Group Theory I 1100-3`TG1
The course will present basic concepts of group theory ant theory of group representations as well as more advanced topics directed towards applications in physics. Covered material will also serve as basis for more advanced topics offered in the spring semester.
Program:
1. Basics of group theory
(group, subgroup, homomorphism, normal subgroup, quotient space and quotient group, product of groups, semidirect product of groups, group actions, homogeneous spaces, group algebra)
2. Representations of finite and compact groups
(representation, subrepresentation, irreducible representation, Peter-Weyl theorem, character of a representation, finding all irreducible representations, decomposition of the group algebra, example: representations of the symmetric group, extension to compact groups)
3. Commutative groups, Pontriagin duality (Rn and Zn)
(dual group of a commutative group, Fourier transformation)
4. Induced representations
(induced representation, elements of Mackey's theory of representations of semidirect product of a commutative group, example: Poincare group)
5. Lie groups and Lie algebras
(Lie group, Lie algebra of a Lie group, examples: classical matrix groups: GL(n), SU(n), SO(n), SL(n,C), morphisms of Lie groups and Lie algebras, adjoint representation, exponential map, Maurer-Cartan form)
Student's effort
Lectures: 30 h -- 1 ECTS
Exercise classes: 30 h -- 1 ECTS
Preparation for the lectures and classes: 30 h -- 1 ECTS
Homework problems and preparation for the test: 30 h - 1 ECTS
Preparation for the exam: 30 h -- 1 ECTS
Mode
Prerequisites (description)
Course coordinators
Learning outcomes
Knowledge: Familiarity with basic group theory and theory of group representations.
Skills: Ability to solve simple problems of group theory and the theory of group representations, in particular involving the semisimple product, characters of representations, decomposition into irreducible components for finite groups
Attitude: Appreciation of the beauty, depth and importance of group theory, especially in the context of its applications in physics.
Assessment criteria
Every student has to get a positive grade based on the performance at exercise classes and pass the written and oral exams.
Practical placement
Does not apply
Bibliography
1. A. Trautman "Grupy oraz ich reprezentacje" (lecture notes WF UW)
2. J.P. Serre "Reprezentacje liniowe grup skończonych"
3. A. Barut, R. Rączka "Theory of group representations and applications"
4. B.Simon, "Representations of finite and compact groups"
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: