What is Lie theory? Here is the big picture. | Lie groups, algebras, brackets #3

Part 4:    • Can we exponentiate d/dx? Vector (fields)?...   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. 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: 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" 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!