# Contents

## Idea

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

## Properties

###### Theorem

$(N^* \dashv N) : LieAlg_k^\Delta \stackrel{\overset{N^*}{\leftarrow}}{\underset{N}{\to}} dgLieAlg_k$

between simplicial Lie algebras (over a field $k$) and dg-Lie algebras, where $N$ acts on the underlying simplicial vector spaces as the Moore complex functor.

This is (Quillen, prop. 4.4).

###### Remark

There is a standard structure of a category of 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 0 the corresponding derived functors constitute an equivalence of categories between the corresponding homotopy categories

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

of 1-connected objects on both sides.

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

## References

An early account is in part I, section 4 of

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