The course is devoted to the structure theory and classification of reductive algebraic groups over fields. We will see how the structure theory and classification of algebraic groups over algebraically closed fields extends to non-algebraically closed fields. The main tools here are Galois cohomology and Galois descent. We demonstrate their power by classifying Tits indices (generalized Satake diagrams from Lie group theory). We will also discuss Tits’ theorem on simplicity of groups of points of isotropic simple algebraic groups and its relation to the classification of finite simple groups.
It is desirable to have preliminary knowledge of simple algebraic groups over algebraically closed fields and/or of simple Lie groups. Alternatively, this course can be run as a seminar, see 060167 Reductive Groups/Редуктивные группы.
A. Borel, Linear algebraic groups, 2nd ed., Springer-Verlag, New York, 1991.
J. S. Milne, Algebraic groups. The theory of group schemes of finite type over a field, Cambridge University Press,
Cambridge, UK, 2017.
J. Tits, “Classification of algebraic semisimple groups”, Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence RI, 1966, 33–62.
J. Tits, Algebraic and abstract simple groups, Ann. of Math. 80 (1964), 313–329.