Lie Algebras Generated by Indecomposables
Lie Algebras Generated by Indecomposables
复制标题
DOI：
10.1006/jabr.1994.1351
复制
发表时间：
1994-12
影响因子：
0.9
通讯作者：
Christine Riedtmann
中科院分区：
文献类型：
--
作者：
Christine Riedtmann
设A是一个有限维结合C-代数，它的单位元只有1/2个不可分解的（左）模，直到同构。本文在由不可分解1-模的同构类自由生成的Z-模L（A）上定义了一个Z-李代数结构。若（）是Dynkin图A，D，E，E，或E，且C是一个具有底层图Q的图，则C-代数CQ只有100个不可合成的图，且它们与对应于Q的单C-Lie代数的正根成双射（见[4]）。我们证明了L（CQ）同构于这个单李代数的Z-形式的正部分。本文的一部分是对Ringel在[12]和斯科菲尔德在[13]中所得到的一些结果的一种不同的方法; Ringel在[12]中表明，仅用表示论就可以恢复对应于Q的单李代数的两个正根的括号。更一般地，他定义了由A-模的同构类生成的自由Z-模上的结合代数（称为Hall代数）的结构，并证明了不可分解模上的自由Z-模是李子代数，条件是1是某个域上的有限维代数，该域（i）只允许1/2个不可分解模，（ii）是定向的（定义见第5节）。
Let A be a finite dimensional associative C-algebra with unit element which has only finitely many indecomposable (left) modules, up to isomorphism. We define a Ž-Lie algebra structure on the Z-module L (A) freely generated by the isomorphism classes of indecomposable 1-modules. If () is a Dynkin diagram A, D, E, E, or E, and if C is a quiver with underlying graph Q, the quiver algebra CQ has only finitely many inde-composables, and they are in bijection with the positive roots of the simple C-Lie algebra corresponding to Q (see [4]). We prove that L (CQ) is isomorphic to the positive part of a Z-form of this simple Lie algebra. Part of this article is a different approach to some of the results obtained by Ringel in [12] and by Schofield in [13]; Ringel shows in [12] that it is possible to recover, by means of representation theory only, the bracket of two positive roots of the simple Lie algebra corresponding to Q. More generally, he defines the structure of an associative algebra (called Hall algebra) on the free Z-module generated by the isomorphism classes of A-modules and shows that the free Z-module on the indecomposables is a Lie subalgebra, provided that 1 is a finite dimensional algebra over some field which (i) admits only finitely many indecomposable modules and (ii) is directed (see Section 5 for the definition).