Lie groups & Lie algebras
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Lie Groups & Lie Algebras — scope drawn from open Lie theory notes and typical US graduate Lie groups syllabi.
Lie groups and Lie algebras
- Lie groups: definitions, examples (GL(n), SO(n), SU(n)) — Lie groups: definitions, examples (GL(n), SO(n), SU(n))
- Lie algebras: bracket, homomorphisms, and ideals — Lie algebras: bracket, homomorphisms, and ideals
- Exponential map — Exponential map
- Baker–Campbell–Hausdorff formula (introduction) — Baker–Campbell–Hausdorff formula (introduction)
- Adjoint representation and Killing form — Adjoint representation and Killing form
- Semisimple and simple Lie algebras — Semisimple and simple Lie algebras
Structure theory
- Cartan subalgebras and root systems — Cartan subalgebras and root systems
- Weyl group and root reflections — Weyl group and root reflections
- Dynkin diagrams — Dynkin diagrams
- Classification of simple Lie algebras — Classification of simple Lie algebras
- Highest weight representations (overview) — Highest weight representations (overview)
- Compact Lie groups and maximal tori — Compact Lie groups and maximal tori
Advanced topics
- Universal enveloping algebras (introduction) — Universal enveloping algebras (introduction)
- Haar measure and integration on compact groups — Haar measure and integration on compact groups
- Symmetric spaces (preview) — Symmetric spaces (preview)
- Lie cohomology (optional) — Lie cohomology (optional)
- Applications to differential geometry and physics — Applications to differential geometry and physics
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Lie groups and Lie algebras
Lie groups: definitions, examples (GL(n), SO(n), SU(n))
Coming soon - Structure theory
Cartan subalgebras and root systems
Coming soon - Advanced topics
Universal enveloping algebras (introduction)
Coming soon
Notes
Graduate Lie Groups & Lie Algebras — scope drawn from open Lie theory notes and typical US graduate Lie groups syllabi. Topic outline: `content/topics/graduate/lie_groups.json`.