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.

scholarly journals Groups of the virtual trefoil and Kishino knots

2018 ◽  
Vol 27 (13) ◽  
pp. 1842009
Author(s):  
Valeriy G. Bardakov ◽  
Yuliya A. Mikhalchishina ◽  
Mikhail V. Neshchadim
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Lower Central Series ◽  
Central Series ◽  
Virtual Link ◽  
Virtual Knots ◽  
Formal Power

In the paper [13], for an arbitrary virtual link [Formula: see text], three groups [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] were defined. In the present paper, these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also, we prove that [Formula: see text] distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence, we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots.

WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA

2004 ◽  
Vol 15 (10) ◽  
pp. 987-1005 ◽  
Author(s):  
MAHMOUD BENKHALIFA
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Integral Domain ◽  
Free Lie Algebra ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Equivalent Class

Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.

scholarly journals Free Lie algebras as modules for symmetric groups

1999 ◽  
Vol 67 (2) ◽  
pp. 143-156 ◽  
Author(s):  
R. M. Bryant ◽  
L. G. Kovács ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebras ◽  
Symmetric Groups ◽  
Free Lie Algebra ◽  
Prime Characteristic ◽  
Specht Modules ◽  
Generating Set ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Free Lie Algebras ◽  
Direct Decompositions

AbstractLet r be a positive integer, F a field of odd prime characteristic p, and L the free Lie algebra of rank r over F. Consider L a module for the symmetric group , of all permutations of a free generating set of L. The homogeneous components Ln of L are finite dimensional submodules, and L is their direct sum. For p ≤ r ≤ 2p, the main results of this paper identify the non-porojective indecomposable direct summands of the Ln as Specht modules or dual Specht modules corresponding to certain partitions. For the case r = p, the multiplicities of these indecomposables in the direct decompositions of the Ln are also determined, as are the multiplicities of the projective indecomposables. (Corresponding results for p = 2 have been obtained elsewhere.)

The Schreier Technique for Subalgebras of a Free Lie Algebra

1997 ◽  
Vol 49 (3) ◽  
pp. 600-616 ◽  
Author(s):  
Shmuel Rosset ◽  
Alon Wasserman
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Free Group ◽  
Finite Codimension ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Free Lie Algebras

AbstractIn group theory Schreier's technique provides a basis for a subgroup of a free group. In this paper an analogue is developed for free Lie algebras. It hinges on the idea of cutting a Hall set into two parts. Using it, we show that proper subalgebras of finite codimension are not finitely generated and, following M. Hall, that a finitely generated subalgebra is a free factor of a subalgebra of finite codimension.

THE EQUATION [x,u] + [y,v] = 0 IN FREE LIE ALGEBRAS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1165-1187 ◽  
Author(s):  
VLADIMIR REMESLENNIKOV ◽  
RALPH STÖHR
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Solution Space ◽  
Free Generator ◽  
Finite Codimension ◽  
Free Lie Algebra ◽  
Free Lie Algebras ◽  
Free Generators ◽  
Bilinear Equation

We investigate equations of the form [x,u] + [y,v] = 0 over a free Lie algebra L. In the case where u and v are free generators of L, we exhibit two series of solutions, we work out the dimensions of the homogeneous components of the solution space, and we determine its radical. In the general case we show that the results on free generator coefficients are sufficient to obtain the solution space up to finite codimension. As an application we determine the radical of the bilinear equation [x1,x2] + [x3,x4] = 0.

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.

Sign in / Sign up

Export Citation Format

Share Document