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 ${\mathcal {Lie}}(x_{1},\dots ,x_{n})$ 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.

Koszul-Dual

The Koszul-dual of ${\mathcal {Lie}}$ 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, MR 1301191

External links

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