Nil restricted Lie algebras of oscillating intermediate growth

[摘要] The research is motivated by a construction of groups of oscillating growth by Kassabov and Pak[25] and a description of possible growth functions of finitely generated associative algebras by Bell and Zelmanov[9]. In this paper we address both, the question of possible growth functions in case of Lie algebras, and the Kurosh problem, because our examples of restricted Lie algebras have a nil p-mapping, which is an analogue of nillity for associative algebras or periodicity for groups. Namely, for any field of positive characteristic, we construct a family of 3-generated restricted Lie algebras of intermediate oscillating growth. We call them Phoenix algebrasbecause, for infinitely many periods of time, the algebra is almost dying by having a quasi-lineargrowth, namely the lower GelfandKirillov dimension is one, more precisely, the growth is of type [GRAPHICS] , where q is an element of N, k > 0 are constants. On the other hand, for infinitely many n the growth function has a rather fast intermediate behaviorof type exp(n/(lnn)(lambda)), lambda being a constant determined by characteristic, for such periods the algebra is resuscitating. Moreover, the growth function is bounded and oscillating between these two types of behaviour. These restricted Lie algebras have a nil p-mapping, thus addressing the Kurosh problem as well. (c) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-15

[关键词] Restricted Lie algebras;p-groups;Growth;Self-similar algebras;Nil-algebras;Graded algebras;Lie superalgebra;Lie algebras of differential operators;Kurosh problem