Quiver Representation And Comodule Structure

It is well known that given a representation of Q = (Q₀, Q₁, s, t: Q₁â†'Q₀)(denoted by Q below) we can construct a left module over path algebra ;given a left module over path algebra we can construct a representation of Q,furthermore,the category of representation of Q is equivalent to the category of left module over the path algebra by two construction. This paper is devoted to studying the path coalgebra from the view of locally finite category ,the representation of Q, the comodule over the path coalge-bra,and the relation between the category of representation of Q ,the category of left module over the path algebra and the category of the comodule over the path coalge-bra.Being based on [4] we give the relation between the simplest quantized enveloping algebra U_q{sl₂) (where q is not a unit root) and the vector of upper triangular matrices as coalgebra .In chapter One, we mainly introduce the background knowledge of representations of Q and our research goals, present the preliminary knowledge which we need in our article and explicitly formulate the idea and method of giving questions and studying questions.In chapter Two, we mainly give the representation of Q and the properties of the path algebra and prove relavent questions;we also give some new propositions on the base of old questions and correspoinding proof.Chapter Three is the first main content of the whole article .we start from the view of locally finite category and introduce the defination of the path coalgebra P(C) such as:as a vector ,P(C) has basis all the paths in Q,where coproduct A and counit e is as follows:...