Ideals and primitive elements of some relatively free Lie algebras

Let F be a free Lie algebra of finite rank over a field K. We prove that if an ideal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\langle \tilde{v}\right\rangle $$\end{document}v~ of the algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F/\gamma _{m+1}\left( F^{\prime }\right) $$\end{document}F/γm+1F′ contains a primitive element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{u}$$\end{document}u~ then the element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{v}$$\end{document}v~ is primitive. We also show that, in the Lie algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F/\gamma _{3}\left( F\right) ^{\prime }$$\end{document}F/γ3F′ there exists an element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar{v}$$\end{document}v¯ such that the ideal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\langle \bar{v}\right\rangle $$\end{document}v¯ contains a primitive element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{u}$$\end{document}u¯ but, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{u}$$\end{document}u¯ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{v}$$\end{document}v¯ are not conjugate by means of an inner automorphism.

for free metabelian groups. Does a similar result, as in L, holds for the Lie algebras F /γ m+1 F ′ and F /γ 3 (F ) ′ ? In the group case this question was answered by Timoshenko (1997). In the present paper we answer this question. We obtain an affirmative answer for the Lie algebra F /γ m+1 F ′ . In contrast to the case of free metabelian Lie algebras and free Lie algebras of the form F /γ m+1 F ′ , for the Lie algebra F /γ 3 (F ) ′ we prove that the question has a negative answer. Our main results are similar to the result of Timoshenko (1997) in the case of groups but there are some essential differences.

Preliminaries
Let F be the free Lie algebra generated by a set X = {x 1 , . . . , x n } over a field K of characterisitic zero, U (F ) be the universal enveloping algebra of F and its augmentation ideal, that is, the kernel of the natural homomorphism σ : U (F ) −→ K defined by σ (x i ) = 0, 1 ≤ i ≤ n . For a given subalgebra R of F we denote by R the left ideal of U (F ) generated by the subalgebra R. In the case where R is an ideal of F, R becomes a two-sided ideal of U (F ). In fact R is the kernel of the natural homomorphism U (F ) −→ U (F /R). For any element u of F we denote by u the ideal of F generated by the element u. Fox (1953) gave a detailed account of the differential calculus in a free group ring. We introduce here free derivations ∂ ∂x i : It is an obvious consequence of the definitions that ∂ ∂x i (1) = 0 . The ideal is a free left U (F )-module with a free basis X and the mappings ∂ ∂x i are projections to the corresponding free cyclic direct summands. Thus any element f ∈ can be uniquely written in the form For any elements g 1 , . . . , g n of U (F ) we can always find an element f of U (F ) such that Let ∂f be the column vector ∂f ∂x 1 , . . . , ∂f ∂x n T , where T indicates transpose. For any Lie algebra G, the lower central series Let R be an ideal of F. If u is an element of F, then we denote the images of u under the natural homomorphisms as follows: by u in F / R, by u in F /R ′ and by u in F /γ m+1 (R), where m ≥ 1.
In (Umirbaev 1993), Umirbaev has defined the right derivatives in the algebras F /R ′ and F /γ m+1 (R). We give a summary here referring to (Umirbaev 1993). Let be the natural componentwise homomorphism, i.e., where f 1 , . . . , f n T is the transpose of the vector f 1 , . . . , f n .

Consider the composition mapping
This mapping induces the mappings Since the kernel of the mapping ∂ is R ′ /γ m+1 (R) (see Umirbaev 1993 for details) then it induces the mapping ∂ : For any element f of F the components ∂ f of the vectors are called the partial derivatives of f , f and f respectively. Here we use left derivatives instead of right derivatives.
Hence the linear mapping is well defined and it is an inner automorphism of We need the following technical lemmas. The first lemma is an immediate consequence of the definitions.

Lemma 1 Let J be an arbitrary ideal of
The next lemma can be found in Yunus (1984).

Main results
Let F be the free Lie algebra generated by a set X = {x 1 , . . . , x n }, n ≥ 2, over a field K of characteristic zero and let R be a non-trivial verbal ideal of F.
For an element f of F the vector ∂f ∂x 1 , . . . , ∂f ∂x n is called unimodular, if there exist a 1 , . . . , a n ∈ U (F ) such that Umirbaev (1993) has proved a criterion of primitiveness for a system of elements in a finitely generated free Lie algebra of the form F /γ m+1 (R), where m ≥ 1 and R = F ′ . Umirbaev's criterion for the primitivity of an element of the algebra F /γ m+1 (R) is stated below.
We are going to consider the case R = F ′ .

Proposition 4 An element f of the free metabelian Lie algebra F /F ′′ is primitive if and only if the image f is primitive in the free nilpotent-by-abelian Lie algebra
Proof Suppose that the element f of F /F ′′ is primitive. If we put m = 1 in Proposition 3 we have that the vector ∂ f ∂x 1 , . . . , ∂ f ∂x n is unimodular in U F /F ′ , that is, there exist a 1 ,…,a n ∈ U F /F ′ such that n i=1 a i ∂ f ∂x i = 1. Let H = F /γ m+1 F ′ /F ′′ /γ m+1 F ′ . We calculate the derivative ∂ f ∂x i by using the natural homomorphism θ : F /γ m+1 F ′ → H , the isomorphism ϕ : H → F /F ′′ and the chain rule for derivatives: . Hence by Proposition 3, f is primitive in a 1 ∂f ∂x 1 + · · · + a n ∂f ∂x n = 1.
Now suppose that f is primitive element of the algebra F /γ m+1 F ′ . By definition it can be extended to a free generating set Y = f = f 1 , . . . , f n of F /γ m+1 F ′ . Clearly Y is linearly independent modulo F /γ m+1 F ′ ′ . Therefore the image θ (Y ) of Y in the algebra H is linearly independent modulo H ′ . As a simple application of theorem 4.2.4.9 of Bahturin (1987) we see that θ (Y ) freely generates the algebra H. Hence the image of θ (Y ) under the isomorphism ϕ : H → F /F ′′ generates the algebra F /F ′′ . That is, the algebra F /F ′′ is freely generated by the set ϕ(θ (Y )) = f = f 1 , . . . , f n . Thus, f is a primitive element of the algebra F /F ′′ .
As a consequence of the result of Chirkov and Shevelin (2001), we obtain the following proposition. Although its proof is given in Ersalan and Esmerligil (2014), our proof is more explicit. The idea of the proof is similar to the idea of the proof of Proposition 2 of the paper by Timoshenko (1997) for groups.

Proposition 5 Let u be a primitive element of the algebra
Assume that u is primitive and that it is contained in the ideal v of F /γ m+1 F ′ . By Proposition 4, u is primitive. In the view of Proposition 4, it sufficies to the prove that the element v of F /F ′′ is primitive.
Since we have that is, From the result of Chirkov and Shevelin (2001), we obtain that the elements u and v are conjugate by means of an inner automorphism. Therefore v is primitive. Hence the result follows.
The mapping : F → F /R can be extended to the mapping : U (F ) → U (F /R) for which we preserve the same notation.
The following lemma will play a crucial role in proving our main result.
Lemma 6 Let R be a verbal ideal of F , r ∈ R and let v ∈ F . Then r + R ′ ∈ �v� + R ′ if and only if there exist an element α ∈ U (F /R) and an element β i ∈ � v , such that ∂r ∂x i = α ∂v ∂x i + β i , where i = 1, . . . , n and v is the ideal generated by the element v in the algebra U (F /R).
Proof Let r be an element of the ideal R, v ∈ F and r ∈ �v�. Then r ∈ �v� modR ′ , where v is the ideal of F generated by v. Any element of the ideal v can be written as linear combinations of commutators of F depending on the element v. Applying the Jacobi identitiy and the anticommutativity, these commutators can be rewritten as linear combinations of commutators of the form If r ≡ v modR ′ then clearly ∂r ∂x i = ∂v ∂x i , i = 1, . . . n. Now assume that the element r is written as a linear combination of elements of the form (2). Without loss of generality we may assume that By straightforward calculations we see that the form of the derivatives ∂r ∂x i are Hence the element r − αv of F can be written as r − αv = h + z, where h ∈ v , z ∈ R . By Lemma 2 we get h ∈ �v� ′ and z ∈ R ′ . Hence r + R ′ = αv + h + R ′ . This completes the proof.
In contrast to the case of free metabelian Lie algebras we can show that there exists an element v of the algebra F /γ 3 (F ) ′ such that the ideal v of F /γ 3 (F ) ′ contains a primitive element u, but u and v are not conjugate by means of an inner automorphism.
Theorem 7 There is an element v in the algebra F /γ 3 (F ) ′ such that the ideal v of F /γ 3 (F ) ′ contains the element x 1 , but the elements v and x 1 are not conjugate modulo γ 3 (F ) ′ by means of an inner automorphism.
Proof We consider the element which is an analogue of the element given in Fox (1953) We have Now consider the images ∂w ∂x i under the homomorphism Then Clearly ∂w ∂x i ∈ � x 1 = � v . In the above equalities if we set α = 0 and ∂x i , i = 1, 2, then we see that By Lemma 6 w + γ 3 (F ) ′ ∈ �v� +γ 3 (F ) ′ . Therefore we have Now we are going to verify that the element w can not be written in the form [x 1 , u] in the algebra F /γ 3 (F ) ′ .
Assume that the rank of F equal to 2, u ∈ γ 3 (F ) and Let us calculate the derivative ∂ ∂x 1 of both sides of (3). We have Taking the image under the homomorphism : U (F ) −→ U (F /γ 3 (F )) we get It is well known that the set x 1 , x 2 , x 1 , x 2 is a basis of F /γ 3 (F ). Therefore by Poincare-Birkhoff-Witt's theorem the algebra U (F /γ 3 (F )) is a free K-module generated 1 and the all ordered monomials of the form Thus every element of U (F /γ 3 (F )) can be uniquely written as : U (F ) −→ U (F /γ 3 (F )), i = 1, 2.