Representation Theory Of The Triangular Matrix Algebras

This paper is divided into three parts.In the first part, we introduce a class of triangular matrix algebras, in which each algebra is called a normally triangular matrix algebra, and characterise Goren-stein projective modules over this class of algebras. Moreover, a sufficient condition of strongly Gorenstein projective modules over normally triangular matrix algebras is given. The importance of normally triangular matrix algebras for us is that it includes the so-called path algebras of quivers over algebras and generalized path algebras. Due to this, we characterize Gorenstein projective modules and strong-ly Gorenstein projective modules over path algebras of quivers over algebras and generalized path algebras as applications of the main results on normally triangu-lar matrix algebras. At last, we give an example to show how all indecomposable Gorenstein projective modules over a given algebra are constructed by the result on generalized path algebras.In the second part, we build a new representation-theoretic realization of fi-nite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g. that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matri-ces. Meanwhile, it means the corresponding results for some other important class-es of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.In the third part, we study cluster algebras of skew-symmetrizable case via generalized path algebras. Firstly, using the isomorphism of generalized path al-gebras and path algebras, we prove that they still keep isomorphism after some mutations. Then using the 1-1 corresponding between non-initial cluster variables and indecomposable rigid modules in symmetric case, we build a corresponding be- tween non-initial cluster variables and indecomposable locally free rigid modules in symmetrizable case.