66696 - Algebra Complements

Learning outcomes

Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory.

The goal of this course is to give an introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases.

 

Course contents

Basic notions of representation theory: algebras, irreducible representations, tensor product of representations, dual representations, semisimple representations

General results of representation theory: The density theorem, Representations of direct sums of matrix algebras,  Filtrations, Finite dimensional algebras, Characters of representations, The Jordan-Holder theorem. The Krull-Schmidt theorem

Representations of finite groups: Maschke’s theorem. Characters.Frobenius determinant

Hints on Quiver representations

Readings/Bibliography

P. Etingof. Introduction to representation theory

https://math.mit.edu/~etingof/repb.pdf

Teaching methods

Lessons at the blackboard, discussion sessions

Assessment methods

Oral exam. Exercises sheets will be given to students and corrected in class

Teaching tools

Blackboard

Office hours

See the website of Nicoletta Cantarini

See the website of Luca Migliorini