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Course module: WISM414
WISM414
Lie groups
Course infoSchedule
Course codeWISM414
ECTS Credits7.5
Category / LevelM (Master)
Course typeCourse
Language of instructionDutch
Offered byFaculty of Science; Teaching institute mathematics;
Contact personprof. dr. E.P. van den Ban
Telephone+31 30 2531518
E-mailE.P.vandenBan@uu.nl
Lecturers
Lecturer
prof. dr. E.P. van den Ban
Other courses by this lecturer
Contactperson for the course
prof. dr. E.P. van den Ban
Other courses by this lecturer
Teaching period
Unknown
Teaching period in which the course begins
BLOK 3
Time slot-: Not in use
Study mode
Full-time
Enrolling through OSIRISYes
Enrolment open to students taking subsidiary coursesYes
Pre-enrolmentNo
Waiting listNo
Course placement processniet van toepassing
Aims
A Lie group is a group with the additional structure of a differentiable manifold for which the group operation is differentiable. The name Lie group comes from the Norwegian mathematician M. Sophus Lie (1842-1899) who was the firstto study these groups systematically in the context of symmetries of partial differential equations. The theory of Lie groups has developed vastly in the course of the previous century. It plays a vital role in the description of symmetries in Physics (quantum physics, elementary particles), Geometry and Topology, and Number Theory (automorphic forms). In this course we will begin by studying the basic properties of Lie groups. Much of the structure of a connected Lie group is captured by its Lie algebra, which may be defined as the algebra of left invariant vector fields. The exponential map will be introduced, and the relation between the structure of a Lie group and its Lie algebra will be investigated. Actions of Lie groups will be studied. After this introduction we will focus on compact Lie groups and the integration theory on them. The groups SU(2) and SO(3) will be discussed as basic examples. We will study representation theory and its role in the harmonic analysis on a Lie group. The classifiction of the irreducible representations of SU(2) will be studied. The final part of the course will be devoted to the classification of compact Lie algebras. Key words are: root system, finite reflection group, Cartan matrix, Dynkin diagram. The course will be concluded with the formulation of the classification of irreducible representations by their highest weight and with the formulation of Weyl's character formula.
Content

A Lie group is a group with the additional structure of a differentiable manifold for which the group operation is differentiable. The name Lie group comes from the Norwegian mathematician M. Sophus Lie (1842-1899) who was the firstto study these groups systematically in the context of symmetries of partial differential equations.
The theory of Lie groups has developed vastly in the course of the previous century. It plays a vital role in the description of symmetries in Physics (quantum physics, elementary particles), Geometry and Topology, and Number Theory (automorphic forms).
In this course we will begin by studying the basic properties of Lie groups. Much of the structure of a connected Lie group is captured by its Lie algebra, which may be defined as the algebra of left invariant vector fields.
The exponential map will be introduced, and the relation between the structure of a Lie group and its Lie algebra will be investigated. Actions of Lie groups will be studied.
After this introduction we will focus on compact Lie groups and the integration theory on them.
The groups SU(2) and SO(3) will be discussed as basic examples.
We will study representation theory and its role in the harmonic analysis on a Lie group. The classifiction of the irreducible representations of SU(2) will be studied.
The final part of the course will be devoted to the classification of compact Lie algebras. Key words are: root system, finite reflection group, Cartan matrix, Dynkin diagram.
The course will be concluded with the formulation of the classification of irreducible representations by their highest weight and with the formulation of Weyl's character formula.
 

Competencies
-
Entry requirements
-
Required materials
-
Recommended materials
Dictation
lecture notes E.P. van den Ban, see http://www.math.uu.nl/people/ban/lie2009/lie2009.html.
Instructional formats
Hoorcollege

Tests
Tentamen
Test weight100
Minimum grade6

Assessment
calculation of final grade:
hand-in exercises: 50%, final exam 50%.

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Kies de Nederlandse taal