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.
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 {\mathcal {Lie}}(n)\subset {\mathcal {Lie}}(x_{1},\dots ,x_{n})}
the subspace spanned by all the bracket monomials containing each
The symmetric group
by permutations of the generators and, under that action,
The operadic composition is given by substituting expressions (with renumbered variables) for variables.
is the commutative-ring operad, an operad whose algebras are the commutative rings over k.
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