34733 - Elementary Algebra from an Advanced Standpoint

Learning outcomes

At the end of the module, the student has an advanced knowledge of topics such as numerical sets, polynomials and polynomial equations, algebraic structures, and is able to use various aids to support teaching.

Course contents

Symmetry appears everywhere: in Platonic solids, in patterns in the plane, in crystal structures, in highly regular geometric configurations, and, at a deeper level, in Lie algebras. This course offers a journey through these symmetries from a modern and unifying perspective.

The central concept will be that of a root system. A root system is a finite configuration of vectors in a Euclidean space satisfying strong symmetry properties. At first sight, it may look like a simple geometric object; in fact, it is one of the fundamental languages through which mathematics describes and classifies highly symmetric structures.

We will begin with concrete examples in the plane: vector configurations, reflections, dihedral groups, and symmetries of regular polygons. We will then see how these examples fit into a general theory leading to the classification of root systems and their associated Dynkin diagrams. These diagrams encode a remarkable amount of geometric and algebraic information in a very compact form.

Root systems also provide a natural way to construct and study lattices, which appear in Euclidean geometry, in the theory of discrete symmetries, and in crystallography. The symmetries generated by the reflections associated with roots give rise to Weyl groups, among the most important finite groups of symmetries in mathematics.

In the final part of the course, we will see how the same structure appears in the study of semisimple Lie algebras. Every semisimple Lie algebra has an associated root system, and remarkably the classification of simple Lie algebras reduces to the classification of root systems. This leads to the classical families and to the celebrated exceptional Lie algebras.

Main topics: root systems in the Euclidean plane; reflections and symmetry groups; Weyl groups; classification of root systems; Dynkin diagrams; lattices and crystallography; discrete symmetries; semisimple Lie algebras; classical and exceptional simple Lie algebras.

 

Preliminary knowledge: basic courses in linear algebra and abstract algebra.

Readings/Bibliography

During the course notes will be handed and/or will be given exact references from textbooks.

Some useful textbooks (although they contain far more material than what will be covered in the course)

- James Humphreys, Introduction to Lie algebras and representation theory (chapter III in particular)

- William Fulton, Joe Harris, Representation theory, A first course.

Teaching methods

Lectures and exercises.

Assessment methods

Exam consisting of a written test and an oral test.

Students are admitted to the oral exam only if the written exam is sufficient (i.e.16/30). Both tests must be held in the same exam session.

The written exam consists of some exercises.

The final mark for the course Elementi di Algebra e Geometria da un punto di vista superiore is the average of the marks in Elementi di Algebra and in Elementi di Geometria. The mark in one of the two parts remains valid for the following 3 exam sessions (summer, winter and autumn)

Office hours

See the website of Enrico Fatighenti