Some Researches On The Extension Algebras

The researches on algebraic extensions and relevant properties of extensionalgebras have been the basic problem in the study of algebrads. The dissertationdiscusses the automorphisms and generic modules of T(ψ₁,ψ₂)(A) and the specialmodules of the tensor product of algebras,and obtains many interesting results,whichhave sharpen part results of M.Saorin,X.DU and S. Liu etc.This dissertation is divided into four chapters.In chapter one, we carry on the introduction to the research directions and thetrends of the development related to dissertation, and sum up the groundwork ofthis text.In chapter two, we introduce some basic knowledge of the extension algebraT(ψ₁,ψ₂)(A) and its galois cover (?)(ψ₁,ψ₂), point out the important fact that theinfinite dimensional k-algebra (?)(ψ₁,ψ₂) isn't associative and give the sufficientand necessary condition for the associativity of (?)(ψ₁,ψ₂). As the study of the au-tomorphism groups is one of the hotspots in the algebraic researches, in the sectionthree, wc give the whole structure of the automorphism groups of the extension al-gebra T(ψ₁,ψ₂)(A),induce the characteristic of its units and the concrete structureof the inner automorphisms, characterize the automorphisms that fix the vertices ofT(ψ₁,ψ₂)(A) as well, which have generalized the results of M. Saorin. Besides, asGeneric module is an important kind of modules with infinite length and plays animportant role in the representation theory of the finite dimensional algebras, we getthe Generic A(?)M -modules from Generic A-modules by the functor-(?)_A A(?)Mand investigate the relationship between Generic A-modules and Generic A(?)M-modules in the section four.In chapter three,based on introducing some basic knowledge of the tensor prod-uct of algebras, we discuss the tensor product of triangular Monomial algebras and its quiver characterization.In chapter four, we pay close attension to the relationship between the specialmodules of the extension algebra A (?)_k B and ones of the original algebras A,B, such as projective modules, indecomposable modules, tilting modules and soon, characterize the indecomposable projective A (?)_k B-modules, and obtain the im-portant result that the tensor product of m- tilting A- modules and n- tiltingB-modules is m+n-tilting A(?)_k B-modules.