00006 - Advanced Algebra

Learning outcomes

At the end of this course students should have increased and deepened their knowledge of advanced algebraic notions which are both important on their own and relevant to other fields of mathematics.

Course contents

Lie algebras: basic definitions and examples; classical Lie algebras.
Omomorphisms theorems. Solvable and nilpotent Lie algebras. Engel's Theorem.
Lie's Theorem. Cartan's solvability criterion. Killing form.
Decomposition of a semisimple Lie algebra into the direct sum of simple ideals.
Schur's Lemma. The Casimir element. Weyl's Theorem.
Representations of sl(2,C).
Toral subalgebras of a semisimple Lie algebra. Eigenspace decomposition of a semisimple Lie algebra with respect to a maximal toral subalgebra.
The root system associated to a semisimple Lie algebra.
Root systems and Weyl groups. Dynkin diagrams.
Basis of a root system. Weyl group action.
Cartan matrices.
Classification theorem of irreducible root systems.
Classification of simple finite-dimensional Lie algebras over the complex numbers.
Examples of infinite dimensional Lie algebras.

Readings/Bibliography

Introduction to Lie algebras and representation theory (Graduate Texts in Mathematics), J.E. Humpreys -1994 - Springer

Teaching methods

Lectures and exercises will be alternated in order to explain theoretical concepts through a large number of examples. Students will be asked to actively participate in the lessons.

Assessment methods

Written and oral exam

Teaching tools

Exercises, both solved and unsolved, will be suggested to students

Office hours

See the website of Nicoletta Cantarini

See the website of Fabrizio Caselli