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
Let denote the free Lie algebra (over some field) with the generators and the subspace spanned by all the bracket monomials containing each exactly once. The symmetric group acts on by permutations and, under that action, is invariant. Hence, is an operad.[1]
The Koszul-dual of is the commutative-ring operad, an operad whose algebras are commutative rings.
Notes
- Ginzburg & Kapranov 1994, § 1.3.9.
gollark: They can, but they're nice and don't.
gollark: Our truth cuboids, which are right, generated them, though.
gollark: According to our monitoring, they are currently secretly running over 35% of ATech™ operations.
gollark: You vastly underestimate their powers.
gollark: Of course, they cannot be trusted.
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
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