Nilpotency of Some Lie Algebras Associated with p-Groups

1999 ◽  
Vol 51 (3) ◽  
pp. 658-672 ◽  
Author(s):  
Pavel Shumyatsky
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Locally Finite ◽  
Residually Finite ◽  
Involutory Automorphism ◽  
Graded Lie Algebra

AbstractLet L = L0 + L1 be a 2-graded Lie algebra over a commutative ring with unity in which 2 is invertible. Suppose that L0 is abelian and L is generated by finitely many homogeneous elements a1,...,ak such that every commutator in a1,...,ak is ad-nilpotent. We prove that L is nilpotent. This implies that any periodic residually finite 2ʹ-group G admitting an involutory automorphism ϕ with CG(ϕ) abelian is locally finite.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

Orthogonal decompositions of Lie algebras over finite commutative rings

2021 ◽  
pp. 2350006
Author(s):  
Songpon Sriwongsa
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Orthogonal Decomposition ◽  
Necessary Condition ◽  
Special Linear ◽  
Orthogonal Lie Algebra ◽  
Orthogonal Decompositions ◽  
Finite Commutative Ring

Let [Formula: see text] be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over [Formula: see text]. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over [Formula: see text].

scholarly journals Rings with involution whose symmetric elements are central

1980 ◽  
Vol 3 (2) ◽  
pp. 247-253
Author(s):  
Taw Pin Lim
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Semisimple Lie Algebra ◽  
Direct Sum ◽  
Rings With Involution ◽  
Torsion Free

In a ringRwith involution whose symmetric elementsSare central, the skew-symmetric elementsKform a Lie algebra over the commutative ringS. The classification of such rings which are2-torsion free is equivalent to the classification of Lie algebrasKoverSequipped with a bilinear formfthat is symmetric, invariant and satisfies[[x,y],z]=f(y,z)x−f(z,x)y. IfSis a field of char≠2,f≠0anddimK>1thenKis a semisimple Lie algebra if and only iffis nondegenerate. Moreover, the derived algebraK′is either the pure quaternions overSor a direct sum of mutually orthogonal abelian Lie ideals ofdim≤2.

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.

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.

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.

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.

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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