# Lie operad

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 denote the free Lie algebra (over some field) with the generators and the subspace spanned by all the bracket monomials containing each exactly once. The symmetric group acts on by permutations and, under that action, is invariant. Hence, is an operad.[1]

The Koszul-dual of is the commutative-ring operad, an operad whose algebras are commutative rings.

## Notes

- Ginzburg & Kapranov 1994, § 1.3.9.

## 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

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