The Ringel-Hall Algebra And Hall Polynomial Of Tubular Category

The representation theory of finite dimensional algebras has been developed for over three decades. Its important ideas and techniques turn out to be very useful in many branchs of mathematics. Not only the application of the representation theory provides new insight and approach into these areas,but also gets many new algebras in the same time. Ringel introduce a new algebra which is called Ringel-Hall algebra in 1990. These algebras provide a new approach into the studying for Lie algebra and quantum group. The main purpose of this thesis is to study the Ringel-Hall algebra of tubular catagory by using combinatorial tools.Firstly,we study the simple properties of modules which in the category of locally nilpotent representations. From studying these simple properties,we get a subset of module category, and define some operators on this subset. Moreover,we get a invariable value of modules which independence from the way of the operators action.Secondly,in the category of locally nilpotent representations, for the semisim-ple submodule N, we give a formula of the Hall polynomial F_M~L ,N which is the another form given by Guo.J.Y. For a fixed Hall polynomial F_M~L ,N and module M. N, we get the number of modules L which have this Hall polynomial and satisfy some other conditions.These results can be duality to semisimpe factor modules naturaly.On the other hand,we get a formula of the Hall polynomial F_M~L ,N for indecomposable module N.Finally,in the category of locally nilpotent representations, we get some centralizers of composition algebra of Ringel-Hall algebra, and we proof these are the central elements of Ringel-Hall algebra.Moreover.we proof that these...