Fix a base field k and let ${\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})}$  denote the free Lie algebra over k with generators ${\displaystyle x_{1},\dots ,x_{n}}$  and ${\displaystyle {\mathcal {Lie}}(n)\subset {\mathcal {Lie}}(x_{1},\dots ,x_{n})}$  the subspace spanned by all the bracket monomials containing each ${\displaystyle x_{i}}$  exactly once. The symmetric group ${\displaystyle S_{n}}$  acts on ${\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})}$  by permutations of the generators and, under that action, ${\displaystyle {\mathcal {Lie}}(n)}$  is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then, ${\displaystyle {\mathcal {Lie}}=\{{\mathcal {Lie}}(n)\}}$  is an operad.^[1]

The Koszul-dual of ${\displaystyle {\mathcal {Lie}}}$  is the commutative-ring operad, an operad whose algebras are the commutative rings over k.

• Ginzburg, Victor; Kapranov, Mikhail (1994), "Koszul duality for operads", Duke Mathematical Journal, 76 (1): 203–272, doi:10.1215/S0012-7094-94-07608-4, MR 1301191

• Todd Trimble, Notes on operads and the Lie operad
• https://ncatlab.org/nlab/show/Lie+operad