∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
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. . 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.
The functor from simplicial Lie algebras to dg-Lie algebras from def. has a left adjoint
This is (Quillen 69, prop. 4.4).
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.
For a field of characteristic zero the corresponding derived functors constitute an equivalence of categories between the corresponding homotopy categories
of 1-connected objects on both sides.
This is in the proof of (Quillen, theorem. 4.4).
An early account is in
See also
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)
Last revised on February 22, 2017 at 18:27:38. See the history of this page for a list of all contributions to it.