RESTRICTED LIE ALGEBRAS OF POLYCYCLIC GROUPS

2006 ◽  
Vol 05 (05) ◽  
pp. 571-627
Author(s):  
A. I. LICHTMAN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Restricted Lie Algebras ◽  
Graded Lie Algebra ◽  
Polycyclic Groups

We consider some classes of polycyclic groups which have a p-series such that the restricted graded Lie algebra associated to this p-series is free abelian. We also study p-series and restricted Lie algebras associated to them in arbitrary polycyclic groups.

ON THE STRUCTURE OF SOLUBLE GRADED LIE ALGEBRAS

2011 ◽  
Vol 10 (04) ◽  
pp. 597-604 ◽  
Author(s):  
PAVEL SHUMYATSKY ◽  
CARMELA SICA
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Graded Lie Algebras ◽  
Derived Length ◽  
Elementary Group ◽  
Graded Lie Algebra

Let A be the elementary group of order 2n and L an A-graded Lie algebra with L0 = 0. Assume that L is soluble with derived length k. It is proved that L has a series of ideals of length n all of whose quotients are nilpotent of {k, n}-bounded class.

THE FOX PROBLEM FOR FREE RESTRICTED LIE ALGEBRAS

2008 ◽  
Vol 18 (02) ◽  
pp. 271-283 ◽  
Author(s):  
HAMID USEFI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Free Groups ◽  
Enveloping Algebra ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ωn(L)ω(R) in terms of R only. This problem is the analogue of the Fox problem for free groups.

scholarly journals Subalgebras of free restricted Lie algebras

2005 ◽  
Vol 72 (1) ◽  
pp. 147-156 ◽  
Author(s):  
R.M. Bryant ◽  
L.G. Kovács ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Main Step ◽  
Free Lie Algebra ◽  
Generating Set ◽  
Restricted Lie Algebra ◽  
Corrected Proof ◽  
Restricted Lie Algebras ◽  
Remarkable Result ◽  
Algebra Over A Field

A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.

scholarly journals Restricted Lie algebras of maximal class

1999 ◽  
Vol 59 (2) ◽  
pp. 217-223
Author(s):  
D.M. Riley
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Maximal Class ◽  
Infinite Dimensional ◽  
Restricted Lie Algebras ◽  
Infinite Dimensional Lie Algebra

Let L be a possibly infinite-dimensional Lie algebra of maximal class. We show that if L admits the structure of a Lie p-algebra then the dimension of L can be at most p + 1. Furthermore, this bound is best possible.

scholarly journals Nijenhuis operators on pre-Lie algebras

2019 ◽  
Vol 21 (07) ◽  
pp. 1850050 ◽  
Author(s):  
Qi Wang ◽  
Yunhe Sheng ◽  
Chengming Bai ◽  
Jiefeng Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Regular Representation ◽  
Complex Structure ◽  
Geometric Interpretation ◽  
Dual Representation ◽  
New Approach ◽  
Lie Bracket ◽  
Graded Lie Algebra ◽  
Dendriform Algebras

First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.

scholarly journals Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes

2017 ◽  
Vol 60 (3) ◽  
pp. 470-477 ◽  
Author(s):  
Urtzi Buijs ◽  
Yves Félix ◽  
Aniceto Murillo ◽  
Daniel Tanré
Keyword(s):  
Simplicial Complex ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simplicial Complexes ◽  
Connected Components ◽  
Homotopy Groups ◽  
Graded Lie Algebras ◽  
Simplicial Sets ◽  
Graded Lie Algebra ◽  
Finite Simplicial Complex

AbstractIn a previous work, we associated a complete diòerential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete diòerential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

Restricted Lie Algebras and Their Envelopes

1995 ◽  
Vol 47 (1) ◽  
pp. 146-164 ◽  
Author(s):  
D. M. Riley ◽  
A. Shalev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Codimension ◽  
Augmentation Ideal ◽  
Nilpotent Ideal ◽  
Enveloping Algebra ◽  
Group Of Units ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Algebra Over A Field

AbstractLet L be a restricted Lie algebra over a field of characteristic p. Denote by u(L) its restricted enveloping algebra and by ωu(L) the augmentation ideal of u(L). We give an explicit description for the dimension subalgebras of L, namely those ideals of L defined by Dn(L) - L∩ωu(L)n for each n ≥ 1. Using this expression we describe the nilpotence index of ωU(L). We also give a precise characterisation of those L for which ωu(L) is a residually nilpotent ideal. In this case we show that the minimal number of elements required to generate an arbitrary ideal of u(L) is finitely bounded if and only if L contains a 1-generated restricted subalgebra of finite codimension. Subsequently we examine certain analogous aspects of the Lie structure of u(L). In particular we characterise L for which u(L) is residually nilpotent when considered as a Lie algebra, and give a formula for the Lie nilpotence index of u(L). This formula is then used to describe the nilpotence class of the group of units of u(L).

GROWTH OF SUBALGEBRAS FOR RESTRICTED LIE ALGEBRAS AND TRANSITIVE ACTIONS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1151-1168 ◽  
Author(s):  
V. M. PETROGRADSKY
Keyword(s):  
Group Theory ◽  
Finite Field ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finitely Generated ◽  
Subgroup Growth ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Transitive Actions

We study a growth of subalgebras for restricted Lie algebras over a finite field 𝔽q. This kind of growth is an analog of the subgroup growth in the group theory. Let L be a finitely generated restricted Lie algebra. Then an(L) is the number of restricted subalgebras H ⊂ L such that dim 𝔽q L/H = n, n ≥ 0. We compute the numbers an(Fd) explicitly and find asymptotics, where Fd is the free restricted Lie algebra of rank d, d ≥ 1. As an important instrument, we use the notion of transitive L-action on coalgebras and algebras.

On certain subgroups of the McCool group

2020 ◽  
Vol 30 (05) ◽  
pp. 1081-1096
Author(s):  
C. E. Kofinas
Keyword(s):  
Automorphism Group ◽  
Positive Integer ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Free Group ◽  
Graded Lie Algebras ◽  
Natural Embedding ◽  
Inner Automorphism ◽  
Graded Lie Algebra

For a positive integer [Formula: see text], with [Formula: see text], let [Formula: see text] be a free group of rank [Formula: see text] and let [Formula: see text] be the subgroup of the automorphism group of [Formula: see text] consisting of all automorphisms which induce the identity on the abelianization of [Formula: see text]. We write [Formula: see text] and [Formula: see text] for the upper McCool group and the partial inner automorphism group, respectively. We show that [Formula: see text] is isomorphic to the quotient of [Formula: see text] by its center and we prove similar results for their associated graded Lie algebras and their Andreadakis–Johnson Lie algebras. Furthermore, we give a presentation of the associated graded Lie algebra over the integers of [Formula: see text] and we prove that it admits a natural embedding into the Andreadakis–Johnson Lie algebra of [Formula: see text]. Although the latter results are known, we present proofs based on different methods.

scholarly journals Formality properties of finitely generated groups and Lie algebras

2019 ◽  
Vol 31 (4) ◽  
pp. 867-905 ◽  
Author(s):  
Alexander I. Suciu ◽  
He Wang
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nilpotent Groups ◽  
Direct Products ◽  
Finitely Generated ◽  
Field Extensions ◽  
Finitely Generated Groups ◽  
Graded Lie Algebra ◽  
Fibered Manifolds ◽  
Torsion Free

Abstract We explore the graded-formality and filtered-formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the aforementioned Lie algebras. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.

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