SUPERIOR ALGEBRA
cod. 1006007

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 one of the most fascinating and profound mathematical structures, fundamental in various areas of mathematics and theoretical physics. Originating from the analysis of continuous transformation groups (Lie groups), they are now indispensable in the study of differential geometry, dynamical systems theory, and symmetries in physical theories. This course provides students with the tools to understand the key concepts of Lie algebra theory and their representations, offering a solid foundation for advanced developments and interdisciplinary applications.

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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