nLab
simplicial Lie algebra

Context

\infty-Lie theory

∞-Lie theory

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.

Properties

Theorem

There is an adjunction

(N *N):LieAlg k ΔNN *dgLieAlg k (N^* \dashv N) : LieAlg_k^\Delta \stackrel{\overset{N^*}{\leftarrow}}{\underset{N}{\to}} dgLieAlg_k

between simplicial Lie algebras (over a field kk) and dg-Lie algebras, where NN 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 kk a field of characteristic 0 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) : 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.

Revised on April 14, 2013 00:09:25 by Urs Schreiber (89.204.139.110)