L-infinity algebra

Definition

Let $F$ be a field of characteristic $0$. Let $L = \bigoplus_{i \in \Z} L^i$ be a $\Z$-graded vector space over $F$, and write $\abs{x} = i$ for $x \in L^i$. For homogeneous $x_1, \ldots, x_n$ and a permutation $\sigma$ in the symmetric group $S_n$, let $\chi(\sigma) = \chi(\sigma; x_1, \ldots, x_n)$ be the sign defined by two rules. The sign of a transposition of neighbours $x_k, x_{k+1}$ is $-(-1)^{\abs{x_k}\abs{x_{k+1}}}$, and $\chi$ is multiplicative along products of such transpositions. An $(i, n - i)$-unshuffle is a permutation $\sigma \in S_n$ with $\sigma(1) < \cdots < \sigma(i)$ and $\sigma(i + 1) < \cdots < \sigma(n)$.

An $L_\infty$-algebra structure on $L$ is a family of multilinear maps $l_k \colon L^{\otimes k} \to L$, for $k \ge 1$, such that:

1. $l_k$ has degree $2 - k$, that is, it maps $L^{i_1} \otimes \cdots \otimes L^{i_k}$ into $L^{i_1 + \cdots + i_k + 2 - k}$;
2. $l_k(x_{\sigma(1)}, \ldots, x_{\sigma(k)}) = \chi(\sigma)\, l_k(x_1, \ldots, x_k)$ for all homogeneous $x_j$ and all $\sigma \in S_k$;
3. for each $n \ge 1$ and all homogeneous $x_1, \ldots, x_n$, $$\sum_{i + j = n + 1} \sum_{\sigma} \chi(\sigma) (-1)^{i(j - 1)}\, l_j\big(l_i(x_{\sigma(1)}, \ldots, x_{\sigma(i)}), x_{\sigma(i + 1)}, \ldots, x_{\sigma(n)}\big) = 0,$$ where $i, j \ge 1$ and $\sigma$ runs over the $(i, n - i)$-unshuffles.