ( g w) = F (g) (f (w)) for all g ∈ G ′ , w ∈ g ′ . An action of a Lie pair (G, g) on a graded algebra A is a pair (a (left) action of G on A by graded algebra automorphisms, an action of g on A) such that ( g w)x = gwg −1 x for any g ∈ G, w ∈ g, x ∈ A. Note that then the algebra A g defined by (8. 1.2) is G-invariant. Indeed, if g ∈ G and x ∈ A g , then gx ∈ A g because for all w ∈ g, An action of (G, g) on A composed with a morphism of Lie pairs (G ′ , g ′ ) → (G, g) yields an action of (G ′ , g ′ ) on A. An action of a Lie pair on A induces an action of this Lie pair on Com(A) in the obvious way. Any coalgebra M gives rise to a Lie pair (G = Aut(M ), g = coder(M )) where G acts on g by g w = gwg −1 : M → M for g ∈ G and w ∈ g. The above-defined actions of G and g on A M determine an action of the Lie pair (G, g) because ( g w)x = gwg −1 x for any g ∈ G, w ∈ g, x ∈ A. Since g w and gwg −1 act as derivations in A M , it is enough to check this for the generators x = a α ∈ A M where a ∈ A, α ∈ M . We have g w(a α ) = a ( g w(α)) = a gwg −1 (α) = ga wg −1 (α) = gw(a g −1 (α) ) = gwg −1 (a α ).  Proof. For any homogeneous a, b ∈ A, where the last equality holds because p 21 µ(θ) = µ(θ). Thus, tr θ ([A, A]) = 0. The equality p 21 µ(θ) = µ(θ) implies that each inner coderivation δ of M annihilates θ. Therefore δ(a θ ) = a δ(θ) = 0 for all a ∈ A. Hence tr Proof. Since the map α → v(θ ⊗ α) = ε(α) is the counit of M , Therefore θ is symmetric: Proof. Pick any a, b ∈ A and letǎ,b be their projections toǍ. Then The following lemma gives more information about A(θ) for some θ. Proof. The first claim is obvious. To prove the second claim, we must show that u (a θ ) = a θ for all u ∈ U and a ∈ A. Pick any α ∈ M and observe that The next claim yields a rich source of adjoint elements of cyclic bilinear forms.
Lemma 9.5. Let A be a unital algebra whose underlying module is free of finite rank. Let M = A * be the dual coalgebra and let θ ∈ M be a trace-like element. Then θ is adjoint to the cyclic bilinear form v θ on M defined in Section 6.2.
10.2. The actions. Given a unital graded algebra A and a counital coalgebra M , the action of Aut(M ) on A M defined in Section 7.1 induces an action of Aut(M ) on A + M by graded algebra automorpisms fixing the unit e. We check the compatibility with the normalization relations: for g ∈ Aut(M ), α ∈ M , This makes E into a commutative graded algebra with unit 1 E = e. This graded algebra appears in Hamiltonian reduction as follows.
Theorem 10.2. Let B be a graded algebra and let p : A → B be a graded algebra epimorphism whose kernel is generated, as a 2-sided ideal of A, by a set M ⊂ A −n satisfying the following condition: for every ξ ∈ M there is a scalar k ξ ∈ K such that for all a ∈ A, Proof. The uniqueness of {−, −} is obvious, and we need only to prove the existence. We begin by considering some ξ ∈ A −n and k ∈ K such that for all a ∈ A, We claim that for any α ∈ M , x ∈ A + M , (10. 3.1) {ξ α , x} + v = kδ α (x) where the overbar stands for the cyclic structure M → M * determined by v. Both sides of (10.3.1) are derivations in x (here we use that ξ ∈ A −n ). Hence, it is enough to prove (10.3.1) for each generator x = a β ∈ A + M where a ∈ A, β ∈ M . We have In this case, ξ is called a moment map, cf. [8,5]. For n = 0, one considers multiplicative moment maps η ∈ A 0 such that for all a ∈ A, cf. [8,5]. Then the 1-element set M = {ξ} with ξ = η − 1 A satisfies the conditions of Theorem 10.2 for k ξ = 2.
Now, each term on the right-hand side of (10.5.1) contributes a summand to Q. The resulting eight summands of Q are equal to Here, in the computation of x 2 , x 3 , etc., we use the relation (1 A ) α = ε(α)1 A + M . Using (8.3.1) we obtain that Similarly, Analogous computations allow us to deduce further that Summing up, we obtain that Similarly, starting from the expression we obtain that These expansions for Q and R yield an expansion of {a α , b β , c γ } = Q − R. Now, using the fact that the Jacobi form is a derivation in the first and second variables (cf. (4.2.15)) we deduce that for any  [8] for M and v as in Example 6.3.4.
In the notation of Section 4.1 this gives Next, we define a left action of G on A M by graded algebra automorphisms. Each g ∈ G acts on the generators by g a α = (ga) gα for a ∈ A, α ∈ M . We check the compatibility with the defining relations of A M . The compatibility with the bilinearity relations is obvious. Consider the multiplicativity relation (ab) α = a α 1 b α 2 with homogeneous a, b ∈ A and α ∈ M . If g ∈ G 0 , then g(ab) α = (g(ab)) gα = (g(a)g(b)) gα = (ga) gα 1 (gb) gα 2 = g(a α 1 )g(b α 2 ) = g(a α 1 b α 2 ).
Summing up over all g ∈ G, we obtain that 11.5. The involutive case. We focus now on the case where G is a cyclic group of order two with generator ι and G 0 ⊂ G is the trivial subgroup. To turn a unital graded algebra A into a G-algebra, one needs only to fix an involutive graded antiautomorphism ι of A such that ι(1 in A G+ M . Suppose that A = K[π] is the group algebra of a group π. We treat A as a graded algebra concentrated in degree zero and equip it with the involutive antiautomorphism ι inverting all elements of π. Consider a counital G-coalgebra (M, ι : M → M ) whose underlying module is free of finite rank. For any (non-graded) unital algebra B, the convolution algebra H M (B) = B ⊗ M * is a unital G-algebra with involutive antiautomorphism id B ⊗ι * . We identify the set Hom G−GA + (A, H M (B)) with the set of group homomorphisms from π to the group U ι (B ⊗ M * ) consisting of all invertible u ∈ B ⊗ M * such that u −1 = (id B ⊗ι * )(u). The bijection (11.4.1) exhibits A G+ M as the coordinate algebra of the functor B → Hom(π, U ι (B ⊗ M * )) where B runs over unital commutative algebras.
For example, consider a symmetric Frobenius algebra A equipped with an involutive algebra automorphism ∆ preserving the Frobenius pairing. Pick an invertible matrix J ∈ Mat N (A) with N ≥ 1 such that J t = ∆(J) where t is the matrix transposition and ∆ applies to matrices entry-wise. (One can take J to be the unit matrix or, more generally, a diagonal matrix whose diagonal entries are invertible and fixed by ∆.) The antiautomorphism X → J∆(X t )J −1 of Mat N (A) induces an involutive coalgebra antiautomorphism ι of the coalgebra M = (Mat N (A)) * and makes M into a counital G-colagebra. The cyclic form v N on M defined in Example 6.3.5 is equivariant for any choice of J. This form can be used to derive an n-graded biderivation in A G+ M from an equivariant double bracket in A. Note that for a unital commutative algebra B, the group U ι (B ⊗ M * ) ⊂ GL N (B ⊗ A) consists of all invertible (N × N )-matrices X over the ring R B = B ⊗ A such that J = XJ∆(X t ). The group U ι (B ⊗ M * ) is nothing but the group of automorphisms of the free R B -module (R B ) N of rank N preserving the R B -valued sesquilinear form determined by J. For ∆ = id A , we recover the group of invertible (N × N )-matrices X over R B such that J = XJX t , i.e., the group of automorphisms of the module (R B ) N preserving the R B -valued bilinear form determined by J.
These constructions apply to the fundamental group π of an oriented surface with base point in the boundary. Indeed, the double quasi-Poisson bracket in A = K[π] constructed in [5] satisfies (11.5.1). This bracket and the cyclic form v N induce a biderivation in A G+ M which restricts to a Poisson bracket in E ⊂ A G+ M . As in Section 10.6, the equivariant Hamiltonian reduction yields a Poisson bracket in algebras associated with oriented surfaces with empty boundary.