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simplicial Lie algebra

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A simplicial Lie algebra is a simplicial object in the category of Lie algebras.

Definition

Definition

Let kk be a field. Write LieAlg kLieAlg_k for the category of Lie algebras over kk. Then the category of simplicial Lie algebras is the category (LieAlg k) Δ op(LieAlg_k)^{\Delta^{op}} of simplicial objects in Lie algebras.

Definition

Let (𝔤,[,](\mathfrak{g}, [-,-] be a simplicial Lie algebra according to def. . Then the normalized chains complex NmmathfrakgN \mmathfrak{g} of the underlying simplicial abelian group becomes a dg-Lie algebra by equipping it with the Lie bracket given by the following composite morphisms

[,] N𝔤(N𝔤) k(N𝔤)N(𝔤 k𝔤)N([,])N(𝔤) [-,-]_{N \mathfrak{g}} \;\; (N \mathfrak{g}) \otimes_k (N \mathfrak{g}) \overset{\nabla}{\longrightarrow} N (\mathfrak{g} \otimes_k \mathfrak{g}) \overset{N([-,-])}{\longrightarrow} N (\mathfrak{g})

where the first morphism is the Eilenberg-Zilber map.

This construction extends to a functor

N:LieAlg k Δ opdgLieAlg k N \;\colon\; LieAlg_k^{\Delta^{op}} \longrightarrow dgLieAlg_k

from simplical Lie algebras to dg-Lie algebras.

(Quillen 69, (4.3))

Properties

Theorem

The functor NN from simplicial Lie algebras to dg-Lie algebras from def. has a left adjoint

(N *N):LieAlg k Δ opNN *dgLieAlg k. (N^* \dashv N) \;\colon\; LieAlg_k^{\Delta^{op}} \underoverset {\underset{N}{\longrightarrow}} {\overset{N^*}{\longleftarrow}} {\bot} dgLieAlg_k \,.

This is (Quillen 69, prop. 4.4).

Remark

There is a standard structure of a category with weak equivalences on both these categories, hence there are corresponding homotopy categories. (See also at model structure on simplicial Lie algebras and model structure on dg-Lie algebras.) The following asserts that the above adjunction is compatible with this structure.

Theorem

For kk a field of characteristic zero the corresponding derived functors constitute an equivalence of categories between the corresponding homotopy categories

(LN *N˜):Ho(LieAlg Δ) 1N˜LN *Ho(dgLieAlg) 1 (L N^* \dashv \tilde N) \;\colon\; Ho(LieAlg^\Delta)_1 \stackrel{\overset{L N^*}{\leftarrow}}{\underset{\tilde N}{\to}} Ho(dgLieAlg)_1

of 1-connected objects on both sides.

This is in the proof of (Quillen, theorem. 4.4).

References

An early account is in

See also

On the homotopy theory of simplicial Lie algebras see also