The Nichols algebra of a semisimple Yetter-Drinfeld module
The Nichols algebra of a semisimple Yetter-Drinfeld module
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DOI：
10.1353/ajm.2010.a404140
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发表时间：
2008-03
影响因子：
1.7
通讯作者：
N. Andruskiewitsch;I. Heckenberger;H. Schneider
中科院分区：
文献类型：
--
作者：
N. Andruskiewitsch;I. Heckenberger;H. Schneider
研究了半简单Yetter-Drinfeld模的Nichols代数，引入了包含实根和Weyl群的新不变量。关键因素是定义在任意尼科尔斯代数上的“反射”。我们的构造将量子化Kac-Moody代数的Lusztig自同构的限制推广到幂零部分。作为一个直接应用，我们完成了${\Bbb S}_3$上有限维点Hopf代数和${\Bbb S}_4$上有限维Nichols代数的分类。这一理论在具有类群元的非阿贝尔群的有限维点Hopf代数的分类中得到了令人惊讶的新结果。
We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a "reflection" defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig's automorphisms of quantized Kac-Moody algebras to the nilpotent part. As a direct application we complete the classifications of finite-dimensional pointed Hopf algebras over ${\Bbb S}_3$, and of finite-dimensional Nichols algebras over ${\Bbb S}_4$. This theory has led to surprising new results in the classification of finite-dimensional pointed Hopf algebras with a non-abelian group of group-like elements.