MA3407 Introduction to Lie Theory - Fall 2022

(*) There will be two lectures per week (most likely Wed+Fri). We will probably try to use the last time slot on Thu as an exercise class, but for a couple of weeks it will be a lecture (as I will be away for one of the other days). (Also note that the lecture schedule may still change before the semester starts!)

In previous courses you have seen that group actions describe symmetries. For example, the symmetry group of a square under orientation-preserving rigid motions is the cyclic group of order 4, while the corresponding symmetry group for a circle is the special orthogonal group SO(2) of rotations. In the latter case it is natural to think of the rotation by an angle θ as depending continuously or even smoothly (that is, infinitely differentiably) on the angle, and to make sense of this we want to equip the group with a smooth structure. Such "smooth" groups are called Lie groups (after Sophus Lie (1842-99), arguably the "greatest" Norwegian mathematician), and they play an important role in many areas of mathematics, including analysis, topology, differential geometry, representation theory, and even number theory, as well as in physics.

The foundations for the general theory of Lie groups relies on a lot of prerequisite material from analysis, differential geometry, and topology. In this course, we will therefore focus on the special case of matrix groups. Roughly speaking, these are groups consisting of some class of matrices, with the group operation given by matrix multiplication. For such groups we can develop a much more elementary version of the basic theory, relying mainly on linear algebra and a bit of analysis. Moreover, the class of matrix groups actually includes most of the interesting examples of Lie groups.

In the first part of the course, our main goal is to show that many aspects of a (matrix) Lie group are controlled by a seemingly much simpler structure, its Lie algebra, which is a vector space with a certain type of multiplication operation that encodes the "infinitesimal" behaviour of the group. In the second part we will look at representations of Lie groups and Lie algebras (that is to say their linear actions on vector spaces), in particular in the special case of semisimple Lie algebras; we will also sketch the remarkable classification of semisimple Lie algebras in terms of root systems and Dynkin diagrams, which will reveal that we already know all the examples except for 5 "exceptional" cases. Finally, we will hopefully have time to look more explicitly at the representations of some of the "classical" semisimple Lie algebras/groups.

(See also the study handbook for the official course description.)

The absolute prerequisites are a good grasp of linear algebra and basic group theory; we will also need a bit of analysis and topology.

The first lecture will be on 24 August. (The following schedule is very much subject to change!)

We will not directly follow a particular textbook, but the first part of the course will loosely follow chapters 1-8 of:

• [H] Brian C. Hall, Lie groups, Lie algebras, and representations (Download)

At the end of the course we will hopefully have time to look at the representations of some of the classical groups. A good source for this is:

• [FH] William Fulton and Joe Harris, Representation theory: a first course (Download)

Note that both books can be downloaded (legally!) as PDFs from SpringerLink if you are on the university network (either on campus or via VPN).

Some alternative introductory books:

• [R] Wulf Rossmann, Lie groups: an introduction through linear groups
• [S] John Stillwell, Naive Lie theory (Download)

Some more advanced books:

• [H] James E. Humphreys, Introduction to Lie algebras and representation theory (Download)
• [K] Anthony W. Knapp, Lie groups beyond an introduction (Download)