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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 www.mathemaniac.co.uk/download and enter the following password: whyJacobiidentity
Previous videos are compiled in the playlist: • Lie groups, algebras, ...
Individually:
Part 1: • Why study Lie theory? ... (intro and motivation)
Part 2: • How to rotate in highe... (on SO(n), SU(n) notations)
Part 3: • What is Lie theory? He... (overview of Lie theory)
Part 4: • Can we exponentiate d/... (exponential map on exotic objects)
Part 5: • Matrix trace isn't jus... (on visualising trace)
Videos from other channels that overlap with my previous ideas:
• Dirac's belt trick, To... [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 ... [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) people.reed.edu/~jerry/332/pr... basically what I say, without the vector field visualisations]
(2) www.damtp.cam.ac.uk/user/ho/S... [focus on Q2: a much more tedious approach to motivate Jacobi identity]
(3) en.wikipedia.org/wiki/Directi... [actually quite useful, touches upon many ideas in the video series]
(4) 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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