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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. 1. 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. 2 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

  • 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 KK-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)

Revised on February 22, 2017 13:27:38 by Urs Schreiber (83.208.22.80)