MA8402 Lie groups and Lie algebras – Spring 2016

Information

29 februar Dei fem første kapittel av Kirillovs book (siste utgåve) er pensum.
24 januar Vi har rom 922 reservert til møte/seminar i kurset på mandagar kl.16.15-18.
12 januar Kan alle interesserte møte ved møtearealet til geometri/topologi i 12 etg. kl.11 på fredag 15 januar ?!!?
12 januar Dette kurset blir erstatta i neste studieår med eit tilsvarande kurs MA 3xxx(?): Introduksjon til Lie-teori. Men i dette semester skal iflg. studiehandboka kurset MA8402 bli gitt. Det er uklart pr. idag om kurset kan bli undervist med regulære forelesningar, eventuelt redusert undervisning. Men det har alltid vore mulig å ta kurset som lesekurs. I løpet av denne veke skal vi finne ut kva som vil bli gjort dette semester. Følg med!!

Literature

Here are some texts.

  • [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 book is here Note:This is not the last edition.The last edition (2008) is at :

http://www.math.stonybrook.edu/~kirillov/liegroups/liegroups.pdf

  • [7] J.J.Duistermaat, A.C.Kolk; Lie Groups. Springer.
  • [8] A.W.Knapp; Lie Groups Beyond an Introduction. Birkhauser

The booklet [3] and article [4] can be obtained from the course coordinator.

List of testing problems :test yourself

Possible topics to be chosen

  • 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 must decide upon an appropriate curriculum choice rather soon.

Final choice of curriculum

  • Spring 2016 : The first 5 chapters of Kirillov's book (last edition).
  • June 2013: The first four on the above list of seven topics.
  • December 2013: See the listhere. Kirillovs book is used, together with [3].
2016-04-11, Eldar Straume