Definition of Lie algebras, small-dimensional examples, some classical groups and their Lie algebras (treated informally). Ideals, subalgebras, homomorphisms, modules. Nilpotent algebras, Engel's theorem; soluble algebras, Lie's theorem. Semisimple algebras and Killing form, Cartan's criteria for solubility and semisimplicity, Weyl's theorem on complete reducibility of representations of semisimple Lie algebras. The root space decomposition of a Lie algebra; root systems, Cartan matrices and Dynkin diagrams. Discussion of classification of irreducible root systems, and semisimple Lie algebras.
FSF3604 Lie Algebras 7.5 credits
Course offerings are missing for current or upcoming semesters.
Headings with content from the Course syllabus FSF3604 (Spring 2019–) are denoted with an asterisk ( )
Content and learning outcomes
Course contents
Intended learning outcomes
Students will learn how to utilise various techniques for working with Lie algebras, and they will gain an understanding of parts of a major classification result.
Literature and preparations
Specific prerequisites
Master's or Master's degree with at least 30 credits in mathematics.
Suitable prerequisites are courses in differential geometry and representation theory. Very good knowledge of linear algebra.
Recommended prerequisites
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Equipment
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Literature
Brian C. Hall, "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction "(Graduate Texts in Mathematics) 2nd ed. 2015 Edition.
See http://www.amazon.com/Lie-Groups-Algebras-Representations-Introduction/dp/3319134663/ref=sr_1_1?ie=UTF8&qid=1460893363&sr=8-1&keywords=lie+groups+hall
Examination and completion
If the course is discontinued, students may request to be examined during the following two academic years.
Grading scale
G
Examination
- SEM1 - Seminars, 7.5 credits, grading scale: P, F
Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.
The examiner may apply another examination format when re-examining individual students.
Oral examination
Other requirements for final grade
Continuous oral examination
Opportunity to complete the requirements via supplementary examination
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Opportunity to raise an approved grade via renewed examination
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Examiner
Ethical approach
- All members of a group are responsible for the group's work.
- In any assessment, every student shall honestly disclose any help received and sources used.
- In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.
Further information
Course room in Canvas
Registered students find further information about the implementation of the course in the course room in Canvas. A link to the course room can be found under the tab Studies in the Personal menu at the start of the course.
Offered by
Main field of study
This course does not belong to any Main field of study.
Education cycle
Third cycle
Add-on studies
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Contact
Pär Kurlberg (kurlberg@kth.se)