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Lie algebras visualized: why are they defined like that? Why Jacobi identity?

Автор: Mathemaniac

Загружено: 2024-04-09

Просмотров: 97032

Описание:

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:    • Lie groups, algebras, brackets  

Individually:
Part 1:    • Why study Lie theory? | Lie groups, algebr...   (intro and motivation)
Part 2:    • How to rotate in higher dimensions? Comple...   (on SO(n), SU(n) notations)
Part 3:    • What is Lie theory? Here is the big pictur...   (overview of Lie theory)
Part 4:    • Can we exponentiate d/dx? Vector (fields)?...   (exponential map on exotic objects)
Part 5:    • Matrix trace isn't just summing the diagon...   (on visualising trace)

Videos from other channels that overlap with my previous ideas:

   • Dirac's belt trick, Topology,  and Spin ½ ...   [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]

   • The Mystery of Spinors   [specifically the “homotopy classes” part]

   • Spinors for Beginners 18: Irreducible Repr...   [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/pr... basically what I say, without the vector field visualisations]

(2) https://www.damtp.cam.ac.uk/user/ho/S... [focus on Q2: a much more tedious approach to motivate Jacobi identity]

(3) https://en.wikipedia.org/wiki/Directi... [actually quite useful, touches upon many ideas in the video series]

(4) https://projecteuclid.org/journals/jo... [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:   / mathemaniacyt  
Instagram:   / _mathemaniac_  
Twitter:   / mathemaniacyt  
Patreon:   / 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!

Lie algebras visualized: why are they defined like that? Why Jacobi identity?

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