Seminar über Halbeinfache Algebren (SS16)

Seminar on Semisimple Algebras

with Dr. Heer Zhao

The topic of the seminar is semisimple algebras, more precisely, semisimple finite-dimensional associative algebras over a field $K$. We do not assume that the algebras are commutative. Interesting examples are division algebras over $K$, i.e., algebras in which every non-zero element is invertible. A concrete example is the $\mathbb R$-algebra of quaternions constructed by Hamilton. We will see many other examples in the seminar.

This is a classical topic with many connections to other areas, and is a core part of “modern mathematics”. Towards the end of the seminar, we will make the connection with local class field theory, the main topic of the course Algebraic Number Theory 2. (So you get most out of it if you do both. It is possible, however, to take part in the seminar without attending the course, or vice versa.)

Credit points: The seminar is a combined Bachelor/Master seminar. If you successfully give one of the talks marked Ba, you get 6 Credit Points. If you successfully give one of the talks marked Ma, you obtain 9 CP (Master seminar) or 6 CP (Bachelor Seminar) at your choice.

Prerequisites: Linear Algebra, Algebra. If you want to give a talk in the second half, some knowledge on local fields (e.g., finite extensions of $\mathbb Q_p$) is needed; in this case, it is strongly recommended that you attend the course “Algebraic Number Theory 2”.

Seminar program: pdf

Time and place: Tue, 4-6pm, S-U-3.02. Starting date: April 12.

If you would like to participate in the seminar, but have missed the organizational meeting, please contact me at: ulrich.goertz@uni-due.de.

References: We will mainly follow the following sources (for detailed pointers see the seminar program):

  • J. Milne, Class field theory, Chapter IV
  • F. Lorenz, Algebra 2, §§30-33 (available in German and English)

See also:

  • N. Bourbaki, Algèbre, Ch. 8
  • P. Gille, T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, Ch. 1, 2
  • I. Kersten, Brauergruppen
  • J. P. Serre, Corps locaux (in English: Local Fields, Springer GTM)

These topics are also, at least partially, treated in many other advanced algebra text books, e.g. Lang, Algebra; Hungerford, Algebra.

Talks

12.4. Semisimple algebras: Basic notions U. Görtz
19.4. Decomposition of a semisimple algebra as a product of simple algebras H. Zhao
26.4. Wedderburn’s Theorem S. Melzer
3.5. Representations of finite groups A. Stechemesser
10.5. Tensor products and central simple algebras U. Görtz
24.5. The theorem of Skolem and Noether and the Brauer group of a field H. Zhao
7.6. Existence of splitting fields U. Görtz
14.6. Cohomological description of the Brauer group G. Grossi
28.6. $\mathop{\rm Br}(L/K)$ for a cyclic extension $L/K$ T. Kreutz
5.7. The Brauer group of a local field 1 M. Lekić
12.7. The Brauer group of a local field 2 H. Zhao
(19.7.) Connection with Local Class Field Theory 1 N. N.
() Connection with Local Class Field Theory 2 N. N.