Jump to content

Lie operad

From Wikipedia, the free encyclopedia

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

[edit]

Fix a base field k and let denote the free Lie algebra over k with generators and the subspace spanned by all the bracket monomials containing each exactly once. The symmetric group acts on by permutations of the generators and, under that action, is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then, is an operad.[1]

Koszul-Dual

[edit]

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

Notes

[edit]

References

[edit]
  • 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
[edit]