TAILIEUCHUNG - Đề tài " Automorphism groups of finite dimensional simple algebras "

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. 1. Introduction In this paper, ‘algebra’ over a field means ‘nonassociative algebra’, ., a vector space A over this field with multiplication given by a linear map A ⊗ A → A, a1 ⊗ a2 → a1 a2 , subject to no a priori conditions; cf. . | Annals of Mathematics Automorphism groups of finite dimensional simple algebras By Nikolai L. Gordeev and Vladimir L. Popov Annals of Mathematics 158 2003 1041 1065 Automorphism groups of finite dimensional simple algebras By Nikolai L. Gordeev and Vladimir L. Popov 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 0k K where A is a finite dimensional simple algebra over k. 1. Introduction In this paper algebra over a field means nonassociative algebra . a vector space A over this field with multiplication given by a linear map A 0 A A a1 0 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. As usual 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 we denote by VE the vector space V 0F 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 is defined over k that is always the case if k is perfect cf. Sp then for each field extension F k the full automorphism group Aut AF of F-algebra AF is the group Aut AK F of F-rational points of the algebraic group Aut AK . Both authors were supported in part by The Erwin Schrodinger International Institute for Mathematical Physics Vienna Austria . 1042 NIKOLAI L. GORDEEV AND VLADIMIR L. POPOV Let Ak be the class of linear algebraic fc-groups Àut A .

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