nLab van Est isomorphism

Idea

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)

References

The original article is

• Willem van Est, Group cohomology and Lie algebra cohomology in Lie groups. I, II. Nederl. Akad. Wetensch. Proc. Ser. A. 56, Indagationes Math. 15, (1953). 484–492, 493–504. (MR0059285)

A comprehensive statement is around theorem 3.7 in

• Christoph Wockel, Topological group cohomology of Lie groups and Chern-Weil theory for compact symmetric spaces, ZMP-HH/14-1, Hamburger Beiträge zur Mathematik Nr. 498 (arXiv:1401.1037)

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)