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. This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. Files for download: Go to https://www.mathemaniac.co.uk/download and enter the following password: whyJacobiidentity Previous videos are compiled in the playlist: https://www.youtube.com/playlist?list=PLDcSwjT2BF_WDki-WvmJ__Q0nLIHuNPbP Individually: Part 1: https://www.youtube.com/watch?v=IlqVo3sJFLE (intro and motivation) Part 2: https://www.youtube.com/watch?v=erA0jb9dSm0 (on SO(n), SU(n) notations) Part 3: https://www.youtube.com/watch?v=ZRca3Ggpy_g (overview of Lie theory) Part 4: https://www.youtube.com/watch?v=9CBS5CAynBE (exponential map on exotic objects) Part 5: https://www.youtube.com/watch?v=B2PJh2K-jdU (on visualising trace) Videos from other channels that overlap with my previous ideas: https://www.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] https://www.youtube.com/watch?v=b7OIbMCIfs4 [specifically the “homotopy classes” part] https://www.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) https://www.damtp.cam.ac.uk/user/ho/S3prob.pdf [focus on Q2: a much more tedious approach to motivate Jacobi identity] (3) https://en.wikipedia.org/wiki/Directional_derivative [actually quite useful, touches upon many ideas in the video series] (4) https://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: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk Ko-fi: https://ko-fi.com/mathemaniac [for one-time support] For my contact email, check my About page on a PC. See you next time!