Algebra II (MATH 101b, Spring 2012)

Day & Time: Tuesday and Friday 12:30 — 1:50pm

Location: Goldsmith 0226

Instructor: Joel Bellaiche

Office hours: Tuesdays, 3pm — 4pm

Grade: 40% exercises in class, 40% personal project, 20% final exam

Textbooks:

  • Serre: linear representations of finite groups
  • Fulton-Harris: Representation theory (a first course)
  • Lam: A first course in non-commutative rings
  • Weibel: An intorduction to homological algebra
  • Lam: Lectures on rings and modules
  • Curtis-Reiner: Representation theory of finite groups and associative algebras,

    Program:

  • First part: Representation theory and non-commutative rings
  • Second part: Homological algebra

    Schedule of Lectures

    Class Topic Suggested Reading Exercises
    Jan 17 Introduction.
    Jan 20 Representations theory of finite groups: first definitions Serre, chapters 1 and 2; Fulton-Harris, chapters 1 and 2 Exercises 1
    Jan 24 Representations theory of finite groups: Complete reducibilty, Schur lemma, characters Serre, chapters 1 and 2; Fulton-Harris, chapters 1 and 2
    Jan 27 Orthogonality of irreducible characters Serre, chapters 1 and 2; Fulton-Harris, chapters 1 and 2
    Jan 31 Consequences of the orthogonality of characters. The case of S_3 Serre, chapters 1 and 2; Fulton-Harris, chapters 1 and 2 Exercises 2
    Feb 3
    Feb 7
    Feb 10
    Feb 14
    Feb 28 (Non-Commutative) Rings. Left and right modules Exercises 3
    Mar 2 Examples of rings. Basic properties of modules
    Mar 6 Simple modules and Schur Lemma. Uniqueness of the decomposition as a sum of simple modules
    Mar 9 Semi-simple modules
    Mar 13 No class
    Mar 16 Two sided-ideals and quotient. Annihilator of a module. Theorems of Jacobson and Burnside Exercises 4
    Mar 20 No class
    Mar 23 No /td>
    Mar 27
    Mar 30
    Mar 27
    Apr 3
    Apr 17 Exercises 5