General Prerequisites: A thorough understanding of the material of Part A Linear Algebra, and of basic notions of abstract algebra such as group actions, rings, ideals, quotients etc..
The Michaelmas term Part B course ”Introduction to Representation Theory” is recommended. Although the results of that course are not largely not logically needed for the Lie algebras course, the representation theory of Lie algebras mirrors that of groups, and so some familiarity with the concepts of the Michaelmas course is very helpful. Some notions are also introduced in the Part A Group Theory course.
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 outline the techniques and structures that go into this classification theorem, which shows that semisimple Lie algebras are encoded in finite sets of highly symmetric vectors in a Euclidean vector space known as root systems, which in turn are classified by a kind of graph known as a Dynkin diagram.
Learning Outcomes: By the end of the course, students will be able to identify the basic classes of Lie algebras - nilpotent, solvable and semisimple, give examples of each, and appreciate the role they play in understanding the structure of Lie algebras. They should be able to use basic notions such as ideals and representations to analyse the structure of Lie algebras, and employ the Cartan criteria. They should also be able to analyse concrete examples of semisimple Lie algebras and identify the associated Dynkin diagram.
Although not all of the key theorems are proved in this course, should they need to, a student who has internalised the techniques used in these lectures should not have too much difficulty filling in these gaps using any of the standard textbooks on the subject.
Course Synopsis: Definition of Lie algebras, small-dimensional examples, some classical groups and their Lie algebras (treated informally). Ideals, subalgebras, homomorphisms, isomorphism theorems.
Basics of representation theory: \(\lieg)\-modules (or equivalently \(lieg)\-representations). Irreducible and indecomposable representations, semisimplicity. Composition series and the Jordan-H(\"o)\lder theorem for modules. Operations on representations: subrepresentations, quotients, Homs and tensor products.
Composition series for Lie algebras. Short exact sequences and the notion of an extension. Split sequences and semi-direct products. Definition of solvable and nilpotent Lie algebras. A representation in which every element of the Lie algebra acts nilpotently has the trivial representation as its only composition factor. Engel’s theorem.
Representations of solvable Lie algebras over an algebraically closed field of characteristic zero including Lie’s theorem. Decomposition of representations of nilpotent Lie algebras into generalised weight spaces.
Cartan subalgebras and the Cartan decomposition. Trace forms and Cartan’s criterion for solvability. The solvable radical and Cartan’s criterion for semisimplicity. Semisimple and simple Lie algebras. The Jordan decomposition. The representations of \(\lsl_{2}\) (to be done through problem set questions – only the classification of irreducibles is needed for use elsewhere in the course).
The Cartan decomposition of a semisimple Lie algebra and the structure of the root system using the representation theory of \(\lsl_{2}\). The abstract root system attached to a semisimple Lie algebra and the reduction of the classification of abstract root systems to the classification of Cartan matrices/Dynkin diagrams. Statement of the classification theorem and informal discussion of the proof of the classification of semisimple Lie algebras.
