Please see below for lecture summaries, homework and other study material. For your grades, please see Canvas.
| Date | Topics |
| Jan.10. | Review of groups and group homomorphisms. Notes. |
| Jan.12. | Review of group isomorphism theorems, rings and fields. Notes. Homework (tex). |
| Jan.14. | Review of vector spaces and linear algebra. Notes. |
| Jan.17. | Martin Luther King Holiday, no class. |
| Jan.19. | Facts about vector space bases. Definition of algebras, ideals and subalgebras. Notes. Homework (tex). |
| Jan.21. | Examples of algebras. Notes. |
| Jan.24. | k embeds into k-algebras. Homomorphisms. "End_k(V) \cong M_n(k)". Factor algebras. Notes. |
| Jan.26. | More on factor algebras, including review on well-definedness. Isomorphism theorems for algebras. Notes. Homework (tex). |
| Jan.28. | Free vector spaces. Group algebras. Path algebras of quivers. Notes. |
| Jan.31. | Examples of algebra isomorphisms. Principal ideals. Notes. |
| Feb.02. | Class cancelled due to inclement weather. No homework due next week. |
| Feb.04. | Ideals of k[t]. Modules of algebras: definition, remarks, and examples. Notes. Homework (due on Feb.16, finalized on Feb.09) (tex). |
| Feb.07. | Two more natural modules. Equivalence of modules and representations. Notes. |
| Feb.09. | Induced representations/modules. Submodules, quotient modules, and module homomorphisms. Notes. (See Feb.04 summary for homework due next Wednesday.) |
| Feb.11. | Isomorphism/correspondence theorems for the modules. External and internal direct sums of modules. Notes. |
| Feb.14. | The recognition theorem for direct sums. Modules of k[x]. Notes. |
| Feb.16. | Projector woes (no new lecture slides). Homework discussion. Homework (tex) |
| Feb.18. | Modules of an algebra A vs. modules of quotients of A. The case where A=k[x]. Notes. |
| Feb.21. | Preservation of scaling actions. Representations of groups vs. group algebras. Notes. |
| Feb.23. | From group actions to group representations. Definition and examples of simple modules. Notes. Homework (tex) |
| Feb.25. | Lemma 3.3: a simplicity test. More examples and non-examples of simple modules. Notes. |
| Feb.28. | Simplicity of quotient modules. Simple modules of quotients of k[x]. Midterm announcement. Notes. |
| Mar.02. | Simple modules of path algebras of acyclic quivers. Notes. Homework (tex) |
| Mar.04. | Definitions, examples and statements of results related to composition series. Notes. |
| Mar.07. | Homework discussion. Finite dimensional modules have finite length. Inheritance of composition series. Notes. |
| Mar.09. | Midterm review.
Notes. (No homework will be due next week.) Midterm 1, available on Canvas from 5pm Mar.09 to 5pm Mar.10. |
| Mar.11. | Proof of the Jordan--Hölder Theorem. Properties of module lengths. Notes. |
| Mar.16. | Schur's lemma and its applications. Preview on semisimple modules and algebras. Notes. Homework (tex) |
| Mar.18. | Examples of non-semisimple modules. Equivalent definitions of semisimple modules and some consequences. Notes. |
| Mar.28. | More properties of semisimple modules. All modules of semisimple algebras are semisimple. Notes. |
| Mar.30. | Homework discussion. Properties of semisimple algebras. Notes. Homework (tex) |
| Apr.01. | More properties of semisimple algebras. Introduction to Jacobson radicals. Notes. |
| Apr.04. | Properties of Jacobson radicals and their proofs. Notes. |
| Apr.06. | More properties of Jacobson radicals: Jacobson radicals and semisimplicity. Notes. |
| Apr.08. | Homework discussion. Jacobson radical and (non)-semisimplicity of path algebras. Notes. |
| Apr.10. | The Artin--Wedderburn Theorem: statement and proof outline. Notes. |
| Apr.13. | Proof of the Artin--Wedderburn Theorem: endomorphism algebras, matrices of morphisms. Notes. Homework (tex) |
| Apr.15. | Proof of the Artin--Wedderburn Theorem: endomorphisms of direct sums, from endomorphism to matrices. Notes. |
| Apr.18. | The Artin--Wedderburn Theorem: finishing the proof, revisiting the statements. Notes. |
| Apr.20. | Statement of Maschke's Theorem. Example AW decompositions of semisimple algebras. Notes. Homework (tex) |
| Apr.22. | More on AW decompositions of group algebras. Proof of Maschke's Theorem: the "only if" direction, a lemma for the "if" direction. Notes. |
| Apr.25. | Finishing the proof of Maschke's Theorem. Numbers of 1-dimensional simples for finite groups. Notes. |
| Apr.27. | Review for final exam. Notes. |