Lie groups

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M3
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Lie group and the associated Lie algebra. Exponential map. Adjoint action.
Lie theory.
Homogeneous spaces. Slice theorem.
Compact Lie groups. Haar measure. Maximal tori.
Other possible topics::
Weyl group. Root spaces. Representations of compact Lie groups. Solvable, nilpotent and semisimple Lie groups and Lie algebras. Harmonic analysis on a Lie group.

Readings

J. F. Adams: Lectures on Lie Groups, W. A. Benjamin, New York-Amsterdam, 1969.
F. W. Warner: Foundations of Differentiable Manifolds and Lie Groups, Springer, New York-Berlin, 1983.
J. P. Serre: Lie Algebras and Lie Groups, 2nd edition, Springer, Berlin, 2006.
T. Bröcker, T. T. Dieck: Representations of Compact Lie Groups, Springer, New York, 2003.
•J. J. Duistermaat, J. A. C. Kolk: Lie Groups, Springer, Berlin, 2000.

Objectives and competences

Student gets familiar with the basic concepts of Lie group with the associated Lie algebra, and with Lie theory. In particular, the student learns the basic theory of representations of compact Lie groups and homogeneous spaces. Lie groups are a central concept of differential geometry and are applied in many areas of mathematics and mathematical physics.

Intended learning outcomes

Knowledge and understanding: Knowledge and
understanding of basic concepts and definitions
of the theory of Lie groups.
Application: Solving problems using the theory.
Reflection: Understanding of the theory from
the applications.
Transferable skills: Skills in using the literature and other sources, the ability to identify and solve the problem, critical analysis.

Learning and teaching methods

lectures, exercises, homeworks, consultations

Assessment

2 midterm exams instead of written exam, written exam or homework
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Franc Forstnerič:
FORSTNERIČ, Franc, ROSAY, Jean-Pierre. Approximation of biholomorphic mappings by automorphisms of C [sup] n. Inventiones Mathematicae, ISSN 0020-9910, 1993, let. 112, št. 2, str. 323-349. [COBISS-SI-ID 9452121]
FORSTNERIČ, Franc. Runge approximation on convex sets implies the Oka property. Annals of mathematics, ISSN 0003-486X, 2006, vol. 163, no. 2, str. 689-707. [COBISS-SI-ID 13908825]
FORSTNERIČ, Franc. Actions of (R,+) and (C,+) on complex manifolds. Mathematische Zeitschrift, ISSN 0025-5874, 1996, let. 223, št. 1, str. 123-153. [COBISS-SI-ID 6928729]
Janez Mrčun:
MRČUN, Janez. On isomorphisms of algebras of smooth functions. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2005, vol. 133, no. 10, str. 3109-3113. [COBISS-SI-ID 13782361]
MOERDIJK, Ieke, MRČUN, Janez. Introduction to foliations and Lie groupoids, (Cambridge studies in advanced mathematics, 91). Cambridge, UK: Cambridge University Press, 2003. IX, 173 str., ilustr. ISBN 0-521-83197-0. [COBISS-SI-ID 12683097]
MOERDIJK, Ieke, MRČUN, Janez. On the integrability of Lie subalgebroids. Advances in mathematics, ISSN 0001-8708, 2006, vol. 204, iss. 1, str.101-115. [COBISS-SI-ID 14074201]
Pavle Saksida:
SAKSIDA, Pavle. Maxwell-Bloch equations, C Neumann system and Kaluza-Klein theory. Journal of physics. A, Mathematical and general, ISSN 0305-4470, 2005, vol. 38, no. 48, str. 10321-10344. [COBISS-SI-ID 13802073]
SAKSIDA, Pavle. Neumann system, spherical pendulum and magnetic fields. Journal of physics. A, Mathematical and general, ISSN 0305-4470, 2002, vol. 35, no. 25, str. 5237-5253. [COBISS-SI-ID 11920217]
SAKSIDA, Pavle. Lattices of Neumann oscillators and Maxwell-Bloch equations. Nonlinearity, ISSN 0951-7715, 2006, vol. 19, no. 3, str. 747-768. [COBISS-SI-ID 13932377]