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 RU-vid 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 RU-vid 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 😳🤯🤓🖖🎶
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
@@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
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.
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
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?
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 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 :)
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.
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
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 ☺️
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!
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!!
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....
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.
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!
"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"
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.
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.
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.
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!
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!
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.
@@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.
I'm impressed and not for the reason you might think. Out of the hundreds of people on RU-vid 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.
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
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..❤
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.
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).
I'm curious about something. Towards the end of this video, you mentioned simple groups and simple Lie algebras and exceptional Lie algebras. I've known about all of these for some time, but now I wonder if there's more of a connection between them than just a useful analogy. I'm intrigued by the facts that simple groups can be divided into 18 infinite families and 26 or 27 sporadic ones, and similarly, simple Lie algebra can be divided into 4 infinite families and 5 exceptional ones. Are there any deeper connections between simple groups and simple Lie algebras than the ones you mentioned? For instance, is there a deep connection between the monster group and E8, and does this have anything to do with Monstrous Moonshine?
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"?
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 :)
If you want to see how exceptional E8 actually is, check out Skip Garibaldi's survey "E8 the most exceptional group". Skip is basically the godfather of algebraic groups next to Tits and Borovoi and has provided countless results in the field, especially about E8. Needless to say, that the survey is absolutely hardcore compared to this video.
And manipulations on Shirley's Surface give renormalization as being a single sided surface it connects minima and maxima through an inversion. The radially symmetric Klein bottle? Surface(cos(u/2)cos(v/2),cos(u/2)sin(v/2),sin(u)/2),u,0,2pi,v,0,4pi
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 wouldn't say that I know a lot about Tits group, but from my understanding, it is kind of in an embarrassing position which isn't exactly generated the same way as one of these infinite families, but very closely related. I doubt it is really to do with the jokes of the name if some authors decided to include it in one of the infinite families.
Nice video! For WP attribution creative commons copyright - the E8 Petrie projection image in the opening and elsewhere was created by me. Fixed it for you....
