# Lie theory, part 10 | Daniel Tubbenhauer

## Метаданные

- **Канал:** VisualMath
- **YouTube:** https://www.youtube.com/watch?v=R5q2keTyTKw
- **Дата:** 18.04.2026
- **Длительность:** 9:19
- **Просмотры:** 341

## Описание

🎓 Lie Theory | Daniel Tubbenhauer

What is a Lie group? What is a Lie algebra? And why does “continuous symmetry” come with such a precise algebraic shadow? In this series, we build Lie theory from the ground up: starting with concrete matrix groups and gradually developing the core dictionary between geometry (groups, manifolds, flows) and algebra (brackets, exponentials, and representations).
The goal is conceptual clarity with hands-on examples. We’ll compute with classical matrix groups like SO(n), SU(n), and SL(n)​, learn how Lie algebras capture local structure, and then lean hard into representation theory: because once symmetry acts on vector spaces, it becomes something you can actually organize, compare, and (sometimes) classify.

💡 Keywords: Lie groups, Lie algebras, exponential map, adjoint action, commutators, representation theory, characters, highest weights, applications
💬 Comments welcome! Corrections and suggestions are very welcome (email is best).

Contents will roughly orbit around:
    1. Matrix Lie groups: examples, first properties, and why they matter
    2. Lie algebras: tangent spaces, brackets, and “infinitesimal symmetry”
    3. The exponential map and one-parameter subgroups
    4. Structure via the adjoint action (and what it reveals)
    5. Representations: basic language, examples, decompositions
    6. Characters / weights / highest-weight ideas (as far as we want to go)
    7. Applications and “why care?”: symmetry in geometry, physics-flavored examples, and other places Lie theory shows up

About me.
Hi, I’m Daniel Tubbenhauer (but feel free to call me Dani, they/them). I’m a mathematician working around algebra, topology, and representation theory, with a soft spot for conceptual explanations and concrete computations.

🌐 Website: http://www.dtubbenhauer.com
📁 TeX and slides: https://github.com/dtubbenhauer/My-TeX-files
🧵 #lietheory #liegroups #liealgebras #representationtheory #mathematics

## Содержание

### [0:00](https://www.youtube.com/watch?v=R5q2keTyTKw) Segment 1 (00:00 - 05:00)

Okay, welcome everyone to this continuation of Lee theory. Not the theory of life as usual. — [snorts] — The exponential map in more details. I really would like to talk about the exponential. representation theory of Lee algebras because it's so good. I overall Lee groups is kind of the same. Hopefully eventually I will convince you that's kind of the same Lee algebras and Lee groups because it's really good and it appears everywhere. Oh my God, the physicists love that. They actually discovered it before the math simultaneously to the mathematicians. Classical quantum what is it? Quantum mechanics is built a lot on Lee theory. That's roughly when also representation theory of Lee algebras and Lee groups became very popular or was studied due to Hermann Weyl. So that was think about the 1920s so the theory the algebraic theory of Lee groups and Lee algebras was a bit older. Representation theory of it is roughly 1920s although things were not before. Yeah. That's roughly what it is. But anyway, that's what I want to talk about but we still need to that's that sounds like you should click on this video. Maybe you should but it will be hopefully have some fun pictures. The exponential map in more details and indeed I think I have really good pictures which I stole. So here's a link by the way here up here the link. So this is a really good picture I think it's really beautiful because it really tells you that here's our little manifold whatever a sphere whatever you want a Lee group in some way or form and the tangent space is like a local picture of it which loses some information right? So here's a little curvature and [snorts] here everything is just flattened little circles. But it also well keeps some information. It's literally like a map of the world. Of course the map is not perfect. will lose a little bit of information but it's actually not so bad. That's why everyone uses maps. So that's how I want to think about the relationship between Lee groups and the algebras and the way to go from one to the other is you take the obvious definition of the exponential. Yeah. Hopefully this is the obvious definition. Let's have a look. I just have the exponential map here. So let's see whether Wikipedia agrees that this is the obvious definition. Exponential map I want the exponential function. Fine. I don't care. Let's have a look at the exponential function. That thing. So let's hope Wikipedia agrees that this is I know what the graph is. There you go. Wikipedia agrees that this is the standard definition. I just replace X the real number by a matrix and just hope for the best. And of course it's the exponential so it will be fine. — [snorts] — And it's really we have a good nice picture right? The curve on the next slide. Again same creator. Really nice pictures. The curve is a flow generated by the infinitesimal direction. That's literally what it is. Right? So the Lee algebras again are these tiny motions and the exponential map turns it into like group things. I mean my cool relationship right? Exponential map works like that. And it creates a group type object. A motion type object if you want. That's our little Lee group. So here's an even better picture somewhat. It's a bit overloaded. So this is kind of a good starter. Same creator. Excellent picture. So this is literally what it is. So on the Lee algebra you have some motion type thing something curved something going somewhere and on the Lee sorry on the Lee group and here's the Lee algebra. you have something curved something whatever and on the Lee algebra everything is of course just flat because it's very boring. It's a vector space. And the logarithm which is kind of not the great map but the logarithm gets you from here to here and exponential the motion picture on the manifold. Really nice picture. Hopefully that makes some sense. Yeah. So near the identity here you go or near here which are mapped to one another the essentially the exponential map is just change of coordinates because around the identity our little very curvy group is actually very flat. That's all point. Smooth approximation first order approximation around the point is actually not too bad. Not globally as you can literally see here. It need not to be injective not needed not even to be surjective at all. Around the identity it kind of captures the geometry of the group. That's literally our little picture. An example which I really want to do a practical example. If you have rotations so rotations really just described by an angle. The way to do this usually is to write these as two by two matrices. As a two by two grid so you write them as two by two matrices and you usually call them something like SO2. Special orthogonal. Orthogonal is always something rotation thing. And the Lee algebra SO2 is skew-symmetric matrices something like

### [5:00](https://www.youtube.com/watch?v=R5q2keTyTKw&t=300s) Segment 2 (05:00 - 09:00)

that. Yeah. Skew-symmetric. It's not symmetric so if you reflect along the diagonal you don't get the same but you get the same up to a sign. And that should somehow map to something rotation becomes it because it somehow is supposed to come from a special orthogonal group. A rotation matrix usually looks like that. Cosine sine cosine. But sines and cosines turns up in rotations. Yeah. As you can see here in this picture here's somewhere a sine and here's somewhere a cosine on this stripe. Literally that's what's going on. And if you just plot this kind of a nice exercise plot this A into our favorite formula here you actually recover that thing in the exponential. It's kind of a nice thing. Put in this little T here for the rotation angle T. So this is literally how the exponential map works. You have some Lee alge- Lee group object the rotation matrix and the Lee algebra object which is just a boring skew-symmetric matrix here and the exponential map gets you from one to the other. It's kind of a nice example here. And the derivative at zero is the following. So exponential map we had this already is like I plus TA plus higher order terms and that at T equals zero this is exactly like I right? The tangent vector at I. Right? So the differential of that thing at zero is identity map on the Lee algebra and this is the Lee algebra data really encodes the local geometry of the group. Again you should have this picture in mind so there will be some loss somewhere. But you always hope that the loss will be reasonably small for let's say reasonably nicely behaved Lee groups. Yeah. And why do we care? Well essentially it boils down to products are sums. Now let's have a look at this funny picture again here. Well this is an exponential this is a logarithm. So let's have a look. How can we do that? Logarithm logarithm. Can I spell logarithm? The inverse function of the exponential. Yeah, absolutely fantastic. Let's try I want blah blah blah blah blah what is the origin of logarithm? log? Bash. — [snorts] — Let's see. Searching blah blah. Yeah, exactly. It was to simplify calculations because it turns multiplication into addition. Exactly that's why we are playing this game. So Napier and Burgi I guess is a Swiss mathematician. They kind of wanted those logarithms. Yeah. And that's why we have them here. They turn something difficult into something easier. Usually it's kind of a nice idea. Just if you have difficult mathematics just take the logarithm makes it easier. That's essentially what we're doing here. So easy case so essentially products are harder than sums and products are Lee groups and sums are Lee algebras. Yeah. So for example if A and B commute then you have the usual formula here. But if A and B don't commute then you get something more difficult which is this famous Baker-Campbell-Hausdorff formula which is a really why I made this video because I want to have one video just explaining that formula which explains the logarithm of the product of nearly identity. You should expect that to be not super nice because it's some kind of I mean it's a logarithm essentially but turns out it's not too bad for a non- as a non-commuting logarithm that can't be too great but it's actually not too bad. So here's a nice example of um order matters so commutativity is very rare. So if you first rotate around the X axis and then around the Z axis or you first rotate then around the X axis you actually get something different. Again this is a mass example why commutativity matters. Everyone knows that commutativity matters. It matters in what type of order you do operations every day. But somehow in math we are kind of trained that everything is commutative. I don't know where that started. It's probably because people identify uh calculating with numbers and mathematics. Anyway, I hope you enjoyed this video and I also hope to see you next time.

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*Источник: https://ekstraktznaniy.ru/video/51423*