仿射合成代数的三角分解

设是一个仿射箭图,它的极小虚单根为ｎ．设ｋ是一个有限域,记Ａ＝ｋ为ｋ上关于箭图的路代数,而记Ｃ(Ａ)为关于Ａ的合成代数．由Ｃ．Ｒｉｎｇｅｌ和Ｊ．Ｇｒｅｅｎ的工作,Ｃ(Ａ)揭示了Ａ的表示与量子群有密切的关系．文１１]证明了对应于Ａ的不可分解表示可以分成预投射,正则,和预内射三个部分,Ｃ(Ａ)具有一个三角分解．１１]中的证明需要假设维数向量为ｎ的拟单模存在,而对于|ｋ|＝２,是ｎ型和ｍ型(ｍ＝６,７,８)的情形,此假设不满足,本文的目的是给出一个简化的,而且不需要前面所提假设的证明．由此,得到一个与域ｋ无关的Ｃ(Ａ)的三角分解．

TRIANGULAR DECOMPOSITION OF AFFINE COMPOSITION ALGEBRAS

Let A = k A be the path algebra of an affine quiver A over a finite field k, with minimal imaginary root n, and C(A) the corresponding composition algebra. By the work of C. Ringel and J. Green, C(A) relates the representation theory of A with the. quantum groups. It has been proved in 11] that C(A) has a triangular decomposition corresponding to the division of the indecomposables into the preprojectives, the regulars, and the preinjectives. However, the proof there needs the assumption that there exists a homogeneous quasi-simple with dimension vector n, and hence it fails for |k| = 2 and of type Dn and Em (m = 6,7,8). The aim of this paper is to give a new proof of this result in which the assumption can be dropped. In this way, a triangular decomposition of C(A) independent of k are obtained.