Basic of Representation Theory of Lie Algebras

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Code	Completion	Credits	Range	Language
01TRLA	ZK	2	2+0	Czech

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

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

Requirements:

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 .

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