Frobenius Morphism And Fixed-point Algebra Of The Extension Algebra A(?)s B

Algebra extension is a process of forming new kind of algebras from some know algebras according to some certain rules. Algebra extension is the basic subject in the studying algebras. This thesis focus on the automorphisms of the extension algebras A (?)_S B. This dissertation is organized as follows. In chapter one, we present the background of our studying, and describe the main results. In chapter two, we introduce the notion of the extension algebra A (?)_S B. After giving some examples, we describe some properties of A (?)_S B. The chapter three is the main part of this thesis. We first shows that the quiver automorphism of extension algebra A (?)_S B is determined by the quiver automorphisms of A and the quiver automorphisms of B. Then we prove that the Frobenius morphism of A (?)_S B is determined by the Frobenius morphism of A and the Frobenius morphism of B completely. We recall the notion of a Frobeneus fix-point algebra and describe the clous relationship between the fix-point algebra and modulated quiver. Then we prove the fixed-point algebra of A (?)_S B is isomorphisic to the tensor of the fixed-point algebra of A and the fixed-point algebra of B.