中文摘要：

首先研究建立在任意域是上的A∞型路代数kA∞的有限维模范畴，给出了kA∞的有限维模范畴与A∞的有限子quiver所对应的路代数上的有限维模范畴之间的关系，特别的具体的给出了所有的不可分解有限维kA∞模，精确的刻画了不可分解模之间的模扩张；然后给定有限域k，研究了建立在有限维kA∞模范畴上的Ringel—Hall代数H（kA∞）．证明了H（kA∞）恰好是当n趋向∞时H（kA∞）的正向极限，特别的找到了H（kAv）的一个PBW基，并且证明H（kA∞）恰好与它的合成子代数相符合．

英文摘要：

The category of the finite-dimensional representations of kA∞ was studied first, with all its indecomposable objects and their extenswere were given explicitly, the Ringel-Hall algebra H（kA∞） was investigated for a finite field k was investigated. The main viewpoint of this investigation is to regard H（kA∞） as the direct limit of the Ringel-Hall algebra H（kA∞）. In particular, a PBW-basis of H（kA∞） was gotten. The investigation shows that H（kA∞） coincides with its composition subalgebra.