Research On Some Problems In The Theory Of Algebras And Coalgebras

There are some interesting questions about algebras and coalgebras, which hasattracted many mathematicians working and studying on it continuously. Amongthose questions, how to study the algebra and coalgebra by using the methodologyof representation theory of algebras, is the hot topic in recent years. Representa-tion theory of algebras has been a new branch of algebras since 1980s. Its basiccontent is to study the modules of artin algebras. During the past more than20 years, this theory had been well developed. Its major research includes thebasic theory of Hall algebra and its methodology, and emphasizes on realizing aKac-Moody Lie algebra and its relative quantum enveloping algebra by utilizingthis theory and methodology. Another major research is about qusi-hereditaryalgebra, its representation theory and the connection relationship between thetheory of complex semisimple Lie algebras and representation theory of algebraicgroups. In the same time, some mathematicians had studied the structure ofalgebras and coalgebras by using the methodology of representation theory, aswell as the structure and character of di?erent types of algebras and coalgebras,such as semiperfect coalgebras, qusi-coFrobenius coalgebras, hereditary coalgebras,pure-semisimple coalgebras and serial coalgebras, hereditary algebras, Frobeniusalgebras, serial algebras, incidence algebras, etc.In this dissertation, our study focuses on the following three aspects. Firstly,we introduce the concept of (*)-serial coalgebras and make research on it. As weknow, it is considered in serial coalgebras that its lattice of subcomodules of any injective indecomposable comodules is a chain (finite or infinite). While for thesituation of two chains, that is, there are only two maximal chains in its lattice ofsubcomodules of any injective indecomposable comodules of a coalgebra. Whatis the structure of coalgebras ? And what are the di?erent properties comparingwith serial coalgebras? The same questions are presented for associative algebras.This is the first topic we discuss in this dissertation. Secondly, we introduce theconcept of the primary subcoalgebras and discuss such coalgebras. As we know,simple subcoalgebras only control the vertices of the quiver, however they do notcontrol the arrows . For this reason we are interested in a generalization of simplesubcoalgebras: the prime subcoalgebras and the primary subcoalgebras. Moreover,we introduce the concept of Krull dimension of coalgebras and study it by means ofthe prime subcoalgebras. Thirdly, the problem, which is devoted to studying theproperties of formal triangular matrix algebras from di?erent angles, is concernedby many mathematicians. In chapter 5, on the basis of the investigations by somescholars, we make the research on formal triangular matrix algebras, and get someinteresting results.The major conclusions of this dissertation are as following:In chapter 2, we give the definition of (*)-serial coalgebra and it is a naturalgeneralization of serial coalgebras. Further, we make investigation on them. It isshown that there exists such a (*)-serial coalgebra by exhibiting some examples.Meanwhile, it is well-known that the localization theory plays an important rolein the structure of coalgebras and algebras. Localization is a canonical theorywhich has been developed by many authors with di?erent points of view. Amongthem, the most renowned procedure is the localization in rings as a systematic method of adding multiplicative inverses to a ring. Also Goodearl, Warfied andothers investigate the localization in the noncommutative case and obtain manynice results. In a higher level of abstraction, P. Gabriel describes the localizationin Abelian and Grothendieck categories. In the process of localization, one usuallyapplies a functor onto a new category, the quotient category, which has a rightadjoint, the section functor. G. Navarro elaborates Gabriel's ideas in comodulecategories (Grothendieck categories of finite type). The key-point of the theorylies in the fact that the quotient category becomes a comodules category, andthen it is better understood than in the case of modules over an arbitrary algebra.Therefore, we study and characterize the localization of right (*)?serial coalgebras.We get the following main results:Theorem 2.2.1 The following statements about a coalgebra C are equiva-lent:(1) C is (*)-serial.(2) Each finite dimensional right C-comodule is a direct sum of uniserialcomodules or biserial comodules.(3) Each finite dimensional left C-comodule is a direct sum of uniserial co-modules or biserial comodules.(4) Each finite dimensional indecomposable right C-comodule is uniserial orbiserial.(5) Each finite dimensional indecomposable left C-comodule is uniserial orbiserial.(6) Each finite dimensional subcoalgebra of C is (*)-serial. Theorem 2.2.2 A basic coalgebra C is right (*)-serial if and only if foreach indecomposable injective C-comodule E, there exists at most one positiveinteger m such that socmE/socm?1E is a sum of two simple C-comodules andsocm+1E/socmE is simple; meanwhile, sociE/soci?1E is zero or simple for anypositive integer i = m.Theorem 2.2.3 If a basic coalgebra C is right (*)-serial, then each vertexSi of the right valued Gabriel quiver (QC,dC) is at most the sink of two arrowsand, if such arrows exist, they are of the formfor some vertex Sj,Sj and some positive integers d and d . In particular, if Cis pointed and C is right (*)-serial, then each vertex in the (non-valued) Gabrielquiver of C is at most the sink of two arrows.In chapter 3, we give another generalization of simple subcoalgebras (that is,the prime subcoalgebras)??primary subcoaglebras, and introduce the notion ofprimary subcoalgebras. Further more, we do the basic research of the structureand properties for primary subcoalgebras. It is significant to research prime andprimary subcoalgebras. It can help us study the control issue of arrows of pathcoalgebras with quiver furtherly and deeply. Therefore, we can well understandthe structure of coalgebras from them. Moreover, we investigate the question ofKrull dimension of coalgebras by prime subcoalgebras and localization method,and as a consequence, we continue to study some properties of coagebras in termof Krull dimension. We get the following main results: Theorem 3.2.1 Let D be a finite dimensional subcoalgebra of kQ. Thefollowing statements are equivalent:(a) D is primary.(b) The coalgebra eDe ? eCe is primary for any non-zero idempotent elemente∈(kQ)?, defined as the characteristic function of a set of two vertices.Theorem 3.3.1 Let D be an arbitrary prime subcoalgebra in a coalgebraC, and e a semicentral idempotent element in C? associated to soc D. Then wehaveTheorem 3.3.3 Let C be a coalgebra and M be a C-comodule. If thecoe?cient subcoalgebra cf(M) is noetherian, then K dim(M) = 0.In chapter 4, we give the notion of (*)-serial algebras, and it is a naturalgeneralization of serial algebras. In the same time, we explore investigation onthem. It approves that there exists such (*)-serial coalgebra by exhibiting someexamples. We get the following main results:Theorem 4.2.2 A finite dimensional, associative algebra A is right (*)-serialif and only if for any indecomposable projective right A-module P in mod-A, thereexists at most one positive integer n such that radnP contains only two maximalsubmodules (i.e. radn+1P is the intersection of two maximal submodules in radnP)and radiP contains the unique maximal submodule for any i = n.In chapter 5, on the base of the previous research, we make discussions on thecharacterization of formal triangular matrix algebras to be PP-algebras, general-ized PP-algebras, hereditary algebras and semi-hereditary algebras. Meanwhile, we also study the su?cient and necessary conditions of which the formal triangularmatrix algebras are serial or (*)-serial. We get the following main results:Theorem 5.4.1 Let the formal triangular matrix algebra T = MA B0 .Then T is a right serial algebra if and only if(a) A and B are both right serial algebras.(b) MA is a direct sum of uniserial modules.Theorem 5.4.2 Let the formal triangular matrix algebra T = MA B0where BM is an indecomposable module. If T is a right (*)-serial algebra, then Aand B are both right (*)-serial algebras.