MATH 5112 (Spring 2024)

• Welcome to the homepage for MATH 5112: Advanced Algebra II!
• Send questions to eric.marberg@gmail.com.
• For office hours, send me an email to arrange an appointment.
• For information about prerequisites, grades, and so forth, consult the syllabus.

Lectures:

• Time: Tuesdays and Thurdays from 1:30PM to 2:50PM.
• Location: Room 2612B, near Lift 31-32.

Textbooks:

• Our main textbook is Introduction to Representation Theory by Etingof et al.

Schedule:

• Week 1:
  • Lecture 1: algebras and representations (Sections 2.1–2.3)
• Week 2:
  • Lecture 2: ideals, quotients, generators, relations (Sections 2.4–2.7)
  • Lecture 3: quivers and Lie algebras (Sections 2.8–2.9)
• Week 3:
  • Lecture 4: tensor products (Sections 2.11–2.15)
• Week 4:
  • Lecture 5: semisimple representations, density theorem (Sections 3.1–3.2)
  • Lecture 6: matrix algebras, filtrations, finite-dimensional algebras (Sections 3.3–3.5)
• Week 5:
  • Lecture 7: semisimple algebras, characters, two general theorems (Sections 3.5–3.7)
  • Lecture 8: Krull-Schmidt theorem, tensor products of algebras (Sections 3.8–3.10)
• Week 6:
  • Lecture 9: Group representations, Maschke's theorem, characters (Sections 4.1–4.4)
  • Lecture 10: orthogonality relations, unitary representations (Sections 4.5–4.7)
• Week 7:
  • Lecture 11: character tables, Frobenius determinants (Sections 4.8–4.10)
  • Lecture 12: Frobenius-Schur indicators, algebraic numbers (Sections 5.1–4.2)
• Week 8:
  • Lecture 13: Frobenius divisibility, Burnside's theorem (Sections 5.3–5.4)
  • Lecture 14: product and virtual representations, restriction and induction (Sections 5.6–5.9)
• Week 9:
  • Lecture 15: Frobenius reciprocity, representations of symmetric groups (Sections 5.10–5.12)
• Week 10:
  • (no lectures)
• Week 11:
  • Lecture 16: more representations of symmetric groups (Sections 5.12-5.13)
  • Lecture 17: complexification, Frobenius character formula (Sections 5.14-5.17)
• Week 12:
  • Lecture 18: Schur-Weyl duality (Sections 5.18-5.19)
  • Lecture 19: Schur polynomials, Artin's theorem (Sections 5.21-27)
• Week 13:
  • Lecture 20: Gabriel's theorem (Sections 6.1-6.9)
  • Lecture 21: category theory basics (Sections 7.1-7.4)
• Week 13:
  • Lecture 22: representable functors, adjoint functors, abelian categories (Sections 7.5-7.7)
  • Lecture 23: exact functors, projective modules, Ext and Tor, lifting idempotents (Sections 7.8-8.2)
• Week 14:
  • Lecture 24: projective covers, blocks, finite abelian categories, Morita equivalence (Sections 9.1-9.7)
  • Lecture 25: course recap

Assignments:

• Homework 0: lecture transcription
  • Each student must transcribe 2-3 lectures in TeX (sign-up sheet in class).
  • Start with this template (change the file name as appropriate).
  • Use the handwritten slides posted above or your own notes from class.
  • Please finish each transcription within one week of the lecture.
• Homework 1 due Tuesday, February 20
• Homework 2 due Tuesday, February 27
• Homework 3 due Tuesday, March 5
• Homework 4 due Tuesday, March 19
• Homework 5 due Tuesday, April 9
• Homework 6 due Thursday, April 25
• Homework 7 due Thursday, May 9