Mathemaniac | Lie algebras visualized: why are they defined like that? Why Jacobi identity? @mathemaniac | Uploaded 6 months ago | Updated 2 hours ago
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 mathemaniac.co.uk/download and enter the following password: whyJacobiidentity
Previous videos are compiled in the playlist: youtube.com/playlist?list=PLDcSwjT2BF_WDki-WvmJ__Q0nLIHuNPbP
Individually:
Part 1: youtube.com/watch?v=IlqVo3sJFLE (intro and motivation)
Part 2: youtube.com/watch?v=erA0jb9dSm0 (on SO(n), SU(n) notations)
Part 3: youtube.com/watch?v=ZRca3Ggpy_g (overview of Lie theory)
Part 4: youtube.com/watch?v=9CBS5CAynBE (exponential map on exotic objects)
Part 5: youtube.com/watch?v=B2PJh2K-jdU (on visualising trace)
Videos from other channels that overlap with my previous ideas:
youtube.com/watch?v=ACZC_XEyg9U [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]
youtube.com/watch?v=b7OIbMCIfs4 [specifically the “homotopy classes” part]
youtube.com/watch?v=Q_RUDQkDsE0 [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) https://people.reed.edu/~jerry/332/projects/venkatamaran.pdf basically what I say, without the vector field visualisations]
(2) 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:
https://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: facebook.com/mathemaniacyt
Instagram: instagram.com/_mathemaniac_
Twitter: twitter.com/mathemaniacyt
Patreon: 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!
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 mathemaniac.co.uk/download and enter the following password: whyJacobiidentity
Previous videos are compiled in the playlist: youtube.com/playlist?list=PLDcSwjT2BF_WDki-WvmJ__Q0nLIHuNPbP
Individually:
Part 1: youtube.com/watch?v=IlqVo3sJFLE (intro and motivation)
Part 2: youtube.com/watch?v=erA0jb9dSm0 (on SO(n), SU(n) notations)
Part 3: youtube.com/watch?v=ZRca3Ggpy_g (overview of Lie theory)
Part 4: youtube.com/watch?v=9CBS5CAynBE (exponential map on exotic objects)
Part 5: youtube.com/watch?v=B2PJh2K-jdU (on visualising trace)
Videos from other channels that overlap with my previous ideas:
youtube.com/watch?v=ACZC_XEyg9U [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]
youtube.com/watch?v=b7OIbMCIfs4 [specifically the “homotopy classes” part]
youtube.com/watch?v=Q_RUDQkDsE0 [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) https://people.reed.edu/~jerry/332/projects/venkatamaran.pdf basically what I say, without the vector field visualisations]
(2) 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:
https://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: facebook.com/mathemaniacyt
Instagram: instagram.com/_mathemaniac_
Twitter: twitter.com/mathemaniacyt
Patreon: 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!
