Please see below for lecture summaries, homework and other study material.
| Date | Topics |
| Jan.15. | Review: groups, rings, fields and vector spaces. Notes. |
| Jan.18. | MLK Holiday; no class. |
| Jan.20. | Linear algebra review. Free vector spaces via universal property. Notes. Homework (tex). |
| Jan.22. | More on universal properties. Definition and first example of algebras. Notes. |
| Jan.25. | Polynomial and endomorphism algebras. Notes. |
| Jan.27. | Monoid algebras. Free algebras. Path algebras of quivers. Notes. Homework (tex). |
| Jan.29. | Examples of path algebras. Ideals of algebras. Notes. |
| Feb.01. | Quotient algebras. The first isomorphism theorem for algebras. Notes. |
| Feb.03. | Facts about k[x]. Definition and examples of modules. Notes. Homework (tex). |
| Feb.05. | Submodules. Simplicity of column modules. Quotients modules. Notes. |
| Feb.08. | "Induced" modules. Direct products. External and internal direct sums. Notes. |
| Feb.10. | More on module homomorphisms and isomorphisms. Notes. Homework (tex). |
| Feb.12. | Representations. "Representations are the same as modules". Modules of k[x]. Notes. |
| Feb.15. | More on modules of k[x] and its quotients. Notes. |
| Feb.17. | Wellness Day; no class. Homework (tex). |
| Feb.19. | Modules from group actions. Simple modules. Notes. |
| Feb.22. | Simple vs. cyclic modules. Simpleness of quotient modules. Composition series. Notes. |
| Feb.24. | More examples of composition series. Existence and uniqueness of composition series. Notes. Homework (tex). |
| Feb.26. | Proof of the Jordan--Hölder Theorem. Properties of module lengths. Notes. |
| Mar.01. | Classifying simple modules of path algebras of acyclic quivers. Notes. |
| Mar.03. | Representations of quivers vs. modules of path algebras. Notes. Homework (tex). |
| Mar.05. | Midterm review. Midterm I (available on Canvas from 5pm of Mar. 5 to 5pm of Mar. 6). |
| Mar.08. | More on representations of quivers. Notes. |
| Mar.10. | Bounded quiver algebras. Schur's Lemma. Notes. Homework (tex). |
| Mar.12. | Applications of Schur's Lemma. Simple modules of quotients of k[x]. Notes. |
| Mar.15. | Class cancelled due to winter storm. |
| Mar.17. | Overview of semisimple modules, semisimple algebras, and Maschke's Theorem. Notes. Homework (tex). |
| Mar.19. | Equivalent definitions of semisimple modules. Operations preserving sesisimplicity. Notes. |
| Mar.22. | Semisimple algebras. Operations preserving semisimplicity of algebras. Notes. |
| Mar.24. | Properties of Jacobson radicals, Part 1. Notes. Homework (tex). |
| Mar.26. | Properties of Jacobson radicals, Part 2. Notes. |
| Mar.29. | Examples and applications of Jacobson radicals. Notes. |
| Mar.31. | Spring Pause (not very well defined, don't ask) Lecture I: homework discussion. |
| Apr.02. | Spring Pause Lecture II: some applications of algebra representations. |
| Apr.05. | The Artin-Wedderburn Theorem: statement, proof ingredients, and proof strategy. Notes. |
| Apr.07. | The Artin-Wedderburn Theorem: endomorphism algebras, matrices of morphisms. Notes. Homework (tex). |
| Apr.09. | The Artin-Wedderburn Theorem: finishing the proofs. Notes. |
| Apr.12. | The Artin-Weddeburn Theorem: summary, corollaries and first examples. Notes. |
| Apr.14. | Maschke's theorem. Notes. Homework (tex). |
| Apr.16. | Combining Artin-Wedderburn and Maschke. Notes. |
| Apr.19. | More on Artin-Wedderburn decompositions for finite groups. Notes. |
| Apr.21. | Homework discussion. Final Exam announced (available on Canvas from 11:59 am of May 1 to 11:59 pm of May 2). Notes. Homework (tex). |
| Apr.23. | Group characters. Notes. |
| Apr.26. | Properties of characters and character tables. Notes. |
| Apr.28. | Statement of the Krull-Schmidt theorem. Course review. Notes. |