${\cal U}_q(g)$在其正部分${\cal U}_q^ (g)$上的作用
Action of ${\cal U}_q(g)$ on Its Positive Part ${\cal U}_q^ (g)$

DOI：10.3770/j.issn:1000-341X.2011.04.011

 作者 单位 王志华 扬州大学数学科学学院，江苏 扬州 225002 南京师范大学泰州学院数学科学与应用学院，江苏 泰州 225300 李立斌 扬州大学数学科学学院，江苏 扬州 225002

本文介绍了一类Nichols代数上的两种斜导子,然后揭示了这两种导子之间的关系,特别这两种导子满足量子Serre关系.因此由这些导子及相应的自同构所生成的代数为Drinfeld-Jimbo量子包络代数$\mathcal {U}_q(g)$的同态像,因而证明了该Nichols代数为$\mathcal {U}_q(g)-$模代数.但是文中的Nichols代数实际上为Drinfeld-Jimbo量子包络代数$\mathcal {U}_q(g)$的正部分$\mathcal {U}_q^ (g)$,因此$\mathcal {U}_q^ (g)$为$\mathcal {U}_q(g)-$模代数.

In this paper, two kinds of skew derivations of a type of Nichols algebras are introduced, and then the relationship between them is investigated. In particular they satisfy the quantum Serre relations. Therefore, the algebra generated by these derivations and corresponding automorphisms is a homomorphic image of the Drinfeld-Jimbo quantum enveloping algebra ${\cal U}_q(g),$ which proves the Nichols algebra becomes a ${\cal U}_q(g)$-module algebra. But the Nichols algebra considered here is exactly ${\cal U}_q^ (g),$ namely, the positive part of the Drinfeld-Jimbo quantum enveloping algebra ${\cal U}_q(g),$ it turns out that ${\cal U}_q^ (g)$ is a ${\cal U}_q(g)$-module algebra.