Automorphisms of affine algebraic groups

Abstract

As is well-known, it is generally not possible to endow the automorphism group of an affine algebraic group with the structure of an algebraic group. The typical illustration of this is the example of an algebraic toroid of dimension greater than 1. Here, we analyze the automorphism groups of connected affine algebraic groups over algebraically closed fields of characteristic 0, to an extent sufficient for obtaining a structural characterization of those groups whose automorphism groups have a natural structure of affine algebraic group that is fully compatible with that of the given afline algebraic group. It turns out that the only obstructions in this direction arise from the presence of certain algebraic toroids in the given group. The compatibility demands we make on the algebraic structure of the automorphism group of a group motivate our definition of a consemati~e group, on which we base our present investigation. \Vc say that an affine algebraic group is conservative if the action of its automorphism group on its algebra of polynomial functions is locally finite. If the group G is conservative, then its automorphism group has a natural and fully satisfactory structure of affine algebraic group. In fact, the property of being conservative is precisely what is needed for having the holomorph of G inherit an affine algebraic group structure with which it is the semidirect product, in the sense of affine algebraic groups, of G and its automorphism group. Throughout, we work in the category of afline algebraic groups over a fixed base field, the groups consisting of points rational over the base field. We. presuppose a knowledge of the basic classical results for these groups, such as are developed in [Z]. On the other hand, we take the invariant point