An Introduction to Algebraic Geometry and Algebraic Groups

Algebraic geometry, in its classical form, is the study of algebraic sets in affine or projective space. By definition, an algebraic set in kn (where k is a field) is the set of all common zeros of a collection of polynomials in n variables. Algebraic groups are both groups and algebraic sets, where the group operations are given by polynomial functions. For example, the special linear group SLn(k) consisting of all n × n matrices with determinant 1 is an algebraic group.

Historically, these groups were first studied in an analytic con- text, where the ground field k is R or C. This is the classical theory of ‘Lie groups’ (see Chevalley (1946) or Rossmann (2002), for example), which plays an important role in various branches of mathematics. Through the fundamental work of Borel and Chevalley in the 1950s, it is known that this theory also makes sense over an arbitrary alge- braically closed field. This book contains an introduction to the theory of ‘groups of Lie type’ over such a general ground field k; con- sequently, the main flavour of the exposition is purely algebraic. In the last chapter of this book, we will even exclusively study the case where k is an algebraic closure of a finite field of characteristic p > 0. Then the corresponding algebraic groups give rise to various fami- lies of finite groups. By the classification of the finite simple groups, every non-abelian finite simple group arises from an algebraic group over a field of characteristic p > 0, except for the alternating groups and 26 sporadic simple groups; see Gorenstein et al. (1994). This is one reason why algebraic groups over fields of positive characteristic also play an important role.