王志华,李立斌.${\cal U}_q(g)$在其正部分${\cal U}_q^ (g)$上的作用[J].数学研究及应用,2011,31(4):665~674
${\cal U}_q(g)$在其正部分${\cal U}_q^ (g)$上的作用
Action of ${\cal U}_q(g)$ on Its Positive Part ${\cal U}_q^ (g)$
投稿时间:2009-10-21  最后修改时间:2011-04-18
DOI:10.3770/j.issn:1000-341X.2011.04.011
中文关键词:  Nichols代数  Yetter-Drinfeld 模  斜导子  量子群.
英文关键词:Nichols algebra  Yetter-Drinfeld module  skew derivation  quantum group.
基金项目:国家自然科学基金(Grant No.10771182).
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 
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中文摘要:
      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.
英文摘要:
      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.
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