MU03262 Introduction to the Theory of Lie Groups

Mathematical Institute in Opava
Summer 2015
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Guaranteed by
prof. RNDr. Artur Sergyeyev, Ph.D., DSc.
Mathematical Institute in Opava
Prerequisites
MU/03038
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives (in Czech)
Předmět slouží k získání základní představy o struktuře obecné Lieovy grupy a o její akci na varietě. Předmět je zakončen zkouškou a zápočtem.
Syllabus
  • - The concept of a Lie group. Analytical, continuous and smooth groups. Hilbert's fifth problem.
    - Local theory of Lie groups.
    - Lie algebras. Tangent Lie algebra of a Lie group. Classification of simple Lie algebras.
    - General linear group and its subgroups. Linear representations. The Ado theorem.
    - The Baker-Campbell-Hausdorff formula.
    - Differential geometry of Lie groups. Left- and right-invariant vector fields and differential forms. One-dimensional Lie subgroups. Solution of the Maurer-Cartan equations. Exponential map.
    - The global theory of Lie groups. The Cartan theorem. Construction of all Lie groups for a given tangent Lie algebra. Lie groups which have no faithful linear representations.
    - Transformation groups of manifolds and their actions. The fundamental vector fields. Principal bundles.
Literature
    recommended literature
  • K. Erdmann, M. Wildon. Introduction to Lie algebras. Springer, 2006. info
  • C. Isham. Modern Differential Geometry for Physicists. Singapore, 1999. info
  • P.J. Olver. Equivalence, Invariants and Symmetry. 1995. info
  • M. M. Postnikov. Gruppy i algebry Li. Nauka, Moskva, 1982. info
  • N. Bourbaki. Lie groups and Lie algebras. Herman, Paris, 1975. info
  • L. S. Pontrjagin. Nepreryvnye gruppy. Nauka, Moskva, 1973. info
  • N. Jacobson. Lie algebras. J. Wiley-Interscience, London, 1962. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
Teacher's information
Oral exam; further requirements to be specified in the course of the semester.
The course is also listed under the following terms Summer 2000, Summer 2001, Summer 2002, Summer 2003, Summer 2004, Summer 2005, Summer 2006, Summer 2007, Summer 2008, Summer 2009, Summer 2010, Summer 2011, Summer 2012, Summer 2013, Summer 2014, Summer 2016, Summer 2017, Summer 2018, Summer 2019.
  • Enrolment Statistics (Summer 2015, recent)
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