Lecture Course: Algebra I

Prof. Dr. Catharina Stroppel, Dr. Olaf Schnürer.

Time: Thursdays, 14.15-16.00, and Fridays, 8.15-10.00.
Place: Kleiner Hörsaal, Wegeler Str. 10

The re-examinations will be oral exams.

Exercise sheets

  • Sheet 1 (pdf)
  • Sheet 2 (pdf)
  • Sheet 3 (pdf)
  • Sheet 4 (pdf)
  • Sheet 5 (pdf)
  • Sheet 6 (pdf)
  • Sheet 7 (pdf)
  • Sheet 8 (pdf)
  • Sheet 9 (pdf)
  • Sheet 10 (pdf)
  • Sheet 11 (pdf)
  • Exam (pdf)

  • Tutorial problems

  • Tutorial Problems 1 (pdf)
  • Tutorial Problems 2 (pdf)

  • Tutorials

  • Wednesdays, 10.15-12, N 0.007, Marc Sauerwein.
  • Wednesdays, 16.15-18, N 0.003, Thomas Poguntke.
  • Thursdays, 16.15-18, N 0.003, Arik Wilbert.
  • Fridays, 12.15-14, SR 1.008, Tomasz Przezdziecki.
  • If a public holiday falls on your tutorial please feel free to attend a different one.

    The course gives an introduction into important concepts in algebra. On the way it will give some basic notions and definitions from category theory, always illustrated by the concrete examples. One main focus and large part of the course will be the study of actions of groups on sets and rings and their invariants. As a special case actions of groups on polynomial rings will appear. Hereby we will deal with finite groups, but also with groups admitting the extra structure of an affine algebraic variety, so-called affine algebraic groups.

    Prerequisites: Basic knowledge in group theory and rings and modules. For instance the course taught by Prof. Schwede or the book "Algebra" by Bosch.
    Most of the tools needed will be introduced again as required, but it is assumed that the students have worked with groups, rings and modules before.

    The course does not follow a single book, but good references for the material are

    1) Lang: Algebra
    2) Knapp: Advanced algebra
    2) Procesi: Lie groups: An approach through invariants (only the more basic parts)
    3) Goodman, Wallach: Representations and invariants of the classical groups (only the more basic parts)
    4) Kunz: Einführung in die algebraische Geometrie

    Here are a few topics which will be covered in the course

    I) (Linear) actions of groups and invariants
    2) Symmetric polynomials and the main theorem of invariant theory
    3) Classical linear algebraic groups
    4) Ring of invariants
    4) Representations and complete reducibility
    5) Semisimplicity (Maschke's theorem, Wedderburn theorem)
    6) Double centralizer theorem and density theorem