Part 4:    • Can we exponentiate d/dx? Vector (fie...  

A bird's eye view on Lie theory, providing motivation for studying Lie algebras and Lie brackets in particular.

Basically, Lie groups are groups and manifolds, and thinking about them as manifolds, we know that we want to understand Lie algebras; and thinking about them as groups, we know what additional structure we want on the Lie algebras - the Lie bracket.

YouTube, please do not demonetise this video for me saying “Tits group”. This is an actual mathematical object named after a French mathematician Jacques Tits.

Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: so3embeddedin5dim

SO(3) embedded in R^5: http://at.yorku.ca/b/ask-an-algebraic...

https://en.wikipedia.org/wiki/Whitney... (n-dim manifold can be properly embedded in R^(2n): if you only want “the overall picture”, but perhaps distances are distorted)

https://en.wikipedia.org/wiki/Nash_em... (n-dim Riemannian manifold can be isometrically embedded in n(3n+11)/2 dim if compact, n(n+1)(3n+11)/2 dim if not compact: if you want everything to remain intact, i.e. distances are preserved)

BCH formula (why Lie brackets are useful): https://en.wikipedia.org/wiki/Baker%E...

Finite simple groups as building blocks: https://en.wikipedia.org/wiki/Composi...

Classification of finite simple groups: https://en.wikipedia.org/wiki/Classif...

Levi decomposition (the more precise meaning of “building blocks” in Lie algebra): https://en.wikipedia.org/wiki/Levi_de...

E8 (the monster group of Lie algebras):
https://aimath.org/E8/e8.html
https://en.wikipedia.org/wiki/E8_(mat...)
https://en.wikipedia.org/wiki/An_Exce...

Video chapters:
00:00 Introduction
01:26 Lie groups - groups
05:41 Lie groups - manifolds
10:23 Lie algebras
14:16 Lie brackets
18:03 The "Lie theory picture"

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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.

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