Confusion about signs in the definition of an $A_\infty$-algebra

Question

We are trying to understand the definition of $A_\infty$-algebras. But we are puzzled by what appear to be two different sign conventions (and we cannot figure out how these two are equivalent)

We see that an $A_\infty$-algebra $A$ consists of a graded vector space $A=\underset{n\in \mathbb Z}{\bigoplus}A_n$ along with maps

$$ m_k:A^{\otimes k}\longrightarrow A$$

of degree $(2-k)$. But what is the relation that these $m_k$ satisfy?

(1) See, for instance, page 6 of Keller (https://arxiv.org/pdf/math/9910179.pdf)

$$ \sum (-1)^{pq+r} m_{p+1+r}(1^{\otimes p}\otimes m_q\otimes 1^{\otimes r})=0$$

where the sum is taken over all decompositions of each $n=p+q+r$

(2) See, for instance, page 25 of Kenji Lefevre-Hasegawa (https://arxiv.org/pdf/math/0310337.pdf) where we find the equation

$$\sum (-1)^{p+qr} m_{p+1+r}(1^{\otimes p}\otimes m_q\otimes 1^{\otimes r})=0$$

This difference in signs $(-1)^{pq+r}$ and $(-1)^{p+qr}$ seems to happen at many places in the literature. Is there some way to tinker with the indexing of the graded space, or put signs in front of the $m_k$'s to turn one definition into the other and vice-versa?

Thanks a lot for your time!

Answer 1

Alexander Polishchuk has written a Field Guide to $A_\infty$ Sign Conventions which I have found quite helpful (despite some typos and imprecisions). Interestingly, Raf Bocklandt gives this as the main reference for sign conventions in his recent book A Gentle Introduction to Homological Mirror Symmetry.

Btw, I think you mixed up indices in (1): The conventions in Keller and Lefèvre-Hasegawa agree. In Polishchuk's field guide, this is sign convention $\epsilon_1$.