The primary goal of the course is to provide an introduction to the theory of reductive algebraic groups over arbitrary fields and local regular rings. The main objectives are to give some basic material on their structure and classification.

Lecturers: V. Chernousov and N. Semenov

Background:
Linear algebraic groups have been investigated for over 100 years. They first appeared in a paper of Picard related to differential equations. The subject was later developed by Cartan, Killing, Weyl and others who studied and classified semisimple Lie groups and Lie algebras over the complex and real numbers. With the development of algebraic geometry, it became important to study algebraic groups in a more general setting. The fundamental work of Weil and Chevalley in the 1940s and 1950s initiated the development of the theory of algebraic groups over arbitrary fields. Over the next thirty years, the foundations of this theory (and of the even more general theory of group schemes) over arbitrary fields and rings led to many important results by Borel, Chevalley, Grothendieck, Demazure, Serre, Springer, Steinberg, Tits and others. The motivation for this generalization was to establish a synthesis between different parts of mathematics such as number theory, the theory of finite groups, representation theory, invariant theory, the theory of Brauer groups, the algebraic theory of quadratic forms, and the study of Jordan algebras. Indeed, using the language of the theory of algebraic groups, many outstanding problems and conjectures can be reformulated in a uniform way. Nowadays this branch of mathematics is a very interesting mixture of group theory and algebraic geometry. Over finite fields it classifies almost all simple finite groups, over number fields it studies important arithmetic properties of different algebraic objects such as quadratic and hermitian forms, central simple algebras, arithmetic groups, discrete subgroups, modular forms, over real numbers it clarifies the theory of Lie groups, and so on.
The primary goal of the course is to provide an introduction to the theory of reductive algebraic groups over arbitrary fields and local regular rings. The main objectives are to give some basic material on their structure and classification.

The course will begin with an overview of some notions and objects in algebraic groups over algebraically closed fields and their properties (part I) such as: subgroups, homomorphisms, Lie algebras, semisimple and unipotent elements, tori, solvable groups, semisimple and unipotent elements, Jordan decomposition. After that it will pass to the Borel fixed-point theorem concerning the action of a solvable group on a quasi-projective variety. They lead to the important conjugacy theorems and from them to the long road of the classification of reductive groups over algebraically closed fields in terms of root systems.

Then the main direction of the course will shift to the theory of algebraic groups over arbitrary fields (part II). This will be based on the celebrated paper by Tits on the classification of semisimple linear algebraic groups and the Book of Involution by Knus, Merkurjev, Rost and Tignol. As was shown by Tits, any semisimple group G over a field is determined by its anisotropic kernel and a combinatorial datum, called the Tits index. In the course these two concepts will be systematically studied. In particular, the notions of an inner/outer, strongly inner forms of linear algebraic groups will be introduced together with explicit links to the theory of central simple algebras, Jordan algebras and quadratic forms.

Prerequisites: The main prerequisite is some familiarity with Lie algebras and algebraic geometry, like for example the first part of the book Linear Algebraic Groups by James E. Humphreys.

Course structure: The course will run from mid-January until the beginning of March so that students are well-prepared to follow the remainder of the thematic program. Both parts will have approximately 20 hours. Arrangements will be made so that the course can be taken for credit by participating students. The final grade will be based on homework assignments. The solutions of the homework problems will be discussed in tutorials.