Lie operad

In mathematics, the Lie operad is an operad whose algebras are Lie algebras. The notion (at least one version) was introduced by in their formulation of Koszul duality.

Definition à la Ginzburg–Kapranov
Fix a base field k and let $$\mathcal{Lie}(x_1, \dots, x_n)$$ denote the free Lie algebra over k with generators $$x_1, \dots, x_n$$ and $$\mathcal{Lie}(n) \subset \mathcal{Lie}(x_1, \dots, x_n)$$ the subspace spanned by all the bracket monomials containing each $$x_i$$ exactly once. The symmetric group $$S_n$$ acts on $$\mathcal{Lie}(x_1, \dots, x_n)$$ by permutations of the generators and, under that action, $$\mathcal{Lie}(n)$$ is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then, $$\mathcal{Lie} = \{ \mathcal{Lie}(n) \}$$ is an operad.

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