法式tubular代数与仿射Kac-Moody代数

Canonical Tubular Algebras and Affine Kac-Moody Algebras

彭联刚-肖杰等利用有限维遗传代数A的根范畴的Ringel-Hall李代数实现所有可对称化Kac-Moody代数,其中Ringel-Hall李代数的李乘不完全由Hall积提供.本文通过新方法实现仿射Kac-Moody代数,李代数L(A)1C/I的李乘完全由Hall积给出.对任意D4(1),E6(1),E7(1)或E8(1)型扩张Dynkin图Δ,在型为Δ的法式tubular代数A的退化合成李代数L(A)1C上构造它关于一个具体李理想I的商代数L(A)1C/I,证明商代数L(A)1C/I同构于对应的Δ型仿射Kac-Moody代数.这将有助于利用法式tubular代数的模范畴研究仿射Kac-Moody代数.

Peng and Xiao had realized every complex symmetrizable Kac-Moody algebra as a Ringel-Hall Lie algebra of the socalled root category of a finite-dimensional hereditary algebra, where the Lie multiplication was not completely given by Hall multiplication. In new method of realizing the complex affine Kac-Moody algebra in this paper, the Lie bracket was completely given by the Hall multiplication. For each simply-laced extended Dynkin graph △ of type D4^（1） ,E6^（1） ,E7^（1） or E8^（1） ,the complex affine Kac-Moody algebra of type △ is realized as a quotient algebra of the complex degenerate composition Lie algebra L（A）1^C of a canonical tubular algebra A of type △ by some ideal I of L（A）1^C which is defined via the Hall algebra of A,and give an explicit form of L It would be of great benefit to study affine Kac-Moody algebras via the module categories of canonical tubular algebras.