This biannual learning unit is being organized in 20222023
Teacher(s)
Language
French
Prerequisites
It is recommended that the student be familiar with the fundamental notions of linear algebra, as developed for example in the courses LMAT1131 or LEPL1101, as well as those of multilinear algebra and group theory, as developed for example in the course LMAT1231.
It is interesting but not essential that the student be familiar with the notion of ring, in particular with polynomial rings, as developed for example in the course LMAT1331, as well as with the basic notions of Euclidean and differential geometry, as developed for example in the course LMAT1141.
It is interesting but not essential that the student be familiar with the notion of ring, in particular with polynomial rings, as developed for example in the course LMAT1331, as well as with the basic notions of Euclidean and differential geometry, as developed for example in the course LMAT1141.
Main themes
Lie algebras, classification of semisimple Lie algebra, groups associated to semisimple algebras and their generalisations.
Learning outcomes
At the end of this learning unit, the student is able to :  
Recognise and understand a basic foundation of mathematics.


Show evidence of abstract thinking and of a critical spirit. He will have made progress in :


Analyse a mathematical problem and suggest appropriate tools for studying it in depth.  
Course Specific Learning Outcomes : By the end of this activity, students will be able to show his mastering of fundament concepts in classical Lie theory, including those concerning semisimple algebras (Killing form, root space decomposition, root system and Weyl group), and to illustrate them on examples. They will also be able to explore a more advanced topic of Lie theory. 

Content
Part on Lie algebras :
 Lie algebras: motivation, definitions and examples
 Nilpotent Lie algebras
 Solvable Lie algebras
 Levi decomposition
 Killing form and Cartan criteria
 Semisimple Lie algebras
 Representations of semisimple algebras
 Classification of semisimple complex algebras
 Serre's therorem
 Affine algebraic groups
 Affine groups schemes
 Hopf algebras
 Representations of affine groups schemes
 Lie algebra of an affine group scheme
 Semisimple groups over an algebraically closed field
Teaching methods
Lecture courses (and exercise sessions). Possibility (noncompulsory) for the students to hand over solutions to exercises during the course period, so as to get feedbacks as well as collect points towards the final exam.
Evaluation methods
Oral exam, with lecture notes if needed.
Other information
LaTextyped lecture notes will be made available.
Bibliography
 J. E. Humphreys, Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, 9. SpringerVerlag, New YorkBerlin, 1972. xii+169p.
 Waterhouse, William C., Introduction to affine group schemes. Graduate Texts in Mathematics, 66. SpringerVerlag, New YorkBerlin, 1979. xi+164p.
Faculty or entity