In mathematics, the Lie operad is an

operad In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...

whose algebras are

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

s. The notion (at least one version) was introduced by in their formulation of

Koszul duality In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant co ...

Definition à la Ginzburg–Kapranov

Fix a base field ''k'' and let

\mathcal(x_1, \dots, x_n)

denote the

free Lie algebra In mathematics, a free Lie algebra over a field ''K'' is a Lie algebra generated by a set ''X'', without any imposed relations other than the defining relations of alternating ''K''-bilinearity and the Jacobi identity. Definition The definition ...

over ''k'' with generators

x_1, \dots, x_n

and

\mathcal(n) \subset \mathcal(x_1, \dots, x_n)

the subspace spanned by all the bracket monomials containing each

x_i

exactly once. The

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

S_n

acts on

\mathcal(x_1, \dots, x_n)

by permutations of the generators and, under that action,

\mathcal(n)

is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then,

\mathcal = \

is an operad.

Koszul-Dual

The Koszul-dual of

\mathcal

is the commutative-ring operad, an operad whose algebras are the commutative rings over ''k.''

Notes

References

External links

*Todd Trimble
Notes on operads and the Lie operad
*https://ncatlab.org/nlab/show/Lie+operad Algebra {{algebra-stub