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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.
Last update: T_MUUK (13.05.2015)
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Study of non-commutative analysis. Last update: T_MUUK (13.05.2015)
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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.
Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (25.09.2023)
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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
Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (22.02.2019)
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Lecture and exercise. Last update: T_MUUK (13.05.2015)
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We test definitions and theorems and its application in clearly arranged situations. Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (22.02.2019)
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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).
Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (25.09.2023)
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