A Galois-Type Correspondence Theory for Actions of Finite-Dimensional Pointed Hopf Algebras on Prime Algebras

Abstract

The objective of this paper is to present a Galois-type correspondence theory for X-outer actions of finite-dimensional Hopf algebras over a field k on prime algebras R. Noncommutative Galois theory was initiated by Noether in her work on inner automorphisms of simple algebras and was w x further studied by Jacobson J1 , who established a Galois correspondence w theory for a finite group of automorphisms acting on a division ring. In J2, x Chap 6 Jacobson used the product of the ring and the group of automorphisms to develop a Galois theory for a ring of linear transformations of a vector space. A major advance is due to Kharchenko, who developed a Galois correspondence theory for actions of automorphisms and derivaŽ . tions on prime semiprime rings. His definitions of inner and outer actions, which were named X-outer and X-inner actions, involve the Martindale ring of quotients of a prime ring R. Montgomery and Passman presented the X-outer case separately and used trace forms of minimal length to show in an elegant way Kharchenko’s Galois correspondence