Group Theory I

Analysis and algebra

Classroom

The course is devoted to basic group theory, representations of groups, theory of Lie groups and algebras. It will discuss some applications of groups to physics

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

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"

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.

The final grade is based on one midterm test, a written exam and an oral exam.

Does not apply