What is Lie theory? Here is the big picture. | Lie groups, algebras, brackets #3
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- Опубликовано: 27 май 2024
- Part 4: • Can we exponentiate d/...
A bird's eye view on Lie theory, providing motivation for studying Lie algebras and Lie brackets in particular.
Basically, Lie groups are groups and manifolds, and thinking about them as manifolds, we know that we want to understand Lie algebras; and thinking about them as groups, we know what additional structure we want on the Lie algebras - the Lie bracket.
RUclips, please do not demonetise this video for me saying “Tits group”. This is an actual mathematical object named after a French mathematician Jacques Tits.
Files for download:
Go to www.mathemaniac.co.uk/download and enter the following password: so3embeddedin5dim
SO(3) embedded in R^5: at.yorku.ca/b/ask-an-algebraic...
en.wikipedia.org/wiki/Whitney... (n-dim manifold can be properly embedded in R^(2n): if you only want “the overall picture”, but perhaps distances are distorted)
en.wikipedia.org/wiki/Nash_em... (n-dim Riemannian manifold can be isometrically embedded in n(3n+11)/2 dim if compact, n(n+1)(3n+11)/2 dim if not compact: if you want everything to remain intact, i.e. distances are preserved)
BCH formula (why Lie brackets are useful): en.wikipedia.org/wiki/Baker%E...
Finite simple groups as building blocks: en.wikipedia.org/wiki/Composi...
Classification of finite simple groups: en.wikipedia.org/wiki/Classif...
Levi decomposition (the more precise meaning of “building blocks” in Lie algebra): en.wikipedia.org/wiki/Levi_de...
E8 (the monster group of Lie algebras):
aimath.org/E8/e8.html
en.wikipedia.org/wiki/E8_(mat...)
en.wikipedia.org/wiki/An_Exce...
Video chapters:
00:00 Introduction
01:26 Lie groups - groups
05:41 Lie groups - manifolds
10:23 Lie algebras
14:16 Lie brackets
18:03 The "Lie theory picture"
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
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If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
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See you next time!
Originally this was the first video in the series, but I really want to give a convincing enough motivation for the series, and introduce the notation SO(n) and SU(n) beforehand.
P.S. Haven’t had the best of luck with the RUclips algorithm lately, and I honestly don’t know what I could do / what I have done wrong at this point. It seems that you guys really enjoyed it, but RUclips is really reluctant to push out to non-subscribers / less avid subscribers, so the overall performance is much worse than the other recent videos not in this series. If you genuinely enjoy this video series so far (and I promise the series is only going to get better), please do like, **make sure the bell is on**, and share, and perhaps if able, support on Patreon :)
Non subscriber(not anymore) here, really liked the video even tho some stuff went a bid over my head, gonna watch the other videos that come before this and rewatch this one :)
Have you tried tagging one of the videos with #Some3 ? Because Im subscribed to a nunch of channels that post those I get suggestions from channels that didnt
Maybe algorithm dings you for the breaks you take doing school? Anyway, I usually have to stop what I’m doing to catch up when you post something.
Also, my interest in SO and SU were certainly stoked considering EigenChris did several videos on these groups in the midst of his spinor series.
Cheers!
I think it is essential to not go into a thinking mode along the lines of 'have I done something wrong to displease the algorithm'.
It seems to me that the youtube algorithm puts us mortals in the same position as, say, the human protagonists in ancient greek mythology. In ancient greek mythology the Gods are totally unpredictable, acting on whim. The humans have no agency. But the humans keep falling for the belief that somehow they do have some agency. Tragedy ensues.
Presumably: if the working of the algorithm would be public then that would give bad actors opportuniy go all in on gaming the system. Presumably that is why the youtube management doesn't give any information about the algorithm.
All you can do, it seems, is communicate to viewers the thing that you communicated in your pinned message: the trends you see in the youtube analytics. All you can do, it seems, is to try and crowd source support. Individually you have no agency.
Yt demonetises a video because GROUPS?
Nice, group theory. I grew to fancy abstract maths. For 3D animations the abstract approach is very good too.
Always, groups are like sets on steroids.
Now now, I want to watch Riemannian mannigfaltigkeit
Yes we are going there.
Did you cover before Riemannian Surfaces?
Cause this is a little bit of a pickle, anyways, mine is U(1)xSU(2)xSU(3)
DEGREES of freedom yeah, we get 8D Transforming to 9 in atomic nucleus, 8 gluons.
In general relativity, of course it's manifolding every point and we can have the velocities defined by angles, "boosts" in this model. In quantum, basically a special relativity or Lorentz covariant application, in a Hilbert space, differentiable...
Operators, matrices.
Always, linear álgebra and transformations.
Since we have things like spin, with a period of 4pi,i it's easier to work like this.
Hamiltonian=total energy =1, in a vector space we can put operations on, and we have here tangents to represent points. Tangent bundles, those are of course differentiable manifolds, like vector bundles, vector fibers, except each fiber is a vector space 😳🤯🤓🖖🎶
The quality of this series is out of this world
it‘s pretty crazy yes
Well illustrated and explained 😊for non math non nerds!!
You are a god among men. The "quantum leap" that people like you are making for accessible, highly specified education is truly going to transform the world. You are contributing massively to the next generation of highly skilled and motivated mathematicians and physicists. I cannot thank you enough for making this. I'd place this series up there with eigenchris's content which is the single largest complement i currently know how to give. Please continue this series. I will consume every video you make like a swarm of locusts at harvest time. Thank
Thank you so much for the compliment, but I feel quite uneasy when you say "you are a god among men."
@@mathemaniacI think that the ability of explaining math is way more uncommon than mathematical genius. You really are the prometheus of math
@@t0k4m4k7 without the infinite torture, hopefully
it is so ugly to describe a creature as god ! this is so horrible blaspheme , I call you to be a Muslim..to know the purpose of life
@@mathemaniac
You should reject this kind of hyperbole praise
This is a blatant transgression against the status of the Creator, Glory be to Him..and it is so ugly blaspheme
I take this opportunity to invite you to look at the evidence of Islam, and I promise you that you will discover the truth in it brighter than the sun, with one condition: that you look at Islam and its evidence without previous stereotypes.
I mean without biased preconception
Great video. Well done.
A thing I thought of.
When you introduced the concept of a manifold, I noticed that you gave a couple of examples. A tip in such situations is to also consider giving a counter example, like a space with at least one point that is not deformable to a line or a plane etc.
Counter examples can be just as important to learning a concept as positive examples and that’s an instance where I would have found it very useful.
Just a friendly tip, use it if it resonates with your vision, otherwise feel free to ignore it!
Once again, great work here!
Yes, I originally thought about that, using a "cross" / "double cone", but for some reason just couldn't fit into the script in a natural way.
To be very honest, the definition of manifolds would be one of the least important things here, because, well, we usually wouldn't even try to prove the Lie groups are manifolds in a very rigorous way. To actually prove that SO(n) or SU(n) is a manifold, you most likely need to learn differential geometry to a certain level (say knowing preimage theorem), which I don't think is too necessary for understanding Lie theory anyway.
@@mathemaniac Thanks for sharing a bit of your thinking on the matter!
You’re the best person to decide on what level you want to do things.
To add a bit to the conversation, for each subtopic you have a few different pieces to decide the level of depth to aim for and how much time to spend.
For manifolds there is the definition itself; how to work with it or how to motivate or prove specific instances; and finally example instances to give a boost to intuition.
One can probably vary the level of depth for each of those pieces somewhat independently.
Say, with the informal definition “it locally looks like line/plane/hyperplane” a counter example might be as you said a cone or a point connecting three curves or whatever which wouldn’t have to take a lot of time in a video (though might take time to put in).
That might help give the right intuition for the informal definition without needing to aim for rigor.
A more formal definition might be a choice for another type of video; and maybe showing how to prove that a certain topological space (which wasn’t even mentioned) is a topological manifold is yet a different kind of video.
Even then it would probably be overkill to do proofs for all relevant standard examples since the work would be very repetitive.
I think it’s good that there can be videos online at a whole range of different levels on all those points and you’ve clearly considered where to put the effort.
Thanks again!
This is a very beautiful explanation of Lie Algebra
Simply fantastic. Finally youtube mathematics at a higher level
Came for lie theory, stayed for tits group
Comment for the algorithm
The explanation of the relationship between the Lie Algebra and Lie Group and how the tangent map and exponentiation are used is brilliant. I never really did get all this back when I was studying Quantum, but this explanation alone was immensely insightful. Thank you for the fantastic work, eagerly looking forward to the rest of the series.
It makes me so happy that 3b1b's video style is everywhere now
surprisingly, mathemaniac doesn't even use manin
I don’t like, or comment, or share anything on any platform ever… but this series/work is outstanding and exactly what I needed to understand before applying to graduate school. I’ll do whatever you need to widen its reach! Thank you for this incredible work of art🎉🎊
Thank you so much for taking the time to make this series. I've seen bits and pieces of this theory a lot of places, but never an overview of how it all fits together. Looking forward to more!
Exceptional video. Came in contact with Lie theory a couple of years ago. If I were to have seen this back then, it would definitely have helped in clearing up the big picture 10x faster than I did.
This is absolutely fantastic, as someone working on quantum information theory this gives so much insight and makes things so much clearer than any book. Cant wait for the rest of the series!!
Im doing a lot of group theory and lie algebra for my robotics project and this video is full of big and small "eureka" moments for me. You've just earned a subscriber, sir
Really looking forward to the rest of this series! I was trying to learn about Lie theory earlier this summer, and there was not many resources online to do so, but this is great!
The Jacobi Identity is very poorly motivated in most textbooks I've read. It's usually just handed to you.
This is the first time something has forced me to recognise that it comes from the requirement of the exponential map to generate a group.
So thanks
I had skimmed over some videos about Lie theory before, but it all flew over my head and seemed too complicated. This was very accessible and gave me a clear idea of what the Lie algebra actually is.
Thank you very much and I look forward to the rest of the series :)
This is absolutely amazing. I am taking a course on Lie Groups and Lie Algebras at the moment and was struggling to see the big picture of it. This was just perfect. Thank you!!!
I rarely give the "oh, this made things so clear!!1" comments on videos, because usually they don't fit me (though they can communicate things in a new and interesting way). This video is an exception. I'd been exposed to Lie groups and Lie algebras before and had some idea of the Lie bracket, but i couldn't for the life of me understand the actual connection between a Lie group and its associated Lie algebra. That changed today with your video.
Of course the actual topic is so much simpler than it's usually described. The part about the Lie algebra being the tangent space actually made things harder to understand for me since I didn't realize it was specifically the tangent space at the identity. In fact, since the identity isn't actually contained in the Lie algebra, I think it would honestly make more sense to me to just give the two as completely separate manifolds, with the exponential as the map between them. The key point in the Lie theory is then little more than the generalization of the power law to non-commutative Lie groups, and the bracket is just a primitive used to define said generalization. Then you can do algebra on the curved Lie group without leaving the Lie algebra (though it does still seem to require an infinite sum, so there'd still be value in working in the Lie group).
I had been wondering about the infinite sum in the Lie algebra - does it cause any problems in practise with questions of convergence, or even computing its value?
I've been watching maths videos on youtube for many years now and from many different maths youtubers, having done 2 A-Levels in Maths back at college just over 20 years ago, For some reason youtube has never promoted any of your videos to me before, as far as I know, even though you've been a channel for about 4 years. I think this Lie Group series might be going a bit outside of my comfort zone in terms of my level of maths, although I was able to grasp a fair bit of what you were explaining, but I see there are at least a few other videos that I think I would be able to better follow, so I'll be sure to getting watching them as and when I can. I also subscribed and rang the bell etc, having previously not only been a non-subscriber but one who was completely unaware of your existence.
The series everyone has been waiting for! So great!
Thank you! I've been interested in this subject for some time, but can only get superficially deep with my current background. This video was a wonderful synopsis of everything I've been able to find so far, presented in a much more digestible and intuitive way. I'm looking forward to exploring it more deeply... hopefully we casual learners can still keep up as you zoom in.
Loving this mini-series! Lots! And ... what seems really really good is tying together different cultures in mathematics such as, of course, algebra and group theory and manifolds and topology and analysis and (best of all?) differential geometry
Thank you for providing very interesting explanations of wonders of math.
I agree on your "best of all".
Excellent video. I think it's hard to learn Lie Theory coming from a purely "calculatory" (i.e. physics) context because you miss the motive for its original inception. Your over-arcing metaphor -- the utility of creating a coordinate system for group transformations by implementing manifold theory -- is a perfect introductory frame. And you illustrate it simply and beautifully. Really appreciate you!
Wow, a whole series is coming! That's highLie appreciated.
i barely have any higher maths learning but you're still able to explain and prove in ways where it makes sense and i (despite having only vague/hazy visual imagination) can even figure out how to animate what I'm seeing, finding out a few moments later that you've animated them the same way i anticipated.
so your words and proofs are buttressing visual/representative "math sense" in me despite not only the information gap between me and you, but also an ability gap (I'm autistic, "Level 2" so my intellectual domains vary distinctively in terms of limits and strengths).
You're doing a great thing, skillfully.
Loved the perfect rundown of groups and manifolds
Thank you so much for explaining the exponential map, i spent hours looking for an explanation of the name or how it should be understood, and the best i got was "it is called the exponetial map in analogy to the exponential function," and it wasn't until this video that I actually had a good understanding of what was happening. So thank you
I'm studying particle physics and you make my life easier on a daily basis. Thanks for your perfect videos
Your explanation is unmatchable!!!! 🔥🔥🔥🔥🔥🔥
Wow, this video totally enhanced my understanding of Lie theory. I was always puzzled through books. But now, many things are clear. Thank you so much. Looking forward to seeing the rest of the videos ☺️
Wow! ❤ I had been thinking of something discrete that looked like this, and now you've connected my mundane efforts to all this richness of expression!!!
I began with Galois fields and equal subdivisions of the straight angle to make the points I needed to say in the context of my research. All the while, I was talking in terms of Lie groups but at the foundational level.
Awesome! Looking forward to the rest of the videos in the series.
Thank you! Super clear explanation! Can't wait for the next video!
Perfectly timed for @Eigenchris’s video!
This is the best presentation I have ever seen on the Lie groups and Lie algebras.
this was amazing! thank you for the ride through math castle!
This was a really lovely video. As a physcist who once had to join between more maths or more physics, perhaps had I watched something like this back then it might have changed my choice. Can't wait for next one!!
Well I am a mathematician turned physicist :)
Thank you for the amazing video, and all the references in the description!
I am excited for the rest of this series!
This is an amazing video! Please let me know when the next Lie group video is online!
seriously, reading math topics on wiki is the most intimidating thing in the world
Thanks! Excellent presentation.
Literally the first time I have given a shit about lie algebras, after 20 years of studiously ignoring them and doing applied category theory in my software development/computational geometry work. Now I wonder what all I've been missing! Subscribed, and ready for more amazing lectures!
This is one of my favorite series and is a fun part of maths to learn.
Exceptionally well explained!
🏆🙏
Jesus man you surprise me again with such a simplified view of this topic, where one sees only symbols after symbols a collective ugly mess, you make it delightful!! I don't believe this!! Spectacular. Watching your videos for me special occasion, switch off all light, put the headphone, start the video for a beautiful journey....
Thank you so much. Very helpful. Looking forward to the rest of these videos
Well made video, was accessible to me as someone who is not familiar with manifolds or lie algebras. Waiting for the next video in the series =)
Excellent video, I am a student studying Lie Theory and it's really satisfying seeing E8 explained
More work needs to be done on this important topic. Thanks for simplifying.
Thx. Tried and failed to understand this in the past. Good motivator overview.
One of the greatest videos.....awesome...thank you for your contribution..👌👌👏👏👏👏👏👍👍👍
One of my favorite channels :) 😊
A video of the highest quality of this kind!
As a PhD phsyics student, thank you so much for helping visualize this
Such a revelation! Thank you!
You are definitely a descendant of the great Marius Sophus Lie...good sir!
Your exposition & pedagogical skills deserve all the plaudits one can bestow.
Very glad this came up as a recommendation....worhty of subscription indeed!
Exceptional work! Could you show the connection of SU(3) and Gell-Mann's baryon octets?
Excellent explanation of Lie groups and Lie algebras! Like most physics grad students, I was introduced to these back in physics grad school about 40 years ago, but they were never explained that well to me and ever since then, I never felt I had a good handle on them. However, now I think I do, due to your very clear and intuitive explanation. Great job!
Damn!! You must be really old...
@@apoorvmishra6992 I'm 61, though I wouldn't consider that "really old" these days! How old are you?
Yeah I was just kidding. No one is old, until they consider themselves one.
I'm 22 and preparing for entrances to get into a good university to pursue Masters in physics.
Nice knowing another physicist.
@@apoorvmishra6992 Same here! Best of luck to you! I'm happy to help you with physics if you need a tutor.
@@dcterr1 Thanks for the offer Sir. I can really use your help as I've self learned physics so far and want to study Lie Theory to get a deeper understanding of Spinors. Maybe you teaching me would make the process faster.
Beautiful exposition, thank you...
Amazing video! Thank you for sharing
Wow! This Nicely explains many things from prelim Quantum mechanics. I realised the connection between Generators of rotation and the rotations themselves and why the generators are exponentiated..❤
Cool 😎
I am glad you started this series 🙂👏
Truly insightful. I wonder if an analogy can be made between the motivation for Lie algebra and that for linear algebra/matrices. A lot of textbooks on linear algebra start with matrices & determinants and delay the discussion of linear transforms, possibly because matrix calculations are more "tangible" while linear transforms are more "abstract" in some way. You can easily calculate things to get meaningful results once you grasp the basic rules. But it's the linear transforms that are the true objects of interest, while the matrices seem to be the tools invented to study linear transforms. As I delved deeper into the study of physics, my appreciation of the significance of linear transforms grew. I have not studied Lie theory, but from what I learned in this video, it seems to follow the same logic. Lie groups (like linear transforms) are difficult to deal with on their own, so we turn to Lie algebra (like matrices) which allow for easier and more straightforward calculation. Is this so?
Wonderful video! Thank you for making this!
Nice material! Many concepts to follow. 💭
I came across this vid by accident, understood 40% of it at best. Had a blast
Awesome video as always!
However, I'd like to add one small detail. When talking about Lie algebras around 11:30 one must be careful to not confuse the way shown with 'simply taking the imaginary part'. There is a reason why he said: we *correspond* it to a point on the Lie group.
This detail can be a stumbling block for those not listening carefully like me for example.
This was interesting, I remember first trying to look into these topics and being amazed, this just reinforces that! Higher level physics and theoretical physicist use these, or at least have it in their tool box.
This is such a well made video!!
What a wonderful video!!!
Outstanding. Thank you.
Sir, please make a playlist about *Essence of Real Analysis* your video is helpful for us.
❤
Beautifully explained presentation! @mathemaniac on this subject, have you ever come across the paper by Doran, Hestenes, Sommen, and Acker titled "Lie groups as Spin Groups", where apparently the authors show that "every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group"?
This video was very helpful for me!
I'm impressed and not for the reason you might think. Out of the hundreds of people on RUclips that fail using the word "basically" in the correct way, YOU used it in the correct way. It means you REALLY DO know what you are talking about! BTW I already subscribed long ago but I never made a comment until now.
Excellent work
Very nice video!!!! An obvious idea for a future video: "defining" the Lie bracket using the BCH makes the Lie bracket look like something pretty hard to grasp precisely, but in reality it's not. You can draw Lie brackets of vector fields on a manifold in general, then (left or whatever) invariant vector fields on the Lie group. Everything in this story is sufficiently geometric to be drawable. It's challenging, but with your animation skills I think it can be done and I think it would be of interest both to people who are learning the basics and to people who already know this stuff pretty well.
While here I used the BCH as a motivation for why we consider the Lie algebra, I didn't plan to use BCH to introduce it in that future video. Many people use commutators, which I think isn't strong enough of a motivation because it isn't immediately obvious why commutators are useful.
Yes, I understand that we can use vector fields on a manifold, because Lie brackets are also just something that we could use on a manifold, but I am not planning to use it, but rather, if we are thinking about matrix Lie groups, we have a bit of nice intuition for what [X,Y] should be. This intuition, by the way, would tell you why tr(AB) = tr(BA).
Brilliant explanation 👏
If you are able to build an intuitive understanding of the monster group, you’ll be my hero.
really love this vid
Great video! Thank you
We are waiting for the next installment ❤❤
well, SO(3) as a ball... well, you just have to imagine such a ball, that has each pair of it's opposite points glued together. For example, rotation around Z axis for Pi clockwise and rotation for Pi counterclockwise - is the same rotation; so in this model you have to glue the point at top of the ball with the point at the bottom; and the same goes for each other possible rotation axis. Quite an interesting ball :) Something resembeling a projective plane :)
Pictures are scary but we're familiar with unit circle which has natural coordinates. The way we can make sense of these groups is via coordinates.
Math ppl could come up with any random name composed of and i would totally buy that it's an actual field of study. Graph theory, group theory, knot theory, field theory, ring theory, etc. Wouldn't be surprised if a Pumpkin theory existed
Beautiful teach
genius man continue please
Great video! Commenting to boost the video
Waw . This is genius visual presentation
Please make more vidéos . This really help for hard maths
i love these videos
waiting for the next video ! ! !
"Nice explanation, even for a layman"
This reminds me of Quote:
"If you can't explain it to a six year old, you don't understand it yourself ~ Albert Einstein"
Gracias, soy estudiante de pregrado con interes en estos temas matemáticas y físicos. Muy buena explicación.
Pdst: Espero para el siguiente capítulo..
Thank you very much!
You changed the world🎉❤
thank you so much!
I love this man
Thank you 🙏!