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.
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Week 1:
- Lecture 1: algebras and representations (Sections 1.1–1.3)
- Lecture 2: ideals, quotients, generators, relations (Sections 1.4–1.7)
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Week 2:
- Lecture 3: quivers and Lie algebras (Sections 1.8–1.9)
- No lecture on Thursday
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Week 3:
- Lecture 4: tensor products (Sections 1.10–1.15)
- Lecture 5: semisimple representations, density theorem (Sections 2.1–2.2)
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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)
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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)
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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)
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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)
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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)
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Week 9:
- Lecture 16: representations of symmetric groups (Sections 4.12-4.13)
- No lecture on Thursday
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Week 10:
- No lecture on Tuesday
- Lecture 17: complexification, Frobenius character formula (Sections 4.14-4.17)
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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)
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Week 12:
- Lecture 20: Gabriel's theorem (Sections 5.1-5.9)
- Lecture 21: category theory basics (Sections 6.1-6.4)
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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)
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Week 14:
- Lecture 24: lifting idempotents, projective covers (Sections 7.2-7.3)
- Lecture 25: blocks, finite abelian categories, Morita equivalence
Assignments: