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Course

Code:	WISM 414
Subject:	Mathematics
Topic:	Lie Groups
Views:	62,741

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Name:	Utrecht University (Utrecht)
Type:	University

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Instructor

Name

Erik P. van den Ban

Lie Groups and Lie Algebra

Video Lectures

Displaying all 32 video lectures.

Lecture 1 Play Video	Lec 1A - Introduction to Lie Groups
Lecture 2 Play Video	Lec 1B - Lie Groups Definitions and Basic Properties
Lecture 3 Play Video	Lec 2A - Invariant Vector Fields & The Exponential Map
Lecture 4 Play Video	Lec 2B - The Lie Algebra of a Lie Group
Lecture 5 Play Video	Lec 3A - The Lie Algebra of a Lie Group II
Lecture 6 Play Video	Lec 3B - Commuting Elements & Component of the Identity
Lecture 7 Play Video	Lec 4A - Commutative Lie Groups
Lecture 8 Play Video	Lec 4B - Lie Subgroups & Analytic Subgroup Theorem
Lecture 9 Play Video	Lec 5A - Closed Subgroups
Lecture 10 Play Video	Lec 5B - The Groups SU(2) and SO(3)
Lecture 11 Play Video	Lec 6A - Group Actions and Orbit Spaces
Lecture 12 Play Video	Lec 6B - Actions of Principal Fiber Bundle Type & Smoothness Principle
Lecture 13 Play Video	Lec 7A - Proper and Free Actions & Coset Spaces Section 12: existence of Thm 12.5; principal fiber bundles Section 13: proper and free actions Section 14: coset spaces skipped Section 15: orbits of smooth actions. Section 16: the Baire property: background reading Section 17: normal subgroups, the isomorphism theorem
Lecture 14 Play Video	Lec 7B - Baire Property, Normal Subgroups, and Isomorphism Theorem Section 12: existence of Thm 12.5; principal fiber bundles Section 13: proper and free actions Section 14: coset spaces skipped Section 15: orbits of smooth actions. Section 16: the Baire property: background reading Section 17: normal subgroups, the isomorphism theorem
Lecture 15 Play Video	Lec 8A - Densities and Integration A few comments on Section 15: orbits of smooth actions. Prepare by reading this section in advance. Section 18: skipped Section 19: Densities and integration
Lecture 16 Play Video	Lec 8B - Densities and Integration II A few comments on Section 15: orbits of smooth actions. Prepare by reading this section in advance. Section 18: skipped Section 19: Densities and integration
Lecture 17 Play Video	Lec 9A - left Haar measure, bi-invariant Haar measure Section 19: left Haar measure, bi-invariant Haar measure Section 20: finite dimensional continuous representations: basic notions: invariant subspace, intertwining operator, equivalence, irreducibility, unitary representation, unitarizability
Lecture 18 Play Video	Lec 9B - Finite Dimensional Continuous Representations Section 19: left Haar measure, bi-invariant Haar measure Section 20: finite dimensional continuous representations: basic notions: invariant subspace, intertwining operator, equivalence, irreducibility, unitary representation, unitarizability
Lecture 19 Play Video	Lec 10A - Schur's Lemma & Orthogonality Section 20: Schur's lemma, example: irreducible representations for SU(2) Section 21: Schur orthogonality: for matrix coefficients, done in class. Section 23: Formulation of the Peter-Weyl theorem. Sections 24 and 25: proof of Peter-Weyl not done in class. Read on your own!
Lecture 20 Play Video	Lec 10B - Formulation of the Peter-Weyl theorem Section 20: Schur's lemma, example: irreducible representations for SU(2) Section 21: Schur orthogonality: for matrix coefficients, done in class. Section 23: Formulation of the Peter-Weyl theorem. Sections 24 and 25: proof of Peter-Weyl not done in class. Read on your own!
Lecture 21 Play Video	Lec 11A - Characters and Multiplicities Corollary 21.5 and equivariant form of Peter-Weyl theorem Section 22: characters and multiplicities Sections 26, 27: class functions and classical Fourier series. Read by yourself: Section 28: 28.1, 28.2, 28.3, 28.4, 28.6: classification of the irreducible representations of SU(2)
Lecture 22 Play Video	Lec 11B - Class Functions and Classical Fourier Series Corollary 21.5 and equivariant form of Peter-Weyl theorem Section 22: characters and multiplicities Sections 26, 27: class functions and classical Fourier series. Read by yourself: Section 28: 28.1, 28.2, 28.3, 28.4, 28.6: classification of the irreducible representations of SU(2)
Lecture 23 Play Video	Lec 12A - Lie Algebra Representations Section 29: Lie algebra representations Section 30: irreducible representations of sl(2,C) Section 31: roots and weights
Lecture 24 Play Video	Lec 12B - Irreducible Representations of sl(2,C) Section 29: Lie algebra representations Section 30: irreducible representations of sl(2,C) Section 31: roots and weights
Lecture 25 Play Video	Lec 13A - Highest Weight of Irreducible Representation Section 31: highest weight of irreducible representation Section 32, just result Section 35, compact Lie algebras: decomposition into simple ideals.
Lecture 26 Play Video	Lec 13B - Compact Lie Algebras Section 31: highest weight of irreducible representation Section 32, just result Section 35, compact Lie algebras: decomposition into simple ideals.
Lecture 27 Play Video	Lec 14A - Automorphisms and Derivations Section 33, automorphisms and derivations (study yourselves). Section 34, characterization of compact semisimple Lie algebras by Killing form. Section 36: root systems for compact algebras; up to Proposition 36.6. Read the proof of Prop. 36.6 by yourself.
Lecture 28 Play Video	Lec 14B - Characterization of Compact Semisimple Lie Algebras by Killing form Section 33, automorphisms and derivations (study yourselves). Section 34, characterization of compact semisimple Lie algebras by Killing form. Section 36: root systems for compact algebras; up to Proposition 36.6. Read the proof of Prop. 36.6 by yourself.
Lecture 29 Play Video	Lec 15A - Reflections and the Weyl group Section 36: additional properties of root systems: reflections and the Weyl group. Section 37: classification of irreducible representations by highest weight and Weyl's formulas. In the lecture notes, only the results are discussed. Read this on your own. Section 38.1, up to Lemma 38.4.
Lecture 30 Play Video	Lec 15B - Classification of Irreducible Representations Section 36: additional properties of root systems: reflections and the Weyl group. Section 37: classification of irreducible representations by highest weight and Weyl's formulas. In the lecture notes, only the results are discussed. Read this on your own. Section 38.1, up to Lemma 38.4.
Lecture 31 Play Video	Lec 16A - Cartan Integers (Final lecture) Section 38.1 from Lemma 38.4: Cartan integers. 38.2: positive and fundamental systems 38.3: the rank two root systems 38.5: Dynkin diagrams The classification of root systems, hence of compact semisimple algebras.
Lecture 32 Play Video	Lec 16B - Dynkin Diagrams & Classification of Root Systems (Final Lecture) Section 38.1 from Lemma 38.4: Cartan integers. 38.2: positive and fundamental systems 38.3: the rank two root systems 38.5: Dynkin diagrams. The classification of root systems, hence of compact semisimple algebras.