# MA8402 Lie groups and Lie algebras – Spring 2011

## Information

 19 May The exam day will be Wednesday 8.June. On today's meeting we went through some of the basic results about compact Lie groups G and their linear representations.It would now be appropriate to learn the representation theory of a torus, which, after all, determine the representation theory of a compact Lie group due to Cartan's maximal torus theorem. Then one can take a closer look at classical groups of rank 1 or 2, say(i.e. SU(2), SU(3), SO(3),SO(4),SO(5), SP(2)) and determine their root system by looking at their adjoint representation. This course will not be given as a regular course with lectures. However, it is possible to take the course as a "reading course" based on self study, with an oral exam at the end of the semester. Interested students should contact me (Eldar Straume) as soon as possible, and having your e-mail address I can reach you and make plans for a first meeting.Then we can also consider possible textbooks and an appropriate curriculum. The first meeting will be on 20 January, Thursday at 11.00. We meet in the 12. floor of the math. building (Sentralbygg II), outside room 1250 (my office, ES). More information about the course contents is listed below.The texts [3], [4] can be obtained from the course coordinator. You find the historical part here, which could be a good way to get started.

## Literature

Here are some texts. The booklet [3] and article [4] can be obtained from the course coordinator.The books [1] or [2] are available through the international textbook market as usual.

• [1] A.Baker; Matrix Groups: An introduction to Lie Group Theory, Springer Verlag
• [2] W. Rossmann;Lie Groups. An Introduction Through Linear Groups, Oxford Science Publication.
• [3] E. Straume; Lecture Notes on Lie Groups and Lie Algebras, NTNU
• [4] E.Straume; Lies kontinuerlige og infinitesimale grupper, Normat 4 (160-170), 1992.

### Possible topics to be covered

• 1. A brief introduction to Lies continuous and infinitesimal groups.In particular, the connection between one-parameter groups, vector fields, and flows of dynamical systems cf. [3], [4].
• 2. Matrix groups, linear groups, topological groups, Lie groups. Give precise definitions and examples.cf. [1], [2],[3].
• 3. Basic representation theory : linear representations, and also non-linear group actions. Give precise definitions. Describe in particular how representations of a torus (both real and complex) are classified, in terms of "weights" (or "characters").cf. [3], and perhaps [1].
• 4. Lie groups, Lie algebras and their relationships. Compact classical groups, that is, the compact simple groups which are not exceptional.cf. [1],[2],[3].
• 5. More about characters and weights. Definition of the character (or equivalently the weight system) of a real or complex representation of a compact Lie group.Give examples of representations of rank 1 or 2 groups such as SO(3),SO(4), SU(2),SU(3), etc., cf. [3], possibly [2].
• 6. The root system of classical groups, SO(n), SU(n), Sp(n), of rank 1 or 2. The root system, is by definition, the weight system of the so-called adjoint representation.
• 7. A more specialized study of SU(2) and SO(3) and their representation theory. For example, with emphasis on applications in quantum mechanics or classical mechanics.

In agreement with the course coordinator, the student should decide upon a final curriculum, based upon the following list of topics.

### Final choice of curriculum

The following parts of Rossmann's book:
1. Chapter 1, 2, and 4
3. Section 1 of Chapter 3