In mathematics, the Lie operad is an operad whose algebras are Lie algebras. The notion (at least one version) was introduced by Ginzburg & Kapranov (1994) in their formulation of Koszul duality.

## Definition à la Ginzburg–Kapranov

Let ${\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})}$ denote the free Lie algebra (over some field) with the 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 ${\displaystyle \Sigma _{n}}$ acts on ${\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})}$ by permutations and, under that action, ${\displaystyle {\mathcal {Lie}}(n)}$ is invariant. Hence, ${\displaystyle {\mathcal {Lie}}=\{{\mathcal {Lie}}(n)\}}$ is an operad.[1]

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

## References

• 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