Course: Introduction to the Theory of Lie Groups

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Course title Introduction to the Theory of Lie Groups
Course code MU/03262
Organizational form of instruction Lecture + Lesson
Level of course Master
Year of study not specified
Semester Summer
Number of ECTS credits 6
Language of instruction Czech
Status of course Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Course availability The course is available to visiting students
  • SERGYEYEV Artur, doc. RNDr. Ph.D.
Course content
- 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.

Learning activities and teaching methods
Recommended literature
  • C. Isham. Modern Differential Geometry for Physicists. Singapore, 1999.
  • K. Erdmann, M. Wildon. Introduction to Lie algebras. Springer, 2006.
  • L. S. Pontrjagin. Nepreryvnye gruppy. Nauka, Moskva, 1973.
  • M. M. Postnikov. Gruppy i algebry Li. Nauka, Moskva, 1982.
  • N. Bourbaki. Lie groups and Lie algebras. Herman, Paris, 1975.
  • N. Jacobson. Lie algebras. J. Wiley-Interscience, London, 1962.
  • P.J. Olver. Equivalence, Invariants and Symmetry. 1995.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Mathematical Institute in Opava Mathematical Analysis (1-IVT) Mathematics courses 4 Summer
Mathematical Institute in Opava Mathematical Analysis (1) Mathematics courses 1 Summer