simplicial Lie algebra
∞-Lie theory (higher geometry)
Formal Lie groupoids
A simplicial Lie algebra is a simplicial object in the category of Lie algebras.
Let be a field. Write for the category of Lie algebras over . Then the category of simplicial Lie algebras is the category of simplicial objects in Lie algebras.
Let be a simplicial Lie algebra according to def. 1. Then the normalized chains complex of the underlying simplicial abelian group becomes a dg-Lie algebra by equipping it with the Lie bracket given by the following composite morphisms
where the first morphism is the Eilenberg-Zilber map.
This construction extends to a functor
from simplical Lie algebras to dg-Lie algebras.
(Quillen 69, (4.3))
The functor from simplicial Lie algebras to dg-Lie algebras from def. 2 has a left adjoint
This is (Quillen 69, prop. 4.4).
This is in the proof of (Quillen, theorem. 4.4).
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)
Christian Rüschoff, section 8.3 of relative algebraic -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