Lie groups: Haar measure

This lecture is part of an online graduate course on Lie groups.
We show the existence of a left-invariant measure (Haar measure) on a Lie group. and work out several explicit examples of it.
Correction: At 21:40 There is an exponent of -1 missing: the parametrization of the unitary group is (I+iH)(I-iH)^(-1) not (I+iH)(I-iH).
For the other lectures in the course see • Lie groups

[@annaclarafenyo8185]

Your series leads one to discover the wonderful skewy world of nilpotent lie groups, so I started to wonder why they never show up in physics. Your video here gives the most abstract answer possible: the lattice path integration for them doesn't make sense unless they are commuting.

[@nikolavelov7628]

Thank you so much for making your wonderful lectures available on UA-cam! At my institution there are only a few courses where I can learn some of the material your lectures cover and your lectures have helped me become a much better mathematician.

[@vitjastrum]

The Cayley parametrization (around ~21:40) should read as (I+iH)(I-iH)^(-1), I think. Otherwise, multiplying out you get a Hermitean matrix, not a unitary one.

[@richarde.borcherds7998]

Yes, you are right. Thanks.

[@annaclarafenyo8185]

The explicit measure on O(n) can be constructed algorithmically: pick the first column of the matrix as a random unit vector (uniform unit sphere measure), then the next column as an orthogonal vector (uniform measure on a sphere of dimension one less), and so on. This coordinatization by bundles-of-spheres has the advantage of manifest invariance, and the measure is inherited from the spheres, the only thing to check is that the stabilizer of the first n picks acts transitively on each next sphere.
For the unitary group, it's the same thing, except the complex orthogonality condition gives two real perpendicularity requirements, so the dimensions of the spheres with uniform measure skip down by steps of 2. This construction seems to me easier and more useful than a (1+iH)/(1-iH) representation, because everything is inherited from Lebesgue measure with no factors, so it is very easy to implement on a computer. Quotient of iterated bundles of spheres also provides clear intuition on the geometry.

[@bentoomey15]

You mentioned "I always get a little confused why we add x instead of a," when talking about the Haar measure on the group of translations and scalings of the plane (though I'm sure you know, I mention this for anyone else).
We certainly need to write the measure in terms of the element (x,y) --- not in terms of the group element g = (a,b) applied, since we can't have an invariant measure depend on the group element applied --- and since x maps to (ax), putting an x^2 in the denominator introduces the factor of a^2 we want in the denominator under the left action.

[@mamahuhu_one]

Incredible teaching skills. I couldn't find the lectures on dynkin diagrams sadly :(

[@omrizemer6323]

Thank you Proffesor!

[@rohitchatterjee2327]

i love these videos

[@danielmassart]

I know I'm late to the party but what happened to this series ? the introduction promised so much more, e.g. Dynkin diagrams... Btw, your youtube channel is a gift to mankind

[@misakiohta]

Thank you very much!!

[@BosonCollider]

24:00 Noooo this ends in a cliffhanger (not really but the playlist is incomplete)

[@Byron_Vega]

Yeah it is so sad :(

[@larbitouil5552]

-What is the purpose of the har measure?
-why the tribe of Borel is invariant by the translations

[@migarsormrapophis2755]

yeeeeeeeeeeeeeeeeeeeeee
ye

[@ethanjahan780]

Your comment gives me déjà Vu vibes

[@migarsormrapophis2755]

yeee