The Quiver Method On The Representation Theory Of Tensor Product Algebras And Hereditary Algebras

In this thesis, we mainly apply the quiver technology to investigate the representation type of tensor product algebras and the corank of hereditary algebras. Our methods come from the combination mathematics and graphic technology on the representation theory of finite dimensional associative alge-bras. It consists of the following three parts: 1. Tensor product algebra is an important research objects in the rep-resentation theory of algebras, because it contains some important classes of algebras, such as the enveloping algebra, which play an important role in the Hochschild cohomology, and the triangular matrix algebra Tn(A). Further-more, since each category of bimodules of two algebras is equivalent to the category of the right (left) modules of some tensor product algebra, the tensor product algebras brings convenience on the problem being involved with bi-modules, such as the formal triangular matrix algebras. Applying the quiver to investigate the structure of algebra and its module category, the Gabriel quiver and its admissible ideal is principal. So, we strictly show the form of Gabriel quiver and its admissible ideal of tensor product algebra, and give the quiver prepresentation of bimodule on the corresponding tensor product alge-bras. Finally, Cartan matrix and Coxeter matrix of tensor product algebras.2. The representation type is a fundemental question in the representation theory. For the representation-finite algebras, we easily obtain the AR-quiver of algebras, and understand the whole category of modules in certain sense. In the chapter, we dicuss the representation type of tensor product algebras, and give the complete classification of representation-finite tensor product al-gebras. By the forbidden subquiver, covering theory and one-point extension theory, we obtain some sufficent and necessary conditions which make the ten-sor product algbras of two path algebras and the triangular matrix algebras over Nakayama algebras be representation-finite. Combining with the known conclusion, we answer completely the representation type question of these two tensor product algebras. Finally, we consider some condition of representation-finite generalized tensor product algebras.3. The radical vector and the corank are two important invariants in the representation theory of algebras, especially hereditary algebras. In the chapter four, applying the relation between the radcial vector of Euler quadratic form of hereditary algebras and the additive functions on the underlying quiver of the quiver, we transfer the algebraic question to the graphic question. In the case of the trees, we obtain the structure of hereditary algebras with non-zero corank and the computional formula of corank.