This course introduces fundamental algebra notions. A first part is focused on finite groups and their linear representations (on the set of complex numbers). The main results of the subject are demonstrated:
Schur's lemma, Maschke's theorem, Peter-Weyl theorem, Plancherel's formula, character orthogonality... The adopted perspective is the use of convolution and Fourier transform.
In the second part, we study A rings (not necessarly commutative) and modules on these rings. We develop, under some hypothesis, classification methods (composition series, extension study, etc.). We the case A is and deduce, for example, the classification of fintite-type abelian groups or conjugacy class of GL(n,k).
Required level: It is recommended to have basis algebraic objects (linear algebra, groups, rings, set, quotient notion) and particularly to have validated the MAT451 course (Algebra and Galois theory).
Course language: French
- Teaching coordinator: Beuzart-Plessis Raphael