XylyXylyX: Lie Groups and Lie Algebras
Forty-five lessons from the quaternions to the universal covering group
About this series
Most physics courses introduce Lie groups by asserting that the generators satisfy a commutation relation and moving on. XylyXylyX spends 45 lessons doing it properly: ten on the classical groups as concrete objects before anything is abstracted, then generators, the commutator, and the exponential map back to the group. The last third is topology — parameter spaces, homotopy, fundamental groups — arriving at why SU(2) is the universal cover of SO(3), which is the fact spin actually rests on.
It sits directly under the written Lie groups course, and feeds the rotation group in quantum mechanics and particle physics.
Lesson 1 — Prerequisites
Lesson 2 — Quaternions
Lesson 3 — Classical Groups Part I
Lesson 4 — The Classical Groups Part II
Lesson 5 — The Classical Groups Part III
Lesson 6 — The classical groups part IV (redux)
Lesson 7 — The Classical Groups Part V
Lesson 8 — the Classical Groups part VI
Lesson 9 — The Classical Groups Part VII
Lesson 10 — The Classical Groups part VIII
Lesson 11 — The Classical Groups Part IX
Lesson 12 — The Classical Groups Part X (redux)
About these lectures
The series assumes linear algebra and little else, and takes the classical groups slowly enough that the abstraction later has something to attach to. Lessons 21 is marked optional by the author; lessons 6, 12, 33, 43 and 44 are re-recordings or corrections, which is why their titles say so.
All content is publicly available on YouTube and belongs to its creator. Watch the full series on the original playlist.