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

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Fix a base field k and let   denote the free Lie algebra over k with generators   and   the subspace spanned by all the bracket monomials containing each   exactly once. The symmetric group   acts on   by permutations of the generators and, under that action,   is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then,   is an operad.[1]

Koszul-Dual

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

Notes

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References

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  • 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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