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When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie …

The method of integration that I like most is via the theory of central extensions. To start with note that if $\mathfrak{g}$ is a subalgebra of $\mathfrak{gl}_n$, then the subgroup $\langle\exp(\math …

If $G$ is a (finite-dimensional) Lie group, then the exponential function $\exp\colon\mathfrak{g}\to G$ is injective on some identity neighbourhood. If, moreover, $\mathfrak{g}$ is semi-simple and $\m …

I'm pretty confused about the precise relation of the integral and the real cohomology of the classifying space $BG$ of a compact Lie group $G$. The natural map $H^n(BG;\mathbb{Z})\to H^n(BG;\mathbb{R …

There is a crossed module associated to this situation, which will be trivial if $G \to \pi_0(G)$ splits on the level of gropus. Thus an obstruction for the existence of a splitting is the non-trivial …

Perhaps it is interesting to know that for group cocycles the answer is in general no (even up to coboundaries). For instance the extension $\mathbb{Z} \to \mathbb{R} \to S^1$ is described by a 2-cocy …