论文标题
三叶草限制了准线性生长的代数
Clover nil restricted Lie algebras of quasi-linear growth
论文作者
论文摘要
Grigorchuk和Gupta-Sidki群体在现代小组理论中扮演着基本角色。它们是自然产生的周期性组的自然例子。在特征2的限制代数的情况下，作者构建了它们的类似物，Shestakov和Zelmanov将这种结构扩展到了任意的积极特征。此外，作者还建立了一个由2产生的限制性的谎言代数，其代数为缓慢的多项式增长，并以零$ p $绘制。 现在，我们构建了一个所谓的三叶草3生成的限制性Lie代数$ t（ξ）$的家族，其中一个积极特征的领域是任意的，而$ξ$是一个无限的正整数元组。我们证明，$ 1 \ le \ mathrm {gkdim} t（ξ）\ le3 $，此外，三叶草的gelfand-kirillov dimensions lie代数的gelfand-kirillov dimensions constant constant tume contance tume contance tume contluse contance tume contection $ [1,3] $。我们构建了一个非属性零限制的lie代数$ t（ξ_{q，κ}）$的亚家族，其中$ q \ in \ mathbb n $，$κ\ in \ mathbb r^+$，类型非常慢： $γ_{t（ξ_{q，κ}）}（m）= m \ big（\ ln^{（q）} \！m \ big）^{κ+o（1）} $，为$ m \ to \ to \ infty $。 本研究是由卡萨博夫（Kassabov）和一系列振荡增长组的建造所激发的。作为一个类似物，我们在另一篇论文中构建了中间振荡生长的限制为无限的代数。我们称它们称为“凤凰代数”，因为在无限的许多时期内，代数几乎是“垂死”的，因为它具有上述“准线性”的增长，对于无限的许多$ n $而言，增长功能的作用均以$ \ exp（n/（\ ln n n）^λ）为$ \ exp（n/（\ ln n）^λ）$，在此类时期，对于Algebra is algebra is resuscitating'resuscitating'resuscitating''resusciting''resuscitating''resuscitating''resuscitating''resusciting''目前的3生成零限制的lie代数是准线性增长的代数，是该结果的重要组成部分，负责该结构中较低的准线性增长。
The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2, Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic. Also, the author constructed a family of 2-generated restricted Lie algebras of slow polynomial growth with a nil $p$-mapping. Now, we construct a family of so called clover 3-generated restricted Lie algebras $T(Ξ)$, where a field of positive characteristic is arbitrary and $Ξ$ an infinite tuple of positive integers. We prove that $1\le \mathrm{GKdim}T(Ξ)\le3$, moreover, the set of Gelfand-Kirillov dimensions of clover Lie algebras with constant tuples is dense on $[1,3]$. We construct a subfamily of non-isomorphic nil restricted Lie algebras $T(Ξ_{q,κ})$, where $q\in\mathbb N$, $κ\in\mathbb R^+$, with extremely slow quasi-linear growth of type: $γ_{T(Ξ_{q,κ})}(m)=m\big(\ln^{(q)}\!m\big)^{κ+o(1)}$, as $m\to\infty$. The present research is motivated by a construction by Kassabov and Pak of groups of oscillating growth. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in another paper. We call them "Phoenix algebras" because, for infinitely many periods of time, the algebra is "almost dying" by having a "quasi-linear" growth as above, for infinitely many $n$ the growth function behaves like $\exp(n/(\ln n)^λ)$, for such periods the algebra is "resuscitating". The present construction of 3-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear growth in that construction.
