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Lie operad

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Source updated November 22, 2025

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.

01Definition 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.

02Koszul-Dual

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

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