# Lie Groups, Lie Algebras And Representations

### Accès rapides

Prochain Séminaire de la FIP :
Accéder au programme

Retrouvez toutes les informations pour vos stages :
Stages L3
Stages M1 ICFP

Actualités : Séminaire de Recherche ICFP
du 14 au 18 novembre 2022 :

Retrouvez le programme complet

Contact - Secrétariat de l’enseignement :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr

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Theoretical Physics
Faculty : David Hernandez
Tutor : Achilleas Passias
ECTS credits : 6
Language of instruction : English
Examination : Written
Covid-19 alternative examination : Oral examination on course material and tutorials

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. Compact Lie groups and their representations.

5. Infinite dimensional Lie algebras : fundamental examples.

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.

### Accès rapides

Prochain Séminaire de la FIP :
Accéder au programme

Retrouvez toutes les informations pour vos stages :
Stages L3
Stages M1 ICFP

Actualités : Séminaire de Recherche ICFP
du 14 au 18 novembre 2022 :

Retrouvez le programme complet

Contact - Secrétariat de l’enseignement :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr

r>