Learning objectives
The students will learn the basic definitions, problems, and techniques in the theory of Lie algebras.
Prerequisites
Algebra (groups, rings, fields); Linear Algebra; Differential Geometry (preferable).
Course unit content
Lie algebras are fundamental objects in Mathematics and Physics.
This course is an introduction to Lie algebras and their representation theory. The main goals of the course are the classification theorem of simple complex Lie algebras and the Weyl character formula.
Full programme
Fundamental theory of Lie algebras: basis; representations of Lie algebras; universal enveloping algebras; Poincaré-Birkhoff-Witt theorem.
Semisimple Lie algebras: solvable and nilpotents Lie algebras; Lie's and Engel's Theorems; reductive and semisimple Lie algebras; Cartan's criterion; Killing form and Jordan decomposition; Whitehead's Theorem and Weyl's Theorem of complete reducibility; structure of semisimple Lie algebras; root systems; Weyl groups; classification of irreducible representations; Weyl's character formula.
Bibliography
The main reference for the course is the book "Introduction to Lie Algebras and Representation Theory" by James Humphreys.
Teaching methods
The topics of the course will be discussed during the lectures, together with examples, applications, and exercises. Attendance is highly recommended.
Assessment methods and criteria
The exam is structured in two parts. The first part of the exam will consist of solving 30 exercises chosen from those assigned during the course. The oral exam will consist of a 30-minute seminar for in-depth discussion, accompanied by a brief report, on a topic agreed upon with the instructor. At the end of the presentation, the student will be asked to explain the solution to one of the exercises presented.
Other information
The courses of “Algebra Superiore 1” and “Algebra Superiore 2” are completely independent. However, they are complementary. The attendance of both courses (when possible) may be beneficial, and it is strongly recommended.
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