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) in their formulation of Koszul duality.

Definition à la Ginzburg–Kapranov

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

Koszul-Dual

The Koszul-dual of ${ℒ}{𝒾}{ℯ}$ is the commutative-ring operad, an operad whose algebras are the commutative rings over k.

References

• Ginzburg, Victor; Kapranov, Mikhail (1994), "Koszul duality for operads", Duke Mathematical Journal 76 (1): 203–272, doi:10.1215/S0012-7094-94-07608-4 

External links

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