I'm doing some algebraic manipulations, and I'm getting crazy over a stupid sign error. I think I've located the source of the problem. It should come from an error I'm making (but can't see) in the following:
Consider an algebraic structure on a vector space $V$ given simply by a multiplications that is bilinear and anti-associative, i.e. $(ab)c = -a(bc)$. Let $a,b,c,d\in V$, then we have $$\begin{align} a(b(cd)) = & -a((bc)d) = (a(bc))d = -((ab)c)d,\\ a(b(cd)) = & -(ab)(cd) = ((ab)c)d, \end{align}$$ where I've used anti-associativity in two different ways.
Now, either there's an error, or any multiplication of four elements is zero, and I'm pretty certain that it's not the second. Could someone point out the (probably obvious) mistake I'm doing?
If this can be useful, $V$ is in fact a graded vector space, the multiplication has degree $1$ and is anti-symmetric.
What I'm actually trying to do is to compute the minimal model for the operad $BV$ of Batalin-Vilkoviski algebras. I have found the Koszul dual operad, which is generated by a degree $2$ (in fact $-2$, but it doesn't really matter) operator commuting with all other operations, a degree $0$ Lie bracket, and a multiplication as above (with some compatibility with the bracket). It would be weird (in my opinion/experience so far) if composing the multiplication three times I got zero...
With a hint from my advisor, I think I (partially) understood what is going on. The computation I presented is correct for a degree $0$ operation (and that shows that the operad on a single degree $0$ operation in arity $2$ with the anticommutativity relation is not Koszul). However, as my operation is of degree $1$ there is a further sign appearing by Koszul sign rule (when using the parallel composition relation).
This is on a purely operadic level. I still have to really understand what is going on in an algebra over my operad, and I don't have any concrete examples. I will try to update my answer as soon as I find out more. Let $\mu$ be a degree $1$, arity $2$ anti-associative operation. Then we have $$\begin{align} \mu(\mu\otimes1)(\mu\otimes1\otimes1) = & -\mu(1\otimes\mu)(\mu\otimes1\otimes1)\\ = & \mu(\mu\otimes1)(1\otimes1\otimes\mu)\\ = & -\mu(1\otimes\mu)(1\otimes1\otimes\mu), \end{align}$$ where the first and last equalities are given by antiassociativity: $$\mu(\mu\otimes1) = -\mu(1\otimes\mu),$$ and the second one by Koszul sign rule (we "switch" two degree $1$ operations, getting a minus sign). Here's a drawing, which will hopefully make a bit clearer what is going on by representing the operations as trees: