Lie operad

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

Definition à la Ginzburg–Kapranov

Fix a base field k and let $ℒ𝒾ℯ(x_1,\dots ,x_n)$ denote the free Lie algebra over k with generators $x_1,\dots ,x_n$ and $ℒ𝒾ℯ(n)\subset ℒ𝒾ℯ(x_1,\dots ,x_n)$ the subspace spanned by all the bracket monomials containing each $x_i$ exactly once. The symmetric group $S_n$ acts on $ℒ𝒾ℯ(x_1,\dots ,x_n)$ by permutations of the generators and, under that action, $ℒ𝒾ℯ(n)$ is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then, $ℒ𝒾ℯ=\{ℒ𝒾ℯ(n)\}$ is an operad.

Koszul-Dual

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

External links

• Todd Trimble, Notes on operads and the Lie operad
• https://ncatlab.org/nlab/show/Lie+operad

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