论文标题
无限制的振荡中间生长的代数
Nil restricted Lie algebras of oscillating intermediate growth
论文作者
论文摘要
Grigorchuk和Gupta-Sidki群体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。众所周知，著名的Golod结构产生了有限生成的指数增长的尼尔代数。该方法的最新扩展允许构建多项式和中间生长的有限生成的nil-Elgebras。该论文的另一个动机是构建了卡萨博夫和帕克的振荡生长群体。 对于任何主要的$ p $，我们建立了一个由中间振荡增长的三家限制性谎言代数。我们之所以称它们为Phoenix代数，是因为在无限的许多时期内，代数几乎是通过准线性增长而“垂死的”，即较低的Gelfand-kirillov dimension是一个，更确切地说，增长是$ n \ big（\ ln^{（q）（q）} {（q）} \！n \ big）。另一方面，对于无限的许多$ n $，增长具有$ \ exp（n/（\ ln n）^λ）$类型的中间行为，在此期间，代数正在“恢复”。此外，这两种类型的行为之间的增长函数正在振荡。这些受限制的谎言代数有一个零$ p $绘制。
The Grigorchuk and Gupta-Sidki groups 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. It is known that the famous construction of Golod yields finitely generated associative nil-algebras of exponential growth. Recent extensions of that approach allowed to construct finitely generated associative nil-algebras of polynomial and intermediate growth. Another motivation of the paper is a construction of groups of oscillating growth by Kassabov and Pak. For any prime $p$ we construct a family of 3-generated restricted Lie algebras of intermediate oscillating growth. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is "almost dying" by having a quasi-linear growth, namely the lower Gelfand-Kirillov dimension is one, more precisely, the growth is of type $n \big(\ln^{(q)} \!n\big )^κ$. On the other hand, for infinitely many $n$ the growth has a rather fast intermediate behaviour of type $\exp( n/ (\ln n)^λ)$, for such periods the algebra is "resuscitating". Moreover, the growth function is oscillating between these two types of behaviour. These restricted Lie algebras have a nil $p$-mapping.
