弱交换性与幂零性
Weak commutativity and nilpotency
摘要：
我们继续分析李代数的弱交换性构造。这是由固定李代数的两个同构副本$\mathfrak{g}$和$\mathfrak{g}^ψ$生成的李代数$χ(\mathfrak{g})$，对所有$x \in \mathfrak{g}$都服从关系$[x,x^ψ]=0$。在这篇文章中，我们研究了理想的$x-x^ψ$产生的所有$x \in \mathfrak{g}$的$L =L(\mathfrak{g})$。我们获得了一个（无限）介绍$L$作为李代数，我们表明，在一般情况下，它不能减少到一个有限的。有了这一点，我们研究幂零性的问题。证明了若$\mathfrak{g}$是$c$类的幂零，则$χ(\mathfrak{g})$至多是$c+2$类的幂零，若$\mathfrak{g}$是$2$生成的或$c$是奇数的，则此界可改进为$c+1$.如果$\mathfrak{g}$是类$2$或$3$的自由幂零，我们也得到了$L(\mathfrak{g})$的具体描述（从而也得到了$χ(\mathfrak{g})$的具体描述）.最后，利用Gröbner-Shirshov基方法证明了交换理想$R(\mathfrak{g}) = [\mathfrak{g}, [L, \mathfrak{g}^ψ]]$是无穷维的，如果$\mathfrak{g}$至少$3$是无秩的.
英文摘要：
We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra $χ(\mathfrak{g})$ generated by two isomorphic copies $\mathfrak{g}$ and $\mathfrak{g}^ψ$ of a fixed Lie algebra, subject to the relations $[x,x^ψ]=0$ for all $x \in \mathfrak{g}$. In this article we study the ideal $L =L(\mathfrak{g})$ generated by $x-x^ψ$ for all $x \in \mathfrak{g}$. We obtain an (infinite) presentation for $L$ as a Lie algebra, and we show that in general it cannot be reduced to a finite one. With this in hand, we study the question of nilpotency. We show that if $\mathfrak{g}$ is nilpotent of class $c$, then $χ(\mathfrak{g})$ is nilpotent of class at most $c+2$, and this bound can improved to $c+1$ if $\mathfrak{g}$ is $2$-generated or if $c$ is odd. We also obtain concrete descriptions of $L(\mathfrak{g})$ (and thus of $χ(\mathfrak{g})$) if $\mathfrak{g}$ is free nilpotent of class $2$ or $3$. Finally, using methods of Gröbner-Shirshov bases we show that the abelian ideal $R(\mathfrak{g}) = [\mathfrak{g}, [L, \mathfrak{g}^ψ]]$ is infinite-dimensional if $\mathfrak{g}$ is free of rank at least $3$.
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