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 UA-cam 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 UA-cam 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
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!
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.
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!
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🎉🎊
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!!
after a whole semester studying Lie Algebra, it was these videos that helped me understand the connection between lie groups and lie algebras. THANK YOU MY GOOD MAN
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
This series is simply amazing! I especially appreciate the time spent on the background and motivation of the topics, including the historical overview in the first video. Your style does a fantastic job of building a very natural framework for the subsequent ideas to 'stick to.' I am studying physics and I feel this has given me a whole new perspective on the framework of classical and quantum mechanics.
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.
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!
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?
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.
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
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....
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 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!!
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 ☺️
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!
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!
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.
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.
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 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.
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!
I'm impressed and not for the reason you might think. Out of the hundreds of people on UA-cam 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.
13:03 is it not misleading to have i*theta be the exact same vertical height as e^i*theta? If the 2 lines are the same length e^i*theta should be lower.
They aren't the exact same length, even in the video - but I agree the difference is small. I literally just input that in my software, and they are actually i theta and e^(i theta)! It might just goes to show how good of an approximation sin(theta) ≈ theta is.
E8 was mentioned some years ago, because it's apparently related to string theory in some sense. I didn't understand anything of it and as you said Wikipedia researching Lie Algebras didn't help me much. I'm not sure I understand it now, but at least I got some idea of what a Lie Algebra is. Thank you!
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..❤
I just released a possible breakthrough paper related to 3 manifolds and their Lie algebra/group dynamics & connections to physical theories. Its called Grand Unified Theory Using Thurston's Geometrization Conjecture (researchgate) DOI: 10.13140/RG.2.2.13631.28327 . You said you had a good math background, it is not too technical or abstract. I wonder what you think of it, anyone else please comments are welcome❤
I couldn't follow the technicalities, but the gist of the paper is that matter arises from a particular geometry of an underlying structure, correct? I'm not sure I understood how general relativity is connected to quantum mechanics in this view, or how it "completes the circle".
@@FunkyDexter general relativity established the relation between spacetime curvature ( and the spacetime manifold in general) and the matter that is causing the curvature, only it did so on a macroscopic scale only, mostly dealing with celestial objects. This relation, this deep principle, was completely obscured in the view of quantum mechanics (submicroscale ) And particularly, the matter content curves the space that is present, but on the quantum mechanical scale it is the other way around (so completing the circle so to say). That is something new in the context of my paper I believe, I have spoken to many experts on it. So I think my idea has a thing to say about this, coming with evidence too, the subatomic geometrical spaces itself (and their curvatures), are enabling and creating particles, they are able to since they are naturally so extremely rich in dynamics. Another deep connection is that both dynamics are acting on the same space... (QM and GR) There is a unity of the theories when looking at it with a "manifold"/geometric lense .
@@Unidentifying it's interesting, because in my reading I came across multiple mentions of a modern aether with regards to general relativity. I've long suspected that fundamental particles are really just "knots" in the fabric of reality, and every known property of these particles arise as a particular configuration of the fabric. For example, charge is entirely topological in this view, a "vortex" like those of kelvin. Counter rotating vortices attract, isn't that curious? I've also encountered the hopf fibration multiple times with regards to the electron, and spin being fundamentally a rotation in 4 dimensions. And all the higher particle families, like muon or Tau, simply unstable ephemera, excitations of the most stable structure of the electron. There is just so much circumstantial evidence, but the culprit still eludes me, perhaps because I don't grasps the advanced mathematics required. It's all just intuition. Another gripe I still have is, what is it then this fundamental field whose vibrations and distortions are the energy that make up our existence? What is the 4th dimension, is it really time? If it is, does that mean our perception of time as proceeding from past to the future is an illusion? I firmly believe that if we understood this, it would open up our way to the stars.
it is truly amazing that nearly everything you are saying before your last paragraphs I connected with the mathematics and also wrote down. A slight problem for me as well, is that my paper is lacking the real mathematical rigorous derivations and calculations to check it further (don't feel bad dude the mathematics is truly insane, I didn't make most of that, just made all the connections etc). already have some good eyes on it@@FunkyDexter Let me think about your questions
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?
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).
The statement is that there is such an h, so in a group, if g * h = e, then g * h = e = h * g. But it doesn't mean that g * h = h * g in general (just one very special case that is true).
13:16 This is nitpicking, but I feel it is slightly confusing/misleading to say that general manifolds have exponential maps, as this requires a manifold as well as a metric.
Yes - though technically you need a "connection", slightly weaker than a metric. I said "general" in the sense that "not just Lie groups", but then it sounds like all manifolds.
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 :)
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"?
13:30 - man i always got so confused by these subjects. this intution just makes it so obvious what's been written everywhere. A while ago I had this realization for "pi" showing up everywhere, even whough pi seems to be a lot more wild than exp and log. to be honest all of this e^thing thing to me is like a hack we found in mathematics, now at least I have something to imagine how we're hacking it.
Can't wait but I still think you won't convince me it's not a hack. Calling it a hack mostly means you can use it to "hack" other problems. What I think will happen is I'll discover how does hacking math works xD. Anyway, let's wait and see.
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.
I first read LA as linear algebra... but yes that's true - I am just thinking about it as a coordinate system because it makes a little more sense to me.
After watching the video and actually seeing the connection (and by that I mean "exponential") between the Lie algebras and Lie groups, this does appear to be an accurate description of the topic.
@@sumairahmad9464 so, do you know what it means to linearize a function at a point? It means to find a linear (... or affine) function from R^n to R^m (for some n and m) which, for inputs near that point, gives a good approximation of the original function. This can be done whenever the function is differentiable at that point. Now, in a Lie group, the operation of multiplication of two elements of the Lie group, is a smooth, and therefore differentiable, function. The domain and codomain aren’t (generally) exactly of the form R^n and R^m , but because they are smooth manifolds, they can be locally approximated by such spaces. (In this case, m is the dimension of the Lie group and 2n is twice that) So, this allows us to do the process of “taking a linear approximation of a function at a point” to the function that sends any pair of Lie group elements (g,h) , to the Lie group element g•h . Specifically, we can do this where the point where we take this linear approximation, is the identity element of the Lie group. .... Uh... except that’s... not *quite* what we want to do, I think...? (I am not as familiar with the theory of Lie groups as either of them) Rather, to obtain the Lie bracket, I think we want to consider something like, for each element g of the Lie group, the function that sends any element h of the Lie group to g•h•(g^(-1)), and then take the linear approximation of that, for h near the identity, then, we want to take the linear approximation of the function sending elements g to (the linear approximation functions described above). That, I think, should give us our Lie bracket. For concreteness, suppose that our Lie group is a group of matrices. We know how to define an exponential of a matrix; it is defined by the power series for the exp(x)=e^x function, applied to the matrix. Because I have made a probably bad choice of variable names, I’m going to now make a probably worse choice of variable names, because I’m writing this on phone and don’t want to rewrite, or think of better names. Sorry. suppose that the matrix g is of the form exp(dg) , where “dg” is the name of a matrix. While this name is inspired by differentials, I don’t actually mean for dg to be a differential, it is just a matrix. Again, sorry for the bad choice of names. Similarly, let h be of the form exp(dh) for some matrix dh. g^(-1) = exp(-dg). So, g•h•(g^(-1)) = exp(dg) • exp(dh) • exp(-dg) To take a linear approximation of this near h the identity matrix, well, that should be near dh the zero matrix, And, well, the gradient of exp(dh) with respect to dh (see how bad a choice of variable name?) is... well... at dh=0, the gradient in direction... let’s say “ dh’ “, is just dh’ . So, if we take the gradient of exp(dg) • exp(dh) • exp(-dg) with respect to dh, at dh=0, in the direction dh’ , we get exp(dg) • dh’ • exp(-dg) . If we now do the same thing to this, but now taking the gradient of dg , at dg=0, in the direction dg’ , well, we have to do the product rule, and also because of that minus sign, also the chain rule. we end up with dg’ • dh’ • 1 + 1• dh’ • (- dg’) = dg’ • dh’ - dh’ • dg’ which is the commutator [dg’ , dh’ ] of the matrices dg’ and dh’ . Now, more generally in Lie Theory, the groups and algebras don’t need to be matrices, but a similar kind of thing will work, and, the thing they are talking about, with the linearization, is rather like the thing I described in the special case of matrices. ( I guess whenever I took those gradients, I should maybe have added the identity matrix to the result, so that it actually forms a linear approximation? But, the commutator (not the commutator plus identity matrix) is the part that corresponds to the Lie bracket (or, *is* the Lie bracket in these cases with matrix groups), and there’s not really a need for the “add on the identity element” part generally, because we can choose the coordinate chart for the manifold to have the identity element of the group correspond to the 0 of the vector space... ) Because linearization, if dh and dg are sufficiently small, then using them in the expression 1 + (dg • dh - dh • dg) (with dg replacing dg’ and dh replacing dh’) (Where 1 is the identity matrix) will be a good approximation to g•h•(g^(-1)) = exp(dg) • exp(dh) • exp(-dg) . So... I hope that answers your question? Edit: corrected a small bit talking about exponential of a matrix
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.
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.
This is a video series (the third part), so you can go check out the previous parts if you want to. There are also a video series on group theory that I did in the past on this channel as well.
"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"
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
Just one point is not obvious. We have said that each vector in tangent space correspond to a point in manifold. Is this only valid in a small region around the point through which the tangent space passes, or is it valid for all tangent vectors and all points in manifold?
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 UA-cam 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 UA-cam 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
Simply fantastic. Finally youtube mathematics at a higher level
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
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.
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!
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🎉🎊
Thanks!
Thanks! Excellent presentation.
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!!
after a whole semester studying Lie Algebra, it was these videos that helped me understand the connection between lie groups and lie algebras. THANK YOU MY GOOD MAN
Came for lie theory, stayed for tits group
Comment for the algorithm
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
This series is simply amazing! I especially appreciate the time spent on the background and motivation of the topics, including the historical overview in the first video. Your style does a fantastic job of building a very natural framework for the subsequent ideas to 'stick to.' I am studying physics and I feel this has given me a whole new perspective on the framework of classical and quantum mechanics.
Such a revelation! Thank you!
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.
"this group is called the Tits group"
me: hah
"who actually died a year ago"
me: :-(
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 series everyone has been waiting for! So great!
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?
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.
Wow, a whole series is coming! That's highLie appreciated.
I'm studying particle physics and you make my life easier on a daily basis. Thanks for your perfect videos
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
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....
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.
Please continue this series! It's helping me so much
This is the best presentation I have ever seen on the Lie groups and Lie algebras.
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 :)
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 ☺️
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!
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!
Loved the perfect rundown of groups and manifolds
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.
really love this vid
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.
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".
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.
19:15 - The French pronunciation is more like "teets", to rhyme with "sweets". I tired your pronunciation once in a lecture and someone corrected me.
Oh right - I should probably check my eyes when I glossed over the IPA given - it was /i/ and not /ɪ/.
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!
This is an amazing video! Please let me know when the next Lie group video is online!
As a PhD phsyics student, thank you so much for helping visualize this
this was amazing! thank you for the ride through math castle!
Hi Trevor, this video is utterly beautiful. Great, great work you should be really proud. You’ve done very well
I am excited for the rest of this series!
Excellent video, I am a student studying Lie Theory and it's really satisfying seeing E8 explained
Thx. Tried and failed to understand this in the past. Good motivator overview.
This is one of my favorite series and is a fun part of maths to learn.
Exceptional work! Could you show the connection of SU(3) and Gell-Mann's baryon octets?
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!
I'm impressed and not for the reason you might think. Out of the hundreds of people on UA-cam 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.
I came across this vid by accident, understood 40% of it at best. Had a blast
Your explanation is unmatchable!!!! 🔥🔥🔥🔥🔥🔥
Thank you for the amazing video, and all the references in the description!
seriously, reading math topics on wiki is the most intimidating thing in the world
More work needs to be done on this important topic. Thanks for simplifying.
One of my favorite channels :) 😊
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 =)
13:03 is it not misleading to have i*theta be the exact same vertical height as e^i*theta? If the 2 lines are the same length e^i*theta should be lower.
They aren't the exact same length, even in the video - but I agree the difference is small. I literally just input that in my software, and they are actually i theta and e^(i theta)! It might just goes to show how good of an approximation sin(theta) ≈ theta is.
@@mathemaniac Wow that really is a good approximation. Thank you
E8 was mentioned some years ago, because it's apparently related to string theory in some sense. I didn't understand anything of it and as you said Wikipedia researching Lie Algebras didn't help me much. I'm not sure I understand it now, but at least I got some idea of what a Lie Algebra is. Thank you!
Thank you so much. Very helpful. Looking forward to the rest of these videos
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..❤
Very nice explanation of the exponential map
A video of the highest quality of this kind!
I just released a possible breakthrough paper related to 3 manifolds and their Lie algebra/group dynamics & connections to physical theories. Its called Grand Unified Theory Using Thurston's Geometrization Conjecture (researchgate) DOI: 10.13140/RG.2.2.13631.28327 . You said you had a good math background, it is not too technical or abstract. I wonder what you think of it, anyone else please comments are welcome❤
I couldn't follow the technicalities, but the gist of the paper is that matter arises from a particular geometry of an underlying structure, correct?
I'm not sure I understood how general relativity is connected to quantum mechanics in this view, or how it "completes the circle".
@@FunkyDexter general relativity established the relation between spacetime curvature ( and the spacetime manifold in general) and the matter that is causing the curvature, only it did so on a macroscopic scale only, mostly dealing with celestial objects. This relation, this deep principle, was completely obscured in the view of quantum mechanics (submicroscale )
And particularly, the matter content curves the space that is present, but on the quantum mechanical scale it is the other way around (so completing the circle so to say). That is something new in the context of my paper I believe, I have spoken to many experts on it. So I think my idea has a thing to say about this, coming with evidence too, the subatomic geometrical spaces itself (and their curvatures), are enabling and creating particles, they are able to since they are naturally so extremely rich in dynamics.
Another deep connection is that both dynamics are acting on the same space... (QM and GR)
There is a unity of the theories when looking at it with a "manifold"/geometric lense .
@@FunkyDexter you are correct, please let me know if I can explain further in details, thank you for the message
@@Unidentifying it's interesting, because in my reading I came across multiple mentions of a modern aether with regards to general relativity. I've long suspected that fundamental particles are really just "knots" in the fabric of reality, and every known property of these particles arise as a particular configuration of the fabric. For example, charge is entirely topological in this view, a "vortex" like those of kelvin. Counter rotating vortices attract, isn't that curious? I've also encountered the hopf fibration multiple times with regards to the electron, and spin being fundamentally a rotation in 4 dimensions. And all the higher particle families, like muon or Tau, simply unstable ephemera, excitations of the most stable structure of the electron. There is just so much circumstantial evidence, but the culprit still eludes me, perhaps because I don't grasps the advanced mathematics required. It's all just intuition.
Another gripe I still have is, what is it then this fundamental field whose vibrations and distortions are the energy that make up our existence? What is the 4th dimension, is it really time? If it is, does that mean our perception of time as proceeding from past to the future is an illusion?
I firmly believe that if we understood this, it would open up our way to the stars.
it is truly amazing that nearly everything you are saying before your last paragraphs I connected with the mathematics and also wrote down. A slight problem for me as well, is that my paper is lacking the real mathematical rigorous derivations and calculations to check it further (don't feel bad dude the mathematics is truly insane, I didn't make most of that, just made all the connections etc). already have some good eyes on it@@FunkyDexter Let me think about your questions
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?
Shouldn't at 9:39, it be n-dimensional Euclidean space , the last line?
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).
At 5:03 shouldn’t the inverse property just be g*h = e? the g*h = h*g isn’t necessarily true unless the group is abelian
The statement is that there is such an h, so in a group, if g * h = e, then g * h = e = h * g. But it doesn't mean that g * h = h * g in general (just one very special case that is true).
13:16 This is nitpicking, but I feel it is slightly confusing/misleading to say that general manifolds have exponential maps, as this requires a manifold as well as a metric.
they will have a naturally endowed metric coming from the particular geometry
Yes - though technically you need a "connection", slightly weaker than a metric. I said "general" in the sense that "not just Lie groups", but then it sounds like all manifolds.
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 :)
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"?
13:30 - man i always got so confused by these subjects. this intution just makes it so obvious what's been written everywhere.
A while ago I had this realization for "pi" showing up everywhere, even whough pi seems to be a lot more wild than exp and log.
to be honest all of this e^thing thing to me is like a hack we found in mathematics, now at least I have something to imagine how we're hacking it.
It isn't actually a hack - there is an actual reason why exponential map is so useful. Stay tuned for the next video.
Can't wait but I still think you won't convince me it's not a hack.
Calling it a hack mostly means you can use it to "hack" other problems. What I think will happen is I'll discover how does hacking math works xD. Anyway, let's wait and see.
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.
Gracias, soy estudiante de pregrado con interes en estos temas matemáticas y físicos. Muy buena explicación.
Pdst: Espero para el siguiente capítulo..
Beautiful exposition, thank you...
Holy shit this is the best math video I've ever watched
Great video. I think of LA as linearizing about a group element, typically the identity
I first read LA as linear algebra... but yes that's true - I am just thinking about it as a coordinate system because it makes a little more sense to me.
@@mathemaniac @cdenn016 Could any of you two explain it a bit more. The only part I got is that LA is Lie Algebra :)
After watching the video and actually seeing the connection (and by that I mean "exponential") between the Lie algebras and Lie groups, this does appear to be an accurate description of the topic.
@@sumairahmad9464 so, do you know what it means to linearize a function at a point? It means to find a linear (... or affine) function from R^n to R^m (for some n and m) which, for inputs near that point, gives a good approximation of the original function. This can be done whenever the function is differentiable at that point.
Now, in a Lie group, the operation of multiplication of two elements of the Lie group, is a smooth, and therefore differentiable, function. The domain and codomain aren’t (generally) exactly of the form R^n and R^m , but because they are smooth manifolds, they can be locally approximated by such spaces. (In this case, m is the dimension of the Lie group and 2n is twice that)
So, this allows us to do the process of “taking a linear approximation of a function at a point” to the function that sends any pair of Lie group elements (g,h) , to the Lie group element g•h .
Specifically, we can do this where the point where we take this linear approximation, is the identity element of the Lie group.
....
Uh... except that’s... not *quite* what we want to do, I think...? (I am not as familiar with the theory of Lie groups as either of them)
Rather, to obtain the Lie bracket, I think we want to consider something like, for each element g of the Lie group, the function that sends any element h of the Lie group to g•h•(g^(-1)),
and then take the linear approximation of that, for h near the identity,
then, we want to take the linear approximation of the function sending elements g to (the linear approximation functions described above).
That, I think, should give us our Lie bracket.
For concreteness, suppose that our Lie group is a group of matrices.
We know how to define an exponential of a matrix; it is defined by the power series for the exp(x)=e^x function, applied to the matrix.
Because I have made a probably bad choice of variable names, I’m going to now make a probably worse choice of variable names, because I’m writing this on phone and don’t want to rewrite, or think of better names. Sorry.
suppose that the matrix g is of the form exp(dg) , where “dg” is the name of a matrix. While this name is inspired by differentials, I don’t actually mean for dg to be a differential, it is just a matrix. Again, sorry for the bad choice of names.
Similarly, let h be of the form exp(dh) for some matrix dh.
g^(-1) = exp(-dg).
So, g•h•(g^(-1)) = exp(dg) • exp(dh) • exp(-dg)
To take a linear approximation of this near h the identity matrix, well, that should be near dh the zero matrix,
And,
well, the gradient of exp(dh) with respect to dh (see how bad a choice of variable name?) is...
well... at dh=0, the gradient in direction... let’s say “ dh’ “, is just dh’ .
So, if we take the gradient of exp(dg) • exp(dh) • exp(-dg) with respect to dh, at dh=0, in the direction dh’ , we get
exp(dg) • dh’ • exp(-dg)
.
If we now do the same thing to this, but now taking the gradient of dg , at dg=0, in the direction dg’ ,
well, we have to do the product rule, and also because of that minus sign, also the chain rule.
we end up with dg’ • dh’ • 1 + 1• dh’ • (- dg’)
= dg’ • dh’ - dh’ • dg’
which is the commutator [dg’ , dh’ ] of the matrices dg’ and dh’ .
Now, more generally in Lie Theory, the groups and algebras don’t need to be matrices,
but a similar kind of thing will work,
and, the thing they are talking about, with the linearization, is rather like the thing I described in the special case of matrices.
(
I guess whenever I took those gradients, I should maybe have added the identity matrix to the result, so that it actually forms a linear approximation?
But, the commutator (not the commutator plus identity matrix) is the part that corresponds to the Lie bracket (or, *is* the Lie bracket in these cases with matrix groups),
and there’s not really a need for the “add on the identity element” part generally, because we can choose the coordinate chart for the manifold to have the identity element of the group correspond to the 0 of the vector space...
)
Because linearization, if dh and dg are sufficiently small, then using them in the expression
1 + (dg • dh - dh • dg) (with dg replacing dg’ and dh replacing dh’)
(Where 1 is the identity matrix)
will be a good approximation to
g•h•(g^(-1)) = exp(dg) • exp(dh) • exp(-dg)
.
So...
I hope that answers your question?
Edit: corrected a small bit talking about exponential of a matrix
10:45 flat map image actually should be more distorted due to projection issues.
I deliberately chose the "equirectangular projection", which actually was used well before Mercator.
11:39 shouldn't "distance" be "radius" here? I got confused and thought it's gonna be chord length lol
Nice material! Many concepts to follow. 💭
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.
I've honestly wondered if the Tits group is included in the infinite families to cut down on the 'giggle' factor involved in mentioning it.
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.
@@mathemaniac I'm certain your knowledge of the situation far exceeds mine; I'm more of an analysis guy than algebraist.
Can you do a video that's a prequel to this video? I didn't fully understand a lot of things.
This is a video series (the third part), so you can go check out the previous parts if you want to. There are also a video series on group theory that I did in the past on this channel as well.
Don't worry about subs, just keep cranking. Good things will come.
"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"
Exceptionally well explained!
🏆🙏
One of the greatest videos.....awesome...thank you for your contribution..👌👌👏👏👏👏👏👍👍👍
Cool 😎
I am glad you started this series 🙂👏
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
Waw . This is genius visual presentation
Please make more vidéos . This really help for hard maths
Just one point is not obvious. We have said that each vector in tangent space correspond to a point in manifold. Is this only valid in a small region around the point through which the tangent space passes, or is it valid for all tangent vectors and all points in manifold?