scholarly journals Cup products, lower central series, and holonomy Lie algebras

2019 ◽  
Vol 223 (8) ◽  
pp. 3359-3385 ◽  
Author(s):  
Alexander I. Suciu ◽  
He Wang
Keyword(s):  
Lie Algebras ◽  
Lower Central Series ◽  
Central Series ◽  
Cup Products

GROWTH OF FINITELY GENERATED POLYNILPOTENT LIE ALGEBRAS AND GROUPS, GENERALIZED PARTITIONS, AND FUNCTIONS ANALYTIC IN THE UNIT CIRCLE

1999 ◽  
Vol 09 (02) ◽  
pp. 179-212 ◽  
Author(s):  
V. M. PETROGRADSKY
Keyword(s):  
Unit Circle ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Central Series ◽  
Precise Asymptotics ◽  
Finitely Generated ◽  
Asymptotic Growth ◽  
Enveloping Algebra ◽  
Finitely Generated Groups ◽  
Kirillov Dimension

Recently, the author has suggested a series of dimensions of algebras which includes as first terms dimension of a vector space, Gelfand–Kirillov dimension, and superdimension. These dimensions enabled us to describe the change of the growth in the transition from a Lie algebra to its universal enveloping algebra. In fact, this is a result on some generalized partitions. In this paper, we obtain more precise asymptotics for generalized partitions. As a main application, we obtain more precise asymptotics for the growth of free polynilpotent finitely generated Lie algebras. As a corollary, we specify the asymptotic growth of the lower central series ranks for free polynilpotent finitely generated groups. We essentially use Hilbert–Poincaré series and some facts on the growth of functions analytic in the unit circle. By the growth of such functions, we mean their growth when the variable tends to 1. Finally, we study two kinds of p-central series for free polynilpotent finitely generated groups. We obtain asymptotics for the ranks of these series, in one case we have an example of a polynomial, but not rational growth.

Automorphisms of the completion of relatively free Lie algebras

2018 ◽  
Vol 28 (06) ◽  
pp. 1091-1100
Author(s):  
C. E. Kofinas
Keyword(s):  
Automorphism Group ◽  
Power Series ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Free Lie Algebra ◽  
Central Series ◽  
Dense Subgroup ◽  
Free Lie Algebras ◽  
Finite Set ◽  
Formal Power

Let [Formula: see text] be a relatively free Lie algebra of finite rank [Formula: see text], with [Formula: see text], [Formula: see text] be the completion of [Formula: see text] with respect to the topology defined by the lower central series [Formula: see text] of [Formula: see text] and [Formula: see text], with [Formula: see text]. We prove that, with respect to the formal power series topology, the automorphism group [Formula: see text] of [Formula: see text] is dense in the automorphism group [Formula: see text] of [Formula: see text] if and only if [Formula: see text] is nilpotent. Furthermore, we show that there exists a dense subgroup of [Formula: see text] generated by [Formula: see text] and a finite set of IA-automorphisms if and only if [Formula: see text] is generated by [Formula: see text] and a finite set of IA-automorphisms independent upon [Formula: see text] for all [Formula: see text]. We apply our study to several varieties of Lie algebras.

On the Andreadakis conjecture restricted to the “lower-triangular” automorphism groups of free groups

2017 ◽  
Vol 16 (05) ◽  
pp. 1750099 ◽  
Author(s):  
Takao Satoh
Keyword(s):  
Cohomology Group ◽  
Homology Group ◽  
Automorphism Groups ◽  
Lower Central Series ◽  
Affirmative Answer ◽  
Central Series ◽  
Homology Groups ◽  
Cup Product ◽  
Lower Triangular ◽  
Cup Products

In this paper, we consider a certain subgroup [Formula: see text] of the IA-automorphism group of a free group. We determine the images of the [Formula: see text]th Johnson homomorphism restricted to [Formula: see text] for any [Formula: see text] and [Formula: see text]. By using this result, we give an affirmative answer to the Andreadakis conjecture restricted for [Formula: see text]. Namely, we show that the intersection of the Andreadakis–Johnson filtration and [Formula: see text] coincides with the lower central series of [Formula: see text]. In a series of this research, we obtain additional results on the integral (co)homology groups of [Formula: see text]. In particular, we determine the first homology group, and study the cup product of first cohomologies of [Formula: see text]. Furthermore, we construct nontrivial second homology classes of [Formula: see text] by observing its generators and relators, and show that the second cohomology group is not generated by cup products of the first cohomology groups.

scholarly journals Homotopy Lie algebras, lower central series and the Koszul property

2004 ◽  
Vol 8 (3) ◽  
pp. 1079-1125 ◽  
Author(s):  
Ştefan Papadima ◽  
Alexander I Suciu
Keyword(s):  
Lie Algebras ◽  
Lower Central Series ◽  
Central Series ◽  
Koszul Property

scholarly journals A sandwich in thin lie algebras

2022 ◽  
pp. 1-10
Author(s):  
Sandro Mattarei
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Zero Element ◽  
Homogeneous Component ◽  
Central Series ◽  
Two Dimensional ◽  
Dimension One ◽  
Positive Integers ◽  
Modular Lie Algebras

Abstract A thin Lie algebra is a Lie algebra $L$ , graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$ , and such that each non-zero ideal of $L$ lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of $L$ (that is, the next diamond past $L_1$ ) occurs in degree $k$ . We prove that if $k>5$ , then $[Lyy]=0$ for some non-zero element $y$ of $L_1$ . In characteristic different from two this means $y$ is a sandwich element of $L$ . We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.

NOTTINGHAM LIE ALGEBRAS WITH DIAMONDS OF FINITE TYPE

2004 ◽  
Vol 14 (01) ◽  
pp. 35-67 ◽  
Author(s):  
A. CARANTI ◽  
S. MATTAREI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Type ◽  
Lower Central Series ◽  
Central Series ◽  
Graded Lie Algebras ◽  
Graded Lie Algebra

We study a class of positively graded Lie algebras with a pattern of homogeneous components similar to that of the graded Lie algebra associated to the Nottingham group with respect to its lower central series.

scholarly journals Diamond distances in Nottingham algebras

2021 ◽  
Author(s):  
M. Avitabile ◽  
S. Mattarei
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Central Series ◽  
Graded Lie Algebras ◽  
Infinite Dimensional ◽  
Graded Lie Algebra ◽  
The Difference ◽  
Dimension One

Nottingham algebras are a class of just-infinite-dimensional, modular, [Formula: see text]-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree [Formula: see text], and the second occurs in degree [Formula: see text], a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to [Formula: see text]. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra [Formula: see text] can be assigned a type, in such a way that the degrees and types of the diamonds completely describe [Formula: see text]. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals [Formula: see text]. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type [Formula: see text].

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