Researches On The Structures And Categorification Of The Quotient Algebras Of Double Quiver Algebras

Double quivers arise as a means of translating combinatorial problems in graph theory into the algebraic framework of quivers.Double quiver algebras play an im-portant role in the structures of quantum groups,Lie algebras,and vector bundles et al.This thesis studies on the structures and categorification of the quotient alge-bras of double quiver algebras which consist of Leavitt path algebras,preprojective algebras,and dual extension algebras.The concrete structure of the thesis is as follows:At the beginning of this thesis,we make a detailed introduction of recent devel-opments in the quotient algebras of double quiver algebras which consist of Leavitt path algebras,preprojective algebras and dual extension algebras,recollement of Abelian categories and derived categories,the mutations of quivers,higher-rank graphs and cartesian product of quivers,and the categorification of algebras.At the end,we show the research content and framework of this thesis.Based on the work of Abrams and Pino,in the first chapter,we introduce the definition and examples of Leavitt path algebras.Then character Leavitt path algebras and its endomorphism rings which are Zorn rings and(semi)perfect rings.Note that the researches on the gluing of quivers are one of the basic topic in the graph theory.In the second chapter,we firstly introduce the definition of recollement of Abelian categories.Then using the isomorphism between Leavitt path algebras and triangular matrix algebras,we get the recollement of module categories and derived categories over Leavitt path algebras associated to a special quiver.Further,we depict a recollement of module categories of Leavitt path algebras through the recollement of module categories over local unit rings.The mutations of quivers are the basic elements of cluster algebras,then one introduced the mutation equivalence in order to study the finite type of cluster algebras.In the third chapter,we focus on the sufficient conditions on the mutation equivalence of ring theorists of Leavitt path algebras,such as,strongly graded rings,Zorn rings,exchange rings,locally finite rings and simple rings.Higher-rank graphs are combinatorial structures which are higher-rank dimen-sional analogues of quivers.In the fourth chapter,we introduce the definition of Kumjian-Pask algebras,and give a construction of recollement of module categories over Kumjian-Pask algebras,then give some examples to illustrate it.Further,prove that Kumjian-Pask algebras of the cartesian product of higher-rank graphs are i-somorphic to the tensor algebras of Kumjian-Pask algebras of higher-rank graphs.However,it isn't true for Leavitt path algebras.Then we depict the relationships between the Leavitt path algebras of the cartesian product of two quivers and the tensor algebras of two Leavitt path algebras from two weak conditions.Based on the reduction algorithm of the bases of the quotient algebras,in the fifth chapter,we focus on the structures of the quotient algebras of double quiver algebras.Here we commit to the bases of dual extension algebras,then give formulas to calculate the dimensions of dual extension algebras of Dynkin quivers.Then compare the dimensions among Leavitt path algebras,preprojective algebras,and dual extension algebras of tree Dynkin quivers.In the sixth chapter,we define a quotient category of a double category for a given preadditive category,and research the finiteness conditions on it.We initial achieve the categorification of the quotient algebras of double quiver algebras,and offer a new way to study them.Finally,a brief conclusion and a prospect are made in the last chapter,it points out that many researches maybe extended to the structures and categorification of the quotient algebras of double quiver algebras.