Linear Algebraic Group and Finite Groups of Lie Type
Let k be an algebraically closed field. A linear algebraic group over k is a closed subgroup of the General Linear Group over k. In this course we will first cover the basic concepts about linear algebraic groups and their morphisms, examples of algebraic groups, connectedness, dimension, Jordan decomposition, unipotent subgroups. We will classify commutative linear algebraic groups via their Jordan decomposition. Then we will cover tori, characters and cocharacters. Then we will go on with the structure of connected solvable groups and Lie-Kolchin Theorem, actions of linear algebraic groups, existence of rational representations, properties of the Borel subgroup and Borel fixed point theorem. We will define the Lie algebra of a linear algebraic group and the adjoint representation. In the second chapter we will introduce root systems and the classification theorem of Chevalley to study the structure of reductive and semisimple linear algebraic groups. Then we will study BN pairs and Bruhat decomposition, parabolic subgroups and Levi decomposition, subgroups of maximal rank, Borel-de Siebenthal theorem. We will prove some results about centralizers and conjugacy classes in simple linear algebraic groups. In the third chapter we will deal with endomorphisms of linear algebraic groups and then finite groups of Lie type, as fixed points of Steinberg endomorphisms. We will classify simple groups of Lie type. We will cover Weyl groups, root systems and root subgroups. We will end the course with a discussion on maximal subgroups of finite classical groups, and theorems of Liebeck, Seitz and Aschbacher.