Matrix Lie group representations
Code  Completion  Credits  Range 

02REP  Z  2  2+0 
 Garant předmětu:
 Jiří Hrivnák
 Lecturer:
 Jiří Hrivnák
 Tutor:
 Jiří Hrivnák
 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.Oneparameter 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.Finitedimensional representations of Lorentz group, tensor product of representations.
8.Representations of SU(3), GellMann 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.Oneparameter 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.Finitedimensional representations of Lorentz group, tensor product of representations.
8.Representations of SU(3), GellMann 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 finitedimensional 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:
 Timetable for winter semester 2023/2024:
 Timetable is not available yet
 Timetable for summer semester 2023/2024:
 Timetable is not available yet
 The course is a part of the following study plans:

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