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### Selected topics in representation theory

 Code: 160588 ECTS: 5.0 Lecturers in charge: prof. dr. sc. Filip Najman

### 1. komponenta

Lecture typeTotal
Lectures 45
Description:
COURSE AIMS AND OBJECTIVES: The purpose of the course is to present some basic notions, concepts and results of the representation theory for groups. In particular, the representations of both finite and compact groups will be studied. Thus we will introduce the students to the theory of Lie groups and Lie algebras.

COURSE DESCRIPTION AND SYLLABUS:
1. Basic notions. Group representations; invariant subspaces; reducibility; irreducibility; Schur lemma; direct sum of representations; complete reducibility; tensor product.
2. Representations of finite groups. Regular representation; group algebra; orthogonality relations; characters; central functions; decomposition of an arbitrary finite-dimensional representation; Induced representation; representations of symmetric group.
3. Representations of compact groups. Haar measure; complete reducibility; matrix coefficients; orthogonality relations; characters; Peter-Weyl theorem.
4. Lie groups and Lie algebras. Lie groups; Lie algebras; the exponential map; relationship between the continuous finite-dimensional representations of a Lie group and the representations of its Lie algebra; groups SO(3), SU(2), SU(3) and their representations.

TEACHING AND ASSESSMENT METHODS:
Students' obligations during classes: Lecture attendance and active participation in tutorials, solving homework problems, passing 2 mid-term exams.
Signature requirements: Recorded activity at 70% of lectures and tutorials, submission of 70% of written homework problems, passing grade at all mid-term exams.
Taking of exams: The final examination is in a written or oral form. The final grade is based on solving homework problems results, the grades for mid-term exams and the grade for the final examination.
Literature:
1. S. Kurepa: Konačno dimenzionalni vektorski prostori i primjene
2. J.-P. Serre: Représentations linéaires des groupes finis
 1. semester Not active Izborni predmet 1, 2 - Regular study - Theoretical Mathematics 2. semester Izborni predmet 1, 2 - Regular study - Theoretical Mathematics 3. semester Not active Izborni predmet 3, 4 - Regular study - Theoretical Mathematics 4. semester Izborni predmet 3, 4 - Regular study - Theoretical Mathematics
Consultations schedule: