Representation theory

A.Y. 2021/2022
9
Max ECTS
73
Overall hours
SSD
MAT/02
Language
Italian
Learning objectives
The aim of the course is to present the basic Ideas of Representation Theory for finite groups (in the 6-credits part) and of Lie algebras (In the advanced 3-credits part).
Expected learning outcomes
Knowledge of the basic ideas of Representation Theory for finite groups (in the 6-credits part) and of Lie algebras (In the advanced 3-credits part).
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
- Both lectures and exercise classes will be delivered through the application Zoom, and there will be the possibility to recover them at any time because they will be recorded and posted on ARIEL.
Prerequisites for admission
Basics of Algebra, in particular of Group Theory.
Assessment methods and Criteria
The exam is divided into a 6-credit part (related to group representations) and a 3-credit part (optional, related to Lie algebras). There is no temporal or order restriction between the performance of the two parts.
Teoria della rappresentazione (prima parte)
Course syllabus
1. Definitions and examples. Irreducible and completely reducible representations of finite groups.
2. Representations and modules. Simple and semisimple modules.
3. Applications to the group algebra. Maschke's Theorem.
4. Characters of finite groups: basic definitions and properties, irreducible characters, orthogonality relations, linear characters.
5. Character tables. Examples.
6. Applications of Character Theory. Solubility criteria, Burnside's Theorem, existence of normal subgroups and how to determine them.
7. Product of representations.
8. Induced representations and characters. Frobenius' Theorem.
9. Representations of symmetric groups. Partitions and Young tableaux, degrees of the irreducible representations of S_n.
Teaching methods
Blackboard lectures
Teaching Resources
C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras", Interscience Publ. New York (1962).

I.M. Isaacs, "Character Theory of finite groups", Academic Press (1976).

M.P. Malliavin, "Les groupes finis et leurs représentations complexes", Volume 1. Masson, 1981
Teoria della rappresentazione mod/2
Course syllabus
Smooth manifolds, tangent space and tangent bundle of a smooth manifold, smooth vector fields and associated Lie algebra. Lie groups and associated Lie algebras: left invariant smooth vector fields. Functor between the category of Lie groups and that of Lie algebras.
Examples of Lie algebras. Adjoint representation. Ideals. Solvable, nilpotent and semisimple algebras. Theorems by Engel and Lie.
Killing form and characterization of semisimple algebras. Modules for Lie algebras and Weyl's Theorem on complete reducibility of modules for a semisimple Lie algebra. Modules of sl(2,C). Toral subalgebras and Cartan decomposition; roots systems.
Teaching methods
Blackboard lectures
Teaching Resources
J.E. Humphreys, "Introduction to Lie Algebras and Representation Theory", Springer (1972).

W. Fulton, J. Harris, "Representation Theory: A First Course", Springer (1991).

J.M. Lee, "Introduction to Smooth Manifolds", Springer (2012).
Teoria della rappresentazione (prima parte)
MAT/02 - ALGEBRA - University credits: 6
Practicals: 24 hours
Lessons: 28 hours
Teoria della rappresentazione mod/2
MAT/02 - ALGEBRA - University credits: 3
Lessons: 21 hours
Professor: Andreatta Fabrizio