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Lie groups, Lie algebras and representations
The theory of groups and their representations is a central topic which studies symmetries in various contexts occurring in pure or applied mathematics as well as in other sciences, most notably in physics.
Lie theory (i.e. the study of Lie groups and Lie algebras) has played an important role in mathematic ever since its introduction by the Norwegian mathematician Sophus Lie in the 19th century. It has had a profound impact in physics as well.
The aim of this course is to provide an introduction, from the mathematical perspective, of the classical concepts and techniques of Lie theory. The course will in particular deal with Lie groups, Lie algebras (of finite dimension) and their representations, and include the study of numerous examples.
Syllabus
1. Lie Groups.
- Quick introduction to differential geometry (manifolds, bundles, covering spaces,...)
- Lie groups and representations
- The Lie algebra of a Lie group, the exponential map
- Lie's Theorems
- Compact Lie groups
- Classical examples of Lie groups
2. Finite dimensional Lie algebras.
- Generalities on Lie algebras, representations
- Fundamental examples (classical Lie algebras, Heisenberg and Virasoro,...)
- Nilpotent, solvable, semi-simple Lie algebras
- Representations categories, simple representations
- Complete reducibility
- Structure of semi-simple Lie algebras
- Root systems, Weyl group combinatorics
3. Representations of finite-dimensional semisimple Lie algebras.
- The case of sl(2,C)
- Highest weight modules, Verma modules
- Parametrization of simple representations. Jordan-Holder series. Multiplicities
- Finite-dimensional representations. Tensor structure, characters
- Weyl character formula
Prerequisites
There are no real prerequisites, besides linear algebra (including the Jordan decomposition theorem) and standard undergraduate algebra. Some familiarity with basic concepts of differential geometry could prove useful.
Evaluation
Exam + one (graded) homework
Bibliography
(Fulton-Harris is very good for a start) :
- 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.
- J-B. Zuber, Invariances en physique et théorie des groupes,
https://www.lpthe.jussieu.fr/~zuber/Cours/InvariancesTheorieGroupes-2014.pdf