On The Simple Connectedness And Representation Type Of Some Algebras

Finite dimensional algebras consist of finite and infinite representation classes.The representation-infinite algebras can be further divided into two disjoint classes:tame and wild algebras, in which tame algebras are the focus of recent researchinto representation theory of algebras. One way to study the tame algebras isthe research into its integrity. Some good results have been achieved by Drozd,Crawley-Boevey, Krause, de la Pen a, Geiss and many others. However, due tothe complexity of its definition, it is still an open problem to find a less complexcombinatorial method for determining the tame algebras in general case in finitesteps. The other way is to classify the tame algebras by certain criteria and studyits parts. Although there are good standards for judging the tameness of somespecial classes, the criteria for determining the tameness of other classes such as(*)-serial algebras are still needed to be established. The study conducts theinvestigation into the tameness of (*)-serial algebras.The notion of simple connectedness stemmed from topology. The simply con-nected algebra was first introduced by Bongartz and Gabriel who defined simpleconnectedness for the representation-finite algebras, and then was generalized tothe representation-infinite algebras by Assem and Skowron′ski. Up to now, certainresearch has been conducted and some progress has been made in simply connectedrepresentation-infinite algebras; nevertheless, they mainly focus on some specialclasses such as strongly simple connected algebras, incidence algebras, Schurianalgebras. The research into other types is still needed to acquire a better under-standing of simply connected representation-infinite algebras. The study inves-tigates the simple connectedness of minimal-infinite incidence algebras, biserialincidence algebras,(*)-serial incidence algebras and (*)-serial algebras, andthe strong simple connectedness of minimal representation-infinite algebras.The study shows that: firstly, we classify all the minimal-infinite incidencealgebras, biserial incidence algebras and (*)-serial incidence algebras, and obtainthat they are simply connected if and only if they are strongly simply connected; secondly, we study the strong simple connectedness of minimal representation-infinite algebras; thirdly, we study the simple connectedness of (*)-serial alge-bras, and obtain that it is simply connected if and only if it is Schurian stronglysimply connected; fourthly, we discover the connection between (*)-serial alge-bras and almost strongly acyclic algebras, and obtain that the (*)-serial algebrasis almost strongly acyclic except some cases; fifthly, we obtain that the represen-tation type of simply connected (*)-serial algebras is of polynomial growth;sixthly, we give a positive answer to the tameness of (*)-serial algebras byGalois covering techniques.