Lie theory MOC

Haar measure of a compact Lie group

Let be an -dimensional Compact Lie group, be its corresponding Lie algebra¹ with basis , and be a coördinate chart with . For each define so that

where both sides are clearly elements of . Then for any Borel set , the unique left and right Haar measure is given by group

where is a normalisation constant. The Haar measure is defined for the whole of by translating the chart (enabled by invariance).

Proof this is a Haar measure

Let define a chart with , and let be a transition map such that for all . Then

i.e. and thus

as required.

For we define the chart

then

which gives left-invariance.

For each let so that . Letting , i.e. , then

i.e. . But

and since is compact all integrals are finite, thus the , i.e. is unimodular. Therefore is right-invariant.

Proof of uniqueness

Let both be two sided Haar measures normalised such that . Then for any

thus .

Properties

Properties under integration

Let and be a bijection (e.g. taking the inverse of each group element)

\begin{align*} \int _{G} f(hg) , d\mu(g) = \int _{G}f(g) , d\mu(h^{-1}g) = \int _{G}f(g) , d\mu(g)
\end{align*}

\begin{align*} \int {G}f(\phi(g)) , d\mu(g) = \int{G} f(g) , d\mu(g) \end{align*}

Footnotes

1. Here we use Keppeler’s Lie algebra convention ↩