ABSTRACT
In this chapter we introduce one of the objects central to our attentions: the affine algebraic groups, i.e., the group objects in the category of affine algebraic varieties. We will not attempt to summarize here the historical development of this subject that is rooted in different areas of classical mathematics, mainly invariant theory and the theory of Lie groups and Lie algebras. The reader interested in these developments may look at the excellent survey due to A. Borel, [13, Chap. V–VIII]. We only mention that after the seminal work of L. Maurer and E. Picard around 1880, the subject was taken again mainly by C. Chevalley and E.R. Kolchin in the late 1940’s. Thereafter, developments were manifold, most of them associated with the work of A. Borel and his collaborators as well as with Grothendieck’s school.
