96759 - Representation Theory

Learning outcomes

At the end of the course, the student knows the fundamental notions of finite-dimensional Lie algebras and their representation theory. The student is able to handle the main tools of this theory that can be used to construct mathematical models.

Course contents

Those students interested in the combinatorial aspects of the cours will be able to deepen such aspects in the course Group theory (09346), which will be activated next year and will cover the Coxeter groups.

Lie algebras are fascinating algebraic structures, with many applications in geometry and mathematical physics. In this course, we will study finite dimensional Lie algebras.

After studying some basic classes of Lie algebras (such as the nilpotent and the solvable ones), we will focus on the class of semisimple Lie algebras. We will describe their structure, their classification and finally their representation theory.

In particular we will introduce the fundamental notion of root system, and we will study particular finite reflection groups associated to them, called Weyl groups. We will see how these objects of combinatorial nature allow to classify the semisimple Lie algebras and their irreducible representations.

Weekly exercises will be assigned to guide the students in the learning process.

The only prerequisite of the course is a good knowledge of linear algebra.

Those students intersted in representation theory in more general situations are invited to follow the course Algebraic Combinatorics (96730), which will be held in the same semester, and which will cover the representation theory of the finite groups and of the symmetric groups.

Those students interested in the combinatorial aspects of the cours will be able to deepen such aspects in the course Group theory (09346), which will be activated next year and will cover the Coxeter groups.

Readings/Bibliography

The foundamental reference for this course is the book

• J. Humphreys: Introduction to Lie Algebras and Representation Theory, third edition, Springer, 1990.

Another useful reference (a bit more elementary than the previous one, but not covering the representation theory) is

• K. Erdmann, M. Wildon: Introduction to Lie algebras, Springer, 2006.

Teaching methods

Every week, 4 hours of frontal instruction and one extra hour dedicated to the exercises (to be discussed with the students).

Exercise sheets will be weekly assigned. Student's work on the exercises will be fundamental for the understanding of the theory.

Assessment methods

Oral exam

Teaching tools

The material for the course will be uploaded on the webpage

http://www.dm.unibo.it/~jacopo.gandini/rep_2122.html

Office hours

See the website of Jacopo Gandini