Questions wrt. definition of $L_\infty$- and dg-Lie-algebras

Question

I am trying to understand this definition from nLab (Def. 3.2) of $L_\infty$-algebras, and the following example that is supposed to boil down to dg-Lie algebras.

1. What is the difference between shuffles and unshuffles, as in that definition? According to the entry linked from there, both seem to be the same. This matches at least the following examples (where the sum is spelled out for $i=2, j=2$.

2. How is the sign for $(−1)^{|v_2|(|v_1|+|v_3|)}[[v_1,v_3],v_2]$ obtained? According the the definition just above, I would have guessed it's $-(-1)^{|v_2||v_3|}$ instead.

3. Can that graded signature $\chi$ of a permutation completely be avoided by appealing to some Koszul sign rule?

Answer 1

1. An unshuffle is the inverse of a shuffle. In other words, if $\sigma$ is a shuffle, then $\sigma^{-1}$ is an unshuffle and vice-versa. It turns out that $\mathrm{Sh}_{1,2} = \mathrm{Sh}^{-1}_{1,2}$ but this is not true in general.
2. I'm not sure the signs are correct in Example 3.5. It should indeed just be $-(-1)^{|v_2| |v_3|}$ as far as I can tell. Compare with the definition of a dg Lie algebra.
3. I don't know what you mean by "avoided": the signs are there, if you remove them then the definition is not correct. But using the Koszul sign rule you can remember the signs. For example in the previous equation, $l_2$ is of degree $0$ so you can forget it. Then all you need to remember is that $l_2$ is a biderivation with respect to itself (that's what the Jacobi identity means) and that it is antisymmetric, and any time you switch two elements you put a sign.