This is the website for the course Math 8174. Topics in Algebra. The course is an introduction to Lie algebras and representation theory. Office hours are held on Mondays 3-5 pm. or by appointment. Short lecture summaries, notes, and homework will be posted below.
| Date | Summary |
| Aug.24. | Definition of Lie algebras. General linear algebras. The adjoint representation. Ado's Theorem. Notes. |
| Aug.26. | The classical Lie algebras of types A, B, C, D. Notes. |
| Aug.28. | Derivations. Ideals of Lie algebras. Simplicity of sl2. Isomorphism theorems. Notes. Homework 1. |
| Aug.31. | More on adjoint representations. Tensors of representations. Notes. |
| Sep.02. | The Chevalley involution of sl2. Finite dimensional irreps of sl2: construction. Notes. |
| Sep.04. | Finite dimensional irreps of sl2: classification. Symmetric powers of Lie algebra representations. Notes. |
| Sep.07 | Labor day, no class. |
| Sep.09. | Homework discussion. Finishing classification of sl2(C) irreps. Notes. |
| Sep.11. | Basic results on solvable and nilpotent Lie algebras. Notes. Homework 2. |
| Sep.14. | Engel's Theorem and Lie's Theorem: statements and reductions. Notes. |
| Sep.16. | Engel's Theorem and Lie's Theorem: finishing the proofs. Notes. |
| Sep.18. | Corollaries of Engel's and Lie's Theorems. Cartan's solvability criterion. Notes. |
| Sep.21. | Some linear algebra: Killing forms and Jordan Decompositions. Notes. |
| Sep.23. | Proof of Cartan's criteria. Notes. |
| Sep.25. | Class rescheduled to October 1. Homework 3. |
| Sep.28. | Applications of Cartan's criteria on semisimple algebras. Notes. |
| Sep.30. | Abstract Jordan decompositions. Progress summary. Notes. |
| Oct.01. | Homework help session. |
| Oct.02. | Definition and examples of weight spaces. Notes. |
| Oct.05. | Toral and Cartan subalgebras. Definition of Cartan decompositions. Notes. |
| Oct.07. | Cartan subalgebras are self-centralizing. Notes. |
| Oct.09. | sl2 triples in Cartan decompositions. Notes. Homework 4. |
| Oct.12. | Adjoint actions of sl2 triples. Integrality properties of root systems. Notes. |
| Oct.14. | More on integrality properties. Working towards an inner product space. Notes. |
| Oct.16. | The promised inner product space. Abstract root systems. Notes. |
| Oct.19 | Overview of classification of semismiple Lie algebras. First steps towards classification of root systems. Notes. |
| Oct.21. | Pairs of roots. Bases of root systems. Notes. |
| Oct.23. | More on bases. Weyl chambers. Notes. Homework 5. |
| Oct.26. | Class canceled due to inclement weather. |
| Oct.28. | Weyl groups. Notes. |
| Oct.30. | Actions of the Weyl group on Weyl chambers and bases. Notes. |
| Nov.02. | Examples: root systems in R^2, root systems of sl(n). Notes. |
| Nov.04. | Root systems of sp(2l). Dynkin diagrams. Notes. |
| Nov.06. | Irreducible root systems and their classification. Notes. Homework 6. |
| Nov.09. | Back to Lie algebras: classification of complex semisimple Lie algebras. Notes. |
| Nov.11. | More on the correspondence between semisimple Lie algebras and root systems. Conjugacy theorems. Notes. |
| Nov.13. | Serre relations. Serre's Theorem. The Existence and Uniqueness/Isomorphism Theorem. Notes. |
| Nov.16. | Algebra review. Universal enveloping algebras. Notes. |
| Nov.18. | Associated graded algebras. Two versions of the PBW theorem. Notes. |
| Nov.20. | Consequences of the PBW theorems. Equivalence of the theorems. Bergman's Diamond Lemma. Notes. Homework 7. |
| Nov.23. | Proof the PBW basis theorem via the diamond lemma. Free Lie algebras. Notes. |
| Nov.25. | The language of bialgebras. Universal enveloping algebras as Hopf algebras. Notes. |
| Nov.27. | Thanksgiving holiday; no class. |
| Nov.30. | More on Hopf algebras. Notes. |
| Dec.02. | Classification of f.d. irreps. of s.s. Lie algebras: definitions, highest weights, and statement of the result. Notes. |
| Dec.04. | Properties of highest weight modules. Verma modules. Notes. Homework 8. |
| Dec.06. | Proof of the classification of f.d. irreps. Examples of irr. and Verma modules in type A. Concluding remarks. Notes. |