Semisimple Lie algebras

Lie algebras and their representations, from brackets and complete reducibility to roots, Serre presentations, highest weights, characters, compact forms and category O.

18 English lessons, 73 solved exercises, 8 figures and editable Markdown sources. Lesson exposition and checks: GPT-6.1 Sol (OpenAI), Codex, Ultra; opening foundations and integration: GPT-6 Astra (OpenAI), Codex, Ultra. Supplementary unproved statements and prerequisites remain explicit; no human review is claimed.

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1. Lie algebras: definitions, examples and first constructions
2. Nilpotent and solvable Lie algebras: Engel's and Lie's theorems
3. The Killing form and Cartan's criteria
4. Complete reducibility: Casimir elements and Weyl's theorem
5. Representations of sl(2)
6. Low-degree cohomology, Whitehead's lemmas and the Levi decomposition
7. The root space decomposition of a semisimple Lie algebra
8. Root systems and their Weyl groups
9. Cartan matrices, Dynkin diagrams and the classification of root systems
10. Cartan subalgebras and their conjugacy
11. The isomorphism theorem and Serre's theorem
12. The simple Lie algebras: classical models and the exceptional algebras
13. The universal enveloping algebra and the Poincaré–Birkhoff–Witt theorem
14. Weights, Verma modules and the theorem of the highest weight
15. The centre of the enveloping algebra and Harish-Chandra's theorem
16. Weyl's character formula and the multiplicity formulas
17. Compact real forms and Weyl's unitary trick
18. Category O and the Kazhdan–Lusztig conjecture