Theory of representations 2

Course guarantor:

Lecturer:

Tutor:

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Department of Mathematics

Synopsis:

Basic course presenting fundamental notions and results in theory of algebras, groups and their representations, with emphasis given to compact groups, Lie groups, Lie algebras. The course includes the use of groups and their representations in various concrete situations, especially in physics.

Requirements:

Theory of representations 1 (01TR1).

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:

Knowledge: basic theory of compact groups and their representations, classification of representations of Lie algebras, other types of representations.

Skills: use of groups and their representations in various problems arising in physics.

Study materials:

Key references:

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

Recommended references:

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. Zhelobenko: Compact Lie groups and their representations, AMS 1973

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

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: