Lie operad

Remarks

The space $\mathrm{Lie}(n)$ has dimension $(n - 1)!$. A basis is given by the brackets $[\ldots[\,[x_1, x_{\sigma(2)}], x_{\sigma(3)}], \ldots, x_{\sigma(n)}]$, where $\sigma$ runs over the permutations of $\set{2, \ldots, n}$. For example, $\mathrm{Lie}(1) = F x_1$, $\mathrm{Lie}(2) = F [x_1, x_2]$ is the sign representation of $S_2$, and $\mathrm{Lie}(3)$ has basis $[\,[x_1, x_2], x_3]$ and $[\,[x_1, x_3], x_2]$. The third bracket $[\,[x_2, x_3], x_1]$ is a combination of these by the Jacobi identity. An algebra over $\mathrm{Lie}$ is a vector space $V$ with maps $\mathrm{Lie}(n) \otimes V^{\otimes n} \to V$, and these are determined by the image of $[x_1, x_2]$, which is a Lie bracket on $V$.