Basic of Representation Theory of Lie Algebras

The course is not on the list Without time-table
Code Completion Credits Range Language
01TRLA ZK 2 2+0 Czech
Garant předmětu:
Department of Mathematics

Lie algebra is an integral part of many theories in natural sciences. The lecture formulates the fundamentals of Lie algebras and their representations.


Basic course of Calculus and Linear Algebra (in particular, the courses 01MA1, 01MAA2-4, 01LAP, 01LAA2 held at the FNSPE CTU in Prague).

Syllabus of lectures:

1. Definitions and examples of Lie algebras. 2. The relationship between Lie group and Lie algebra . 3. The definition of Lie algebra representations , adjoint representation. 4. Enveloping algebra, Casimir elements. 5. Structural theory, subalgebras and ideals of Lie algebras. 6. Direct and semi-direct product of Lie algebras. 7.Nilpotent, simple, and semi-simple Lie algebras. 8. Cartan decomposition, construction of Lie algebra from Cartan matrix. 9. Kac-Moody algebras . 10. Superalgebra. 11. Examples of using in mathematics and mathematical physics.

Syllabus of tutorials:
Study Objective:

Knowledge: to acquire the mathematical basis of the representation theory of Lie algebras. Abilities: able to use the representation in applications.

Study materials:

Key references: [1] Serre, Jean-Pierre (1965), Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University, Lecture notes in mathematics, 1500, Springer, ISBN 3-540-55008-9 .

[2] Adams, John Frank (1969), Lectures on Lie Groups, Chicago Lectures in Mathematics, Chicago: Univ. of Chicago Press, ISBN 0-226-00527-5 . [2] Borel, Armand (2001), „Essays in the history of Lie groups and algebraic groups“, History of Mathematics (American Mathematical Society) 21, ISBN 0-8218-0288-7 . [3] Bourbaki, Nicolas, Elements of mathematics: Lie groups and Lie algebras . Chapters 1-3 ISBN 3-540-64242-0, Chapters 4-6 ISBN 3-540-42650-7, Chapters 7-9 ISBN 3-540-43405-4. [4] Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, MR1153249, ISBN 978-0-387-97527-6, ISBN 978-0-387-97495-8. [5]Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, ISBN 0-387-40122-9 .

Further information:
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Data valid to 2024-05-18
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