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Course, academic year 2019/2020
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Advanced Course of Group Theory for Physicists - NMAF038
Title in English: Pokročilé partie z teorie grup pro fyziky
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: doc. RNDr. Svatopluk Krýsl, Ph.D.
Classification: Physics > Mathematics for Physicists
Annotation -
Last update: T_KMA (15.05.2008)
An advanced course of group theory for physicists. It is following to the basic course of mathematics for physicists.
Aim of the course -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (07.02.2018)

Learn basics of the representation theory of Lie groups.

Course completion requirements -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (13.10.2017)

The examination is oral with a written preparation.

Literature -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (22.02.2019)

A. U. Klimyk, N. Ya. Vilenkin, Representations of Lie groups and speciál functions, Kluwer, Dordrecht, 1991.

W. Fulton, J. Harris, Representation Theory, A first course, Springer, Heidelberg, 1991.

D. P. Želobenko, Compact Lie groups and their representations, Translations of Mathematical Monographs, 40, AMS, Providence, 1973.

V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, Courant Lecture Notes, AMS, Providence, 2004.

L. Frappat, A. Sciarrino, P. Sorba, A dictionary of Lie algebras and superalgebras, Academic Press, 2000.

M. Sepanski, Compact Lie Groups, Springer, 2007.

Teaching methods -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (13.10.2017)

Lectures based on literature available.

Requirements to the exam -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (13.10.2017)

In the oral exam, we test the knowledge of definitions, theorems and their proofs that were presented in the lecture.

Syllabus -
Last update: T_KMA (15.05.2008)
Harmonic analysis on homogeneous spaces.
Transitive groups of transformations. Invariant measures. Homogeneous spaces. Induced representations. Spherical functions, zonal functions and asociated spherical functions.

Branching rules, Gelfand-Zetlin bases.

Representation of groups and speciál functions.
Representation of groups SO(3) (resp. SU(2)), quasi-regular reprezentations, zonal spherical fynctions and Gegenbauer (Legendre, Jacobi) spherical polynomials. Decomposition of functions on the spere, spherical functions and the Laplace operator. Group of Euclidean transformations in the plane, cylindric functions (generating functions, recurrent relations). Group of unimodular matrices SL(2,R) and hypergeometric functions. Hermite polynomials. Applications in physics.

Lie super-algebras, supersymetry.
Super-vektor spaces. Super-algebras. Lieovy super-algebras. Basic type sof simple Lie super-algebras Super-Poincare algebra Representations, Applications in physics.

 
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