论文标题
Zassenhaus的猜想和CPA结构
A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras
论文作者
论文摘要
在较短的CPA结构中，已在特征零的字段中研究了Lie代数上的Lie后代数结构，尤其是出于几何学动机的真实和复数。特征零的完美谎言代数只能承认CPA琐碎的结构。在本文中，我们研究了特征$ p> 0 $领域的这些结构。我们表明，特征性的每个完美的模块化谎言代数$ p> 2 $，具有可解的外部衍生代数仅接受琐碎的CPA结构。这涉及汉斯·扎森豪斯（Hans Zassenhaus）的猜想，说一个简单的模块化谎言lie代数$ \ mathfrak {g} $的外部推导代数$ {\ rm out}（\ mathfrak {g}）$可解决。我们尝试总结Zassenhaus猜想的已知结果，并使用premet的简单模块化谎言代数分类，并证明了一些新的结果，并为特征性$ p> 3 $的代数闭合字段进行premet和条带。作为推论，我们可以获得每个中央简单的模块化谎言特征的代数$ p> 3 $仅接受琐碎的CPA结构。
Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic $p>0$. We show that every perfect modular Lie algebra in characteristic $p>2$ having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation algebra ${\rm Out}(\mathfrak{g})$ of a simple modular Lie algebra $\mathfrak{g}$ is solvable. We try to summarize the known results on the Zassenhaus conjecture and prove some new results using the classification of simple modular Lie algebras by Premet and Strade for algebraically closed fields of characteristic $p>3$. As a corollary we obtain that that every central simple modular Lie algebra of characteristic $p>3$ admits only the trivial CPA-structure.
