scholarly journals Clover nil restricted Lie algebras of quasi-linear growth

2020 ◽  
pp. 2250057
Author(s):  
Victor Petrogradsky
Keyword(s):  
Lie Algebras ◽  
Linear Growth ◽  
Positive Characteristic ◽  
Intermediate Growth ◽  
Type Formula ◽  
Periodic Groups ◽  
Restricted Lie Algebras ◽  
Self Similar ◽  
Characteristic 2 ◽  
Positive Integers

The Grigorchuk and Gupta–Sidki groups play a 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 [V. M. Petrogradsky, Examples of self-iterating Lie algebras, J. Algebra 302(2) (2006) 881–886], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [I. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebras, J. Eur. Math. Soc. (JEMS) 10(2) (2008) 391–398]. Now, we construct a family of so called clover 3-generated restricted Lie algebras [Formula: see text], where a field of positive characteristic is arbitrary and [Formula: see text] an infinite tuple of positive integers. All these algebras have a nil [Formula: see text]-mapping. We prove that [Formula: see text]. We compute Gelfand–Kirillov dimensions of clover restricted Lie algebras with periodic tuples and show that these dimensions for constant tuples are dense on [Formula: see text]. We construct a subfamily of nil restricted Lie algebras [Formula: see text], with parameters [Formula: see text], [Formula: see text], having extremely slow quasi-linear growth of type: [Formula: see text], as [Formula: see text]. The present research is motivated by construction by Kassabov and Pak of groups of oscillating growth [M. Kassabov and I. Pak, Groups of oscillating intermediate growth. Ann. Math. (2) 177(3) (2013) 1113–1145]. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in [V. Petrogradsky, Nil restricted Lie algebras of oscillating intermediate growth, preprint (2020), arXiv:2004.05157 ]. 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 [Formula: see text] it has a rather fast intermediate growth of type [Formula: see text], for such periods the algebra is “resuscitating”. The present construction of three-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear bound in that construction.

Intermediate growth in finitely presented algebras

2017 ◽  
Vol 27 (04) ◽  
pp. 391-401 ◽  
Author(s):  
Dilber Koçak
Keyword(s):  
Lie Algebras ◽  
Associative Algebras ◽  
Intermediate Growth ◽  
Type Formula ◽  
Growth Types ◽  
Finitely Presented

For any integer [Formula: see text], we construct examples of finitely presented associative algebras over a field of characteristic [Formula: see text] with intermediate growth of type [Formula: see text]. We produce these examples by computing the growth types of some finitely presented metabelian Lie algebras.

scholarly journals Restricted Lazard elimination and modular Lie powers

2001 ◽  
Vol 71 (2) ◽  
pp. 259-278 ◽  
Author(s):  
Ralph Stöhr
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Positive Characteristic ◽  
Group Actions ◽  
Finite Group Actions ◽  
Free Lie Algebras ◽  
Restricted Lie Algebras

AbstractWe exhibit a variation of the Lazard Elimination theorem for free restricted Lie algebras, and apply it to two problems about finite group actions on free Lie algebras over fields of positive characteristic.

scholarly journals Fermionic realization of twisted toroidal Lie algebras

2020 ◽  
pp. 2150143
Author(s):  
Naihuan Jing ◽  
Chad R. Mangum ◽  
Kailash C. Misra
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Type Formula ◽  
Toroidal Lie Algebra

In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type [Formula: see text] and [Formula: see text] based on the newly found Moody–Rao–Yokonuma-like presentation.

scholarly journals Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type

10.37236/933 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Gregg Musiker ◽  
James Propp
Keyword(s):  
Lie Algebras ◽  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Finite Dimensional ◽  
Combinatorial Interpretations ◽  
Rank 2 ◽  
The Individual ◽  
Affine Type ◽  
Positive Integers ◽  
Two Parameter

Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all $b,c$, each term of the $(b,c)$ sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers $b,c$ satisfy $bc < 4$, the recurrence is related to the root systems of finite-dimensional rank $2$ Lie algebras; when $bc>4$, the recurrence is related to Kac-Moody rank $2$ Lie algebras of general type. Here we investigate the borderline cases $bc=4$, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.

scholarly journals Schubert Class and Cyclotomic NilHecke Algebras

2021 ◽  
Vol 28 (03) ◽  
pp. 379-398
Author(s):  
Kai Zhou ◽  
Jun Hu
Keyword(s):  
Algebraic Method ◽  
Basic Algebra ◽  
Graded Algebra ◽  
Algebra Isomorphism ◽  
Type Formula ◽  
Basis Element ◽  
Schubert Classes ◽  
Positive Integers ◽  
Schubert Class

Let [Formula: see text] and [Formula: see text] be positive integers such that [Formula: see text], and let [Formula: see text] be the Grassmannian which consists of the set of [Formula: see text]-dimensional subspaces of [Formula: see text]. There is a [Formula: see text]-graded algebra isomorphism between the cohomology [Formula: see text] of [Formula: see text] and a natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of the type [Formula: see text] cyclotomic nilHecke algebra [Formula: see text]. We show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class [Formula: see text] coincides with the basis element [Formula: see text] constructed by Hu and Liang by purely algebraic method, where [Formula: see text] with [Formula: see text] for each [Formula: see text], and [Formula: see text] is the [Formula: see text]-multipartition of [Formula: see text] associated to [Formula: see text]. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian [Formula: see text] and the [Formula: see text]'s basis ([Formula: see text] is an [Formula: see text]-multipartition of [Formula: see text] with each component being either [Formula: see text] or empty) of the natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of [Formula: see text] is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.

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