∞-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
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The content of a van Est isomorphism is that the canonical comparison map from Lie group cohomology to Lie algebra cohomology (by differentiation) is an isomorphism whenever the Lie group is sufficiently connected.
In particular when the underlying topological space of a Lie group is a contractible topological space, then Lie group cohomology (both the naive and the correct stacky version) coincides with Lie algebra cohomology vanEst 53, theorem 14.1 (see e.g. the cocycle which gives the Heisenberg group extension)
The original article is
A comprehensive statement is around theorem 3.7 in
The generalizations to Lie groupoids and Lie algebroids is considered in
Alan Weinstein, Ping Xu, Extensions of symplectic groupoids and quantization, Journal für die reine und angewandte Mathematik 417 (1991): 159-190 (EuDML)
Marius Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes (arXiv:0008064)
Last revised on September 23, 2021 at 08:57:28. See the history of this page for a list of all contributions to it.