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.
21:10 I don't understand the right hand side. What does g A g^-1 mean? How do you define multiplication between an element of a Lie Group and an element of a Lie Algebra? Aren't they completely different things? The only thing the Lie Algebra has is addition, scalar multiplication and the Lie Bracket. Since the Lie Group are not scalars, I don't see how you can multiply them like this.
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?
@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?
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 TH-cam - 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 TH-cam 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 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!
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!
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.
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.
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.
Representation theory (Cartan subalgebra, Dynkin diagrams, classification of classical Lie algebras) would be a great future topic not covered in a visual, intuitive way by anyone else.
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 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.
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.
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?
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.
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.
@@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.
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
Thanks for this terrific video. There seems to be a minor error on one screen. At 38:06, the last term on the third line appears as ad_A(B)ad_A(C) when it looks like it should just be ad_B ad_A(C) -- which is what it is on the next page. Keep up the great work!!
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
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!
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.
Very nice video! I have one question, it is mentioned at 21:31 in the video: If A is on Lie algebra and g is a Lie group element, why is gAg^{-1} also staying on Lie algebra?
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.
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.
I just want to add, I recently attended a seminar on so called "Lie Groupoids", which I don't understand very well, but one thing that stuck out to me: In this setting, the antisymmetry of the bracket fails in general, which caused a problem with the Jacobi identity; not every form of it is equivalent! What struck me was that the one that the speaker chose to impose was not the cyclic sum version, but rather the product rule version, because that's really the more fundamental property.
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.
Great video. There’s just one thing I don’t get from chapter 4. You have two vector fields Z and X, then you apply a rotation exp(tX) with t = 1 to the vector field Z. Then you get the rate of change and conclude that it’s the vector field -Y. However, you use the rate of change for t = 0 to get this new vector field. Shouldn’t we use the rate of change for t = 1?. Why are we using the tangent vector at t = 0 instead of t = 1? Or maybe I didn’t understand what actually happened. Thanks for such a great video.
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 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.
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.
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 TH-cam 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?
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 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 :)
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.
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.
Certainly very good illustrations. This is may be a bit personal: As a German speaker with a healthy accent in my EN myself i found the voiceover a little painful to listen to. The accentuation is very strong but a bit random (why always “vector FIELD” for example?) and there is something with the Rs and the Ts that sounds - weird. Again, I have an accent myself and no problem with that in general, but you seem to emulate really posh EN speakers and this imo goes terribly wrong. Might be better to just celebrate your natural accent, no matter which one that is - always better than sounding like King Charles after two weeks of constipation (“agghhr times agghhr”). Bit personal, but i am convinced that would give you a wider audience…
@@jogloran i hear you - but i really have no intention to insult or belittle. It may sound “horrid” to you but it reflects my heartfelt impression. “Posh English” is nice to listen to if it is done well, but in his case he is far, far away from it and the result is absolutely “toe nail-rolling” as we say in German (again, i am far from being native!). Yes this is personal, but on the other hand i am sure that he would significantly increase his viewership due to the high quality content he does if he stopped emulating the UK royalty and just sticked to his natural accent whatever it might be - as they do in the House of Commons for example where everyone celebrates their native accents. This is honest feedback, which sometimes may come out harsh but is meant to help and not to hurt. But granted, you get this kind of feedback likely more around where i come from than the UK:)
@@MGoebel-c8e You seem to be engaging in good faith, so I'm happy to return the courtesy. Have you considered that Mathemaniac's accent isn't putting on airs or an attempt to sound "posh"? There are plenty of countries with a connection to the UK where the English natively spoken by speakers of a certain background sound similar to this. Singapore, Hong Kong, you name it. Putting myself in the shoes of anyone who's had their accent judged is what made me react strongly to your comment. Perhaps things are different where you come from, but this kind of judgement is truly jarring to a lot of people. I truly hope Mathemaniac doesn't change a thing.
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.
21:10 I don't understand the right hand side. What does g A g^-1 mean? How do you define multiplication between an element of a Lie Group and an element of a Lie Algebra? Aren't they completely different things? The only thing the Lie Algebra has is addition, scalar multiplication and the Lie Bracket. Since the Lie Group are not scalars, I don't see how you can multiply them like this.
Thanks for the video. This is a great topic that’s really under-explored in this format. I hope you keep making these
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 TH-cam - 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 TH-cam 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.
Your effort is really appreciated man. We need more people like you.
It's always a great pleasure to watch your video! Many thanks ! "Differential geometry" is my vote!
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!
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!
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.
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.
Thank you so much! You are definitely doing God's work with this one. So many students will benefit from it
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.
Representation theory (Cartan subalgebra, Dynkin diagrams, classification of classical Lie algebras) would be a great future topic not covered in a visual, intuitive way by anyone else.
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.
Yay!!! Hope, you will continue to make videos about differential geometry and GR🎉
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.
Not gonna Lie, this was a good video.
Binge watching this series was one of the most satisfying things I have done :))) Your videos are treasureeeee
Never stop doing videos.... You channel is a gem ❤
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.
I have been loving this series, thank you for it
Thanks!
Thank you for a well prepared and presented series.
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?
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.
I hope that the future entails both your research and diff geo with GR. You produce phenomenal videos. Thank you.
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.
Thanks!
Wonderful work!!! As many others said, i hope you'll go through some videos on DG anf GR.
Огромная благодарность Вам, огромное спасибо Вам. Прошу Вас не останавливайтесь
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
Thanks for this terrific video.
There seems to be a minor error on one screen. At 38:06, the last term on the third line appears as ad_A(B)ad_A(C) when it looks like it should just be ad_B ad_A(C) -- which is what it is on the next page.
Keep up the great work!!
Excellent series, thanks for the intuitive explanation!
Thank you so much for making this series
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
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!
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.
Very nice video! I have one question, it is mentioned at 21:31 in the video:
If A is on Lie algebra and g is a Lie group element, why is gAg^{-1} also staying on Lie algebra?
You've unlocked vectors on spheres for me, now I can do linear algebra on them! Thanks! But only I would do it with bivectors from Clifford Algebra.
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.
Thanks for these videos!
I just want to add, I recently attended a seminar on so called "Lie Groupoids", which I don't understand very well, but one thing that stuck out to me: In this setting, the antisymmetry of the bracket fails in general, which caused a problem with the Jacobi identity; not every form of it is equivalent! What struck me was that the one that the speaker chose to impose was not the cyclic sum version, but rather the product rule version, because that's really the more fundamental property.
Have you considered, in lieu of Lie algebras, doing a series on lattices, tilings, coxeter symmetries, or related kind of stuff?
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.
Great video. There’s just one thing I don’t get from chapter 4. You have two vector fields Z and X, then you apply a rotation exp(tX) with t = 1 to the vector field Z. Then you get the rate of change and conclude that it’s the vector field -Y. However, you use the rate of change for t = 0 to get this new vector field. Shouldn’t we use the rate of change for t = 1?. Why are we using the tangent vector at t = 0 instead of t = 1? Or maybe I didn’t understand what actually happened. Thanks for such a great video.
wonderful series, and nice shoutout to eigenchris
23:52 im confused. is hAh-1 on the lie algebra? if yes why is the curve not an arc of a circle? i thought so3 has det of 1?
oh i see rotating the whole picture messes up length since shorter lie algebra arrows means closer to polar region
I love this video, you are amazing! Thank you!
Wow Great Series ! Quick question. gA rotates A by g, which I understand. But why gAg-1 rotates by g ?
The Math God must necessarily love this!
Amazing series!
Please continue this series
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.
Thanks for this! 🎉 you’re the best!
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.
Great visualizations!
Everyone talks about the lie but one day the truth will win out
I would like to hear your unique insights on Catergory Theory
can you make at least one last video of lie theory connecting it to symmetries of differential equations
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.
I was in need of this.😅
Wow, great job dude
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 TH-cam 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?
Amazing hidden gem..m
excellent...No words
Thanks for a great lecture.
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
I appreciate you love!
Awesome video
Thanks for this
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 :)
Wicked!
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.
Wouldn't that allow for commutative groups? Lie algebras are strictly anti-commutative.
Genius
Differential Geometry and your research would be fun :)
..as doctor Spock , would Say..fascinating.
It is a derivative called the lie derivative.
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.
hi
Much simpler whith geometric algebra ! See David Hestenes
It's all a Lie
+
Eigenchris makes excellent videos but the voiceover is absolutely horrible to listen to so it’s not watchable sadly
Anti-cockwise Rotation was purpose, right? x)
17000 views and you’re not continuing the series … wow I hope everyone unsubscribes
Certainly very good illustrations. This is may be a bit personal: As a German speaker with a healthy accent in my EN myself i found the voiceover a little painful to listen to. The accentuation is very strong but a bit random (why always “vector FIELD” for example?) and there is something with the Rs and the Ts that sounds - weird. Again, I have an accent myself and no problem with that in general, but you seem to emulate really posh EN speakers and this imo goes terribly wrong. Might be better to just celebrate your natural accent, no matter which one that is - always better than sounding like King Charles after two weeks of constipation (“agghhr times agghhr”). Bit personal, but i am convinced that would give you a wider audience…
A horrid take, politely worded, is still a horrid take.
@@jogloran i hear you - but i really have no intention to insult or belittle. It may sound “horrid” to you but it reflects my heartfelt impression. “Posh English” is nice to listen to if it is done well, but in his case he is far, far away from it and the result is absolutely “toe nail-rolling” as we say in German (again, i am far from being native!). Yes this is personal, but on the other hand i am sure that he would significantly increase his viewership due to the high quality content he does if he stopped emulating the UK royalty and just sticked to his natural accent whatever it might be - as they do in the House of Commons for example where everyone celebrates their native accents.
This is honest feedback, which sometimes may come out harsh but is meant to help and not to hurt. But granted, you get this kind of feedback likely more around where i come from than the UK:)
@@MGoebel-c8e You seem to be engaging in good faith, so I'm happy to return the courtesy.
Have you considered that Mathemaniac's accent isn't putting on airs or an attempt to sound "posh"? There are plenty of countries with a connection to the UK where the English natively spoken by speakers of a certain background sound similar to this. Singapore, Hong Kong, you name it.
Putting myself in the shoes of anyone who's had their accent judged is what made me react strongly to your comment. Perhaps things are different where you come from, but this kind of judgement is truly jarring to a lot of people. I truly hope Mathemaniac doesn't change a thing.