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.
Let formula_1 denote the free Lie algebra (over some field) with the generators formula_2 and formula_3 the subspace spanned by all the bracket monomials containing each formula_4 exactly once. The symmetric group formula_5 acts on formula_1 by permutations and, under that action, formula_7 is invariant. Hence, formula_8 is an operad.
The Koszul-dual of formula_9 is the commutative-ring operad, an operad whose algebras are commutative rings.