nLab simplicial Lie algebra

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.

(Quillen 69, (4.3))

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).

Related concepts

• model structure on simplicial Lie algebras

• simplicial group

References

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)

See also

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

• İ. Akça and Z. Arvasi, Simplicial and crossed Lie algebras Homology Homotopy Appl. Volume 4, Number 1 (2002), 43-57.

On the homotopy theory of simplicial Lie algebras see also

• Stewart Priddy, On the homotopy theory of simplicial Lie algebras, (pdf)

• Graham Ellis, Homotopical aspects of Lie algebras Austral. Math. Soc. (Series A) 54 (1993), 393-419 (web)