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Course, academic year 2023/2024
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Non-commutative harmonic analysis - NMAG534
Title: Nekomutativní harmonická analýza
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:3/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: not taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. RNDr. Svatopluk Krýsl, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Geometry
Annotation -
Last update: T_MUUK (13.05.2015)
Harmonic analysis generalizes the classical Fourier analysis of partial differential equations in R^n for other groups than the abelian R^n. Second part of lecture.
Aim of the course -
Last update: T_MUUK (13.05.2015)

Study of non-commutative analysis.

Course completion requirements -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (25.09.2023)

We test the knowledge of definitions, theorems, and their application.

The exam is oral with a written preparation.

Credit is given for active participation, proving easy theorems or computing examples. Credit is not necessary for entering the exam.

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

Goodman, R., Walach, N., Invariants and Representations of Classical Groups, Oxford

Knapp, A., Representation theory of semi-simple Lie groups: An overview based on examples, Princeton

Kirillov, A., Representation theory and Noncommutative Harmonic Analysis I, II, Springer

Dixmier, J., Envelopping Algebras, AMS

Sepanski, M., Compact Lie groups, Springer

Teaching methods -
Last update: T_MUUK (13.05.2015)

Lecture and exercise.

Requirements to the exam -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (22.02.2019)

We test definitions and theorems and its application in clearly arranged situations.

Syllabus -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (25.09.2023)

1) Universal enveloping algebra of a Lie algebra and the theorem of Poincaré--Birkhoff--Witt. Filtration, associated gradation, and the Noether feature of universal enveloping algebras.

2) Verma modules: Recall of representation theory of simple Lie algebras - Cartan subalgebra, roots, co-roots, positive and simple roots, fundamental weights, Weyl group and Bruhat ordering.

Weights) of representations of semi-simple Lie groups, semi-lattice of non-negative weights. Verma modules - definition, weight property, irreducibility characterization. Description of irreducible and finite-dimensional simple Lie algebra modules. Citation of Bernstein--Gelfand--Gelfand theorem on a connection of homomorphisms of Verma modules and Bruhat ordering.

3) Theorem of (Bott--)Borel--Weil (solutions of Laplace equation on complex flag manifolds): smooth locally trivial fibrations - vector, principal and associated fibrations. Holomorphic manifolds and fibrations. Flag manifolds - Borel and compact presentation of flag manifolds: spheres, projective spaces, Grassmannians, especially Gr_2(4, C). Some results of the structure and representation theory of semi-simple Lie groups. Holomorphic sections for Borel presentations. Formulation of the Borel--Weil theorem and its proof for the complex projective line.

Eventually, the unitary dual of SL(2,R).

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