nLab
simplicial Lie algebra

Idea

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

Properties

Theorem

There is an adjunction

$$(N^{*}⊣N):{\mathrm{LieAlg}}_k^{\Delta }\stackrel{\stackrel{N^{*}}{\leftarrow }}{\underset{N}{\to }}{\mathrm{dgLieAlg}}_k$$(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

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

See also

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

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