# 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, finite-dimensional algebras (Sections 2.3–2.5)
• Lecture 7: semisimple algebras, characters, two general theorems (Sections 2.5–2.8)
• Week 5:
• Lecture 8: Krull-Schmidt 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: Frobenius-Schur 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 4.12-4.13)
• No lecture on Thursday
• Week 10:
• No lecture on Tuesday
• Lecture 17: complexification, Frobenius character formula (Sections 4.14-4.17)
• Week 11:
• Lecture 18: Schur-Weyl duality (Sections 4.18-4.19)
• Lecture 19: Schur polynomials, Artin's theorem (Sections 4.21-23, 4.26)
• Week 12:
• Lecture 20: Gabriel's theorem (Sections 5.1-5.9)
• Lecture 21: category theory basics (Sections 6.1-6.4)
• Week 13:
• Lecture 22: representable functors, adjoint functors, abelian categories (Sections 6.5-6.7)
• Lecture 23: exact functors, projective modules, Ext and Tor (Sections 6.8-7.1)
• Week 14:
• Lecture 24: lifting idempotents, projective covers (Sections 7.2-7.3)
• Lecture 25: blocks, finite abelian categories, Morita equivalence

Assignments: