论文标题
对Zassenhaus猜想的简单模块化谎言代数的反例
Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras
论文作者
论文摘要
我们为Zassenhaus的猜想提供了无限的反例，该家族对简单模块化谎言代数的外部导数代数的溶解性。实际上，我们表明，尺寸的简单模块化谎言代数$ h（2;（1，n））^{（2）} $ dimension $ 3^{n+1} -2在特征$ p = 3 $中没有溶解的外部导数代数，用于所有$ n \ ge 1 $。对于$ n = 1 $，这将恢复$ \ mathfrak {psl} _3（f）$的已知反例。我们表明，外部推导代数为$ h（2;（1，n））^{（2）} $是$（\ Mathfrak {\ mathfrak {sl} _2（f）\ ltimes v（2））\ oplus f^{n-1} $的$ v（2）$是$ v（2）$的自然代表。我们还研究了特征三的其他已知的简单谎言代数，但它们没有产生新的反例。
We provide an infinite family of counterexamples to the conjecture of Zassenhaus on the solvability of the outer derivation algebra of a simple modular Lie algebra. In fact, we show that the simple modular Lie algebras $H(2;(1,n))^{(2)}$ of dimension $3^{n+1}-2$ in characteristic $p=3$ do not have a solvable outer derivation algebra for all $n\ge 1$. For $n=1$ this recovers the known counterexample of $\mathfrak{psl}_3(F)$. We show that the outer derivation algebra of $H(2;(1,n))^{(2)}$ is isomorphic to $(\mathfrak{sl}_2(F)\ltimes V(2))\oplus F^{n-1}$, where $V(2)$ is the natural representation of $\mathfrak{sl}_2(F)$. We also study other known simple Lie algebras in characteristic three, but they do not yield a new counterexample.
