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 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.

Quantum Homomorphisms of Associative Algebras

1997 ◽  
Vol 12 (38) ◽  
pp. 2963-2974
Author(s):  
A. E. F. Djemai
Keyword(s):  
Quantum Group ◽  
Associative Algebra ◽  
Associative Algebras ◽  
Type Formula ◽  
Algebra Of Functions ◽  
Inhomogeneous Quantum

Given an associative algebra A generated by {ek, k=1, 2,…} and with an internal law of type: [Formula: see text], we first show that it is possible to construct a quantum bi-algebra [Formula: see text] with unit and generated by (non-necessarily commutative) elements [Formula: see text] satisfying the relations: [Formula: see text]. This leads one to define a quantum homomorphism[Formula: see text]. We then treat the example of the algebra of functions on a set of N elements and we show, for the case N=2, that the resulting bihyphen;algebra is an inhomogeneous quantum group. We think that this method can be used to construct quantum inhomogeneous groups.

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 Decompositions of algebras and post-associative algebra structures

2019 ◽  
Vol 30 (03) ◽  
pp. 451-466
Author(s):  
Dietrich Burde ◽  
Vsevolod Gubarev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Algebra Structure ◽  
Associative Algebras ◽  
Simple Lie Algebra ◽  
Reductive Lie Algebra ◽  
Baxter Operator

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

scholarly journals Presentations for Subrings and Subalgebras of Finite CO-Rank

2019 ◽  
Vol 71 (1) ◽  
pp. 53-71
Author(s):  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Lie Algebra ◽  
Noetherian Ring ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutative Noetherian Ring ◽  
Rank 2 ◽  
Finitely Presented ◽  
Algebra Over A Field

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

Some new results on Gröbner–Shirshov bases for Lie algebras and around

2018 ◽  
Vol 28 (08) ◽  
pp. 1403-1423
Author(s):  
L. A. Bokut ◽  
Yuqun Chen ◽  
Abdukadir Obul
Keyword(s):  
Lie Algebras ◽  
Associative Algebras ◽  
Leibniz Algebras ◽  
Novikov Algebras

We review Gröbner–Shirshov bases for Lie algebras and survey some new results on Gröbner–Shirshov bases for [Formula: see text]-Lie algebras, Gelfand–Dorfman–Novikov algebras, Leibniz algebras, etc. Some applications are given, in particular, some characterizations of extensions of groups, associative algebras and Lie algebras are given.

Sign in / Sign up

Export Citation Format

Share Document