ABSTRACT
This chapter considers the basic aspects of the representation theory of affine algebraic groups. It introduces a crucial linearization process that consists in taking the Lie algebra associated to the group. The chapter shows that the polynomials on the group have a natural Hopf algebra structure, and use that structure to describe the representations as well as the Lie algebra. It defines the concept of Hopf algebra and shows that the algebra of polynomial functions on an affine group has a natural Hopf algebra structure. The chapter discusses the category of rational modules, i.e., the category of representations, of the affine algebraic group, and prove that it is equivalent to the category of comodules over the corresponding Hopf algebra of polynomials on the group. It considers the first properties of invariants and semi-invariants of linear actions. The algebra of polynomial functions on an affine algebraic group is the most typical example of a commutative Hopf algebra.
