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    Lie Groups, Lie Algebras And Representations

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    Undergraduate program 
    Program, courses and calendar 2022 - 2023  2022-2023 Timetable and calendar  Courses  Internships  Program, courses and calendar 2023 - 2024  2022-2023 Timetable and calendar  Courses  Internships  Program, courses and calendar 2021 - 2022  2021-2022 Timetable and calendar  Courses  Internships  Admissions  Recrutement sur concours  Recrutement sur dossier : Concours normalien étudiant  Contacts  Seminars  Seminars  2023-2024 FIP/ICFP seminar Program  Undergraduate program Intranet  Archives  Archives 2015-2016  Archives 2016-2017  Archives 2017-2018  Archives 2018-2019  Archives 2019-2020  Archives 2020-2021 
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    Why choose this program?  Program, courses and calendar Master ICFP 1st year (2023-2024)  2023-2024 Calendar and timetable   Courses  Internships  Program, courses and calendar Master ICFP 1st year (2022-2023)  2022-2023 Calendar and timetable   Courses  Internships  Program, courses and calendar Master ICFP 2nd year 2023-2024  First Semester  Second Semester   "Physics Now" Seminars 2019/2020  "Physics Now" Seminars 2020/2021  "Physics Now" Seminars 2021/2022  "Physics Now" Seminars 2023/2024  "Physics Now" Seminars 2022/2023  Internships  2023 2024 calendar  Program, courses and calendar Master ICFP 2nd year 2022-2023  First Semester  Second Semester   "Physics Now" Seminars 2019/2020  "Physics Now" Seminars 2020/2021  "Physics Now" Seminars 2021/2022  "Physics Now" Seminars 2022/2023  Internships  2022 2023 calendar  Admission and Language Requirements  How to apply  Information for students  Information for International Students  Fees and Scholarships  Careers  Master Intranet  Archives  Erreur ( A supprimer)  Program, courses and calendar Master ICFP 1st year (2021-2022)  Program, courses and calendar Master ICFP 1st year (2021-2022)  Program, courses and calendar Master ICFP 2nd year 2021-2022  Program, courses and calendar Master ICFP 2nd year 2020-2021  Program, courses and calendar Master ICFP 2nd year 2019-2020  Archives 2015-2016  Archives 2016-2017  Archives 2017-2018  Archives 2018-2019  Archives 2019-2020  Archives 2020-2021 
    Master Astronomie, Astrophysique et Ingénierie Spatiale 
    Master Macroscopic Physics 
    Master Quantum Engineering 
    The Doctoral School Physics in Ile de France (EDPIF - ED564)  Physics for all 
    Astrophysical Astronomy of Île-de-France (ED 127)  Agrégation de physique  ENS Diploma  Student handbooks  Contact us 

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    November 14 - 18, 2022 :
    All information about the program

    Contact us - Student support and Graduate School office :
    Tél : 01 44 32 35 60
    enseignement@phys.ens.fr

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    Next student seminar :
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    Here you can find information about your internships:
    Experimental Internship - Undergraduate program
    Master ICFP first year Internship

    News : ICFP Research seminars
    November 14 - 18, 2022 :
    All information about the program

    Contact us - Student support and Graduate School office :
    Tél : 01 44 32 35 60
    enseignement@phys.ens.fr

    Theoretical Physics
    Faculty: David Hernandez
    Tutor: Bruno Le Floch
    ECTS credits: 6
    Language of instruction: English
    There is no special prerequisites, except standard linear and general algebra

    Description

    The theory of groups and their representations is a central subject about symmetries
    in mathematics and in sciences in general, in particular in physics. For instance, Lie theory (Lie groups and Lie algebras) was a central theme in mathematics since its origin in 19th century, with various applications in mathematics and physics.

    The aim of this course is to give an introduction to classical concepts and tools in Lie theory as well as for the theory of their representations. We will study remarkable examples (in particular for the applications in physics).

    Plan of the course

    1. Representations of groups and algebras.
    Generalities.
    Finite groups and their characters.

    2. Groups and Lie algebras of finite dimension.
    Lie groups, Lie algebras.
    Fundamental examples, Heisenberg algebras.
    Semi-simple Lie algebras.
    Categories of representations, irreducible representations.
    Complete reducibility.
    Structure of semi-simple Lie algebras.
    Root systems, Weyl group.

    3. Representations of finite-dimensional Lie algebras.
    Highest weight modules, Verma modules, category O.
    Parametrization of simple modules. Jordan-Holder series, multiplicities.
    Finite-dimensional representations. Tensor structure, characters.

    4. Infinite dimensional Lie algebras : fundamental examples.
    Kac-Moody algebras, integrable representations.
    Loop algebras and central extensions.
    Virasoro algebras.
    Quantum groups and Yang-Baxter equation.

    References for the course (for starters, the Fulton-Harris is fine):

    • V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press.
    • P. Etingof, I. Frenkel and A. Kirillov, Lectures on representation theory and Knizhnik-Zamolodchikov equations, Mathematical Surveys and Monographs, 58. American Mathematical Society.
    • W. Fulton and J. Harris, Representation Theory : a first course, Graduate Texts in Mathematics, 129, Springer-Verlag.
    • J. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9. Springer-Verlag.
    • V. Kac, Infinite-dimensional Lie algebras, Cambridge University Press.
    • Y. Kosmann-Schwarzbach Groupes et symétries : Groupes finis, groupes et algèbres de Lie, représentations, Editions de l’école polytechnique
    • J-P. Serre, Lie algebras and Lie groups, 1964 lectures given at Harvard University,
      Lecture Notes in Mathematics, 1500, (2006)
    • S. Sternberg, Group theory and physics, Cambridge University Press.

    Quick links

    Next student seminar :
    Access to the program

    Here you can find information about your internships:
    Experimental Internship - Undergraduate program
    Master ICFP first year Internship

    News : ICFP Research seminars
    November 14 - 18, 2022 :
    All information about the program

    Contact us - Student support and Graduate School office :
    Tél : 01 44 32 35 60
    enseignement@phys.ens.fr

     
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