- Source: 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
L
i
e
(
x
1
,
…
,
x
n
)
{\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})}
denote the free Lie algebra over k with generators
x
1
,
…
,
x
n
{\displaystyle x_{1},\dots ,x_{n}}
and
L
i
e
(
n
)
⊂
L
i
e
(
x
1
,
…
,
x
n
)
{\displaystyle {\mathcal {Lie}}(n)\subset {\mathcal {Lie}}(x_{1},\dots ,x_{n})}
the subspace spanned by all the bracket monomials containing each
x
i
{\displaystyle x_{i}}
exactly once. The symmetric group
S
n
{\displaystyle S_{n}}
acts on
L
i
e
(
x
1
,
…
,
x
n
)
{\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})}
by permutations of the generators and, under that action,
L
i
e
(
n
)
{\displaystyle {\mathcal {Lie}}(n)}
is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then,
L
i
e
=
{
L
i
e
(
n
)
}
{\displaystyle {\mathcal {Lie}}=\{{\mathcal {Lie}}(n)\}}
is an operad.
Koszul-Dual
The Koszul-dual of
L
i
e
{\displaystyle {\mathcal {Lie}}}
is the commutative-ring operad, an operad whose algebras are the commutative rings over k.
Notes
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