MATH 5112 (Spring 2021)
Overview:
 Welcome to the homepage for MATH 5112: Advanced Algebra II!
 Send questions to eric.marberg@gmail.com or our Discord server.
 For office hours, send me an email to arrange an appointment.
 For information about prerequisites, grades, and so forth, consult the syllabus.
Lectures:
Textbooks:
Schedule:
 Section numbers refer to the online version of the textbook.

Week 1:
 Lecture 1: algebras and representations (Sections 1.1–1.3)
 Lecture 2: ideals, quotients, generators, relations (Sections 1.4–1.7)

Week 2:
 Lecture 3: quivers and Lie algebras (Sections 1.8–1.9)
 No lecture on Thursday

Week 3:
 Lecture 4: tensor products (Sections 1.10–1.15)
 Lecture 5: semisimple representations, density theorem (Sections 2.1–2.2)

Week 4:
 Lecture 6: matrix algebras, filtrations, finitedimensional algebras (Sections 2.3–2.5)
 Lecture 7: semisimple algebras, characters, two general theorems (Sections 2.5–2.8)

Week 5:
 Lecture 8: KrullSchmidt theorem, tensor products of algebras (Sections 2.8–2.10)
 Lecture 9: Group representations, Maschke's theorem, characters (Sections 3.1–3.4)

Week 6:
 Lecture 10: character orthogonality relations, unitary representations (Sections 3.5–3.8)
 Lecture 11: character tables, Frobenius determinants (Sections 3.8–3.10)

Week 7:
 Lecture 12: FrobeniusSchur indicators, algebraic numbers (Sections 4.1–4.2)
 Lecture 13: Frobenius divisibility, Burnside's theorem (Sections 4.3–4.4)

Week 8:
 Lecture 14: product and virtual representations, restriction and induction (Sections 4.6–4.9)
 Lecture 15: Frobenius reciprocity, representations of symmetric groups (Sections 4.10–4.12)

Week 9:
 Lecture 16: representations of symmetric groups (Sections 4.124.13)
 No lecture on Thursday

Week 10:
 No lecture on Tuesday
 Lecture 17: complexification, Frobenius character formula (Sections 4.144.17)

Week 11:
 Lecture 18: SchurWeyl duality (Sections 4.184.19)
 Lecture 19: Schur polynomials, Artin's theorem (Sections 4.2123, 4.26)

Week 12:
 Lecture 20: Gabriel's theorem (Sections 5.15.9)
 Lecture 21: category theory basics (Sections 6.16.4)

Week 13:
 Lecture 22: representable functors, adjoint functors, abelian categories (Sections 6.56.7)
 Lecture 23: exact functors, projective modules, Ext and Tor (Sections 6.87.1)

Week 14:
 Lecture 24: lifting idempotents, projective covers (Sections 7.27.3)
 Lecture 25: blocks, finite abelian categories, Morita equivalence
Assignments: