C2.1 Lie Algebras - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Prof. Nikolay Nikolov
General Prerequisites: 

B2.1 Introduction to Representation Theory is recommended. A thorough knowledge of linear algebra and familiarity with group actions, quotient rings and vector spaces, isomorphism theorems and inner product spaces will be assumed. Some familiarity with the Jordan-Hölder theorem and the general ideas of representation theory will be an advantage.

Course Term: 
Michaelmas
Course Lecture Information: 

16 lectures.

Course Weight: 
1.00 unit(s)
Course Level: 
M

Assessment type:

Course Overview: 

Lie Algebras are mathematical objects which, besides being of interest in their own right, elucidate problems in several areas in mathematics. The classification of the finite-dimensional complex Lie algebras is a beautiful piece of applied linear algebra. The aims of this course are to introduce Lie algebras, develop some of the techniques for studying them, and describe parts of the classification mentioned above, especially the parts concerning root systems and Dynkin diagrams.

Learning Outcomes: 

Students will learn how to utilise various techniques for working with Lie algebras, and they will gain an understanding of parts of a major classification result.

Course Synopsis: 

Definition of Lie algebras, small-dimensional examples, some classical groups and their Lie algebras (treated informally). Ideals, subalgebras, homomorphisms, modules.

Nilpotent algebras, Engel's theorem; soluble algebras, Lie's theorem. Semisimple algebras and Killing form, Cartan's criteria for solubility and semisimplicity, Weyl's theorem on complete reducibility of representations of semisimple Lie algebras.

The root space decomposition of a Lie algebra; root systems, Cartan matrices and Dynkin diagrams. Discussion of classification of irreducible root systems and semisimple Lie algebras.

Reading List: 
  1. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9 (Springer-Verlag, 1972, reprinted 1997). Chapters 1-3 are relevant and part of the course will follow Chapter 3 closely.
  2. B. Hall, Lie Groups, Lie Algebras, and Representations. An Elementary Introduction, Graduate Texts in Mathematics 222 (Springer-Verlag, 2003).
  3. K. Erdmann, M. J. Wildon, Introduction to Lie Algebras (Springer-Verlag, 2006), ISBN: 1846280400.
Further Reading: 
  1. J.-P. Serre, Complex Semisimple Lie Algebras (Springer, 1987). Rather condensed, assumes the basic results. Very elegant proofs.
  2. N. Bourbaki, Lie Algebras and Lie Groups (Masson, 1982). Chapters 1 and 4-6 are relevant; this text fills in some of the gaps in Serre's text.
  3. William Fulton, Joe Harris, Representation theory: a first course, GTM, Springer.