MA8402 Lie groups and Lie algebras – Spring 2013

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.
• [5] R. Gilmore; Lie Groups, Physics and Geometry. Check the web, either buy the book or find some early version of it freely via internet.
• [6] A. Kirillov; Introduction to Lie groups and Lie Algebras boka er her
• [7] J.J.Duistermaat, A.C.Kolk; Lie Groups. Springer.
• [8] A.W.Knapp; Lie Groups Beyond an Introduction. Birkhauser

Suggestion. Either choose a curriculum from book [5], or choose from [2] (as was done in 2011), but supplemented with material from [3], [4]. The first part of [3] gives the origins of Lie groups, and it should be included.

__ List of testing problems :test yourself

• 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, homogeneous spaces. 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.

We decide upon the curriculum choice after some time.

• June 2013: The first four on the above list of seven topics.
• December 2013: See the listhere. Kirillovs book is used, together with [3].