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

Fix a base field k and let ${\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})}$ denote the free Lie algebra over k with 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 ${\displaystyle S_{n}}$ acts on ${\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})}$ by permutations of the generators and, under that action, ${\displaystyle {\mathcal {Lie}}(n)}$ is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then, ${\displaystyle {\mathcal {Lie}}=\{{\mathcal {Lie}}(n)\}}$ is an operad.^[1]

Koszul-Dual

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