Star products and Jacobi Identity

Question

I'm having a "little" problem with one affirmation on Kontsevich's paper. He says that the second order terms $O(\hbar)$ implies, assuming that the associator $A(f,g,h)=0$, that the Jacobi Identity it's satisfied.

That's my problem, I'm stucked here for days. Using $f*g= \sum_{n=0}^{\infty} \hbar^{n}C_{n}(f,g)$. Now, the formula for composition gives $f*(g*h)=\sum_{n=0}^{\infty}\hbar^{n}\sum_{k+l=n}C_{k}(f,C_{l}(g,h))$. Taking the second order terms, I got (assuming that the associator is vanishing):

$fC_{2}(g,h)+C_{1}(f,C_{1}(g,h))+C_{2}(f,gh)=C_{2}(f,g)h+C_{1}(C_{1}(f,g),h)+C_{2}(f,gh)$

Thw two terms in $C_{1}$ already gave $[f,[g,h]]$ and $[h,[f,g]]$. And I now, since $C_{n}$ are derivatives that we can combine some of the terms in order 2 in pairs to give just one new term using Leibniz rule. But how can I get the third term, namely $[g,[h,f]]$.

I've searched internet for papers or something else that could bring some light to this and I've found nothing, all authors always assume this and just made the same affirmation.

Answer 1

Presuming you are looking at

The condition $Q^2 = 0$ can be translated into an infinite sequence of quadratic constraints on polylinear maps $Q_i$. [...] The third constraint means that $Q_2$ satisfies the Jacobi identity up to homotopy given by $Q_3$, etc.

I think that you are misunderstanding what is being said.

In the above passage Kontsevich is talking about the $L_\infty$ structure on a graded vector space. Writing $\{ - \} = d$ and $\{ -, \dotsc, - \}$ for the “higher brackets”, the Jacobi identity in the above passage would look something like $$ \color{red}{\{ \{ a, b, c \} \}} + \{ \{ a, b \}, c \} + (-1)^{bc} \{ \{a, c\}, b \} + (-1)^{ab+ac} \{ \{ b, c \}, a \} \color{red}{{}+\{ \{a\}, b, c \} + (-1)^{ab} \{\{b\}, a, c \} + (-1)^{ac+bc} \{ \{c\}, a, b \}} = 0 $$ where the red terms involving a ternary bracket (and the unary “bracket”, usually called “differential” and denoted $d$) are what is meant by “up to homotopy”.

All of this is happening in what is sometimes called the “dual picture”. (More info here: https://arxiv.org/abs/math/0403135)

Your calculations are not in the dual picture and I don't know how to fix them. One calculation you can do checking that the associativity of $\star$ means that the first term $C_1$ is a Hochschild 2-cocycle.