= Lie group
{c}
{wiki}
The key and central motivation for studying Lie groups and their <Lie algebras> appears to be to characterize <symmetry> in <Lagrangian mechanics> through <Noether's theorem>, just start from there.
Notably <local symmetries> appear to map to forces, and local means "around the identity", notably: <local symmetries of the Lagrangian imply conserved currents>.
More precisely: <local symmetries of the Lagrangian imply conserved currents>.
TODO <Ciro Santilli> really wants to understand what all the fuss is about:
* https://math.stackexchange.com/questions/1322206/lie-groups-lie-algebra-applications
* https://mathoverflow.net/questions/58696/why-study-lie-algebras
* https://math.stackexchange.com/questions/405406/definition-of-lie-algebra
Oh, there is a low dimensional classification! Ciro is <high flying bird vs gophers>[a sucker for classification theorems]! https://en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras
The fact that there are elements arbitrarily close to the identity, which is only possible due to the group being continuous, is the key factor that simplifies the treatment of Lie groups, and follows the philosophy of <continuous problems are simpler than discrete ones>.
Bibliography:
* https://youtu.be/kpeP3ioiHcw?t=2655 "Particle Physics Topic 6: Lie Groups and Lie Algebras" by Alex Flournoy (2016). Good <special orthogonal group>[SO(3)] explicit exponential expansion example. Then next lecture shows why SU(2) is the representation of SO(3). Next ones appear to eventually get to the physical usefulness of the thing, but I lost patience. Not too far out though.
* https://www.youtube.com/playlist?list=PLRlVmXqzHjURZO0fviJuyikvKlGS6rXrb "Lie Groups and Lie Algebras" playlist by XylyXylyX (2018). Tutorial with infinitely many hours
* http://www.staff.science.uu.nl/~hooft101/lectures/lieg07.pdf
* http://www.physics.drexel.edu/~bob/LieGroups.html
\Video[https://www.youtube.com/watch?v=ZRca3Ggpy_g]
{title=What is Lie theory? by Mathemaniac 2023}