论文标题
通勤和努力弱
Weak commutativity and nilpotency
论文作者
论文摘要
我们继续分析谎言代数的弱通勤构建。这是由两个同构副本生成的lie代数$χ（\ Mathfrak {g}）$ $ \ mathfrak {g} $和$ \ mathfrak {g}^ψ$的固定lie代数的代数，均受关系$ [x，x^ψ] = 0 $的关系，在本文中，我们研究了由$ x-x^ψ$生成的理想$ l = l（\ mathfrak {g}）$ in \ mathfrak {g} $。我们以$ l $作为谎言代数获得（无限的）演示文稿，我们表明总体上不能将其简化为有限的代数。有了这一点，我们研究了尼尔特的问题。我们表明，如果$ \ mathfrak {g} $是$ c $的nilpotent，则$χ（\ mathfrak {g}）$最多是$ c+2 $的班级，并且如果$ \ \ \ \ m athfrak {g} $ 2 $ $ 2 $ $ cagenerated或$ c $是奇数。如果$ \ mathfrak {g} $是$ 2 $ 2 $或$ 3 $，我们还获得了$ l（\ mathfrak {g}）$的具体描述（\ mathfrak {g}）$（因此，$χ（\ mathfrak {g}）$）。最后，使用Gröbner-Shirshov基地的方法，我们表明Abelian Ideal $ r（\ Mathfrak {g}）= [\ Mathfrak {g} {g}，[l，\ Mathfrak {g}^ψ] $ IS Infinite-dimente-dimentional-iffinite-dimention-iffinite-diffrak-mathfrak {g} $免费等级。
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 $3$.
