Studies On The Quiver Method In Representation Theory Of Algebras

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In our paper, we focus on sev-eral specific problems in the representation theory of algebras and solve them via some quiver methods. Meantime, we also study some connections between tilting theory and cluster-tilting theory.The main results of this paper are divided into five parts. After the introduction and the preliminaries, in chapter 3 we first discuss that when a certain graph to be both the tilting graph of an hereditary algebra and also the cluster-tilting graph of another hereditary algebra. As a first step, we recall the definitions of the tilting graph and cluster-tilting graph of hereditary algebra and show that for one hereditary algebra, its tilting graph is a proper subgraph of its cluster-tilting graph. Then we give the main result of this part, that is, a graph is both the tilting graph of a hereditary algebra and also the cluster-tilting graph of another hereditary algebra if and only if it is a skeleton graph of Stasheff polytope. For this result, we give two different claims in our paper, one is pure algebraic and the other one is geometric and combinatorial. At the end of this part, we compare the uniformity between the tilting theory and the cluster-tilting theory and show that such uniformity is not always able to be realized by the example of the geometric realization of simplicial complexes associated with tilting modules and cluster-tilting objects respectively.Secondly, since the tensor product of representations of a Hopf algebra is an impor-tant ingredient in the representation theory of Hopf algebras and quantum groups and in particular the decomposition of the tensor product of indecomposable modules into a direct sum of indecomposables has received enormous attention, hence in chapter 4 we calculate the representation rings of the Nakayama truncated algebras precisely. Al-though both its definition and calculation are similar to the work in [15], [49], we find that the coalgebra structures of these three algebras are different. Therefore, we first give the Hopf algebra structure of the Nakayama truncated algebras via the covering quiver in detail and then calculate their representation rings by the Pascal triangle. Moreover, we show that the ring structure of our result is different from the ring structure in [15], [49]. After we characterize the representation rings of the Nakayama truncated alge-bras via the quotient ring of certain polynomial ring precisely, we discuss the following question:can two different Nakayama truncated algebras share the same representation rings? In order to answer this question, we will use some basic knowledge in algebraic geometry, such that variety, radical ideal and coordinate ring. We obtain some simple results about this question.Next, in chapter 5 we will generalize the representation rings theory of chapter 4 into the Green rings of the monodial category. Indeed, we will introduce the shift ring of complex category and the derived ring of bounded derived category in this part re-spectively. At first, we will give the precise definition of the Green ring of the monodial category. Then we obtain that the polynomial characterization of the shift ring of com-plex category and the derived ring of bounded derived category in general respectively. At last, we focus on the Nakayama truncated algebras considered in the last part and calculate the corresponding shift ring in detail and the corresponding derived ring un-der some conjectures.Additionally, in chapter 6 we prove the following result by using the combinatorics properties of the Auslander-Reiten quiver of an hereditary algebra, that is, for a basic module, its decomposition forms a section in the Auslander-Reiten quiver if and only if its minimal additive subcategory is a slice in the module category. At the end of this part, we first study the ratio between the number of separating tilting module and all the tilting modules over an hereditary algebra and then show that the ratio can be arbitrarily small for some certain hereditary algebras, i,e,. decreasing to 0.Finally in chapter 7 we introduce our recent work about cluster subalgebra. At first, we recall the relations between the total positivity matrices and the cluster algebra the-ory invented by S.Fomin and A.Zelevinsky originally and the definitions of the planer networks and the double wiring diagrams. Then we show that any matrix over complex field can be generated by the generalized elementary Jacobi matrices. Finally, we gen-eralize the relations between the total positivity matrices and the cluster algebra theory into the matrices share some "properties", which the positive definite matrices consid- ered here. Actually, we introduce the definition of the cluster subalgebra and built the relations between the positive definite matrices and the cluster subalgebra theory.