Journal of Lie Theory Volume 13 (2003) 329–357 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

K-star Products on Dual of Lie Algebras

Nabiha Ben Amar

Communicated by B. Ørsted

Abstract. We consider a generalization (⋆_{α}) of the Kontsevich family of star products (⋆_{α}^{K}) for linear Poisson structures α. Such a family is characterized by a formal function F. We study some general properties of such families: invariance and covariance, closeness and relativity, symmetry and reality. Finally, we characterize the Kontsevich family (⋆_{α}^{K}) among all them.

1. Introduction

The study of star products i.e. associative deformations of usual multiplication of functions has been introduced by F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer [8] as a tool for the quantization of a symplectic or Poisson [15] manifold.

Recently, M. Kontsevich solved the problem of the existence of star products on any finite dimensional Poisson manifold [13]. He built a star product ⋆_{α}^{K} on ℝ^{d} equipped with any Poisson bracket α. This star product is defined by using oriented graphs \overrightarrow{Γ}.

In this paper, we restrict ourselves to the case of linear Poisson structures α, i.e. to the case of dual of Lie algebras.

Many authors studied the Kontsevich construction in this case and its properties ([13], [1], [4],[11], [17], [2], [3], [10]). In particular, in [4], we defined a generalization α ↦ ⋆_{α} of the Kontsevich family of star products α ↦ ⋆_{α}^{K}, as:

u ⋆_{α} v = ∑_{n=0}^{∞} ∑_{Γ∈G_{n,2}}  ∑_{orientation} a\overrightarrow{Γ}B\overrightarrow{Γ}(α)(u, v) = ∑_{n=0}^{∞} C_{n}(u, v)

(As a matter of fact, we imposed in [4] C_{n}(u, v) = (−1)^{n}C_{n}(v, u) and C_{n}(1, u) = 0, ∀n ≥ 1). Then ⋆_{α} is given by an integral formula:

(u ⋆_{α} v)(ξ) = ∫_{𝔤^{2}} û(X)v̂(Y)\frac{F (X)F (Y)}{F (X ×_{α} Y)}e^{2iπ⟨ξ,X×_{α}Y ⟩}dX dY

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag where X ×_{α} Y is the Baker-Campbell-Hausdorff formula, viewed as a formal mapping from 𝔤^{2} to 𝔤, F (X) = J(X)H(X) = J(X)e^{h(X)} and

H(X) = 1 + ∑_{n=1}^{∞} ∑_{{s_{1},... ,sp \atop |s|=2n}} a_{s_{1}...s_{p}}T r(ad 2iπX)^{s_{1}} … T r(ad 2iπX)^{s_{p}}

if J(X) = det(j(ad X)) and j(x) = [\frac{sh x/2}{x/2}]^{1/2}.

In the present paper, we call such a generalization of (⋆_{α}^{K}) a K-family and we study the properties of these K-families.

We first consider general properties for every K-star product ⋆_{α} in these K-families, proving that they always satisfy

C_{n}(1, u) = C_{n}(u, 1) = 0 (n ≥ 1),

they are graded and characterized by (X, u) ↦ X ⋆_{α} u and (u, X) ↦ u ⋆_{α} X. We know that they are equivalent to the Kontsevich family of star products (⋆_{α}^{K}), the intertwining operator being :

T = Id + ∑_{n=1}^{∞} ∑_{{s_{1}...sp \atop |s|=n}} a_{s_{1}...s_{p}} T_{s_{1}} ∘ … ∘ T_{s_{p}}

where

(T_{s}u)(ξ) = (2iπ)^{s} ∫_{𝔤} û(X)T r(adX)^{s} e^{2iπ⟨ξ,X⟩}dX.

We prove that T_{1} is a derivation for any ⋆_{α}, then they are all invariant and covariant.

We study then the symmetry and reality properties for (⋆_{α}), (⋆_{α}^{K}) is symmetric and real and a (⋆_{α}) is real if and only if \overline{H(X)} = H(−X).

Restricting ourselves to the case of analytic star products i.e. to formal function H converging near 0, we suppose there exists r > 0 such that:

1 + ∑_{n≥1} r^{n} ∑_{|s|=n} |a_{s_{1}...s_{p}}| < ∞.

In this case, we look for closed and relative K-family. The definition of closed star product was introduced by A. Connes, M. Flato and D. Sternheimer. These star products were also studied by G. Felder and B. Shoikhet when they are coming from cyclic formality [11].

If the Lie algebra 𝔤 defined by our linear Poisson structure α is not unimodular, then a K-star product is never closed on 𝔤^{∗}. Thus we shall say that a K-family (⋆_{α}) is closed if, for any unimodular 𝔤, ⋆_{α} is closed on 𝔤^{∗}. We prove that any closed (⋆_{α}) is relative in the sense of [2] and if (⋆_{α}) is strict and relative, it is closed.

Of course (⋆_{α}^{K}) is an analytic, strict and closed K-family.

Finally we prove that the only symmetric, strict and closed (⋆_{α}) is the Kontsevich family (⋆_{α}^{K}), moreover the weights w_{Γ} occurring in the computation of ⋆_{α}^{K} are all rational numbers.

2. Graded star products

Let M be a smooth manifold, a star product ⋆ is a (formal) deformation of the usual point-wise product on the space of C^{∞}(M) ([8]). Such a deformation is a formal series:

(u, v) ↦ u ⋆ v = uv + ∑_{r=1}^{∞} C_{r}(u, v) = C_{0}(u, v) + ∑_{r≥1} C_{r}(u, v)

which is associative in the following sense:

∑_{r+s=n} C_{r}(C_{s}(u, v), w) = ∑_{r+s=n} C_{r}(u, C_{s}(v, w)).

We suppose the C_{r} to be bi-differential operators. In fact it is well known [13] that if ⋆ is such a deformation, then the antisymmetric part of C_{1} is a Poisson bracket on M:

\frac{1}{2} (C_{1}(u, v) − C_{1}(v, u)) = \{u, v\}

i.e an antisymmetric bracket satisfying the Jacobi identity and being a derivation for each argument. Such a bracket defines uniquely an antisymmetric bi-vector α on M such that:

{u, v} = ⟨α, du ∧ dv⟩.

This bi-vector α is the Poisson tensor associated to {, }. If M = ℝ^{d}, M. Kontsevich in [13] gives an explicit construction of a star product for any Poisson tensor α. We first generalize a little bit this construction.

In order to associate to each graph Γ an m-differential operator C_{Γ}(α), Kontsevich considered some oriented admissible graphs \overrightarrow{Γ}. More precisely, let A = {p_{1}, …, p_{n}} and B = {q_{1}, …, q_{m}} be finite sets . Points of A (resp. B) are the vertices of type 1 (resp. 2) of the graph Γ. The edges are arrows \overrightarrow{px} starting from a vertex p of type 1 and ending at a vertex x in A ∪ B (in fact, Kontsevich restricts himself to the case where the edge ends at x distinct from p, but here, we shall allow this sort of edge). From each vertex p in A, there are exactly two arrows starting. Now we define an orientation of Γ by choosing a total ordering ≤ on A and on the set E(Γ) of edges of Γ which are compatible in the sense that:

p ≤ p′ ⟹ \overrightarrow{px} ≤ \overrightarrow{p′x′} ∀p, p′ ∈ A, ∀x, x′ ∈ A ∪ B.

We denote the ordered set of edges by:

E(Γ) = \{\overrightarrow{e_{1}} ≤ \overrightarrow{e_{2}} ≤ … ≤ \overrightarrow{e_{2n}}\} .

Let us denote by End(x) the set of arrows ending at the point x and by ∂_{End(x)} the differential operator:

∂_{End(x)} = ∂_{l_{1}...l_{s}} if End(x) = \{\overrightarrow{e_{l_{1}}} ≤ … ≤ \overrightarrow{e_{l_{s}}}\} .

Denote the edges starting from the vertex p_{i} by e_{ℓ_{1}^{i}} < e_{ℓ_{2}^{i}}. Thus, for any Poisson structure α, the m-differential operator C\overrightarrow{Γ}(α) is by definition:

C\overrightarrow{Γ}(α)(u_{1}, …, u_{m}) = ∑_{1≤ℓ_{1}...ℓ_{2n}≤d} ∏_{i=1}^{n} ∂_{End(p_{i})}α^{ℓ_{1}^{i}ℓ_{2}^{i}} ∏_{k=1}^{m} ∂_{End(q_{k})}u_{k}

Let us choose any total ordering p_{1} < p_{2} < … < p_{n} on A, for any graph Γ, we define the particular orientation lex(Γ) by putting lexicographic ordering on E(Γ):

\overrightarrow{p_{i}p_{j}} ≤ \overrightarrow{p_{i′}p_{j′}} iff i < i′ or i = i′ and j ≤ j′
\overrightarrow{p_{i}p_{j}} ≤ \overrightarrow{p_{i′}q_{k}} iff i ≤ i′
\overrightarrow{p_{i}q_{k}} ≤ \overrightarrow{p_{i′}q_{k′}} iff i < i′ or i = i′ and k ≤ k′

Finally, starting with lex(Γ), we symmetrize Γ by the action of the group S_{n} of all permutations of vertices p_{1}, …, p_{n}. Let Γ^{σ} the graph obtained from Γ by re-labeling p_{σ^{−1}(i)} the vertices p_{i}. Then, with the lexicographic ordering, σ induces a permutation σ̃ on the edges: we extend first σ to the vertices q_{k} by putting q_{σ^{−1}(k)} = q_{k} for all k, then each edge e_{ℓ} = \overrightarrow{p_{i}p_{j}} or e_{ℓ} = \overrightarrow{p_{i}q_{k}} of lex(Γ) becomes the edge e_{σ̃(ℓ)} = \overrightarrow{p_{σ^{−1}(i)}p_{σ^{−1}(j)}} or e_{σ̃(ℓ)} = \overrightarrow{p_{σ^{−1}(i)}q_{σ^{−1}(k)}} of lex(Γ^{σ}).

We put now:

C_{Γ}(α) = ∑_{σ∈S_{n}} \frac{1}{n!}ε(σ̃)C_{lex(Γ^{σ})}(α).

Let V_{n,m} be the space of admissible graphs. If γ = ∑ a_{i}Γ_{i} is a linear combination of graphs Γ_{i} in V_{n,m}, C_{γ} will be by definition

C_{γ} = ∑ a_{i}C_{Γ_{i}}.

Definition 2.1. (K-family on ℝ^{d}) Let (γ_{n})_{n≥1} be an element of ∏_{n≥1} V_{n,2}, the map α ↦ ⋆_{α} defined by:

u_{1} ⋆_{α} u_{2} = u_{1}u_{2} + ∑_{n=1}^{∞} C_{γ_{n}}(α)(u_{1}, u_{2})

is called a K-family if, for any linear Poisson structure α,

∑_{r+s=n} C_{γ_{r}}(α) (C_{γ_{s}}(α)(u_{1}, u_{2}), u_{3}) = ∑_{r+s=n} C_{γ_{r}}(α) (u_{1}, C_{γ_{s}}(α)(u_{2}, u_{3}))

and

C_{γ_{1}}(α)(u_{1}, u_{2}) − C_{γ_{1}}(α)(u_{2}, u_{1}) = 2⟨α, du_{1} ∧ du_{2}⟩.

An element ⋆_{α} of a K-family (⋆_{α}) is called a K-star product.

From now on, we are looking for ‘universal’ star products on dual of any Lie algebra, since a linear α;

α = ∑_{i,j,k} C_{ij}^{k}x_{k}∂_{i} ∧ ∂_{j}

is a Poisson tensor if and only if the C_{ij}^{k} are structure constants of a Lie algebra 𝔤.

We consider thus only graphs Γ with # End(p_{i}) ≤ 1 for all i. Let W_{n,m} be the space of linear combination of such graphs.

For instance dim(W_{1,1}) = 1, the only graph being Γ_{1}:

E(Γ_{1}) = \{\overrightarrow{p_{1}p_{1}}, \overrightarrow{p_{1}q_{1}}\}.

[figure: q_{1}; p_{1}]

C_{Γ_{1}}(α) = C_{lex(Γ)}(α) = ∑_{i,j} C_{ij}^{i}∂_{j}.

Similarly, dimW_{2,1} = 2 the only graphs being:

Γ_{1,1} : E(Γ_{1,1}) = \{\overrightarrow{p_{1}p_{1}}, \overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{2}p_{2}}, \overrightarrow{p_{2}q_{1}}\}
Γ_{2} : E(Γ_{2}) = \{\overrightarrow{p_{1}p_{2}}, \overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{2}p_{1}}, \overrightarrow{p_{2}q_{1}}\}

[figure: q_{1}; p_{1}; p_{2}; q_{1}; p_{1}; p_{2}]

More generally a basis of W_{n,1} is given by the Γ_{P}, where P is a partition of {1, …, n} in a family of disjoint finite sequences:

{1, …, n} = A_{1} ∪ A_{2} ∪ … ∪ A_{ℓ}, A_{i} = (j_{1}^{i}, …, j_{s_{i}}^{i}), s_{i} > 1,

and

A_{i} ∩ A_{j} = Ø if i ≠ j.

For the C_{Γ}, since, up to a sign, C_{Γ} and C_{Γ^{σ}} coincide, we shall only consider the graphs Γ_{s_{1},... ,s_{ℓ}} (1 ≤ s_{1} ≤ s_{2} ≤ … ≤ s_{ℓ}, ∑ s_{j} = n), associated to the subsets

A_{j} = {∑_{i<j} s_{i} + 1, ∑_{i<j} s_{i} + 2, …, ∑_{i<j} s_{i} + s_{j}} (1 ≤ j ≤ ℓ).

For instance the graph Γ_{1,2,2} is defined by:

E(Γ_{1,2,2}) = \{\overrightarrow{p_{1}p_{1}}, \overrightarrow{p_{2}p_{3}}, \overrightarrow{p_{3}p_{2}}, \overrightarrow{p_{4}p_{5}}, \overrightarrow{p_{5}p_{4}}\} ∪ \{\overrightarrow{p_{i}q_{1}}, i = 1, …, 5\}.

[figure: p_{1}; p_{2}; p_{3}; p_{4}; p_{5}; q_{1}]

Let us put T_{s} = (−1)^{s}C_{Γ_{s}}(α), then:

(−1)^{∑ s_{i}}T_{s_{1}} ∘ … ∘ T_{s_{p}} = C_{Γ_{s_{1},... ,sp}}(α).

Now, in [8], the authors consider only star products vanishing on constants.

Definition 2.2. (Star product vanishing on constants) A star product on a Poisson manifold M is said to be vanishing on constants if, for all u in C^{∞}(M) and for all r > 0,

C_{r}(1, u) = C_{r}(u, 1) = 0.

Lemma 2.3. (K-family vanishes on constants)

If (⋆_{α}) is a K-family , then ⋆_{α} vanishes on the constants.

Proof. First, looking at the total number of edges in admissible graphs Γ in W_{n,2}, we see that for each n > 0, there is at most n edges ending on some vertex p_{i}. Since they are 2n > n edges, some of them end on a vertex q_{ℓ}, thus:

1 ⋆_{α} 1 = 1.

Let us set now

1 ⋆_{α} u = L(α)u = u + ∑_{n=1}^{∞} L_{n}(α)(u).

By definition, each L_{n}(α) is a differential operator defined by graphs without any edge ending on q_{1}, thus the preceding discussion gives:

L_{n}(α) = ∑_{{1≤s_{1}≤...≤sp \atop |s|=n}} a_{s_{1}...s_{p}} T_{s_{1}} ∘ … ∘ T_{s_{p}}

where the numbers a_{s_{1}...s_{p}} do not depend on α. Now, by formal associativity:

1 ⋆_{α} (1 ⋆_{α} u) = 1 ⋆_{α} (L(α)u) = L(α)^{2}u = (1 ⋆_{α} 1) ⋆_{α} u = 1 ⋆_{α} u = L(α)u.

Thus we obtain L(α)^{2}u = L(α)u. This implies L(α) = id.

Indeed, suppose L(α) ≠ id and let n be the smallest index for which L_{n}(α) ≠ 0, write L_{n}(α) = ∑_{{1≤s_{1}≤...≤sp \atop |s|=n}} a_{s_{1}...s_{p}}T_{s_{1}} ∘ … ∘ T_{s_{p}}, let p_{0} be the largest p for which one of the a_{s_{1}...s_{p}} does not vanish, fix the lexicographic ordering on the set of p_{0}-tuples (s_{1}, …, s_{p_{0}}) and let (σ_{1}, …, σ_{p_{0}}) be the largest p_{0}-tuple for which a_{s_{1}...s_{p}} does not vanish. Choose finally d = 2p_{0} and the following α_{0} on ℝ^{d}:

α_{0} = ∑_{k=0}^{p_{0}−1} x_{2k+1}∂_{2k+1} ∧ ∂_{2k+2}.

The only (∂_{2})^{σ_{1}} … (∂_{2p_{0}})^{σ_{p_{0}}} term in L_{n}(α_{0}) is:

a_{σ_{1}...σ_{p_{0}}}(∂_{2})^{σ_{1}} … (∂_{2p_{0}})^{σ_{p_{0}}}.

Now, in L(α_{0})^{2} = (id + L_{n}(α_{0}) + …)^{2} the term of order n is 2L_{n}(α_{0}), thus for that α_{0} we get an unique term:

2a_{σ_{1}...σ_{p_{0}}}(∂_{2})^{σ_{1}} … (∂_{2p_{0}})^{σ_{p_{0}}}.

This is impossible, L(α) = id for any α. The same proof holds for u ⋆_{α} 1.

Corollary 2.4. (The C_{1} term of a K-family)

Let (⋆_{α}) be a K-family, then:

C_{1}(α)(u_{1}, u_{2}) = ⟨α, du_{1} ∧ du_{2}⟩ = {u_{1}, u_{2}}.

Proof. The only graphs Γ in W_{1,2} are:

the Λ- graph such that E(Λ) = \{\overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{1}q_{2}}\}

[figure: q_{1}; q_{2}; p_{1}]

and graphs which does not vanish on constants:

Γ_{1} such that E(Γ_{1}) = \{\overrightarrow{p_{1}p_{1}}, \overrightarrow{p_{1}q_{1}}\}; Γ_{2} such that E(Γ_{2}) = \{\overrightarrow{p_{1}p_{1}}, \overrightarrow{p_{1}q_{2}}\}.

[figure: q_{1}; q_{2}; p_{1}; q_{1}; q_{2}; p_{1}]

Since Γ_{1} and Γ_{2} do not occur,

C_{1}(α)(u_{1}, u_{2}) = a_{Λ}C_{Λ}(α)(u_{1}, u_{2}) = a_{Λ}{u_{1}, u_{2}}

and necessarily a_{Λ} = 1.

Let S(𝔤) be the algebra of all polynomial functions on the dual 𝔤^{∗} of a Lie algebra 𝔤. The algebra S(𝔤) is naturally graded. If u is an homogeneous element of S(𝔤), we will note by |u| its degree.

A multi-linear function C:

C : S(𝔤) × S(𝔤) … × S(𝔤) ⟶ S(𝔤)

is said to be homogeneous with degree −n if for u_{1}, …, u_{k} homogeneous elements of S(𝔤), C(u_{1}, …, u_{k}) is homogeneous with degree |u_{1}| + … + |u_{k}| − n.

Definition 2.5. (Graded star products) Let S(𝔤) be the algebra of polynomial functions on 𝔤^{∗} and S^{p} be the space of homogeneous polynomials with degree p. A star product on S(𝔤) is graded if:

∀r, p, q ∈ ℕ, ∀(u, v) ∈ S^{p} × S^{q}, C_{r}(u, v) ∈ S^{p+q−r}.

Let us show that a graded star product is totally defined by X ⋆u and u⋆X where u belongs to S(𝔤) and X belongs to 𝔤.

Lemma 2.6. (Construction of graded star products)

Let C_{n} and C′_{n} be two sequences of bilinear maps:

C_{n} : 𝔤 × S(𝔤) ⟶ S(𝔤) and C′_{n} : S(𝔤) × 𝔤 ⟶ S(𝔤)

such that: 1- C_{0}(X, u) = X.u, C′_{0}(u, X) = u.X, C_{1}(X, u) = {X, u}, C′_{1}(u, X) = {u, X}. 2- If u is homogeneous then C_{n}(X, u) and C′_{n}(u, X) are homogeneous and their degree is |u| + 1 − n. 3- We set:

X ∘ u = ∑_{n=0}^{∞} C_{n}(X, u) and u ∘ X = ∑_{n=0}^{∞} C′_{n}(u, X)

and we suppose that:

a) C_{2}(X, Y) = C′_{2}(X, Y), (X, Y ∈ 𝔤).

b) X ∘ (u ∘ Y) = (X ∘ u) ∘ Y , ∀X, Y ∈ 𝔤, ∀u ∈ S(𝔤).

c) X ∘ (Y ∘ u) − Y ∘ (X ∘ u) = [X, Y] ∘ u , (X, Y ∈ 𝔤, u ∈ S(𝔤)).

d) (u ∘ X) ∘ Y − (u ∘ Y) ∘ X = u ∘ [X, Y], (X,Y ∈ 𝔤, u ∈ S(𝔤)). Then there exists one and only one star product ⋆ such that:

X ⋆ u = X ∘ u and u ⋆ X = u ∘ X, ∀X ∈ 𝔤, ∀u ∈ S(𝔤).

This star product is graded.

Proof. Let v be in S(𝔤), for any u in S(𝔤), we define u ⋆ v by induction on the degree of u starting with:

1 ⋆ v = v, X ⋆ v = X ∘ v, v ⋆ 1 = v and v ⋆ X = v ∘ X.

If u is an homogeneous polynomial function of the form u = Xu′ then there exists a polynomial function u′′ such that:

Xu′ = X ∘ u′ + u′′ and |u′′| ≤ |u| − 1.

We suppose now u_{1} ⋆ v defined for any u_{1} such that |u_{1}| < |u|, we suppose also that:

u_{1} ⋆ (v ⋆ w) = (u_{1} ⋆ v) ⋆ w if |u_{1}| + |v| < |u|.

Then we set:

(Xu′) ⋆ v = X ∘ (u′ ⋆ v) + u′′ ⋆ v.

This formula defines without ambiguity u ⋆ v. In fact if u has the form:

u = X_{1}X_{2}w = X_{1} ∘ (X_{2} ∘ w) + u′′_{1}
= X_{2} ∘ (X_{1} ∘ w) + u′′_{2}

Then:

X_{2} ∘ (X_{1} ∘ w) + u′′_{2} = X_{1} ∘ (X_{2} ∘ w) + u′′_{2} + [X_{2}, X_{1}] ∘ w.

Thus:

u′′_{1} = u′′_{2} + [X_{2}, X_{1}] ∘ w

and:

X_{1} ∘ ((X_{2} ∘ w) ⋆ v) + u′′_{1} ⋆ v = X_{1} ∘ (X_{2} ∘ (w ⋆ v)) + u′′_{2} ⋆ v + ([X_{2}, X_{1}] ∘ w) ⋆ v
= X_{1} ∘ (X_{2} ∘ (w ⋆ v)) + [X_{2}, X_{1}] ∘ (w ⋆ v) + u′′_{2} ⋆ v
= X_{2} ∘ (X_{1} ∘ (w ⋆ v)) + u′′_{2} ⋆ v
= X_{2} ∘ ((X_{1} ∘ w) ⋆ v) + u′′_{2} ⋆ v

The homogeneous term of maximum degree in u ⋆ v is C_{0}(u, v) = u.v then we set:

C_{n}(u, v) = the homogeneous term of degree |u| + |v| − n.

A simple computation shows that ⋆ is a star product. In fact, let us first show that:

C_{1}(u, v) = {u, v}.

It is clear that the term of degree |u| + |v| − 1 in u′′ ⋆ v is {−X, u′}v (coming from u′′v) and, in X ∘ (u′ ⋆ v), it is X{u′, v} + {X, u′.v} thus the term of degree |u| + |v| − 1 in u ⋆ v is the sum of these terms:

X{u′, v} + {X, u′v} − {X, u′}v = X.{u′, v} + u′{X, v}
= {Xu′, v} = {u, v}.

Using the same construction, we can then define similarly u⋆′v by induction ‘on the right side’ on the degree of v. In fact, thanks to a), ⋆ and ⋆′ coincide if |u| = |v| ≤ 1. Now suppose that they coincide for u′ and v′ such that |u′| + |v′| < |u| + |v|, then:

(X ∘ u′) ⋆ (v′ ∘ Y) = X ∘ (u′ ⋆ (v′ ∘ Y))
= X ∘ (u′ ⋆′ (v′ ∘ Y)) = X ∘ ((u′ ⋆′ v′) ∘ Y)
= (X ∘ (u′ ⋆′ v′)) ∘ Y = (X ∘ u′) ⋆′ v′) ∘ Y
= (X ∘ u′) ⋆′ (v′ ∘ Y)

By induction on the degree of u and w, we can moreover show that ⋆ is associative.

By definition u ⋆ (v ⋆ w) = (u ⋆ v) ⋆ w if |u| ≤ 1 and |w| ≤ 1. Then, by induction on |u|, the same holds if |w| ≤ 1 since:

(X ∘ u′ ⋆ v) ⋆ w = (X ∘ (u′ ⋆ v)) ⋆ w = X ∘ ((u′ ⋆ v) ⋆ w) = X ∘ (u′ ⋆ (v ⋆ w))
= (X ∘ u′) ⋆ (v ⋆ w)

and similarly for any w since:

u ⋆ (v ⋆ (w′ ∘ Y)) = u ⋆ ((v ⋆ w′) ∘ Y)
= ((u ⋆ v) ⋆ w′) ∘ Y = (u ⋆ v) ⋆ (w′ ∘ Y).

Finally ⋆ is a graded star product by construction.

Corollary 2.7. (Characterization of graded star product) (see also [14]) If ⋆ is a graded star product, then ⋆ is entirely determined by the mappings:

C : 𝔤 × S(𝔤) ⟶ S(𝔤)
(X, u) ↦ X ⋆ u

and

C′ : S(𝔤) × 𝔤 ⟶ S(𝔤)
(u, X) ↦ u ⋆ X.

In particular all K-star products are graded star products, each of them being totally determined by the giving of X ⋆ u and u ⋆ X.

3. Universal derivation

In [4] we showed that every K-star product is given by an integral formula of the form:

(u ⋆_{α} v)(ξ) = ∫_{𝔤^{2}} û(X)v̂(Y)\frac{F (X)F (Y)}{F (X ×_{α} Y)} e^{2iπ⟨ξ,X×_{α}Y ⟩} dXdY

for all u, v in S(𝔤).

These star products are all equivalent to the fundamental star product ⋆_{α}^{K} built by Kontsevich.

From results of Kontsevich and Shoikhet, we can deduce that ⋆_{α}^{K} can be written as:

(u ⋆_{α}^{K} v)(ξ) = ∫_{𝔤^{2}} û(X)v̂(Y)\frac{J(X)J(Y)}{J(X ×_{α} Y)} e^{2iπ⟨ξ,X×_{α}Y ⟩} dXdY

for all u, v in S(𝔤), (or u, v smooth functions such that û and v̂ are compactly supported with a sufficiently small support) where:

J(X) = det (\frac{sh ad\frac{X}{2}}{ad \frac{X}{2}} )^{\frac{1}{2}}

(see [3] for instance). An equivalence operator T between ⋆_{α}^{K} and ⋆_{α}:

u ⋆_{α} v = T^{−1}(T u ⋆_{α}^{K} T v)

is given by:

(T u)(ξ) = ∫_{𝔤} û(X)H(X) e^{2iπ⟨ξ,X⟩} dX

if:

F (X) = J(X)H(X).

The operator T is a formal series of differential operators, each of them being a linear combination of products of operators T_{s} as defined in the first section. Indeed for each positive integer s,

(T_{s}u)(ξ) = (2iπ)^{s} ∫_{𝔤} û(X)T r(ad X)^{s} e^{2iπ⟨ξ,X⟩} dX.

Thus T_{s} has the following expression:

(T_{s}u) = ∑_{i_{1}...i_{s}} ∑_{j_{1}...j_{s}} C_{i_{1}j_{1}}^{j_{2}}C_{i_{2}j_{2}}^{j_{3}} … C_{i_{s−1}j_{s−1}}^{j_{s}}C_{i_{s}j_{s}}^{j_{1}}∂_{i_{1}...i_{s}}u,

with that expression, we see that T_{s} is the operator associated to the graph Γ_{s}, called a “wheel” by Kontsevich ([13]). Now we can write:

T = Id + ∑_{n=1}^{∞} ∑_{|s|=n} a_{s_{1}...s_{p}}T_{s_{1}} ∘ … ∘ T_{s_{p}}

if

H(X) = 1 + ∑_{n=1}^{∞} ∑_{|s|=n} a_{s_{1}...s_{p}}T r(2iπ ad X)^{s_{1}} … T r(2iπ ad X)^{s_{p}}.

Proposition 3.1. (Annulation of a C_{Γ})

Let P be the operator C_{Γ}(α) associated to the graph Γ in W_{2,2} with edges:

E(Γ) = \{\overrightarrow{p_{1}p_{1}}, \overrightarrow{p_{1}p_{2}}, \overrightarrow{p_{2}q_{1}}, \overrightarrow{p_{2}q_{2}}\}.

Then P is null.

[figure: q_{1}; q_{2}; p_{2}; p_{1}; P = C_{Γ}(α) = 0]

Proof. The operator P associated to Γ can be written:

P (u, v) = − ∑_{k,ℓ} ∑_{i,j} C_{kℓ}^{i}C_{ij}^{j}∂_{k}u∂_{ℓ}v.

Let (E_{i})_{1≤i≤d} be the basis of 𝔤 dual to the canonical basis of ℝ^{d}, then one has:

ad [E_{k}, E_{ℓ}] = ∑_{i} C_{kℓ}^{i} ad E_{i}

Thus:

T r(ad [E_{k}, E_{ℓ}]) = ∑_{i} C_{kℓ}^{i}T r(ad E_{i}) = ∑_{i,j} C_{kℓ}^{i}C_{ij}^{j}.

But:

T r(ad [E_{k}, E_{ℓ}]) = T r(ad E_{k} ∘ ad E_{ℓ} − ad E_{ℓ} ∘ ad E_{k}) = 0.

Finally, for all u and v:

P (u, v) = − ∑_{k,ℓ} T r(ad [E_{k}, E_{ℓ}]) ∂_{k}u∂_{ℓ}v = 0

and P = 0.

Theorem 3.2. (An universal derivation) The ‘wheel’ operator T_{1} associated to the wheel graph Γ_{1} having one vertex p_{0} and one vertex q_{1} and the edges of Γ_{1} are \{\overrightarrow{p_{0}p_{0}}, \overrightarrow{p_{0}q_{1}}\} is an ‘universal’ derivation for K-star products. We have:

T_{1}(u ⋆_{α} v) = T_{1}u ⋆_{α} v + u ⋆_{α} T_{1}v,

for all u, v and for all linear α, all K-star products ⋆_{α}.

[figure: q_{1}; p_{0}]

Proof. By definition, T_{1} is a vector field. Now, for any admissible graph Γ, we say that a vertex p_{i} of Γ is ‘free’ if End(p_{i}) = Ø. Then, for the corresponding Kontsevich operator C_{Γ}(α), if L_{T_{1}} is the Lie derivative, one has:

L_{T_{1}}C_{Γ}(α)(u_{1}, …, u_{m}) = T_{1}(C_{Γ}(α)(u_{1}, …, u_{m})) − C_{Γ}(α)(T_{1}u_{1}, …, u_{m}) + …
… − C_{Γ}(α)(u_{1}, …, T_{1}u_{m})
= C_{γ}(α)(u_{1}, …, u_{m})

where γ is the sum of all the graphs ∑_{free p_{i}} Γ_{1} ∪_{p_{i}} Γ. The vertices of Γ_{1} ∪_{p_{i}} Γ are

\{p_{0}, p_{1}, …, p_{n}, q_{1}, …, q_{m}\} and its edges are \{\overrightarrow{p_{0}p_{0}}, \overrightarrow{p_{0}p_{i}}\}∪E(Γ). For each Γ_{1}∪_{p_{i}} Γ, there is a subgraph P:

E(P) = \{\overrightarrow{p_{0}p_{0}}, \overrightarrow{p_{0}p_{i}}, \overrightarrow{p_{i}x_{1}}, \overrightarrow{p_{i}x_{2}}\}

if the edges starting from p_{i} are \overrightarrow{p_{i}x_{1}} and \overrightarrow{p_{i}x_{2}}.

Then:

C_{Γ_{1}∪_{p_{i}}Γ}(α)(u_{1}, …, u_{m}) = − ∑_{ℓ_{1}...ℓ_{2n}} (∑_{j,k} C_{kj}^{j}C_{ℓ_{2i+1}ℓ_{2i+2}}^{k}) ∏_{s≠i} ∂_{End(p_{s})}

(C_{ℓ_{2s+1}ℓ_{2s+2}}^{r}x_{r}) ∏_{k=1}^{m} ∂_{End (q_{k})}u_{k}

= 0,

since ∑_{j,k} C_{kj}^{j}C_{ℓ_{2i+1}ℓ_{2i+2}}^{k} = 0.

We will frequently use the following corollary.

Corollary 3.3. (Normalized equivalence) Each K-star product is equivalent to the Kontsevich star product ⋆_{α}^{K} through an operator T associated to a formal function F (or H) such that:

F (X) = 1 + ∑_{n≥2} ∑_{|s|=n} a_{s_{1}...s_{ℓ}}T r(adX)^{s_{1}} … T r(adX)^{s_{ℓ}}.

Proof. Indeed, T_{1} being a derivation of our star products, e^{−a_{1}T_{1}} is an automorphism and:

u ⋆_{α} v = T^{−1}(T u ⋆_{α}^{K} T v) = (T ∘ e^{−a_{1}T_{1}})^{−1}(T ∘ e^{−a_{1}T_{1}}u ⋆_{α}^{K} T ∘ e^{−a_{1}T_{1}}v)

and T ∘ e^{−a_{1}T_{1}} is a formal series without any order 1 term.

4. Invariance and Covariance

We recall that all K-star products are equivalent to the Gutt star product ⋆_{α}^{G} [12] associated to the complete symmetrization mapping between the space of polynomial functions S(𝔤) and the universal enveloping algebra U(𝔤) of 𝔤.

(u ⋆_{α}^{G} v)(ξ) = ∫_{𝔤^{2}} û(X)v̂(Y)e^{2iπ⟨ξ,X×_{α}Y ⟩} dXdY.

This equivalence being through an intertwining operator T of the form:

(T u)(ξ) = ∫_{𝔤} û(X)F (X)e^{2iπ⟨ξ,X⟩} dX.

This operator is also composed of ‘wheel’ operators as in the previous section.

Gutt proved in [12] that ⋆_{α}^{G} is invariant under the coadjoint action. The same is true for the Kontsevich star product ⋆_{α}^{K} (see [13]). Let us first recall the definition of the invariance under the coadjoint action Ad^{∗}.

Definition 4.1. (Invariant star product) A star product ⋆ is said to be invariant under the coadjoint action if, for all X in 𝔤:

X^{−}(u ⋆ v) = (X^{−}u) ⋆ v + u ⋆ (X^{−}v)

where:

(X^{−}u)(ξ) = \frac{d}{dt}|_{t=0}u(Ad^{∗}(exp tX)ξ).

Now using the equivalence operator T, we show that all K-star products are invariant under the coadjoint action.

Theorem 4.2. (Invariance of K-star products) All the K-star products are invariant under the coadjoint action.

Proof. S. Gutt showed in [12] that:

\frac{1}{2}(X ⋆_{α}^{G} u − u ⋆_{α}^{G} X) = \{X, u\} = X^{−}u

for each element X of 𝔤 and every polynomial function u. This implies invariance for ⋆_{α}^{G}. Now since T X = X + T_{1}(X) and T_{1}(X) is a constant, we obtain:

\{X, T u\} = \{T X, T u\} = \frac{1}{2}(T X ⋆_{α}^{G} T u − T u ⋆_{α}^{G} T X)

= \frac{1}{2}T (X ⋆_{α} u − u ⋆_{α} X).

Thus to show the invariance of ⋆_{α} it is sufficient to prove X^{−}T u = T X^{−}u or, for all ‘wheel’ operators T_{n},

X^{−}(T_{n}u) = T_{n}(X^{−}u).

But a direct computation gives:

X^{−}u(ξ) = ∫_{𝔤}((L_{X} û)(Z) − T r(ad X)û(Z))e^{2iπ⟨ξ,Z⟩} dZ

where

(L_{X}φ)(Y) = \frac{d}{dt}|_{t=0}φ(Ad(e^{−tX})(Y)).

Then:

(T_{n}(X^{−}u))(ξ) = ∫_{𝔤} (L_{X} û(Z)T r(ad Z)^{n} − T r(ad X)T r(ad Z)^{n}û(Z))e^{2iπ⟨ξ,Z⟩} dZ.

Now let us compute X^{−}(T_{n}u):

X^{−}(T_{n}u)(ξ) = ∫_{𝔤} (L_{X}(û(Z)T r(ad Z)^{n}) − T r(ad X)T r(ad Z)^{n}û(Z)) e^{2iπ⟨ξ,Z⟩} dZ.

But we have:

(L_{X} [ûT r(ad.)^{n}]) (Z) = (L_{X} û)(Z)T r(ad Z)^{n} + û(Z)(L_{X}T r(ad.)^{n})(Z)

and

\frac{d}{dt}|_{t=0} T r (ad(Ad e^{−tX}(Z))^{n}) = nT r((ad Z)^{n−1}ad([Z, X]))
= nT r(ad Z^{n}ad X − ad Z^{n−1}ad Xad Z)
= 0.

Then

∫_{𝔤} û(Z) (L_{X}T r(ad.)^{n}) (Z) dZ = 0.

And finally:

X^{−}(T_{n}u)(ξ) = ∫_{𝔤} ((L_{X} û)(Z)T r(ad Z)^{n} − T r(ad X)T r(ad Z)^{n}û(Z)) e^{2iπ⟨ξ,Z⟩}dZ
= T_{n}(X^{−}u)(ξ).

Now let us recall the definition of a covariant star product.

Definition 4.3. (Covariant star product) A star product on S(𝔤) is called covariant if, for all X and Y in 𝔤,

\frac{1}{2}(X ⋆ Y − Y ⋆ X) = \{X, Y \} = [X, Y].

In the following theorem we shall show that each K-star product is covariant by considering all the possible graphs.

Theorem 4.4. (Covariance of K-star products) Each K-star product ⋆_{α} is covariant and so is strongly invariant in the sense of [6].

Proof. Since a K-star product ⋆_{α} is graded, in order to show its covariance, it is sufficient to prove that for all α,

C_{2}(α)(X, Y) = C_{2}(α)(Y, X) for all X and Y in 𝔤.

But the coefficients C_{2}(α)(u, v) for any u, v in S(𝔤) are given by graphs having two vertices of first kind p_{1}, p_{2} and two vertices of second kind q_{1}, q_{2}. We remark that there are, up to the ordering of the vertices p_{i}, only eight graphs of the above type non vanishing on constants. These graphs are respectively:

[figure: q_{1}; q_{2}; p_{1}; p_{2}; q_{1}; q_{2}; p_{1}; p_{2}]

E(Γ_{1}) = \{\overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{1}q_{2}}, \overrightarrow{p_{2}p_{1}}, \overrightarrow{p_{2}q_{2}}\}, E(Γ_{2}) = \{\overrightarrow{p_{1}p_{2}}, \overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{2}q_{1}}, \overrightarrow{p_{2}q_{2}}\},

[figure: q_{1}; q_{2}; p_{1}; p_{2}; q_{1}; q_{2}; p_{1}; p_{2}]
E(Γ_{3}) = \{\overrightarrow{p_{1}p_{1}}, \overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{2}q_{1}}, \overrightarrow{p_{2}q_{2}}\}, E(Γ_{4}) = \{\overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{1}q_{2}}, \overrightarrow{p_{2}p_{2}}, \overrightarrow{p_{2}q_{2}}\},

[figure: q_{1}; q_{2}; p_{1}; p_{2}; p_{1}; p_{2}; q_{1}; q_{2}]

E(Γ_{5}) = \{\overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{1}q_{2}}, \overrightarrow{p_{2}q_{1}}, \overrightarrow{p_{2}q_{2}}\}, E(Γ_{6}) = \{\overrightarrow{p_{1}p_{2}}, \overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{2}p_{1}}, \overrightarrow{p_{2}q_{2}}\},

[figure: q_{1}; q_{2}; p_{1}; p_{2}; q_{1}; q_{2}; p_{2}; p_{1}]

E(Γ_{7}) = \{\overrightarrow{p_{1}p_{1}}, \overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{2}p_{2}}, \overrightarrow{p_{2}q_{2}}\}, E(Γ_{8}) = \{\overrightarrow{p_{1}p_{1}}, \overrightarrow{p_{1}p_{2}}, \overrightarrow{p_{2}q_{1}}, \overrightarrow{p_{2}q_{2}}\}.

Among these graphs, only the last three are possibly non vanishing in the computation of C_{2}(α)(X, Y) and C_{2}(α)(Y, X) since X and Y are linear functions. The last graph corresponds to the operator P of proposition 2-1 which is null. The two other graphs give symmetric C_{Γ}(α) and thus C_{2}(α)(X, Y) = C_{2}(α)(Y, X).

5. Symmetry and reality

The star products defined by F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer [8] had symmetry properties. Thus we put:

Definition 5.1. (Symmetric star products) A star product on a Poisson manifold M is said to be symmetric if C_{r} is symmetric (resp skew symmetric) if r is even (resp odd) that is:

C_{r}(u, v) = (−1)^{r}C_{r}(v, u) for all u, v ∈ C^{∞}(M).

For instance, the Gutt star product ⋆_{α}^{G} is symmetric for any α and the star product built by Kontsevich ⋆_{α}^{K} is also symmetric for any α. In fact this is a consequence of the action of the symmetry z ↦ −\overline{z} on the graphs and the orientation of the configuration space. But in the linear case, it is also a consequence of the parity of the function J since we have:

Proposition 5.2. (Symmetric K-star products) Let (⋆_{α}) be a K-family given by a function F then each ⋆_{α} is symmetric if and only if:

F (X) = e^{2iπa_{1}T r(ad X)}G(X)

where G is an even function, or:

F (X) = e^{2iπa_{1}T r(ad X)} (1 + ∑_{n=1}^{∞} ∑_{{s_{1}...sp \atop |s|=2n}} b_{s_{1}...s_{p}}T r(2iπ ad X)^{s_{1}} … T r(2iπ ad X)^{s_{p}} ) .

Moreover, ⋆_{α} is thus associated to the even function G itself.

Proof. First the Gutt star product ⋆_{α}^{G} is symmetric, since, for the Campbell-Hausdorff formula,

(−X) ×_{α} (−Y) = −(Y ×_{α} X).

Thus, if CH_{r}(X, Y) is the degree r term in this formula:

X ×_{α} Y = ∑_{r} CH_{r}(X, Y) with CH_{r}(Y, X) = (−1)^{r+1}CH_{r}(X, Y).

The symmetry of ⋆_{α}^{G} is a consequence of this ([7]). Now let (⋆_{α}) be a K-family characterized by:

F (X) = 1 + ∑_{n=1}^{∞}(2iπ)^{n} ∑_{{s_{1}...sp \atop |s|=n}} a_{s_{1}...s_{p}}T r(ad X)^{s_{1}} … T r(ad X)^{s_{p}}

= e^{2iπ a_{1}T r(ad X)} (1 + ∑_{n=2}^{∞}(2iπ)^{n} ∑_{{s_{1}...sp \atop |s|=n}} b_{s_{1}...s_{p}}T r(ad X)^{s_{1}} … T r(ad X)^{s_{p}} )

= e^{2iπa_{1}T r(ad X)}G(X).

Let us set:

u ⋆_{α} v = ∑_{r=0}^{∞} C_{r}(u, v) and u ⋆_{α}^{G} v = ∑_{r=0}^{∞} C_{r}^{G}(u, v).

⋆_{α} is equivalent to ⋆_{α}^{G} through the equivalence operator:

T = e^{a_{1}T_{1}} ∘ (Id + ∑_{n=2}^{∞} ∑_{|s|=n} b_{s_{1}...s_{p}}T_{s_{1}} ∘ … ∘ T_{s_{p}} )
= e^{a_{1}T_{1}} ∘ T′.

Since T_{1} is a derivation for any ⋆_{α}, e^{a_{1}T_{1}} is an automorphism and ⋆_{α} is equivalent to ⋆_{α}^{G} through the equivalence operator T′.

Let us put:

T′ = Id + ∑_{m≥2} T′_{m}.

Now if G is not even and ⋆_{α} is symmetric for each α , let n_{0} be the first odd n such that there exists b_{s_{1}...s_{p}} ≠ 0 with |s| = n_{0}. We select the largest element (s_{1}, s_{2}, …, s_{p_{0}}) for the lexicographic ordering in the set:

{(s_{1}, …, s_{p}), such that |s| = n_{0} and b_{s_{1}...s_{p}} ≠ 0} .

Then we choose 𝔤 as in the proof of lemma 2-3, then T′_{n_{0}} contains an unique term

b_{s_{1}...s_{p_{0}}}(∂_{2})^{s_{1}} … (∂_{2p_{0}})^{s_{p_{0}}}.
We compute now the n_{0} order term in T′(u ⋆_{α} v) = (T′u) ⋆_{α}^{G} (T′v):

∑_{r+s=n_{0}} T′_{r}C_{s}(α)(u, v) = ∑_{m_{1}+m_{2}+m_{3}=n_{0}} C_{m_{1}}^{G}(α)(T′_{m_{2}}u, T′_{m_{3}}v)

or

T′_{n_{0}}(uv)+ ∑_{{r+s=n_{0} \atop r<n_{0}}} T′_{r}C_{s}(α)(u, v) = (T′_{n_{0}}u)v+u(T′_{n_{0}}v)+ ∑_{{m_{1}+m_{2}+m_{3}=n_{0} \atop m_{2}<n_{0},m_{3}<n_{0}}} C_{m_{1}}^{G}(α)(T′_{m_{2}}u, T′_{m_{3}}v).

In these expression s and m_{1} are necessarily odd, thus we can separate the symmetric and the skew symmetric part and get:

T′_{n_{0}}(uv) − (T′_{n_{0}}u)v − u(T′_{n_{0}}v) ≡ 0

and this is impossible since T′_{n_{0}} contains an unique term b_{s_{1}...s_{p_{0}}}(∂_{2})^{s_{1}} … (∂_{2p_{0}})^{s_{p_{0}}} of order n_{0} which is not a vector field. Then G is necessarily an even formal function.

Conversely if G is an even formal function, we have:

F (X) = e^{aT r(ad X)}G(X).

Thus a = (2iπ)a_{1} and

G(X) = 1 + ∑_{n=1}^{∞}(2iπ)^{2n} ∑_{|s|=2n} b_{s_{1}...s_{p}}T r(ad X)^{s_{1}} … T r(ad X)^{s_{p}}.

Then ⋆_{α} is equivalent to ⋆_{α}^{G} through T′:

T′ = Id + ∑_{n=1}^{∞} ∑_{|s|=2n} b_{s_{1}...s_{p}}T_{s_{1}} ∘ … ∘ T_{s_{p}}

= Id + ∑_{m≥1} T′_{2m}.

By induction, we suppose now:

C_{s}(α)(u, v) = (−1)^{s}C_{s}(α)(v, u) for any s < n

and compute

T′(u ⋆_{α} v) = (T′u) ⋆_{α}^{G} (T′v).

We get:

∑_{r+s=n} T′_{r}C_{s}(α)(u, v) = ∑_{m_{1}+m_{2}+m_{3}=n} C_{m_{1}}^{G}(α)(T′_{m_{2}}u, T′_{m_{3}}v)

= C_{n}(α)(u, v) + ∑_{{r+s=n \atop s<n}} T′_{r}C_{s}(α)(u, v).

Thus:

C_{n}(α)(v, u) = ∑_{m_{1}+m_{2}+m_{3}=n} C_{m_{1}}^{G}(α)(T′_{m_{2}}v, T′_{m_{3}}u) − ∑_{{r+s=n \atop s<n}} T′_{r}C_{s}(α)(v, u)

= ∑_{m_{1}+m_{2}+m_{3}=n}(−1)^{m_{1}}C_{m_{1}}^{G}(α)(T′_{m_{3}}u, T′_{m_{2}}v) − ∑_{{r+s=n \atop s<n}}(−1)^{s}T′_{r}C_{s}(α)(u, v)

but the parities of m_{1} and s coincide with the parity of n, then:

C_{n}(α)(v, u) = (−1)^{n} ( ∑_{m_{1}+m_{2}+m_{3}=n} C_{m_{1}}^{G}(α)(T′_{m_{3}}u, T′_{m_{2}}v) − ∑_{{r+s=n \atop s<n}} T′_{r}C_{s}(α)(u, v))
= (−1)^{n}C_{n}(α)(u, v).

And ⋆_{α} is symmetric.

The symmetry is a property related in some way to the reality of star products, this notion was considered by A. Lichnerowicz.

Definition 5.3. (Real star products) Let (⋆_{α}) be a K-family. Then (⋆_{α}) is said to be real if for any α and for any smooth functions φ_{1} and φ_{2} with sufficiently small supported Fourier transform φ̂_{1}, φ̂_{2}:

\overline{φ_{1} ⋆_{α} φ_{2}} = \overline{φ}_{2} ⋆_{α} \overline{φ}_{1}.

Each element ⋆_{α} of a real K-family (⋆_{α}) is said to be a real K-star product. Now accordingly to this definition we can show the following proposition.

Proposition 5.4. (Symmetric and real K-star products) Let (⋆_{α}) be a symmetric K-family associated to a function F. As the term e^{a_{1}T r(adX)} does not play any role, we suppose F even. Then (⋆_{α}) is real if and only if F is a real function.

Proof. Let us compute:

(\overline{φ}_{2} ⋆_{α} \overline{φ}_{1})(ξ) = ∫_{𝔤^{2}} ˆ \overline{φ}_{2}(X)ˆ\overline{φ}_{1}(Y)\frac{F (X)F (Y)}{F (X ×_{α} Y)}e^{2iπ⟨ξ,X×_{α}Y ⟩} dXdY

= ∫_{𝔤^{2}} \overline{φ̂_{2}}(X)\overline{φ̂_{1}}(Y)\frac{F (−X)F (−Y)}{F ((−X) ×_{α} (−Y))}e^{2iπ⟨ξ,(−X)×_{α}(−Y)⟩} dXdY

= ∫_{𝔤^{2}} \overline{φ̂_{2}}(X)\overline{φ̂_{1}}(Y)\frac{F (X)F (Y)}{F (−(Y ×_{α} X))}e^{2iπ⟨ξ,−(Y ×_{α}X)⟩} dXdY

= ∫_{𝔤^{2}} \overline{φ̂_{2}}(X)\overline{φ̂_{1}}(Y)\frac{F (X)F (Y)}{F (Y ×_{α} X)}e\overline{2iπ⟨ξ,(Y ×_{α}X)⟩} dXdY

and

\overline{φ_{1} ⋆_{α} φ_{2}}(ξ) = \overline{∫_{𝔤^{2}} φ̂_{2}(X)φ̂_{1}(Y)\frac{F (X)F (Y)}{F (Y ×_{α} X)}e^{2iπ⟨ξ,(Y ×_{α}X)⟩} dXdY}

Thus (\overline{φ}_{2} ⋆_{α} \overline{φ}_{1})(ξ) = \overline{φ_{1} ⋆_{α} φ_{2}}(ξ) if and only if \overline{F} = F, if and only if

\frac{F (X)F (Y)}{F (Y ×_{α} X)}

is real. If Y = −X, this becomes F (X)^{2} real but then each coefficient in the formal series defining F (X) should be real. Conversely, if F is real, (⋆_{α}) is also real.

However this condition is no more true if (⋆_{α}) is not symmetric.

Proposition 5.5. (Characterization of real K-family)

Let (⋆_{α}) be a K-family (perhaps not symmetric) defined by a function F:

F (X) = 1 + ∑_{n=1}^{∞} ∑_{{s_{1}...sp \atop |s|=n}} a_{s_{1}...s_{p}}T r(2iπ ad X)^{s_{1}} … T r(2iπ ad X)^{s_{p}},

then the following are equivalent:

i) (⋆_{α}) is real.

ii) \overline{F (X)} = F (−X).
iii) a_{s_{1}...s_{p}} is real for all s_{1}, …, s_{p}.

Proof. It is not difficult to show that (i) ⟹ (iii). In fact, (⋆_{α}) is real if and only if for any 𝔤 and for all φ_{1}, φ_{2} smooth such that φ̂_{1} and φ̂_{2} are sufficiently small supported:

∫_{𝔤^{2}} \overline{φ̂_{2}}(X)\overline{φ̂_{1}}(Y)\frac{F (−X)F (−Y)}{F (−(Y ×_{α} X))}e^{2iπ⟨ξ,−(Y ×_{α}X)⟩} dXdY =
= ∫_{𝔤^{2}} \overline{φ̂_{2}(X)φ̂_{1}(Y)\frac{F (X)F (Y)}{F (Y ×_{α} X)}e^{2iπ⟨ξ,(Y ×_{α}X)⟩} dXdY}

but this equality happens if and only if

\overline{\frac{F (X)F (Y)}{F (Y ×_{α} X)}} = \frac{F (−X)F (−Y)}{F (−Y ×_{α} −X)}.

Taking Y = −X, we get \overline{F (X)^{2}} = F (−X)^{2}. This implies that the first perhaps not real coefficient a_{s_{1}...s_{p}} satisfies 2\overline{a_{s_{1}...s_{p}}} = 2a_{s_{1}...s_{p}}, thus it is in fact real.

Now, (iii) implies (ii) and (ii) implies (i) are obvious since F = F_{e} + F_{o} where:

F_{e} = 1 + ∑_{n=1}^{∞} ∑_{|s|=2n} a_{s_{1}...s_{p}}T r(2iπ ad X)^{s_{1}} … T r(2iπ ad X)^{s_{p}}

and

F_{o}(X) = ∑_{n=0}^{∞} ∑_{|s|=2n+1} a_{s_{1}...s_{p}}T r(2iπ ad X)^{s_{1}} … T r(2iπ ad X)^{s_{p}}.

Then \overline{F (X)} = \overline{F_{e}(X)} + \overline{F_{o}(X)} and F (−X) = F_{e}(X) − F_{o}(X). Thus a_{s_{1}...s_{p}} is real for all s_{1}, …, s_{p} implies that \overline{F (X)} = F (−X) and (⋆_{α}) is real.

6. Closed and relative star products

Definition 6.1. (Analytic star products) A K-family defined by a function F (X) = J(X)e^{h(X)} is said to be analytic if the series

h(x) = ∑_{s_{1}...s_{p}} a_{s_{1}...s_{p}}T r(2iπadX)^{s_{1}} … T r(2iπadX)^{s_{p}}

has a strictly positive convergence radius: there exists r > 0 such that:

∑_{s_{1}...s_{p}} |a_{s_{1}...s_{p}}|r^{s_{1}+...+s_{p}} < ∞.

An element ⋆_{α} of an analytic K-family (⋆_{α}) is said to be an analytic K-star product.

Accordingly to this definition we have the following lemma.

Lemma 6.2. (F is holomorphic) Let ⋆_{α} be an analytic K-star product then F is holomorphic on a neighborhood of zero for any Lie algebra 𝔤.

Proof. Let us fix a norm ∥.∥ on 𝔤. As the function trace T r from the space L(𝔤) (of linear mapping from 𝔤 to 𝔤) to ℂ is linear,then there exists a constant K such that:

|T r(A)| ≤ K∥A∥ for all A ∈ L(𝔤).

Then

|T r(ad X)^{s_{i}}| ≤ K∥(ad X)^{s_{i}}∥ ≤ K∥ad X∥^{s_{i}} ∀s_{i} ∈ ℕ.

Thus we obtain:

∑_{s_{1}...s_{p}} |a_{s_{1}...s_{p}}T r(2iπ ad X)^{s_{1}}T r(2iπ ad X)^{s_{2}} … T r(2iπ ad X)^{s_{p}} |

≤ ∑_{s_{1}...s_{p}} |a_{s_{1}...s_{p}}| K^{p} ∥ad X∥^{s_{1}+...+s_{p}}(2π)^{s_{1}+...+s_{p}}.

If K ≤ 1, then:
∑_{s_{1}...s_{p}} |a_{s_{1}...s_{p}}T r(2iπ ad X)^{s_{1}} … T r(2iπ ad X)^{s_{p}} |

≤ ∑_{s_{1}...s_{p}} |a_{s_{1}...s_{p}}| ∥ad X∥^{s_{1}+...+s_{p}}(2π)^{s_{1}+...+s_{p}}.

Now there exists ρ > 0 such that

∥ad X∥ ≤ \frac{r}{2π} if ∥X∥ < ρ,

thus F is holomorphic on the ball B(0, ρ).

If K > 1, then K^{p} ≤ K^{s_{1}+...+s_{p}}. But one has

∥adX∥ < \frac{r}{2πK} if ∥X∥ < \frac{ρ}{K}

and F is holomorphic on the ball B(0, \frac{ρ}{K}).

Proposition 6.3. (An integral relation) Let ⋆_{α} be an analytic K-star product associated to a function F and let φ_{1}, φ_{2} be two smooth functions on 𝔤^{∗} such that φ̂_{1} and φ̂_{2} have sufficiently small support then:

∫_{𝔤^{∗}}(φ_{1} ⋆_{α} φ_{2})(ξ) dξ = ∫_{𝔤^{∗}} φ̂_{1}(X)φ̂_{2}(−X)H(X)H(−X)e^{−1/2T radX} dX
= ∫_{𝔤^{∗}} φ̂_{1}(X)φ̂_{2}(−X)H(X)H(−X)(Δ(X))^{1/2} dX,

where H(X) = F (X)J^{−1}(X) and Δ(X) is the modular function for 𝔤. In particular if 𝔤 is unimodular we get:

∫_{𝔤^{∗}}(φ_{1} ⋆_{α} φ_{2})(ξ) dξ = ∫_{𝔤} φ̂_{1}(X)φ̂_{2}(−X)H(X)H(−X) dX.

Proof. We can write:

(φ_{1} ⋆_{α} φ_{2})(ξ) = ∫_{𝔤^{2}} φ̂_{1}(X)φ̂_{2}(Y)\frac{F (X)F (Y)}{F (X ×_{α} Y)}e^{2iπ⟨ξ,X×_{α}Y ⟩} dXdY

= ∫_{𝔤^{2}} φ̂_{1}(X)φ̂_{2}((−X) ×_{α} Z)\frac{F (X)F ((−X) ×_{α} Z)}{F (Z)}

× ω((−X) ×_{α} Z)^{−1}ω(Z)e^{2iπ⟨ξ,Z⟩} dXdZ

where

ω(X) = det(d exp X) = det (\frac{1 − e^{−ad X}}{ad X} ) .

Then:

∫_{𝔤}(φ_{1} ⋆_{α} φ_{2})(ξ) dξ = ∫_{𝔤} φ̂_{1}(X)φ̂_{2}(−X)F (X)F (−X)ω(−X)^{−1} dX
= ∫_{𝔤} φ̂_{1}(X)φ̂_{2}(−X)J^{2}(X)H(X)H(−X)J^{−2}(X)det (e^{−ad X/2}) dX
= ∫_{𝔤} φ̂_{1}(X)φ̂_{2}(−X)H(X)H(−X)e^{−T r(ad X/2)} dX
= ∫_{𝔤} φ̂_{1}(X)φ̂_{2}(−X)H(X)H(−X)(Δ(X))^{1/2} dX.

A. Connes, M. Flato and D. Sternheimer studied closed star products in [9]. Let us recall their definition.

Definition 6.4. (Closed star products) A star product on 𝔤^{∗} is said to be closed if, for all smooth compactly supported φ_{1}, φ_{2}, one has:

∫_{𝔤^{∗}}(φ_{1} ⋆ φ_{2})(ξ) dξ = ∫_{𝔤^{∗}} φ_{1}(ξ)φ_{2}(ξ) dξ.

Proposition 6.5. (𝔤 has to be unimodular) Let 𝔤 be a not unimodular Lie algebra and let ⋆_{α} be an analytic K-star product on 𝔤^{∗} then ⋆_{α} is not closed.

Proof. Suppose that ⋆_{α} is closed. We can write:

H(X) = e^{h(X)} where h(X) = ∑_{s_{1}...s_{p}} a_{s_{1}...s_{p}}T r(2iπ ad X)^{s_{1}} … T r(2iπ ad X)^{s_{p}}.

Since the function e^{h(X)+h(−X)−1/2T radX} is holomorphic and equal to 1 on some neighborhood of zero then it is identically equal to 1 that is:

h(X) + h(−X) − \frac{1}{2}T radX = 0.

But one has:

h(X) + h(−X) = 2 ∑_{{s_{1}...sp \atop |s|=2n}} a_{s_{1}...s_{p}}T r(2iπ ad X)^{s_{1}} … T r(2iπ ad X)^{s_{p}}

is an even function. Thus we cannot obtain h(X) + h(−X) − 1/2T r(ad X) = 0, since T r(ad X) is a non-everywhere vanishing odd function.

Accordingly to the above proposition, from now on, we shall define a closed K-star product as follows:

Definition 6.6. (Closed K-family) An analytic K-family (⋆_{α}) will be called closed if and only if ⋆_{α} is closed for all unimodular Lie algebra 𝔤.

Now let us recall the definitions of relative star products and strict Kontsevich star products.

Definition 6.7. (Relative K-star-products) A K-family (⋆_{α}) is said to be relative (to the algebra of invariant polynomial functions) if:

u_{1} ⋆_{α} u_{2} = u_{1}.u_{2}

for every α and every invariant polynomial functions u_{1}, u_{2} (u_{i} ∈ S(𝔤)^{G}).

An element ⋆_{α} of a relative K-family (⋆_{α}) is said to be a relative K-star product.

Definition 6.8. (Strict Kontsevich star products) ([3]) A K-family associated to a function F is said to be strict if there exists a function f holomorphic on a neighborhood of 0 such that:

f(0) = 1 and F (X) = det(f(ad X)).

An element ⋆_{α} of a strict K- family (⋆_{α}) is said to be a strict K-star product. An easy corollary of the result of [3] is:

Corollary 6.9. (Strict relative K-star products) A K-star product ⋆_{α} is a strict relative star product if and only if:

F (X) = J(X)e^{∑_{s=0}^{∞} a_{2s+1}T r(ad 2iπ X)^{2s+1}}.

Proposition 6.10. (The unimodular case) Let ⋆_{α} be an analytic K-star product then:

1) If ⋆_{α} is closed then ⋆_{α} is relative.

2) If ⋆_{α} is strict and relative then ⋆_{α} is closed.

Proof. Let 𝔤 be unimodular, thus T r(ad X) = 0 and for any φ_{1}, φ_{2} with sufficiently small supported Fourier transform we have:

∫_{𝔤^{∗}}(φ_{1} ⋆_{α} φ_{2})(ξ) dξ = ∫_{𝔤} φ̂_{1}(X)φ̂_{2}(−X)H(X)H(−X) dX.

Since ⋆_{α} is an analytic K-star product then we choose H(X) = e^{h(X)} where h is an holomorphic function on a neighborhood of zero.

Suppose that ⋆_{α} is closed then e^{h(X)+h(−X)} is an holomorphic function equal to 1 on the neighborhood of zero and so is identically equal to 1. This means that:

h(−X) = −h(X).

Thus ⋆_{α} is a relative star product.

Now suppose that ⋆_{α} is strict and relative then we choose

H(X) = e^{∑_{s=0}^{∞} a_{2s+1}T r(ad 2iπ X)^{2s+1}}

and so H(X)H(−X) = 1. Then we obtain, if × is the usual convolution of functions,

∫_{𝔤^{∗}}(φ_{1} ⋆_{α} φ_{2})(ξ) dξ = ∫_{𝔤} φ̂_{1}(X)φ̂_{2}(−X) dX
= (φ̂_{1} × φ̂_{2})(0) = ∫_{𝔤^{∗}} φ_{1}(ξ)φ_{2}(ξ) dξ.

This concludes the proof of the proposition.

Let us remark that the reality of a K-family (⋆_{α}) implies that:

(φ_{1}, φ_{2}) ↦ ∫_{𝔤^{∗}}(\overline{φ_{1}} ⋆_{α} φ_{2})(ξ) dξ

is a scalar product. This useful property is a consequence of the closeness of ⋆_{α} if 𝔤 is unimodular, but it is still holding for a general 𝔤. In our opinion, it is the good generalization of closeness for any 𝔤.

7. Characterization of ⋆_{α}^{K}

Proposition 7.1. (Properties of ⋆_{α}^{K}) The star product ⋆_{α}^{K} built by Kontsevich is analytic strict relative, closed symmetric and real.

Proof. First ⋆_{α}^{K} is clearly an analytic star product since h(X) = 0. We showed in [3] that ⋆_{α}^{K} is strict and relative. Since

J(X) = det (\frac{sh ad(X/2)}{ad(X/2)})^{1/2}

is an even real function then ⋆_{α}^{K} is symmetric and real. Now by proposition 6-10 we conclude that ⋆_{α}^{K} is closed.

Our goal is now to characterize (⋆_{α}^{K}) among all K-families.

Proposition 7.2. (Characterization of (⋆_{α}^{K})) If (⋆_{α}) is a symmetric strict and closed K-family then (⋆_{α}) = (⋆_{α}^{K}). Or, equivalently, the only K-family strict relative and symmetric is (⋆_{α}^{K}).

Proof. Let (⋆_{α}) be a strict relative symmetric K-family defined by a function F then there exists a function f holomorphic on a neighborhood of 0 such that f(0) = 1 and

F (X) = det(f(ad X)).

Accordingly to the corollary 6-9, we can write:

F (X) = J(X) det e^{h(2iπ ad X)} where h(x) = ∑_{n=0}^{∞} a_{2n+1}x^{2n+1}

is holomorphic near 0.

Now, remark we can replace h(x) by h(x) − a_{1}x since T_{1} is a derivation. Thus we can suppose that:

h(x) = ∑_{n=1}^{∞} a_{2n+1}x^{2n+1}.

Let us suppose a_{3} = a_{5} = … = a_{2n−1} = 0. Thus, for each α, ⋆_{α} is equivalent to ⋆_{α}^{K} through an intertwining operator T given by:

T = Id + b_{2n+1}T_{2n+1} + ∑_{ℓ≥2n+2} b_{ℓ}T_{ℓ}.

We have:

T (u ⋆_{α} v) = T u ⋆_{α}^{K} T v.

Now let us set:

u ⋆_{α} v = ∑_{ℓ≥0} C_{ℓ}(u, v) and u ⋆_{α}^{K} v = ∑_{ℓ≥0} C_{ℓ}^{K}(u, v).

Since ⋆_{α} and ⋆_{α}^{K} are symmetric, one has:

C_{ℓ}(u, v) = (−1)^{ℓ}C_{ℓ}(v, u) and C_{ℓ}^{K}(u, v) = (−1)^{ℓ}C_{ℓ}^{K}(v, u).

Thus we have:

C_{2n+1}(u, v) + b_{2n+1}T_{2n+1}(uv) = b_{2n+1}(T_{2n+1}u)v + b_{2n+1}u(T_{2n+1}v) + C_{2n+1}^{K}(u, v).

Using the antisymmetry of C_{2n+1} and C_{2n+1}^{K}, we obtain:

2b_{2n+1}(T_{2n+1}(uv) − (T_{2n+1}u)v − u(T_{2n+1}v)) = −2b_{2n+1}(δT_{2n+1})(u, v) = 0,

where δ is the Hochschild cobundary operator.But δT_{2n+1} ≠ 0 for all n ≥ 1 (just consider, as usual, ℝ^{2} with α = x_{1}∂_{x_{1}} ∧ ∂_{x_{2}}) thus we can conclude

b_{2n+1} = 0 then a_{2n+1} = 0 ∀n ≥ 1 and (⋆_{α}) = (⋆_{α}^{K}).

Now since (⋆_{α}^{K}) is graded and symmetric, it is entirely determined by the values of X ⋆_{α} u for X in 𝔤 and u in S(𝔤).

Theorem 7.3. (Explicit formula for the Duflo-Kontsevich star product) Let (E_{i}) be a basis of the Lie algebra 𝔤 and C_{ij}^{k} the structure constants for 𝔤. For any u in S(𝔤) (or u with û smooth and compactly supported),

(E_{i} ⋆_{α}^{K} u)(ξ) = ∑_{k≥0} \frac{1}{(2iπ)^{k}} (\frac{B_{k}}{k!} C_{i_{1}i}^{j_{1}}C_{i_{2}j_{1}}^{j_{2}} … C_{i_{k}j_{k−1}}^{j_{k}}ξ_{j_{k}}∂_{i_{1}...i_{k}}u

− ∑_{ℓ=1}^{[k/2]} \frac{B_{k−2ℓ}}{2(k − 2ℓ)!} \frac{B_{2ℓ}}{(2ℓ)!}C_{j_{k}j_{1}}^{j_{2ℓ}}C_{i_{2}j_{2}}^{j_{1}} … C_{i_{2ℓ}j_{2ℓ}}^{j_{2ℓ−1}}C_{i_{2ℓ+1}i}^{j_{2ℓ+1}}C_{i_{2ℓ+2}j_{2ℓ+1}}^{j_{2ℓ+2}} … C_{i_{k}j_{k−1}}^{j_{k}}∂_{i_{2}...i_{k}}u).

here B_{k} is the k^{th} Bernouilli number; Or with a ‘deformation parameter’ ℏ like in [1] and [12]:

E_{i} ⋆_{ℏ} u(ξ) = ∑_{k≥0} 2^{k}ℏ^{k}(\frac{B_{k}}{k!} C_{i_{1}i}^{j_{1}}C_{i_{2}j_{1}}^{j_{2}} … C_{i_{k}j_{k−1}}^{j_{k}}ξ_{j_{k}}∂_{i_{1}...i_{k}}u

− ∑_{l=1}^{[k/2]} \frac{B_{k−2ℓ}}{2(k − 2ℓ)!} \frac{B_{2ℓ}}{(2ℓ)!}C_{j_{k}j_{1}}^{j_{2ℓ}}C_{i_{2}j_{2}}^{j_{1}} … C_{i_{2ℓ}j_{2ℓ}}^{j_{2ℓ−1}}C_{i_{2ℓ+1}i}^{j_{2ℓ+1}}C_{i_{2ℓ+2}j_{2ℓ+1}}^{j_{2ℓ+2}} … C_{i_{k}j_{k−1}}^{j_{k}}∂_{i_{2}...i_{k}}u)

Proof. Let ϕ be a smooth function such that ϕ̂ is smooth with sufficiently small support near 0, we compute E_{i} ⋆_{α}^{K} u as the Fourier transform of a distribution with {0} support. If X = ∑ x_{i}E_{i},

⟨E_{i} ⋆_{α}^{K} u, ϕ⟩ = ⟨u(Y)J(Y), ⟨E_{i}(X)J(X), ϕ̂(X ×_{α} Y)J^{−1}(X ×_{α} Y)⟩⟩

= ⟨û(Y), \frac{1}{2iπ} \frac{∂}{∂x_{i}} (ϕ̂(X ×_{α} Y)\frac{J(X)J(Y)}{J(X ×_{α} Y)})|_{X=0}⟩

The functions (X, Y) ↦ X ×_{α} Y and X ↦ J(X) being holomorphic near 0, this expression holds if the support of ϕ̂ is sufficiently small. Moreover let us recall that :

\frac{∂}{∂x_{i}} (X ×_{α} Y)|_{X=0} = ∑_{k≥0} \frac{B_{k}}{k!} (ad Y)^{k}E_{i}

(see [7] for instance) and that:

\frac{sh x/2}{x/2} = e^{∑_{k≥1} \frac{B_{2k}}{2k(2k)!} x^{2k}}.

(see [16]). Thus:

J(X) = ∏_{k=1}^{∞} e^{\frac{B_{2k}}{4k(2k)!} T r(ad X)^{2k}} = ∏_{k=1}^{∞} e^{\frac{B_{2k}}{4k(2k)!} ∑_{i_{1}...i_{2k}} x_{i_{1}}...x_{i_{2k}} ∑_{j_{1}...j_{2k}} C_{i_{1}j_{1}}^{j_{2k}}C_{i_{2}j_{2}}^{j_{1}}...C_{i_{2k}j_{2k}}^{j_{2k−1}}} .

Then \frac{∂}{∂x_{i}} J(X)|_{X=0} = 0 and

\frac{∂}{∂x_{i}} J(X ×_{α} Y)|_{X=0} = J(Y) ∑_{k=2}^{∞} ∑_{ℓ=1}^{[k/2]} \frac{B_{k−2ℓ}}{(k − 2ℓ)!} \frac{B_{2ℓ}}{2(2ℓ)!} ∑_{i_{2}...i_{k}} y_{i_{2}} … y_{i_{k}}

∑_{j_{1}...j_{k}} C_{j_{k}j_{1}}^{j_{2ℓ}}C_{i_{2}j_{2}}^{j_{1}} … C_{i_{2ℓ}j_{2ℓ}}^{j_{2ℓ−1}}C_{i_{2ℓ+1}i}^{j_{2ℓ+1}}C_{i_{2ℓ+2}j_{2ℓ+1}}^{j_{2ℓ+2}} … C_{i_{k}j_{k−1}}^{j_{k}}.

Or

\frac{∂}{∂x_{i}} (\frac{J(X)J(Y)}{J(X ×_{α} Y)})|_{X=0} = − ∑_{k=2}^{∞} ∑_{ℓ=1}^{[k/2]} \frac{B_{k−2ℓ}}{(k − 2ℓ)!} \frac{B_{2ℓ}}{2(2ℓ)!} ∑_{i_{2}...i_{k}} y_{i_{2}} … y_{i_{k}}

∑_{j_{1}...j_{k}} C_{j_{k}j_{1}}^{j_{2ℓ}}C_{i_{2}j_{2}}^{j_{1}} … C_{i_{2ℓ}j_{2ℓ}}^{j_{2ℓ−1}}C_{i_{2ℓ+1}i}^{j_{2ℓ+1}}C_{i_{2ℓ+2}j_{2ℓ+1}}^{j_{2ℓ+2}} … C_{i_{k}j_{k−1}}^{j_{k}}.

Thus we get:

E_{i} ⋆_{α}^{K} u = ∑_{k≥0} \frac{1}{(2iπ)^{k}} ( ∑_{{i_{2}...i_{k} \atop j_{1}...j_{k}}} \frac{B_{k}}{k!} C_{i_{1}i}^{j_{1}}C_{i_{2}j_{1}}^{j_{2}} … C_{i_{k}j_{k−1}}^{j_{k}}E_{j_{k}}∂_{i_{1}...i_{k}}u

− ∑_{ℓ=1}^{[k/2]} \frac{B_{k−2ℓ}}{(k − 2ℓ)!} \frac{B_{2ℓ}}{2(2ℓ)!} ∑_{{i_{2}...i_{k} \atop j_{1}...j_{k}}} C_{j_{k}j_{1}}^{j_{2ℓ}}C_{i_{2}j_{2}}^{j_{1}} … C_{i_{2ℓ}j_{2ℓ}}^{j_{2ℓ−1}}C_{i_{2ℓ+1}i}^{j_{2ℓ+1}}C_{i_{2ℓ+2}j_{2ℓ+1}}^{j_{2ℓ+2}} … C_{i_{k}j_{k−1}}^{j_{k}}∂_{i_{2}...i_{k}}u).

This is a special case of the formula with deformation parameter ℏ, if we put ℏ = \frac{1}{4iπ}.

Remark 7.4. 1) For each k, the only graphs occurring in the last expression are one graph Γ_{k} without any wheel and graphs Γ_{k,2ℓ}, with 1 ≤ ℓ ≤ [k/2] with exactly one wheel of size 2ℓ.

Γ_{k} has k vertices of first kind p_{1}, …, p_{k}, 2 vertices of second kind q_{1}, q_{2} and its edges are:

\{\overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{1}q_{2}}, \overrightarrow{p_{2}p_{1}}, \overrightarrow{p_{2}q_{2}}, …, \overrightarrow{p_{k}p_{k−1}}, \overrightarrow{p_{k}q_{2}}\} .

Γ_{k,2ℓ} has the same vertices of Γ_{k} and its edges are:

\{\overrightarrow{p_{1}p_{k}}, \overrightarrow{p_{1}p_{2}}, \overrightarrow{p_{2}p_{3}}, \overrightarrow{p_{2}q_{2}}, …, \overrightarrow{p_{2ℓ}p_{1}}, \overrightarrow{p_{2ℓ}q_{2}}, \overrightarrow{p_{2ℓ+1}q_{1}}, \overrightarrow{p_{2ℓ+1}q_{2}}, \overrightarrow{p_{2ℓ+2}p_{2ℓ+1}}, \overrightarrow{p_{2ℓ+2}q_{2}}, …, \overrightarrow{p_{k}p_{k−1}}, \overrightarrow{p_{k}q_{2}}\}.

2) For each k, there is k!2^{k} graphs having the same weight. These graphs are obtained from Γ_{k} or Γ_{k,2ℓ} (1 ≤ ℓ ≤ [k/2]) by using permutations of the vertices of first kind or the edges coming from these vertices.

Corollary 7.5. (Weights of graphs)

a) The weight of the graph Γ_{k} is w_{k} = \frac{B_{k}}{(k!)^{2}} (see also [1] and [14]).

b) The weight of the graph Γ_{k,2ℓ} is:

w_{k,2ℓ} = −\frac{1}{k!} \frac{B_{k−2ℓ}}{(k − 2ℓ)!} \frac{B_{2ℓ}}{2(2ℓ)!} = −\frac{1}{\binom{k}{2ℓ}} \frac{B_{k−2ℓ}}{((k − 2ℓ)!)^{2}} \frac{B_{2ℓ}}{2((2ℓ)!)^{2}}

= −\frac{1}{\binom{k}{2ℓ}}w_{k−2ℓ} \frac{w_{2ℓ}}{2} .

Especially, for the wheel graph Γ_{2ℓ,2ℓ}, we get:

w_{2ℓ,2ℓ} = −\frac{B_{2ℓ}}{2((2ℓ)!)^{2}} = −\frac{1}{2}w_{2ℓ}.

c) The weight of any graph Γ having at least a ‘parachute’ i.e a wheel linked only with the point q_{2} vanishes.

Remark 7.6. 1) Since the edges of Γ_{2ℓ,2ℓ} are \{\overrightarrow{p_{1}q_{1}}, \overrightarrow{p_{1}p_{2}}, \overrightarrow{p_{2}p_{3}}, \overrightarrow{p_{2}q_{2}}, …, \overrightarrow{p_{2ℓ−1}p_{2ℓ}}, \overrightarrow{p_{2ℓ−1}q_{2}}, \overrightarrow{p_{2ℓ}p_{1}}, \overrightarrow{p_{2ℓ}q_{2}}\}, then passing from Γ_{2ℓ,2ℓ} to Γ_{2ℓ} consists only to move the edge \overrightarrow{p_{1}p_{2}} to \overrightarrow{p_{1}q_{2}}. Thus probably the formula

w_{2ℓ,2ℓ} = −\frac{1}{2}w_{2ℓ},

could be deduced directly from this operation.

2) It is easy to prove directly (see [4],[14]) that the only graphs Γ which can occur in E_{i} ⋆_{α}^{K} u are those of our corollary.

Moreover we see directly that the weight of a graph Γ having at least a parachute vanishes. Indeed, let W be a parachute inside Γ with vertices p_{1}, …, p_{ℓ}, we set the parameters of the configuration space C_{k,2} defined by Kontsevich ([13]) by putting q_{2} = 0, |p_{1}| = 1:

C_{k,2} = {(p_{1}, p_{2}, …, p_{k}, q_{1}) ℑ𝔪(p_{i}) > 0, p_{i} ≠ p_{j} if i ≠ j, |p_{1}| = 1, q_{1} < 0}.

Then C_{k,2} is a dense subset in the product U × V , where:

U = {(p_{1}, p_{2}, …, p_{ℓ}), p_{i} ≠ p_{j} if i ≠ j} ⊂ 𝕋 × H^{ℓ−1}

if H is the standard upper half-plane {z, ℑ𝔪(z) > 0} and 𝕋 the one dimensional torus. And

V = {(p_{ℓ+1}, …, p_{k}, q_{1})} ⊂ H^{k−ℓ} × ℝ^{−}.

Thus the form ω_{Γ} associated by Kontsevich to the graph Γ can be written as:

ω_{Γ} = ω_{W} ∧ ω_{Γ\W} ,

ω_{W} depending only of the points p_{1}, …, p_{ℓ} and so ω_{W} can be viewed as a form on U. Now:

w_{Γ} = ∫_{C_{k,2}} ω_{Γ} = ∫_{U} ω_{W} ∫_{V} ω_{Γ\W} = 0

since the dimension of U is 2ℓ − 1 and ω_{W} is an 2ℓ-form.

Corollary 7.7. (Rationality of weights) Let u and v be two polynomial functions on 𝔤^{∗}, then all the weights of graphs occurring in the expression of u ⋆_{α} v are rational numbers.

Proof. We first remark that if X is in 𝔤 and u is in S(𝔤) then an easy consequence of corollary 7.4 is that weights of all graphs which occur in the expression of X ⋆_{α} u are rational numbers.

Now we can show the corollary by induction on the degree of u. If u is an homogeneous polynomial function of the form u = Xu′ then there exists a polynomial function u′′ such that:

Xu′ = X ⋆_{α} u′ + u′′ with degree of u′′ ≤ degree of u − 1.

Then:

u ⋆_{α} v = (Xu′) ⋆_{α} v = X ⋆_{α} (u′ ⋆_{α} v) + u′′ ⋆_{α} v.

But the weights of graphs in X ⋆_{α} (u′ ⋆_{α} v) and in u′′ ⋆_{α} v are all rational numbers. Thus, we conclude the proof of the corollary.

Acknowledgement. I want to thank Professor Didier Arnal for helpful advice and suggestions.

References

[1] Arnal, D., Le produit star de Kontsevich sur le dual d’une algèbre de Lie nilpotente, C. R. Acad. Sc. 327 (1998), 823–826.

[2] Arnal, D. and N. Ben Amar, Kontsevich’s wheels and Invariant Polynomial functions on the dual of Lie algebras, Lett. Math. Phys. 52 (2000), 291– 300.

[3]  —, 𝔤-relative star products, Lett. Math. Phys. 55 (2001), 33–42.

[4] Arnal, D., N. Ben Amar and M. Masmoudi, Cohomology of good graphs and Kontsevich linear star products, Lett. Math. Phys. 48 (1999), 291–306.

[5] Arnal, D., M. Cahen and S. Gutt, Deformation on coadjoint orbits, J. Geom. Phys. 3 (1986), 327–351.

[6] Arnal, D., J. C. Cortet, P. Molin, and G. Pinczon, Covariance and geometrical invariance in ⋆ quantization, J. Math. Phys. 24 (1983), 276–283.

[7] Ben Amar, N., Tangential deformations on the dual of nilpotent special Lie algebras, Pacific J. of Math. 170, 2 (1995), 297–318.

[8] Bayen, F., M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization I, II, Ann. of Phys. 111 (1978), 61– 151.

[9] Connes, A., M. Flato, and D. Sternheimer, Closed star products and cyclic cohomology, Lett. Math. Phys. 24 (1992), 1–12.

[10] Dito, G., Kontsevich star product on the dual of a Lie Algebra, Lett. Math. Phys 48 (1999), 307–322.

[11] Felder, G., and B. Shoikhet, Deformation quantization with traces, QA/0002057 (2000).

[12] Gutt, S., An explicit ⋆-product on the cotangent bundle of a Lie group, Lett. Math. Phys. 7 (1983), 249–259.

[13] Kontsevich, M., Deformation quantization of Poisson manifolds I., QA/9709040 (1997).

[14] Kathotia, V., Kontsevich’s universal formula for deformation quantization and the Campbell-Baker-Hausdorff formula, I, QA/9811174.

[15] Lichnerowicz, A., Variété de Poisson et feuilletage, Ann. Fac. Sci. Toulouse 4 (1982) 15–262.

[16] Magnus, W., F. Oberhettinger, and R. P. Soni, “Formulas and theorems for the special functions of mathematical physics,” Springer-Verlag Berlin, Heidelberg, New York, 1966.

[17] Shoikhet, B., Vanishing of the Kontsevich integral of the wheels, QA/0007080 (2000).

Nabiha Ben Amar Département de Mathématiques, Faculté des Sciences de Sfax, Route de Soukra, BP 802, 3018 SFAX, Tunisie, e-mail: Nabiha.BenAmar@fss.rnu.tn

Received July 15, 2001 and in final form December 12, 2002