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 ther contexts - such as groups and associative algebras - are invited to follow the courses Complementi di Algebra (66696) and Combinatoria Algebrica (96730).

Readings/Bibliography

The fundamental 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 approachable than the previous one, but not covering the representation theory) is

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

Both books are available in ebook version from the catalogue of the departmental library through the student account.

Teaching methods

Every week, 4 hours of frontal instruction and one extra hour dedicated to the exercises (to be decided together 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

Delivery of the exercises assigned during the course and oral exam.

Teaching tools

Frontal teaching at the blackboard, and exercise sheets delivered weekly.

Office hours

See the website of Jacopo Gandini