A∞型Ringel-Hall代数

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(kAn).In particular,a PBW-basis of H(kA∞) was gotten.The investigation shows that H(kA∞) coincides with its composition subalgebra.%首先研究建立在任意域k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(kAn) 的正向极限,特别的找到了H(kA∞)的一个 PBW 基,并且证明H(kA∞)恰好与它的合成子代数相符合.