Representation theory 2

Garant předmětu:

Severin Pošta

Lecturer:

Severin Pošta

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

1. Basics of representations of compact groups, Schur's lemma, orthogonality relations, Casimir operators.

2. Lie groups and algebras, matrix groups, one parametric subgroups, exponential map, group SU(n) and their representations.

3. Decomposition of representations, Clebsh-Gordan coeficients.

4. Gelfand-Tsetlin bases, Verma bases..

5. Representations of groups and special functions.

6. Classification of irreducible representations of semisimple Lie algebras, Cartan subalgebra, roots, weights, lattices, Weyl chambers.

7. Classical and exceptional simple Lie algebras, Dynkin diagrams.

8. Realizations of Lie algebras, Weyl algebras.

9. Representations of Lie superalgerbas, osp(1,2n).

Requirements:

Syllabus of lectures:

1. Basics of representations of compact groups, Schur's lemma, orthogonality relations, Casimir operators.

2. Lie groups and algebras, matrix groups, one parametric subgroups, exponential map, group SU(n) and their representations.

3. Decomposition of representations, Clebsh-Gordan coeficients.

4. Gelfand-Tsetlin bases, Verma bases..

5. Representations of groups and special functions.

6. Classification of irreducible representations of semisimple Lie algebras, Cartan subalgebra, roots, weights, lattices, Weyl chambers.

7. Classical and exceptional simple Lie algebras, Dynkin diagrams.

8. Realizations of Lie algebras, Weyl algebras.

9. Representations of Lie superalgerbas, osp(1,2n).

Syllabus of tutorials:

Study Objective:

Study materials:

Povinná literatura

1. J. Humphreys: Introduction to Lie Algebras and Representation Theory, Springer, 2012.

2. K. Erdmann, M. J. Wildon: Introduction to Lie Algebras, Springer, 2006.

3. B. Hall: Lie Groups, Lie Algebras and Representations: An Elementary Introduction, Springer, 2016.

Doporučená literatura

4. Knapp: Lie Groups: Beyond an Introduction, Springer, 2013.

5. W. Fulton, J. Harris: Representation Theory: A First Course, Springer, 2013.

6. K. Tapp: Matrix Groups for Undergraduates, AMS, 2008.

7. A. Klimyk, N. Vilenkin: Representations of Lie groups and special functions, Kluwer, 1991.

8. D. Želobenko: Compact Lie groups and their representations, AMS, 1973.

9. M. Scheunert: The Theory of Lie Superalgebras: An Introduction, Springer, 2006.

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)