Automorphism groups of finite dimensional simple algebras

Abstract

We show that if a field k contains sufficiently many elements (for instance, if k is infinite), and K is an algebraically closed field containing k, then every linear algebraic k-group over K is k-isomorphic to Aut(A ⊗k K), where A is a finite dimensional simple algebra over k. In this paper, ‘algebra’ over a field means ‘nonassociative algebra’, i.e., av ector space A over this field with multiplication given by a linear map A ⊗ A → A, a1 ⊗ a2 � a1a2, subject to no a priori conditions; cf. [Sc]. Fix a field k and an algebraically closed field extension K/k. Our point of view of algebraic groups is that of [Bor], [H], [Sp]; the underlying varieties of linear algebraic groups will be the affine algebraic varieties over K .A susual, algebraic group (resp., subgroup, homomorphism) defined over k is abbreviated to k-group (resp., k-subgroup, k-homomorphism). If E/F is a field extension and V is a vector space over F ,w edenote by VE the vector space V ⊗F E over E. Let A be a finite dimensional algebra over k and let V be its underlying vector space. The k-structure V on VK defines the k-structure on the linear algebraic group GL(VK). As Aut(AK), the full automorphism group of AK, is a closed subgroup of GL(VK), it has the structure of a linear algebraic group. If Aut(AK )i sdefined over k (that is always the case if k is perfect; cf. [Sp, 12.1.2]), then for each field extension F/k the full automorphism group Aut(AF )o fF -algebra AF is the group Aut(AK)(F )o fF -rational points of the algebraic group Aut(AK).