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 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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