Lie algebras visualized: why are they defined like that? Why Jacobi identity?
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- čas přidán 18. 05. 2024
- Can we visualise Lie algebras? Here we use the “manifold” and “vector field” perspectives to visualise them. In the process, we can intuitively understand tr(AB) = tr(BA), which is one of the “final goals” of this video. The other is the motivation of the Jacobi identity, which seems random, but actually isn’t.
Files for download:
Go to www.mathemaniac.co.uk/download and enter the following password: whyJacobiidentity
Previous videos are compiled in the playlist: • Lie groups, algebras, ...
Individually:
Part 1: • Why study Lie theory? ... (intro and motivation)
Part 2: • How to rotate in highe... (on SO(n), SU(n) notations)
Part 3: • What is Lie theory? He... (overview of Lie theory)
Part 4: • Can we exponentiate d/... (exponential map on exotic objects)
Part 5: • Matrix trace isn't jus... (on visualising trace)
Videos from other channels that overlap with my previous ideas:
• Dirac's belt trick, To... [only referring to the topology part, as I have issues with using the belt trick to explain spin 1/2, see my previous spin 1/2 video description]
• The Mystery of Spinors [specifically the “homotopy classes” part]
• Spinors for Beginners ... [the “higher-spin” representations]
Apart from @eigenchris video, technically the videos are not specifically talking about Lie groups / algebras in general, but the arguments to be presented are too similar to what I have in mind.
Source:
(1) people.reed.edu/~jerry/332/pr... basically what I say, without the vector field visualisations]
(2) www.damtp.cam.ac.uk/user/ho/S... [focus on Q2: a much more tedious approach to motivate Jacobi identity]
(3) en.wikipedia.org/wiki/Directi... [actually quite useful, touches upon many ideas in the video series]
(4) projecteuclid.org/journals/jo... [not related, but since I am likely not continuing the video series, this is a simpler proof of the BCH formula, but only why knowing the Lie algebra is enough]
Video chapters:
00:00 Introduction
00:52 Chapter 1: Two views of Lie algebras
05:29 Chapter 2: Lie algebra examples
14:44 Chapter 3: Simple properties
21:18 Chapter 4: Adjoint action
30:15 Chapter 5: Properties of adjoint
39:30 Chapter 6: Lie brackets
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This video is a big part of the reason why I started this series in the first place, because I finally understood why tr(AB) = tr(BA) intuitively, rather than using matrix components. A 40-minute video again but I can’t really separate into two - if I do, I have to think of two distinct hooks of the videos, which I can’t do. It is one of the things I hate about video series - each video needs a different hook, or else you risk people leaving the video. Not a good look by the algorithmic gods. You can suggest the idea for the next videos on the channel in the comments!
Hi mathemaniac. Great content as always. Personally, I would love a video about applied topology and also about the latest advancements in differential equations.
Commutators imply two paths.
Vectors (contravariant) are dual to co-vectors |(covariant) -- Dual bases or Riemann geometry is dual.
Positive curvature (attraction) is dual to negative curvature (repulsion) -- curvature is dual.
"Always two there are" -- Yoda.
Please continue with this series. I have binged watched all of them in a single day.
Please continue these series. You explain things from a geometric point of view that I've never seen before. Thank you.
That's the reason I might not continue this series - I can't find a "geometric" way of explaining the representation theory. Maybe the topology, yes, but those have already been covered before.
@@mathemaniac thank you so much. I'm grateful for what you've already done.
Actually, I'm trying to understand the book "applications of lie groups to partial differential equations" by Peter. Olver and that's why I came across your amazing Lie theory video, and followed your channel.
I still have difficulty understanding many things in this book, because they seem vague to me.
Do you possibly have any suggestions for better understanding this book?
Actually I have not used this book before - I am learning Lie theory primarily because of physics, so I can't recommend anything here...
@mathemaniac I am not a math/physics guy, but recently I have come across lie theory and spinors, I found it is really hard to make sense of multiplying a vector like a spin 1/2 state vector by a lie algebra representation like the su(2) spin 1/2 ladder operators. And what makes it even more mysterious for me is the spin state vector can also be multiplied by a SU(2) rotation matrix. I wonder if there are some geometric interpretation of it?
@@hellfirebb There is a geometric interpretation detailed in another video specifically on spin 1/2 on my channel.
Most books always leave the "why" out. As a result it becomes a stupid memorization game of axiom, theorem, proof, repeat until the semester is over, provided you don't drop out.
I think there is huge value in hearing the same idea explained a little differently by different people.
If you're considering not doing these videos just because some other creator has already done it, I would encourage you to create the video anyways.
Some people may have never seen the other video and your explanation may click for a viewer in ways that other explanations did not.
Imagine if only one person ever wrote a calculus book.
I enjoy your content and would love to see videos from you about spinor topology and/or representation theory. 🙏
I agree there is value in doing that, but the main point is that the arguments are the same, or at least very similar. It comes down to how I treat CZcams - I myself only watch videos that haven't treated the topic the same way before. Because I would not like videos that deal with the same topic the similar way, I also don't want to create videos with the same topic in a similar way.
In your calculus book analogy, if the approach to the topic in book B is very similar to book A, with very similar argument, I would rather just have one calculus book actually :) And I would like the author of book B (whichever is published later) to direct their attention to some other subjects / some other radically different approach to calculus. Maybe in some sense, I am treating this CZcams thing the same thing as academia, where I would only want new things to appear on the platform.
I definitely disagree with having one calculus book. And youtube is not academia, people have followings and different presentation style, and it just offers less to people who prefer your channel based on how you create content, who care about your channel over another.
besides, most people who use a similar presentation style often have the same view anyway, and really just end up limiting how comprehensive their content can be.
I know it is demotivating to continue once you've satisfied your primary curiosity. You have to enjoy the process of making these. And if your interest has moved elsewhere then you should follow it.
But as a viewer, I'm following this series because I enjoy your presentation style and perspective, not because it's the "only" way to learn about Lie theory or Spin groups. If you do make a final video in this series you can't help but make something unique - even if you feel the concepts have been covered elsewhere. And we'd enjoy watching it!
I cast my vote on differential geometry!!! Of course, learning about your actual research would be lovely too. Anyway, thank you very much for all you've already provided!
Thanks for the video. This is a great topic that’s really under-explored in this format. I hope you keep making these
Not gonna Lie, this was a good video.
It's always a great pleasure to watch your video! Many thanks ! "Differential geometry" is my vote!
Amazing video as always, your explanations are so easy to follow! Regardless of whether or not you’re continuing with the series, I personally think I would enjoy videos on differential geometry or relativity.
Your effort is really appreciated man. We need more people like you.
Yay!!! Hope, you will continue to make videos about differential geometry and GR🎉
This is the MOST OUTSTANDING of anything on the internet! Go with your inspiration as your greatness evolves to expose and fathom the questions all deep students have had.
I got a lot out of this video -- your presentation of the Lie Bracket as a directional derivative made a lot of sense and really helped me build intuition. Thank you!
You should continue the series, because your teaching style is different from eigenchris
it might resonate more with certain people
I think the main point is that it is nothing different, with very similar arguments. Even with the visuals, I don't think it will be different enough to my liking.
@@mathemaniac Same is dual to different.
Vectors (contravariant) are dual to co-vectors (covariant) -- Dual bases.
Lie algebras are dual to Lie groups.
Subgroups are dual to subfields -- the Galois correspondence.
"Always two there are" -- Yoda.
I’m currently taking a class based on the book Naive Lie Theory and this really helped to get another perspective on the subject! Thank you so much for this series!!
I am a physics student but my math is not very good comparing those who are taking pure maths. However, in this video, everything is simplified into linear algebra which seems to make sense to me. Thanks a lot for your video.
That's the power of lie theory. U can study the curvy lie group manifold by studying it's lie algebra which is a linear space.
I have been loving this series, thank you for it
I was recently learning about Lie groups and algebras, I've seen several ways to define so(3) but still I've learned something new from this video. And even if I didn't I think that your visualizations would be valuable because of the intuitions they provide.
I'd be wonderful if you start at this point with the clasification of semisimple lie groups. Its facinating to me how conjugation plays a hidden role in so many places. In the discrete case the conjugation gives you peter weyl and in the smooth one with the derivative clasifies lie algebras.
Also the idea that there is no room for other else but the list we all know is one underrated as in the Berger list. Keep going
Thank you for a well prepared and presented series.
hope you will continue with these lie theory vids.
As said at the end of the video, I just couldn't find unique enough spin on the other topics that I planned. What specifically about Lie theory do you want illustrated?
@@mathemaniac You can still go for representation theory. Yes, eigenchris did it, but a different take will be useful..
There is also the projective special linear group, PSL(2), as a potential topic to be covered.
I just don't think I have a different argument to the usual way of going through the representation theory. Sure, maybe there could be some visuals, but adding visuals / giving better ones has never been a valid point for me to make a new video on this channel, unless it being a prerequisite to something else.
@@mathemaniac Maybe you can do a continuation on PSL(2).
@@sahhaf1234 Yes that would have been a good idea, but to be honest, I will not talk about the representation theory there, just why PSL(2,C) is isomorphic to the Lorentz group. I still don't know how to present the isomorphism directly, but I do have some ideas on that. This might be similar to what I did in the spin 1/2 video where the main point is to present the SO(3) and SU(2)/Z2 isomorphism directly.
Never stop doing videos.... You channel is a gem ❤
Thank you so much! You are definitely doing God's work with this one. So many students will benefit from it
Hey, i loved the video, and i cant believe you made something so abstract so digestible. If you are going to change your topic for next video I would love to learn about your research. I am currently finishing my bachelor in math and i have started to look into what is out there in research. It is beyond my understanding most of the time but it is also very intresting, plus it would be nice to see a concrete example of what someone who studies math researches.
Thanks for these videos!
Amazing series!
Great video! Probably one reason why many textbooks give the axiomatic definition of Lie algebra is because most Lie algebras have no corresponding Lie group. For example, if your base field is characteristic p or most any field other than R or C. Or, taking any closed, oriented smooth manifold, the homology of its free loop space has the structure of a so-called H*(E_2)-algebra; the E_2 operad in the category of topological spaces induces an operad in the category of graded abelian groups. Since E_2 has a graded Lie structure, so will the homology of the free loop space. But the rank of this homology is rarely finite so it wouldn't have an associated Lie group.
Also, if there is an associated Lie group and you want to study its representation theory, this is mostly determined by the Lie algebra, making the Lie algebra the more important object of study. For classical Lie groups (basically those which come from matrices), you can essentially summarize all the representation theory into Dynkin diagrams.
I've always been confounded by Dynkin diagrams. That would be a great topic for this exceptional channel to pursue! Along with all you have well said above.
Thanks for this! 🎉 you’re the best!
Wonderful work!!! As many others said, i hope you'll go through some videos on DG anf GR.
Thanks for a great lecture.
I would really like you to complete the last two in your list especially on spin. Eigenchris is good but you can treat the subject in the context of your previous videos. It would really complete the course.
Something I'm looking into is optics from the perspective of differential geometry. Things like orthotomic curves, catacaustics. The goal is to build foundations towards understanding Manifold Next Event Estimation for signed distance fields. Special cases for planes and spheres have been fun so far
Огромная благодарность Вам, огромное спасибо Вам. Прошу Вас не останавливайтесь
wonderful series, and nice shoutout to eigenchris
Wow, great job dude
Please continue this series
I hope you continue this series, it is among the best in the whole platform, you explain things amazingly. If you want to give it a more unique spin (pun intended) something I've never seen done here in youtube is taking the Lie Group-Lie Algebra and doing some statistics and computations. For example in my research in equivariant neural networks I need to do lifting from R^n to a bigger space, usually an affine group of translations on R^n and rotations and reflections O(n), and then do convolutions on there to find filters using gradient descent. So looking at the space of square integrable functions on a group, how to represent a Lie Group-Lie Algebra, how to do computations in a Lie Algebra and then take it to the Group, how to do a Fourier Transform, take an integral, all of those topics would be completely novel here. Thank you for these videos, they have been invaluable!
@mathemaniac I'd like to encouragingly pushback against the other comments and say you can end the series here. This video is incredible and really helps make all the intuition of Lie theory click! You don't need to milk things further and can continue other projects. Thank you for this wonderful series and hard work, let's leave masterpieces as masterpieces.
Thanks for this
excellent...No words
Everyone talks about the lie but one day the truth will win out
Awesome video
Thanks!
I was in need of this.😅
Amazing hidden gem..m
I greatly enjoyed this! When I learned a bit about Lie algebra for QFT, I mostly constrained myself to the arithmetic, never really getting an appreciation of the bigger picture. Now I feel like I have a much better intuition. Most of all, I'm now stunned that the ideas underlying the Lie group oriented view of QFT are so similar to the ideas underlying GR. I guess modern physics is just differential geometry all the way down.
Btw if you are looking for something at stuns (at least one of) your viewers: fiber bundles. Both differential geometry of GR and Lie algebras feel very fiber bundle. And I hear more and more physicists who are deep in theory talk about their work in terms of them. But it's not the most accessible topic on the internet, especially considering how intuitive the basic idea is.
In terms of specific questions if you end up making the GR video: why a metric tensor? Ie why is a pseudo metric being nice enough for it to have physical meaning a strong enough requirement to reduce it to an einstein sum like that?
I have heard of fibre bundles, but not done anything with regards to it. Seems very similar to gauges. Will consider that if I get around to learning it.
Actually when it comes to a GR video, I have to also think of ways to motivate / cover GR in a very different way from the many videos covering it already.
Wicked!
Genius
For a geometric view of representation theory you might have to take a step back and formalize what you've been doing so far: so far, you have worked with a matrix Lie group G in terms of its canonical left action on a manifold R^n. You pointed out that also the induced vector fields on the manifold form a Lie algebra, and you showed that there are nice correspondences between the abstract Lie group and its actions on R^n (indeed, because the action is a homomorphism from G to Diff(R^n)). You showed how to use this correspondence to gain intuition about the abstract Lie group.
But representation theory asks: what other vector spaces can G act upon, and how? What really makes an actiom, and how free are we to choose them? Are these different actions related? Is there a smallest vector space?
This is a heavy step from using only one type of action for intuition about G, but all prior videos function as an example.
In the grander scope of group actions on manifolds you might even be able to ask, what manifolds can a Lie group act upon and what properties can this action have? What can the representation tell us about the abstract?
I would LOVE a video which gives geometric intuition about symplectic products!
For 2D ok it’s just the signed area signed by the vectors, but then? What is the reason to define it only on even dimensional spaces?
Haven’t seen anything like this on CZcams and all the explanations I can find online reduce to rational mechanics arguments, which is not that satisfying tbh
P.S. Your content is SUPERB and helps me LOTS thank you so much!!!
I think symplectic stuff is mainly motivated by Hamiltonian mechanics, and that's where I heard about it, so I don't think one can choose not to motivate it by classical mechanics. I would go as far as to say that one should understand it from mechanics point of view.
Perhaps I am biased because now I am more focussed on physics - are there any other places where you have heard of anything symplectic?
@@mathemaniac Well, with decent complex manifolds, most of the times you can split the complex form into an inner product and a symplectic one.
The topic is not inhenrenty dynamical, so it should be possibile to visualize sympletic forms without it, correct?
Also they are used to make volume forms, which can been seen as a purely geometric thing.
So why the symplectic product itself should not have any intuitive meaning apart from the classical mechanics context?
I would like to hear your unique insights on Catergory Theory
Maybe you can then explain why the exponentiation of vectors is a generalization of the exponentiation of numbers, and also talk about Killing forms? Thanks.
Exponentiation is fundamentally about breaking down an action into a sequence of smaller actions and applying them in sequence. The usual derivation of e as a result of compound interest has those small actions being multiplying the money in your bank account by a fixed percentage. A rotation is similar, but instead of taking increasing large steps forward, it takes equally sized steps sideways relative to your position from the center.
There's also the exponential as the function that is its own derivative, and if you've learned circular motion in physics, you'd know that the velocity is always at a right angle to the position, as well as being its derivative.
It's also strongly related to Euler's Formula, where any single rotation can be viewed from the right angle to make it look like a 2D rotation in a complex plane.
Things get slightly more odd when dealing with rotation-like transformations that aren't traditional rotations, namely hyperbolic rotations, translations, screw motion, and higher dimensional multi-rotations. They all still follow the same principles though.
can you make at least one last video of lie theory connecting it to symmetries of differential equations
As someone with no knowledge of Lie theory but a love for abstraction, I have a question: in this video, you perform operations such as gAg^{-1}, which makes sense since both A and g are matrices.
However, in an abstract sense, g is an element of a Group and A is an element of the Lie Algebra (so it’s an element of the tangent space at the identity). Would it still make sense to write “gAg^{-1}” in general? What operation would this be, and is it happening in the Group or in the Algebra? Or should I interpret this to be a shorthand notation where g is actually exp(B) for some element B in the Lie Algebra?
For a group ghg^(-1) makes sense. For a Lie group, let's say h is actually a continuous family of Lie group elements, so we write it as h(t). In particular, we set h(t) = I. For every t, g h(t) g^(-1) makes sense. Then the derivative at 0 is just g h'(0) g^(-1). Since h'(0) is a derivative of Lie group at identity, it is an element of the Lie algebra.
For more details, you can search for adjoint representation of Lie groups (and Lie algebras), which is basically what I want to get to in this video.
Yup, gAg^{-1} is basically conjugation in group theory, which is a very important property in abstract algebra. It doesn't matter whether the group is made out of matrices, integers or abstract operations on manifolds.
You probably already know this, but a sphere isn't fantastic for a lie geoup visualization. Maybe trying something closer to the spinning top or abstract circle with double rotations
At 21:00, what's the difference between "generating a rotation" and "being a rotation"? Seems like the former is matrices that are 90 degree rotations and the latter are any rotations?
The former is not even matrices that correspond to 90 degree rotations, because (0, -a; a, 0) can have a to be anything! It can be thought of as the rate of change of the rotation matrices at the identity.
@@mathemaniac Thanks. I'm gonna have to reflect on that.
One is the exponential of the other, while the other is the logarithm of the first.
And yes, many (all?) generators of rotations are also scaled rotations. In angle doubling contexts, the generators are themselves 180° rotations rather than 90° ones, which also makes them line reflections.
That is not the definition of Lie Alegebra I’ve been taught over many years: If you have a set of elements that form a group and a set of transformations acting on the group which, themselves, form a group, then that is an algebra. If the parameter that describes the algebra can vary continuously then that’s a Lie Algebra.
33:41 If B_1 and B_2 are not commutative, is it still appropriate to separate exp(B_1+B_2) = exp(B_1)*exp(B_2) ?
Yes to first order, as said and shown in the video (up to O(t)). This is also shown in the previous section when we deduced that if A and B generate rotation, then A + B also generate rotation.
..as doctor Spock , would Say..fascinating.
At around the 6m47s mark, where a < 0, why is the velocity vector not ..... argh. No wait I get it. 'a' < 0, but the a coordinate (y) must be positive, hence '-a' to make the coord pos. when a is negative. Also, the vector at 1,0 .. should be negative a, cause it is below the x axis, so... it has to be positive, cause a is negative. Okay. okay. I can keep watching now :)
+
hi
It is a derivative called the lie derivative.
Much simpler whith geometric algebra ! See David Hestenes
It's all a Lie
Why should anyone introduce Lie algebras other than as tangent spaces of Lie groups? Its completely unmotivated! I know that people do, and it bothers me. I don't understand it.
In many cases it is easier to study tangent spaces/generators since you have e.g., a finite amount of them. On another note, in quantum mechanics we study Hamiltonian matrices which are like tangent spaces of the unitary time evolution of the wave function.
Because Lie Groups are not so motivated in on themselves? (sometimes) How do they arise? But considering vector fields on manifolds and them as derivations on the ring of functions of the manifold leads quite intuitively to the notion of Lie Algebra... Without necessarily going through Lie Groups! And considering vector fields as derivations is quite straight forward with the notion of directional derivative. I admit I still don't fully understand the connections between all this concepts hahah, but it seems to me there is a bit more to be said!
@@martifontdecabaalba3952 Groups are interesting. Manifolds are interesting. Personally I think it is therefore interesting to study manifolds that are groups (or groups that are manifolds), aka Lie groups. BUT the absolute argument for why it is justified to look at Lie groups is a posteriori: as it turns out Lie groups form a rather exclusive club. There aren't that many. And mathematicians have every since been interested in exceptions: Platonic solids, quaternions, E8 and Leech lattice etc.
Anti-cockwise Rotation was purpose, right? x)
Eigenchris makes excellent videos but the voiceover is absolutely horrible to listen to so it’s not watchable sadly
17000 views and you’re not continuing the series … wow I hope everyone unsubscribes
Thanks!