Mathemaniac
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Lie algebras visualized: why are they defined like that? Why Jacobi identity?
Can we visualise Lie algebras? Here we use the “manifold” and “vector field” perspectives to visualise them. In the process, we can intuitively understand tr(AB) = tr(BA), which is one of the “final goals” of this video. The other is the motivation of the Jacobi identity, which seems random, but actually isn’t.
Files for download:
Go to www.mathemaniac.co.uk/download and enter the following password: whyJacobiidentity
Previous videos are compiled in the playlist: ua-cam.com/play/PLDcSwjT2BF_WDki-WvmJ__Q0nLIHuNPbP.html
Individually:
Part 1: ua-cam.com/video/IlqVo3sJFLE/v-deo.html (intro and motivation)
Part 2: ua-cam.com/video/erA0jb9dSm0/v-deo.html (on SO(n), SU(n) notations)
Part 3: ua-cam.com/video/ZRca3Ggpy_g/v-deo.html (overview of Lie theory)
Part 4: ua-cam.com/video/9CBS5CAynBE/v-deo.html (exponential map on exotic objects)
Part 5: ua-cam.com/video/B2PJh2K-jdU/v-deo.html (on visualising trace)
Videos from other channels that overlap with my previous ideas:
ua-cam.com/video/ACZC_XEyg9U/v-deo.html [only referring to the topology part, as I have issues with using the belt trick to explain spin 1/2, see my previous spin 1/2 video description]
ua-cam.com/video/b7OIbMCIfs4/v-deo.html [specifically the “homotopy classes” part]
ua-cam.com/video/Q_RUDQkDsE0/v-deo.html [the “higher-spin” representations]
Apart from @eigenchris video, technically the videos are not specifically talking about Lie groups / algebras in general, but the arguments to be presented are too similar to what I have in mind.
Source:
(1) people.reed.edu/~jerry/332/projects/venkatamaran.pdf basically what I say, without the vector field visualisations]
(2) www.damtp.cam.ac.uk/user/ho/S3prob.pdf [focus on Q2: a much more tedious approach to motivate Jacobi identity]
(3) en.wikipedia.org/wiki/Directional_derivative [actually quite useful, touches upon many ideas in the video series]
(4) projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-20/issue-1-2/A-new-proof-of-the-Baker-Campbell-Hausdorff-formula/10.2969/jmsj/02010023.full [not related, but since I am likely not continuing the video series, this is a simpler proof of the BCH formula, but only why knowing the Lie algebra is enough]
Video chapters:
00:00 Introduction
00:52 Chapter 1: Two views of Lie algebras
05:29 Chapter 2: Lie algebra examples
14:44 Chapter 3: Simple properties
21:18 Chapter 4: Adjoint action
30:15 Chapter 5: Properties of adjoint
39:30 Chapter 6: Lie brackets
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!
SUBSCRIBE and see you in the next video!
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.
Social media:
Facebook: mathemaniacyt
Instagram: _mathemaniac_
Twitter: mathemaniacyt
Patreon: www.patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: mathemaniac.myspreadshop.co.uk
Ko-fi: ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
Переглядів: 53 120

Відео

Matrix trace isn't just summing the diagonal | Lie groups, algebras, brackets #5
Переглядів 198 тис.5 місяців тому
Part 6: ua-cam.com/video/gj4kvpy1eCE/v-deo.html Can we visualise this algebraic procedure of adding diagonal entries? What is really happening when we add them together? By visualising it, it is possible to almost immediately see how the different properties of trace comes about. Files for download: Go to www.mathemaniac.co.uk/download and enter the following password: traceisdiv The concept of...
Can we exponentiate d/dx? Vector (fields)? What is exp? | Lie groups, algebras, brackets #4
Переглядів 118 тис.5 місяців тому
Part 5: ua-cam.com/video/B2PJh2K-jdU/v-deo.html Can we exponentiate vectors? What does e^(d/dx) mean? Does it make sense to exponentiate a whole bunch of vectors? Well yes! While what these exponentials do seem very different at first, they can be recast into the same framework. Files for download: Go to www.mathemaniac.co.uk/download and enter the following password: expderivativeshift CHAPTER...
What is Lie theory? Here is the big picture. | Lie groups, algebras, brackets #3
Переглядів 294 тис.10 місяців тому
Part 4: ua-cam.com/video/9CBS5CAynBE/v-deo.html 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...
How to rotate in higher dimensions? Complex dimensions? | Lie groups, algebras, brackets #2
Переглядів 63 тис.11 місяців тому
Part 3: ua-cam.com/video/ZRca3Ggpy_g/v-deo.html Around 11:50, can't imagine that this error got in - it should have been SU(n) = {U in U(n), det U = 1}. Orthogonal and unitary groups. Rotational symmetries, real and complex, are particularly useful in the field of Lie theory, because their (complexified) Lie algebras, together with that of the symplectic group Sp(n), are the only infinite famil...
Why study Lie theory? | Lie groups, algebras, brackets #1
Переглядів 73 тис.11 місяців тому
Next video: ua-cam.com/video/erA0jb9dSm0/v-deo.html Lie’s theory of continuous symmetries was originally for differential equations, but turns out to be very useful for physics because symmetries are manifest in many physical systems. This is the start of a series on Lie groups, Lie algebras, and Lie brackets. Files for download: Go to www.mathemaniac.co.uk/download and enter the following pass...
How many terms do we need in the exp series?
Переглядів 47 тис.11 місяців тому
The exponential function is expressed in terms of the series, but practically, how many terms do we need to compute it to some desired accuracy? Well, it turns out the answer to that is quite interesting. Oscar's video on how computers actually compute the exponential series: ua-cam.com/video/NorMJ5PO3Hc/v-deo.html For the files created for this video, please visit www.mathemaniac.co.uk/downloa...
Queuing theory and Poisson process
Переглядів 85 тис.Рік тому
Queuing theory is indispensable, but here is an introduction to the simplest queuing model - an M/M/1 queue. Also included is the discussion on Poisson process, which is the underlying assumption for the M/M/1 queue. Second channel video on variance of Poisson distribution: ua-cam.com/video/xiX1QoX8i6M/v-deo.html To me, this is mainly a "prequel" which serves as a prerequisite for the next vide...
Theorema Egregium: why all maps are wrong
Переглядів 22 тис.Рік тому
Head to squarespace.com/mathemaniac to save 10% off your first purchase of a website or domain using code mathemaniac. Website for files download (remember to use the password shown in the video!): www.mathemaniac.co.uk/download The Mercator projection is the standard world map, but it famously makes Greenland and Africa the same size, but in reality, Greenland is so much smaller. Gall-Peters p...
The biggest misconception about spin 1/2
Переглядів 85 тис.Рік тому
Head to squarespace.com/mathemaniac to save 10% off your first purchase of a website or domain using code mathemaniac. My website link: www.mathemaniac.co.uk/about “If you rotate a spin 1/2 particle by 360 degrees, it doesn’t go back to its original state, rather you need 720 degrees”. This is only technically correct if you interpret the words “rotate” and “state” in the way it’s intended, and...
I use PowerPoint to edit all* videos (and hit 100k subs!)
Переглядів 46 тис.Рік тому
Head to squarespace.com/mathemaniac to save 10% off your first purchase of a website or domain using code mathemaniac. My blog on how to use GeoGebra: www.mathemaniac.co.uk/blog/how-i-use-geogebra-to-make-animations-in-videos-part-i Yes, I use PowerPoint to edit all* my videos... This is actually a much more efficient way of making videos, and yet it can be made to look professional, like most ...
The deeper meaning of matrix transpose
Переглядів 355 тис.Рік тому
100k Q&A: forms.gle/dHnWwszzfHUqFKny7 Transpose isn’t just swapping rows and columns - it’s more about changing perspective to get the same measurements. By understanding the general idea of transpose of a linear map, we can use it to visualise transpose much more directly. We will also heavily rely on the concept of covectors, and touch lightly on metric tensors in special/general relativity, ...
How hard was my Cambridge interview? (ft. @TomRocksMaths)
Переглядів 438 тис.Рік тому
To prove once and for all that Cambridge is better than O*ford 😤 (just kidding) Tom's channel: ua-cam.com/users/TomRocksMathsfeatured 100k Q and A Google form: forms.gle/WtpM3HqhkdPuwsbP9 You might recognize Tom from his Navier-Stokes equations on the Numberphile channel, and he is a tutor at Oxford teaching students at St. Edmund Hall college in Oxford. I just thought it was funny to have a Ca...
Random walks in 2D and 3D are fundamentally different (Markov chains approach)
Переглядів 613 тис.Рік тому
Second channel video: ua-cam.com/video/KnWK7xYuy00/v-deo.html 100k Q&A Google form: forms.gle/BCspH33sCRc75RwcA "A drunk man will find his way home, but a drunk bird may get lost forever." What is this sentence about? In 2D, the random walk is "recurrent", i.e. you are guaranteed to go back to where you started; but in 3D, the random walk is "transient", the opposite of "recurrent". In fact, fo...
The weirdest paradox in statistics (and machine learning)
Переглядів 1,1 млнРік тому
🌏 AD: Get Exclusive NordVPN deal here ➼ nordvpn.com/mathemaniac. It's risk-free with Nord's 30-day money-back guarantee! ✌ Second channel video: ua-cam.com/video/3ne9yghOtw8/v-deo.html Stein's paradox is of fundamental importance in modern statistics, introducing concepts of shrinkage to further reduce the mean squared error, especially in higher dimensional statistics that is particularly rele...
Why you can't solve quintic equations (Galois theory approach) #SoME2
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Why you can't solve quintic equations (Galois theory approach) #SoME2
The simplest reason why π/4 = 1 - 1/3 + 1/5 -...
Переглядів 93 тис.2 роки тому
The simplest reason why π/4 = 1 - 1/3 1/5 -...
The geometric interpretation of sin x = x - x³/3! + x⁵/5! -...
Переглядів 303 тис.2 роки тому
The geometric interpretation of sin x = x - x³/3! x⁵/5! -...
Complex integration, Cauchy and residue theorems | Essence of Complex Analysis #6
Переглядів 320 тис.2 роки тому
Complex integration, Cauchy and residue theorems | Essence of Complex Analysis #6
What if we define 1/0 = ∞? | Möbius transformations visualized
Переглядів 149 тис.2 роки тому
What if we define 1/0 = ∞? | Möbius transformations visualized
What do complex functions look like? | Essence of complex analysis #4
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What do complex functions look like? | Essence of complex analysis #4
The 5 ways to visualize complex functions | Essence of complex analysis #3
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The 5 ways to visualize complex functions | Essence of complex analysis #3
What are complex numbers? | Essence of complex analysis #2
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What are complex numbers? | Essence of complex analysis #2
Why care about complex analysis? | Essence of complex analysis #1
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Why care about complex analysis? | Essence of complex analysis #1
Green's functions: the genius way to solve DEs
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Green's functions: the genius way to solve DEs
From miller to entering Cambridge at age 40 - George Green
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From miller to entering Cambridge at age 40 - George Green
Solved simply: the impossible integral
Переглядів 63 тис.3 роки тому
Solved simply: the impossible integral
What is Jacobian? | The right way of thinking derivatives and integrals
Переглядів 1,8 млн3 роки тому
What is Jacobian? | The right way of thinking derivatives and integrals
The numerical simulation is NOT as easy as you think! - Average distance #2
Переглядів 53 тис.3 роки тому
The numerical simulation is NOT as easy as you think! - Average distance #2
Dream cheating scandal - explaining ALL the math simply
Переглядів 683 тис.3 роки тому
Dream cheating scandal - explaining ALL the math simply

КОМЕНТАРІ

  • @franciscomacedo8895
    @franciscomacedo8895 7 годин тому

    THIS IS SO FREAKING AWESOME!!!!!!!

  • @nathanaeldean6301
    @nathanaeldean6301 9 годин тому

    For every group, there is an element that does literally nothing. Are you talking about math or about school group projects?

  • @kapoioBCS
    @kapoioBCS 15 годин тому

    The video is great, but I don’t see how non math population will really understand what is going on and how every step is motivated.

  • @alejrandom6592
    @alejrandom6592 21 годину тому

    Gotta love'em lolitudes

  • @marcos5136
    @marcos5136 День тому

    ❤❤❤❤❤

  • @wulphstein
    @wulphstein 2 дні тому

    That's great!

  • @Joshua-hb7cs
    @Joshua-hb7cs 2 дні тому

    I don't get why the parallelogram argument works that way geometrically. Why does the square become a rectangle when you move the vertical component of (a c)? At time 8:15 I already understand the determinant argument, but I want to understand it geometrically

  • @sirknightartorias68
    @sirknightartorias68 2 дні тому

    Shouldn't at 9:39, it be n-dimensional Euclidean space , the last line?

  • @John-fo4wp
    @John-fo4wp 5 днів тому

    Arigato Gyro

  • @SonuRaj-er1hn
    @SonuRaj-er1hn 6 днів тому

    Hey, could you make video how you have make this videos presentation in PowerPoint it would be very helpful. Thanks for providing amazing content 👏

  • @user-lz1yb6qk3f
    @user-lz1yb6qk3f 7 днів тому

    You've unlocked vectors on spheres for me, now I can do linear algebra on them! Thanks! But only I would do it with bivectors from Clifford Algebra.

  • @alexalainterieur
    @alexalainterieur 8 днів тому

    not progressive enough , too much material in 1 video , take it more progressively like 3B1B

  • @kummer45
    @kummer45 8 днів тому

    This is simply beautiful. I studied mathematics and physics. I started studying on my own analysis. These videos explains a lot of intuition.

  • @Mufozon
    @Mufozon 9 днів тому

    You have lost me at 1:45, where it says Modified: e^x = g(1) This is too ambiguous. Does it mean that the function g outputs a function? But e^x is a number, you can't equal it to a function, so did you mean e^x = g(1)(x)? Or does g secretly also depend on x? Why is it not stated like g(t,x) then. If g really outputs a function would you maybe also define what do you mean by g'(t), although I can probably fill this in if you make the above clearer.

  • @rockapedra1130
    @rockapedra1130 9 днів тому

    Cool!!

  • @PhiGuy01
    @PhiGuy01 11 днів тому

    PI and PHI

  • @jasonc0065
    @jasonc0065 11 днів тому

    It turns out that some cubics cannot be solved by real radicals.

  • @user-vh4yd2oz6d
    @user-vh4yd2oz6d 12 днів тому

    ‏‪8:14‬‏ What do you mean by "why that arc"? Didn't you just explain why using geometry?

  • @MrBANANAS1235
    @MrBANANAS1235 12 днів тому

    Hi Trevor, this video is utterly beautiful. Great, great work you should be really proud. You’ve done very well

  • @carlosgordilloolivera6994
    @carlosgordilloolivera6994 12 днів тому

    Amazing video! New sub!

  • @Necrozene
    @Necrozene 13 днів тому

    The Math God must necessarily love this!

  • @gyattrizzV
    @gyattrizzV 13 днів тому

    I wish i knew how to make powerpoint like you

  • @zegevlier7076
    @zegevlier7076 14 днів тому

    I've had a really tough time understanding the isomorphism theorem in my group theory course, and this video helped a lot. Thank you for making it.

  • @wily_rites
    @wily_rites 14 днів тому

    Nice perspective on this subject, greatly clarifying for one who loves math but find its study to be difficult in the current paradigm; Many thanks.

  • @umbraemilitos
    @umbraemilitos 15 днів тому

    Very good video.

  • @moua_d7733
    @moua_d7733 15 днів тому

    You just made the best youtube explanation of machine learning svm without even mentioning svm in the title xd

  • @Joshua-hb7cs
    @Joshua-hb7cs 15 днів тому

    Please continue this series! It's helping me so much

  • @jonathanlerner2797
    @jonathanlerner2797 16 днів тому

    I think there’s a simpler, smaller solution to the second question about 1000 consecutive primes. Multiple together all the primes <=1000, then add 2. That number is the beginning of a sequence of 999 composite numbers by construction. Every number is of the form P + n, where P is the product of all the first primes up to 1000 and 2 <= n <= 1000. Then clearly any prime p less than 1000 divides P, and p also divides n

  • @appybane8481
    @appybane8481 16 днів тому

    8:10 :We only need inversion Inversion with circle passing infinity makes reflection 2 reflections make rotation and translation 2 inversions on circle with same center make things larger/smaller

  • @boon6084
    @boon6084 16 днів тому

    Dream fan are. Crying rn lol😅

  • @AlirezaAroundItaly
    @AlirezaAroundItaly 16 днів тому

    Amazing video , thank you so much

  • @koktszfung
    @koktszfung 17 днів тому

    This is inspiring

  • @koktszfung
    @koktszfung 17 днів тому

    The generating function is neat, but it seems rather difficult to come up with it. Do you know beforehand that you can write it as a generating function?

  • @koktszfung
    @koktszfung 17 днів тому

    I immediately understand the inclusion exclusion principle using the venn diagram so I think it very much aided my understanding

    • @koktszfung
      @koktszfung 17 днів тому

      The idea of overshooting when taking away the overlapping region is very easy to understand

  • @wenxy69
    @wenxy69 17 днів тому

    so, was it just linear algebra all along? ..always has been.

  • @ramashama-tw3ly
    @ramashama-tw3ly 18 днів тому

    🙏🙏👍👍

  • @rossholst5315
    @rossholst5315 19 днів тому

    What happens when the roots repeat?

  • @dewetskywalker
    @dewetskywalker 19 днів тому

    Top notch quality right here, extremely underrated amazing job on this video.

  • @bomminenisurrenderreddy830
    @bomminenisurrenderreddy830 19 днів тому

    i think you just made it more complicated

  • @yordangrigorov6399
    @yordangrigorov6399 20 днів тому

    thank you for the video, but I didn't get it. Can you recommend any materials/video on covectors?

  • @elertmarku6895
    @elertmarku6895 20 днів тому

    I just realize that my professor was horrible. I really dumb my ass thinking that I am not good for. And I just really understand without paper, lapiz whole this . These professor are so intimidating that there problems put on students

  • @pauselab5569
    @pauselab5569 20 днів тому

    I think that projective geometry is the best way to motivate mobius transformations instead of just rational linear functions. the mobius transformations are the automorphism of the projective space over C.

  • @diegogaelvazquezreyes4921
    @diegogaelvazquezreyes4921 21 день тому

    nicee