Matrix Lie group representations

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Physics

Synopsis:

1.Group theory, symmetric group, homomorphism, isomorphism, group action, direct product, semidirect product, normal group, simple and semisimple group, factor group, matrix Lie groups, SO(n), SU(n), Lorentz group, Poincaré group.

2.One-parameter group, Lie algebras, Lie group Lie algebra correspondence, exponential map.

3.Universal covering group, relation between SO(3) and SU(2).

4.Representation theory, unitary representation, regular representation, equivalent representation, irreducibility, reducibility, Schur`s lemma, Weyl`s theorem.

5.Lie algebra representation and their connection to Lie group representation, projective representation.

6.Irreducible representations of SO(3) and SU(2), raising and lowering operators, spin representation.

7.Finite-dimensional representations of Lorentz group, tensor product of representations.

8.Representations of SU(3), Gell-Mann matrices, weights and roots.

9.Young tableaux.

Requirements:

Solving of recommended exercices to apply theoretical knowledge.

Syllabus of lectures:

1.Group theory, symmetric group, homomorphism, isomorphism, group action, direct product, semidirect product, normal group, simple and semisimple group, factor group, matrix Lie groups, SO(n), SU(n), Lorentz group, Poincaré group.

2.One-parameter group, Lie algebras, Lie group Lie algebra correspondence, exponential map.

3.Universal covering group, relation between SO(3) and SU(2).

4.Representation theory, unitary representation, regular representation, equivalent representation, irreducibility, reducibility, Schur`s lemma, Weyl`s theorem.

5.Lie algebra representation and their connection to Lie group representation, projective representation.

6.Irreducible representations of SO(3) and SU(2), raising and lowering operators, spin representation.

7.Finite-dimensional representations of Lorentz group, tensor product of representations.

8.Representations of SU(3), Gell-Mann matrices, weights and roots.

9.Young tableaux.

Syllabus of tutorials:

Study Objective:

Acquired knowledge:

Students learn fundamentals of group theory, matrix Lie groups and algebras and their representations.

Acquired skills:

Ability to understand abstract concepts in group theory and representations. The emphasis is put on the construction of finite-dimensional irreducible representations of specific Lie groups.

Study materials:

Key references

[1] Tung, W.-K., Group Theory in Physics, World Scientific Publishing Co., Philadelphia, PA, 1985

[2] Georgi, H., Lie Algebras in Particle Physics: from Isospin to Unified Theories, Frontiers in Physics, Westview Press, Advanced Book Program, Colorado, 1999

Recommended references:

[3] A. O. Barut, R. Rączka: Theory of Group Representations and Applications, World Scientific Publishing Co. Pte. Ltd., Singapore, 1986

[4] M. Fecko, Diferenciálna geometria a Lieovy grupy pre fyzikov, IRIS, Bratislava, 2004

[5] B. C. Hall, Lie Groups, Lie Algebras, and Representations, An Elementary Introduction, Second Edition, Springer International, Heidelberg, 2015

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Jaderná a částicová fyzika (elective course)
• Kvantové technologie (elective course)