MATH 5112 (Spring 2024)
Overview:
 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 3132.
Textbooks:
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, finitedimensional algebras (Sections 3.3–3.5)

Week 5:
 Lecture 7: semisimple algebras, characters, two general theorems (Sections 3.5–3.7)
 Lecture 8: KrullSchmidt 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: FrobeniusSchur 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:

Week 11:
 Lecture 16: more representations of symmetric groups (Sections 5.125.13)
 Lecture 17: complexification, Frobenius character formula (Sections 5.145.17)

Week 12:
 Lecture 18: SchurWeyl duality (Sections 5.185.19)
 Lecture 19: Schur polynomials, Artin's theorem (Sections 5.2127)

Week 13:
 Lecture 20: Gabriel's theorem (Sections 6.16.9)
 Lecture 21: category theory basics (Sections 7.17.4)

Week 13:
 Lecture 22: representable functors, adjoint functors, abelian categories (Sections 7.57.7)
 Lecture 23: exact functors, projective modules, Ext and Tor, lifting idempotents (Sections 7.88.2)

Week 14:
 Lecture 24: projective covers, blocks, finite abelian categories, Morita equivalence (Sections 9.19.7)
 Lecture 25: course recap
Assignments:
 Homework 0: lecture transcription
 Each student must transcribe 23 lectures in TeX (signup 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