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 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, a...   (intro and motivation)
Part 2:    • How to rotate in higher dimensions? C...   (on SO(n), SU(n) notations)
Part 3:    • What is Lie theory? Here is the big p...   (overview of Lie theory)
Part 4:    • Can we exponentiate d/dx? Vector (fie...   (exponential map on exotic objects)
Part 5:    • Matrix trace isn't just summing the d...   (on visualising trace)

Videos from other channels that overlap with my previous ideas:

   • Dirac's belt trick, Topology,  and Sp...   [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...   [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

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