# On Marginal Automorphisms of Free Nilpotent Lie Algebras

Authors: Özge Öztekin
Journal: Journal of Mathematics
 (2020)

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## Abstract

LetL be the free nilpotent Lie algebra of finite rank over a field of characteristic zero. We define the concepts of marginal ideals and marginal automorphisms of L, and we give some results on marginal automorphisms.


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## Body

## 1. Introduction

The concept of marginal subgroups and marginal automorphisms of a groupG has been already studied by many authors (for details, see [1–3]). The group of marginal automorphisms of G is very important in studying its automorphism group AutG. For a suitable nonempty subset W of a free group, one-way construct marginal automorphism group of G carries the required properties. Lie algebra analogues to the marginal subgroup of a group were considered by Stewart [4]. Stewart has given nonassociative algebra versions of the concepts of verbal and marginal subgroups of a group and specialized in Lie algebras to obtain stronger results.In this paper, we give some properties of marginal ideal and marginal automorphism groups of free nilpotent Lie algebra of finite rank.


## 2. Preliminaries

LetF be the free Lie algebra freely generated by a finite set x1,x2,…,xm, m≥2, over a field K of characteristic zero, and let L=F/γnF, where γnF is the nth lower central term of F. It is clear that L is the free nilpotent of class n−1 Lie algebra. For the Lie multiplication, we use the commutator notation.For any elementw∈F, we denote by w¯ the natural image of w in L and we say that xi appears in w if the word w contains xi.Definition 1.
LetG be any Lie algebra and let W be a nonempty subset of F. For w=wx1,…,xm∈W, we introduce the set of values of w in G as(1)wG≔wu1,…,umu1,…,um∈Gand the set of all w values in G as(2)WG≔wu1,…,umw∈W.(1)
The verbal subalgebraAWG of G with respect to W is the subalgebra generated by the set WG(2)
The verbal idealIWG of G with respect to W is ideal which is generated by the set WGDefinition 2.
LetG be any Lie algebra, and let W be a nonempty subset of F. The marginal subspace, SWMG of G with respect to W, is defined to be the following set:(3)SWMG=α∈Lwu1,…,ui+βα,…um=wu1,…,ui,…um,for allw∈W,β∈Kandui∈G,i=1,…,m.(1)
The largest subalgebra ofG contained in SWMG is called the marginal subalgebra with respect to W and it is denoted by AWMG(2)
The largest ideal ofG contained in SWMG is called the marginal ideal with respect to W and it is denoted by IWMG
If the setW has a single element w, then we will write AwG,IwG,AwMG,andIwMG, instead of AWG,IWG,AWMG,andIWMG, respectively.
Throughout this paper, we take the subsetW of F as W=w1,w2,…,wt,x where wr’s r=1,…,t,t≥1 are monomials of F.


## 3. Main Results

In this section, we prove our main results which are analogue to the results of [1, 2].Lemma 1.
Letw=wx1,x2,…,xm be a monomial of F. Assume that x1,x2,…,xm appear in w. Then, α¯∈SwML if and only if wu1¯,…,uı−1¯,α¯,uı+1¯,…,um¯=0 for all uı¯∈L and ı=1,…,m.Proof.
Letα¯ be any element of SwML. If α¯=0¯, then it is clear that(4)wu1¯,…,uı−1¯,0¯,uı+1¯,…,um¯=0.
Letα¯≠0¯. Then,(5)wu1¯,…,uı−1¯,uı¯+βα¯,uı+1¯,…,um¯=wu1¯,…,uı−1¯,uı¯,uı+1¯,…,um¯,where β∈K. This gives wu1¯,…,uı−1¯,α¯,uı+1¯,…,um¯=0.
The “only if” part of the statement is clear.Lemma 2.
LetH and S be subalgebras of L such that S⊂H. Then, a marginal subalgebra of H may not be the marginal subalgebra of S.Proof.
LetH=γkL and S=γk+1L, 2≤k<n, and w=x1,x2,…,xn−1 be a monomial in F. Then, it is clear that AwMH=H and AwMS=S. Therefore, AwMH≠AwMS.Proposition 1.
LetW be a nonempty subset of F. For each wr∈W and b1¯,…,bm¯∈SWML, we have wrb1¯,b2¯,…,bm¯=0.Proof.
Letwr=wrx1,x2,…,xm∈W and let b1¯,b2¯,…,bm¯∈SWML:(6)wrb1¯,b2¯,…,bm¯=wr0+b1¯,b2¯,…,bm¯=wr0,b2¯,…,bm¯=wr0,0+b2¯,…,bm¯⋮=wr0,0,…,0=0.Definition 3.
We call an endomorphismφ of L a marginal endomorphism if φu¯−u¯∈IWML for each u¯∈L. The set of all marginal endomorphisms of L will be denoted by EndWML. If a marginal endomorphism is an automorphism, then it is called a marginal automorphism. The set of all marginal automorphisms of L is a subgroup of AutL, and it is denoted by AutWML.Example 1.
LetF be the free Lie algebra freely generated by a finite set x1,x2 and let L=F/γ6F.
If we choose the setW as W=w1=x1,x2, then IWML=Iw1ML=γ5L and the automorphisms(7)xi⟶xi+∑kakx1,x2,xi1,xi2,xi3are the marginal (with respect to w1) automorphisms of L where i,i1,i2,i3∈1,2 and α∈K.
Letw2=x1,x2,x1; then, one can easily see that the marginal (with respect to w2) ideal Iw2ML of L is γ6L=γ4F/γ6F and the automorphisms defined as(8)xi⟶xi+∑kαkx1,x2,xi1,xi2+∑kβkx1,x2,xi1,xi2,xi3are the marginal (with respect to w2) automorphisms of L, where i,i1,i2,i3∈1,2 and α,β∈K.Lemma 3.
Every marginal endomorphismφ of L fixes all elements of the verbal subalgebra AWL.Proof.
Letu1¯,u2¯,…,um¯∈L and w=wx1,x2,…,xm∈W. By definition of a marginal endomorphism for φ∈EndWML, there is an element αi¯∈IwML such that φui¯=ui¯+αi¯ for i=1,…,m. Now, we have(9)φwu1¯,u2¯,…,um¯=wφu1¯,φu2¯…,φum¯=wu1¯+α1¯,u2¯+α2¯,…,um¯+αm¯=wu1¯,u2¯+α2¯,…,um¯+αm¯⋮=wu1¯,u2¯,…,um¯.
LetG and H be two Lie algebras; we denote by HomG,H the set of all Lie algebra homomorphisms from G to H.Proposition 2.
Let∅≠W⊂F. For each ψ∈HomL,AWML and w∈W, we have ψw=0.Proof.
Letψ∈HomL,AWML,w∈W and u1¯,u2¯,…,um¯∈L. Since ψui¯∈AWML from Proposition 1, we have(10)ψwu1¯,u2¯,…,um¯=wψu1¯,ψu2¯,…,ψum¯=0.Theorem 1.
IfIWML⊆IWL, then(11)HomLIWL,IWML≅HomL,IWML.Proof.
The mapf:ψ⟶fψ defines a homomorphism from HomL/IWL,IWML to HomL,IWML, where fψu¯=ψu¯+IWL for all u¯∈L. It is clear that f is well-defined and it is easy to see that ψ⟶fψ is an isomorphism from HomL/IWL,IWML to HomL,IWML.
IfIWML is contained in the center of L and θ∈AutWML, then for all u¯∈L, one can easily see that the map fθ:u¯⟶θu¯−u¯ is a homomorphism from L into IWML. On the other hand, for every f∈HomL,IWML, the map θf:u¯⟶u¯+fu¯ is a marginal endomorphism of L. Note that the endomorphism θf is an automorphism if and only if fu¯≠−u¯, for all 0≠u¯∈L.Theorem 2.
IfIWML⊆IWL, then AutWML acts trivially on IWML.Proof.
IfIWML⊆IWL, then W⊆Lk such that IWML⊆Ln−k+1 and k≤n−k+1. Now, let u¯=u¯x1,x2,…,xm∈IWML and θ∈AutWML:(12)θu¯x1,x2,…,xm=u¯θx1,θx2,…,θxm=u¯x1+α1¯,x2+α2¯,…,xm+αm¯=u¯x1,x2,…,xm+u¯α1¯,x2+α2¯,…,xm+αm¯+…++u¯x1+α1¯,x2+α2¯,…,αm¯,where αi¯∈IwML,i=1,…,m. Then, u¯x1+α1¯,…,αi¯,…,xm+αm¯=0, i=1,…,m, by Lemma 1. Therefore, we get θu¯x1,x2,…,xm=u¯x1,x2,…,xm.
The next theorem determines the structure ofAutWML.Theorem 3.
Let∅≠W⊆L such that IWML⊆IWL. Then,(13)AutWML≅HomLIWL,IWML.Proof.
The map(14)ψ:θ⟶ψθ,defines an isomorphism from AutWML to HomL/IWL,IWML, where ψθu¯+IWL=θu¯−u¯ for all u¯∈L.(i)
ψθ is well-defined: Let u1¯,u2¯∈L and u1¯+IWL=u2¯+IWL; then, u1¯−u2¯∈IWL. By Lemma 3, we have θu1¯−u2¯=u1¯−u2¯ and θu1¯−u1¯=θu2¯−u2¯. Therefore, ψθu1¯+IWL=ψθu2¯+IWL, as required.(ii)
ψθ∈HomL/IWL,IWML: Let u1¯,u2¯∈L; then,(15)ψθu1¯+IWL+u2¯+IWL=ψθu1¯+u2¯+IWL=θu1¯+u2¯−u1¯+u2¯=θu1¯−u1¯+θu2¯−u2¯=ψθu1¯+IWL+ψθu2¯+IWL,and similarly, we get(16)ψθcu1¯+IWL,u2¯+IWL=cψθu1¯+IWL,ψθu2¯+IWL.(iii)
ψ is a homomorphism: Let θ1,θ2∈AutWML and u¯∈L. Since θ2u¯−u¯∈IWML, Theorem 2 implies that(17)ψθ1θ2u¯+IWL=θ1θ2u¯−u¯=θ1θ2u¯−u¯+u¯−u¯=θ1θ2u¯−u¯+θ1u¯−u¯=θ2u¯−u¯+θ1u¯−u¯=ψθ1u¯+IWL+ψθ2u¯+IWL.(iv)
ψ is injective: Let θ1,θ2∈AutWML such that ψθ1=ψθ2. Then, θ1u¯−u¯=θ2u¯−u¯ for all u¯∈L, and so θ1=θ2.(v)
ψ is onto: Let β∈HomL/IWL,IWML. Define θ:L⟶L by θu¯=u¯+βu¯+IWL for every u¯∈L. It is clear that θu¯−u¯∈IWML and θ∈EndWML:(18)βu¯+IWL=θu¯−u¯=ψθu¯+IWL=ψθu¯+IWL.From this, we have β=ψθ. So ψ is onto.
Therefore,AutWML≅HomL/IWL,IWML.Theorem 4.
Let∅≠W⊆L such that IWML⊆IWL. Then,(19)AutWML≅HomL,IWML.Proof.
By Theorems1 and 3, the result is obtained clearly.



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*Source: 8723090-2020-04-10.xml*