Contents

# Contents

## Idea

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

## Definition

###### Definition

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

###### Definition

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

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

where the first morphism is the Eilenberg-Zilber map.

This construction extends to a functor

$N \;\colon\; LieAlg_k^{\Delta^{op}} \longrightarrow dgLieAlg_k$

from simplical Lie algebras to dg-Lie algebras.

## Properties

###### Theorem

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

$(N^* \dashv N) \;\colon\; LieAlg_k^{\Delta^{op}} \underoverset {\underset{N}{\longrightarrow}} {\overset{N^*}{\longleftarrow}} {\bot} dgLieAlg_k \,.$

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 $k$ a field of characteristic zero the corresponding derived functors constitute an equivalence of categories between the corresponding homotopy categories

$(L N^* \dashv \tilde N) \;\colon\; Ho(LieAlg^\Delta)_1 \stackrel{\overset{L N^*}{\leftarrow}}{\underset{\tilde N}{\to}} Ho(dgLieAlg)_1$

of 1-connected objects on both sides.

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

An early account is in

• Dan Quillen, part I, section 4 of Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (JSTOR)

• Christian Rüschoff, section 8.3 of relative algebraic $K$-theory and algebraic cyclic homology (pdf)