Additive primitive length in relatively free algebras

TL;DR

This paper establishes bounds on the additive primitive length of elements in various relatively free algebras, providing explicit bounds depending on algebra type, number of generators, and field characteristics, with efficient constructive methods.

Contribution

It introduces new bounds for additive primitive length in polynomial and Lie algebras, extending previous results and providing effective algorithms for primitive element decompositions.

Findings

Bound depends on algebra type, generators, and field characteristic.

Primitive decompositions can be found efficiently in polynomial time.

Results generalize recent findings for free metabelian Lie algebras.

Abstract

The additive primitive length of an element $f$ of a relatively free algebra $F_d$ in a variety of algebras is equal to the minimal number $\ell$ such that $f$ can be presented as a sum of $\ell$ primitive elements. We give an upper bound for the additive primitive length of the elements in the $d$-generated polynomial algebra over a field of characteristic 0, $d>1$. The bound depends on $d$ and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free $d$-generated nilpotent-by-abelian Lie algebras is bounded by 5 for $d=3$ and by 6 for $d>3$. If the field has two elements only, then our bound are 6 for $d=3$ and 7 for $d>3$. This generalizes a recent result of Ela Ayd{\i}n for two-generated free metabelian Lie algebras. In all cases considered in the paper the presentation of the elements as sums of primitive can be found effectively in polynomial time.

Equations

$$\phi (x_j)={\beta }_j+\sum _{i=1}^d{\gamma }_{ij}{x}_i,j=1,\dots ,d,$$

$$\vartheta (x_j)={\gamma }_j{x}_j+v_j(x_{j+1},\dots ,x_d),j=1,\dots ,d,$$

$$s(\alpha )=x_1+\sum _{i=2}^d{\alpha }^{(n+1{)}^{i-2}}{x}_i.$$

$$u=\beta +\gamma x_1+\sum _{p=2}^n{\xi }_p{s}^p(\alpha )$$

$$\vartheta (x_1)=\beta +\gamma x_1+\sum _{p=2}^n{\xi }_p{x}_2^p,\vartheta (x_j)=x_j,j=2,\dots ,d,$$

$$\phi (x_1)=x_1,\phi (x_2)=s(\alpha )=x_1+\sum _{i=2}^d{\alpha }^{(n+1{)}^{i-2}}{x}_i,\phi (x_j)=x_j,j=3,\dots ,d.$$

$${\text{plength}}_{+}(f)\le \left(\genfrac{}{}{0ex}{}{n+d-1}{d-1}\right).$$

$$f=f_0+f_1+\cdots +f_n,$$

$$f_p=\sum _{|a|=p}{\mu }_a{x}_1^{a_1}\cdots x_d^{a_d},{\mu }_a\in K,|a|=a_1+\cdots +a_d,$$

$$f=\beta +\delta x_1+f_2+\cdots +f_n,\beta \in K,\delta =0,1.$$

$$\left(\begin{array}{ccc}{\alpha }_1^{p_1}& \dots & {\alpha }_N^{p_1}\\ ⋮& \ddots & ⋮\\ {\alpha }_1^{p_N}& \dots & {\alpha }_N^{p_N}\end{array}\right)$$

$$u_1=\beta +{\xi }_{11}{x}_1+\sum _{p=2}^n{\xi }_{1p}{s}^p({\alpha }_1),$$

$$u_k={\xi }_{k1}{x}_1+\sum _{p=2}^n{\xi }_{kp}{s}^p({\alpha }_k),k=2,\dots ,N,$$

$$f_p=\sum _{|a|=p}{\mu }_a{x}_1^{a_1}\cdots x_d^{a_d}=\sum _{k=1}^N{\xi }_{kp}{s}^p({\alpha }_k)$$

$$=\sum _{k=1}^N{\xi }_{kp}\sum _{|a|=p}\left(\genfrac{}{}{0ex}{}{a_1+\cdots +a_d}{a_1,\dots ,a_d}\right){\alpha }_k^{P_a}{x}_1^{a_1}\cdots x_d^{a_d},$$

$$P_a=a_1+a_2(n+1)+a_3(n+1{)}^2+\cdots +a_d(n+1{)}^{d-2}$$

$$\left(\genfrac{}{}{0ex}{}{a_1+\cdots +a_d}{a_1,\dots ,a_d}\right)\sum _{k=1}^N{\xi }_{kp}{\alpha }_k^{P_a}={\mu }_a,|a|=p,$$

$$[x_{i_1},x_{i_2},x_{i_3},\dots ,x_{i_n}]=[x_{i_1},x_{i_2}]\text{ad}(x_{i_3})\cdots \text{ad}(x_{i_n}),$$

$$\exp (\text{ad}v):x_j\to x_j+[x_j,v],j=1,\dots ,d,v\in F_d^{\prime }({\mathfrak{A}}^2).$$

$$u_1=\gamma x_1+v(x_2,\dots ,x_d),0\ne \gamma \in K,v(x_2,\dots ,x_d)\in F_d({\mathfrak{A}}^2);$$

$$u_2=\gamma x_j+[w,x_j],n\ge 2,0\ne \gamma \in K,w\in F_d^{\prime }({\mathfrak{A}}^2);$$

$$u_3=\sum _{j=1}^d{\alpha }_j{x}_j+\sum _{j=2}^d{\beta }_j[x_j,x_1],{\alpha }_j,{\beta }_j\in K,$$

$$y_1=\sum _{j=1}^d{\alpha }_j{x}_j={\alpha }_1{x}_1+\sum _{j=2}^d{\alpha }_j{x}_j,y_2=\sum _{j=2}^d{\beta }_j{x}_j,y_3=x_1$$

$${\text{plength}}_{+}(f)\{\begin{array}{l}\le 5,\text{ if }|K|>2,\\ \le 6,\text{ if }|K|=2.\end{array}$$

$${\text{plength}}_{+}(f)\{\begin{array}{l}\le 6,\text{ if }|K|>2,\\ \le 7,\text{ if }|K|=2.\end{array}$$

$$f=\delta x_1+\sum _{j=2}^d{\beta }_j[x_j,x_1]+\sum {\mu }_i[x_{i_1},x_1,x_{i_3},\dots ,x_{i_n}]+v(x_2,\dots ,x_d),$$

$$[w_1,x_1]=\sum {\mu }_i^{(1)}[x_{i_1},x_1,x_{i_3},\dots ,x_{i_{n-1}},x_1],$$

$$[w_2,x_2]=\sum {\mu }_i^{(2)}[x_{i_1},x_1,x_{i_3},\dots ,x_{i_{n-1}},x_2],$$

$$[w_3,x_3]=\sum {\mu }_i^{(3)}[x_{i_1},x_1,x_{i_3},\dots ,x_{i_{n-1}},x_3].$$

$$f=\delta x_1+({\beta }_2[x_2,x_1]+{\beta }_3[x_3,x_1])+[w_1,x_1]+[w_2,x_2]+[w_3,x_3]+v(x_2,x_3).$$

Full text

Introduction

Let $F_{d}({\mathfrak{V}})$ be the $d$-generated relatively free algebra in a variety of algebras $\mathfrak{V}$ over a field $K$. An element $u$ in $F_{d}({\mathfrak{V}})$ is primitive if it is an image of $x_{1}$ under an automorphism of $F_{d}({\mathfrak{V}})$, where $X_{d}=\{x_{1},\ldots,x_{d}\}$ is a free generating system of $F_{d}({\mathfrak{V}})$. The additive primitive length $\text{plength}_{+}(f)$ of an element $f$ of $F_{d}({\mathfrak{V}})$ is equal to the minimal $\ell$ such that $f$ can be presented as a sum of $\ell$ primitive elements. If such a presentation is impossible, we define $\text{plength}_{+}(f)=\infty$. The additive primitive width $\text{pwidth}_{+}(F_{d}({\mathfrak{V}}))$ of $F_{d}({\mathfrak{V}})$ is equal to the maximum of $\text{plength}_{+}(f)$, $f\in F_{d}({\mathfrak{V}})$. The notions of additive primitive length and width were introduced by Aydın in her recent paper [1] by analogy with the notions of primitive length and width for free groups defined and studied by Bardakov, Shpilrain, and Tolstykh [4]. Aydın [1] calculated the additive primitive length of the elements of the two-generated free metabelian (i.e., solvable of class 2) Lie algebra $F_{2}({\mathfrak{A}}^{2})$ over a field $K$ of characteristic 0. In particular she described the elements of infinite length in $F_{2}({\mathfrak{A}}^{2})$. Aydın also asked the problem whether the elements of $F_{d}({\mathfrak{A}}^{2})$, $d>2$, are of finite length and, if this is true, what is $\text{pwidth}_{+}(F_{d}({\mathfrak{A}}^{2}))$ and whether it depends on $d$.

The first topic of the present paper is the study of the additive primitive length of the elements of the polynomial algebra $K[X_{d}]$ over a field $K$ of characteristic 0. The case $d=1$ is trivial because the only primitive elements are of the form $\beta+\gamma x_{1}$, $\beta,\gamma\in K$, $\gamma\not=0$. Hence if $\deg(f)>1$, $f\in K[X_{1}]$, then $\text{plength}_{+}(f)=\infty$. Our main result in this direction is that if $d>1$ and $\deg(f)=n>1$, then $\displaystyle\text{plength}_{+}(f)\leq\binom{n+d-1}{d-1}$. In our presentation of $f$ as a sum of primitive elements we use tame primitive elements only, i.e., images of $x_{1}$ under tame automorphisms of $K[X_{d}]$. Unfortunately, our proof does not work when the base field is of positive characteristic. We show that for $K=\mathbb{Q}$ the presentation of $f$ can be found effectively in polynomial time. Hence the problem to find the presentation belongs to the class P of decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time. It is interesting to know whether $\text{plength}_{+}(f)$ is bounded when $f\in K[X_{d}]$. There are some reasons to believe that, at least for $K[X_{2}]$, the additive primitive length of the polynomials is not bounded, but we cannot prove this.

Then we consider the additive primitive length for the elements of the free metabelian Lie algebra $F_{d}({\mathfrak{A}}^{2})$ over an arbitrary field $K$. In the case $d=2$ Aydın [1] uses the description of the automorphisms of $F_{2}({\mathfrak{A}}^{2})$ given by Shmelkin [13] when $\text{char}(K)=0$. But the same description of $\text{Aut}(F_{2}({\mathfrak{A}}^{2}))$ is true for any field $K$, see [7]. We consider the case $d>2$ and $f\in F_{d}({\mathfrak{A}}^{2})$, $\deg(f)>1$. For $d=3$ we show that $\text{plength}_{+}(f)\leq 5$ if $K$ has more than two elements and $\text{plength}_{+}(f)\leq 6$ if $|K|=2$. If $d>3$, then $\text{plength}_{+}(f)\leq 6$ when $|K|>2$ and $\text{plength}_{+}(f)\leq 7$ when $|K|=2$. The primitive elements which appear as summands in our presentation of $f$ are images of $x_{1}$ under tame and inner automorphisms. Finally, we consider the relatively free algebras $F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$ of the variety ${\mathfrak{N}}_{c}{\mathfrak{A}}$ of (nilpotent of class $c$)-by-abelian Lie algebras, $c>1$. As in the case of groups, see Bryant and Gupta [6], an endomorphism of $F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$ is an automorphism if and only if it induces an automorphism on the free metabelian Lie algebra $F_{d}({\mathfrak{A}}^{2})$, see [7]. This easily implies that the additive primitive length of $f\in F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$ is the same as of the image of $f$ in $F_{d}({\mathfrak{A}}^{2})$. Again, the presentation of the elements of $F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$ as a sum of primitive can be found effectively in polynomial time.

For a background on automorphisms of polynomial algebras, free groups, and free algebras we refer to the books by van den Essen [9] and by Mikhalev, Shpilrain, and Yu [12], and on varieties of Lie algebras to the book by Bahturin [2].

1. Polynomial algebras

In this section we fix a field $K$ of characteristic 0 and work in the polynomial algebra $K[X_{d}]$, $d>1$. Recall that the group of tame automorphisms of $K[X_{d}]$ is generated by the affine automorphisms $\varphi$ defined by

[TABLE]

where $\beta_{j},\gamma_{ij}\in K$ and the matrix $(\gamma_{ij})$ is invertible ($\varphi$ is called linear if all $\beta_{j}$ are equal to 0), and the triangular automorphisms $\vartheta$ defined by

[TABLE]

where $0\not=\gamma_{j}\in K$ and $v_{j}(x_{j+1},\ldots,x_{d})\in K[x_{j+1},\ldots,x_{d}]$.

Lemma 1.1**.**

Let $\alpha,\beta,\gamma,\xi_{p}\in K$, $p=2,\ldots,n$, $\alpha,\gamma\not=0$, and let

[TABLE]

Then the polynomial

[TABLE]

is primitive.

Proof.

We define the triangular automorphism $\vartheta$ and the affine automorphism $\varphi$ by

[TABLE]

[TABLE]

Since $u=\varphi\circ\vartheta(x_{1})$, it is primitive. ∎

The following theorem is the first main result in our paper.

Theorem 1.2**.**

Let the base field $K$ be of characteristic [math] and let $d>1$. If $f\in K[X_{d}]$ and $\deg(f)=n>1$, then

[TABLE]

Proof.

We write $f$ in the form

[TABLE]

where

[TABLE]

is the homogeneous component of $f$ of degree $p$. If $f_{1}\not=0$ there is a linear automorphism $\psi$ which sends $f_{1}$ to $x_{1}$. Since the automorphisms do not change the additive primitive length of the elements, we can work with $\psi(f)$ instead with $f$ and may assume that $f=\beta+x_{1}+f_{2}+\cdots+f_{n}$, $\beta\in K$. If $f_{1}=0$, then $f=\beta+f_{2}+\cdots+f_{n}$. Hence in both cases we can work with $f$ presented in the form

[TABLE]

Since the base field is of characteristic 0 we can choose $\displaystyle N=\binom{n+d-1}{d-1}$ pairwise different nonzero elements $\alpha_{1},\ldots,\alpha_{N}\in K$ such that for a fixed $k$ all powers $1,\alpha_{k},\alpha_{k}^{2},\ldots,\alpha_{k}^{n(n+1)^{d-2}}$ are pairwise different. Additionally, we require that the matrices

[TABLE]

are invertible for all nonnegative integers $p_{1}<\cdots<p_{N}\leq n(n+1)^{d-2}$. Such matrices appear in the determinantal presentation of Schur functions, see e.g. [10]. We want to present $f$ as a sum of the primitive elements $u_{1},\ldots,u_{N}$ of the form (2), where

[TABLE]

[TABLE]

and $s(\alpha_{k})$ is defined in (1) for $\alpha=\alpha_{k}$. Here the coefficients $\xi_{kp}$ are unknown elements of $K$. We can choose all $\xi_{k1}\in K$ in such a way that they are different from 0 and $\xi_{11}+\cdots+\xi_{N1}=\delta$, i.e., the affine component $\beta+\delta x_{1}$ of $f$ is equal to the affine component of $u_{1}+\cdots+u_{N}$. By Lemma 1.1 the polynomials $u_{1},\ldots,u_{N}$ are primitive. Now, let $1<p\leq n$. Then for the homogeneous component $f_{p}$ of degree $p$ of $f$ we obtain

[TABLE]

[TABLE]

where

[TABLE]

and $\displaystyle\binom{a_{1}+\cdots+a_{d}}{a_{1},\ldots,a_{d}}=\frac{(a_{1}+\cdots+a_{d})!}{a_{1}!\cdots a_{d}!}$ is the multinomial coefficient. Comparing the coefficients of the $\displaystyle N_{p}=\binom{p+d-1}{d-1}$ monomials $x_{1}^{a_{1}}\cdots x_{d}^{a_{d}}$, we obtain the linear system

[TABLE]

with $N_{p}$ equations and $\displaystyle N=\binom{n+d-1}{d-1}$ unknowns. Since $a_{j}\leq p=\deg(f_{p})\leq\deg(f)=n$, $j=1,\ldots,d$, the integers $P_{a}$ from (4) have presentations in the numerical system with base $n+1$ and are pairwise different. Hence the rows of the matrix of the system (5) are rows of a matrix of the form (3). Since the matrix (3) is invertible, its rows are linearly independent and the system (5) has a solution. This implies that $f$ can be presented as a sum of $N$ primitive polynomials and $\displaystyle\text{plength}_{+}(f)\leq\binom{n+d-1}{d-1}$. ∎

Corollary 1.3**.**

If $f\in{\mathbb{Q}}[X_{d}]$, $\deg(f)>1$, we can find effectively a presentation of $f$ as a sum of primitive polynomials in polynomial time.

Proof.

Following the proof of Theorem 1.2, we can find the presentation of $f$, $\deg(f)=n>1$, as a sum of primitive polynomials in three steps:

(1) We find a linear automorphism $\psi$ of ${\mathbb{Q}}[X_{d}]$ which send $f_{1}$ to $\delta x_{1}$;

(2) We compute $\psi(f)$;

(3) We solve $n-1$ linear systems with $\displaystyle\leq N=\binom{n+d-1}{d-1}$ equations and $N$ unknowns.

Obviously steps (1) and (2) can be performed in polynomial time. In the third step we shall solve the systems applying the Gaussian elimination. It is well known that for a system with $\leq N$ equations and $N$ unknowns we need ${\mathcal{O}}(N^{3})$ arithmetic operations. The number of arithmetic operations measures the computational complexity when the time for each arithmetic operation is approximately the same. This happens when the coefficients of the system are represented by floating-point numbers or when we work in a finite field. For our purposes we need to work with rational numbers represented exactly. In this case the intermediate entries may become exponentially large and the bit complexity is exponential. But there is a variant of Gaussian elimination due to Bareiss [5] which avoids the exponential growth of the intermediate entries. It has the same arithmetic complexity ${\mathcal{O}}(N^{3})$, but its bit complexity is ${\mathcal{O}}(N^{5})$. ∎

Remark 1.4**.**

(1) In Theorem 1.2 we consider the case $\deg(f)>1$ because the case $\deg(f)\leq 1$ is trivial. If $f\in K[X_{d}]$ and $\deg(f)=1$, then $f$ is primitive and $\text{plength}_{+}(f)=1$; if $\deg(f)=0$, then $f=\beta\in K$, $f=(\beta+x_{1})+(-x_{1})$ and $\text{plength}_{+}(f)=2$.

(2) The proof of Theorem 1.2 does not hold for fields of positive characteristic $p$ because some of the multinomial coefficients become 0 modulo $p$.

2. Metabelian Lie algebras

In this section we consider the free metabelian Lie algebra $F_{d}({\mathfrak{A}}^{2})$, $d>2$, over an arbitrary field $K$. It is well known, see e.g., [2], that as a vector space the commutator ideal $F_{d}^{\prime}({\mathfrak{A}}^{2})$ has a basis

[TABLE]

where $i_{1}>i_{2}\leq i_{3}\leq\cdots\leq i_{n}$, $[v_{1},v_{2},\ldots,v_{n-1},v_{n}]=[[v_{1},v_{2},\ldots,v_{n-1}],v_{n}]$, and $v\text{ad}(w)=[v,w]$. For the Lie algebra $F_{d}({\mathfrak{A}}^{2})$ the group of tame automorphisms is generated by the linear and the triangular automorphisms which are defined in the same way as in the case of polynomial algebras. Additionally, we shall consider the inner automorphisms of $F_{d}({\mathfrak{A}}^{2})$ defined by

[TABLE]

(By the theorem of Bahturin and Nabiyev [3] the inner automorphisms of $F_{d}({\mathfrak{A}}^{2})$ are wild, i.e., not tame.)

Lemma 2.1**.**

The following elements of $F_{d}({\mathfrak{A}}^{2})$ are primitive:

[TABLE]

[TABLE]

[TABLE]

if the elements $\displaystyle\sum_{j=2}^{d}\alpha_{j}x_{j},\sum_{j=2}^{d}\beta_{j}x_{j}$, and $x_{1}$ are linearly independent.

Proof.

The elements $u_{1}$ from (6) and $u_{2}$ from (7) are primitive: $u_{1}$ is the image of $x_{1}$ under a triangular automorphism and $u_{2}$ is an image of $x_{j}$ under the automorphism $\displaystyle\gamma\exp\left(-\frac{1}{\gamma}\text{ad}w\right)$. Finally, since the elements

[TABLE]

are linearly independent we can extend them with elements $y_{4},\ldots,y_{d}$ to a basis $Y_{d}=\{y_{1},\ldots,y_{d}\}$ of the vector space spanned by $X_{d}$. Then $u_{3}$ from (8) has the form $u_{3}=y_{1}+[y_{2},y_{3}]$. In this way $u_{3}$ is an image of $y_{1}$ under a triangular automorphism with respect to the free generating system $Y_{d}$ of $F_{d}({\mathfrak{A}}^{2})$ and hence is primitive. ∎

Theorem 2.2**.**

Let $f\in F_{d}({\mathfrak{A}}^{2})$. Then for $d=3$

[TABLE]

When $d>3$, then

[TABLE]

Proof.

As in the polynomial case in Theorem 1.2 we can apply a linear automorphism to bring $f$ in the form

[TABLE]

where $\delta=0,1$, $\beta_{j},\mu_{i}\in K$, $i_{1}>1\leq i_{3}\leq\cdots\leq i_{n}$, $n\geq 3$, $v(x_{2},\ldots,x_{d})\in F_{d}^{\prime}({\mathfrak{A}}^{2})$.

First, let $d=3$. Then in (9) we present $\sum\mu_{i}[x_{i_{1}},x_{1},x_{i_{3}},\ldots,x_{i_{n}}]$ as a sum of three parts

[TABLE]

[TABLE]

[TABLE]

In this way $f$ has the form

[TABLE]

We want to present it as a sum of the following five primitive elements

[TABLE]

[TABLE]

[TABLE]

with unknown coefficients $\xi,\xi_{1},\xi_{2},\xi_{3},\zeta_{1},\zeta_{2},\zeta_{3}\in K$. We require $\xi,\xi_{1},\xi_{2},\xi_{3},\zeta_{2},\zeta_{3}\not=0$. (If $v(x_{2},x_{3})$ or some $w_{i}$ is equal to zero, then $u_{1}$ or $u_{2}^{(i)}$ is still a primitive element.) If $\beta_{2}x_{2}+\beta_{3}x_{3}\not=0$, then we need the linear independence of $\zeta_{2}x_{2}+\zeta_{3}x_{3}$ and $\beta_{2}x_{2}+\beta_{3}x_{3}$. If $\beta_{2}x_{2}+\beta_{3}x_{3}=0$, then $\zeta_{1}x_{1}+\zeta_{2}x_{2}+\zeta_{3}x_{3}$ may be any. In the latter case, if $\zeta_{1}=\zeta_{2}=\zeta_{3}=0$, then simply $u_{3}$ does not participate in the presentation of $f$. The components of $f$ and $u_{1}+u_{2}^{(1)}+u_{2}^{(2)}+u_{2}^{(3)}+u_{3}$ in $F_{3}^{\prime}({\mathfrak{A}}^{2})$ coincide. Comparing the linear components we obtain the system

[TABLE]

This system always has a solution of nonzero $\xi,\xi_{1},\xi_{2},\xi_{3},\zeta_{2},\zeta_{3}$ and some $\zeta_{1}$. When $\beta_{2}x_{2}+\beta_{3}x_{3}\not=0$ and the field $K$ has more than two elements we can vary $\zeta_{2},\zeta_{3}$ to guarantee the linear independence of $\zeta_{2}x_{2}+\zeta_{3}x_{3}$ and $\beta_{2}x_{2}+\beta_{3}x_{3}$. Hence $\text{plength}_{+}(f)\leq 5$. If $|K|=2$ we can add one more primitive summand: When the only solution for $\beta_{2}$ and $\beta_{3}$ of the system (10) implies that $\zeta_{2}x_{2}+\zeta_{3}x_{3}$ and $\beta_{2}x_{2}+\beta_{3}x_{3}$ coincide, we split $u_{3}$ in two parts $u_{3}=u_{3}^{\prime}+u_{3}^{\prime\prime}$, where

[TABLE]

$\zeta_{2}^{\prime}+\zeta_{2}^{\prime\prime}=\zeta_{2}$, $\zeta_{3}^{\prime}+\zeta_{3}^{\prime\prime}=\zeta_{3}$, to make $\zeta_{2}^{\prime}x_{2}+\zeta_{3}^{\prime}x_{3}$ and $\beta_{2}x_{2}+\beta_{3}x_{3}$ different which gives that $u_{3}^{\prime}$ is primitive.

Now, let $d>3$. In (9) we present $\sum\mu_{i}[x_{i_{1}},x_{1},x_{i_{3}},\ldots,x_{i_{n}}]$ as a sum of four parts

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We want to present $f$ as a sum of six primitive elements

[TABLE]

[TABLE]

[TABLE]

where $\xi,\eta_{d-1},\eta_{d},\xi_{d-1},\xi_{d}\not=0$ and if some $\beta_{2},\ldots,\beta_{d}$ is different from 0, then $\zeta_{d-1}x_{d-1}+\zeta_{d}x_{d}$ and $\displaystyle\sum_{j=2}^{d}\beta_{j}x_{j}$ are linearly independent. (If $\beta_{2}=\cdots=\beta_{d}=0$, then it is also allowed $u_{3}=0$.) Since $i_{1}>1\leq i_{2}\leq\cdots\leq i_{n}$, the generator $x_{d-1}$ does not participate in the expression of $w_{3}$. Similarly, $x_{d}$ does not participate in $w_{4}$. By Lemma 2.1 all summands $u_{1}^{(1)},u_{1}^{(2)},u_{1}^{(3)},u_{2}^{(1)},u_{2}^{(2)},u_{3}$ are primitive. We still have to arrange the equality of the linear components. This means to solve the system

[TABLE]

obtained comparing the coefficients of $x_{1},x_{d-1},x_{d}$. As in the case of polynomial algebras when $|K|>2$ we can find a solution of the system (11) which satisfies all the requirements for being different from 0 and for linear independence. If $|K|=2$ we have to add one more linear primitive element to arrange the required restrictions. ∎

Remark 2.3**.**

As in Corollary 1.3, when $K=\mathbb{Q}$ in Theorem 2.2 we can find the presentation of $f\in F_{d}({\mathfrak{A}}^{2})$ as a sum of primitive elements in polynomial time: Repeating the steps of the proof of Corollary 1.3 we have to perform steps (1) and (2) only and then to solve the system (10) when $d=3$ or (11) when $d>3$.

Corollary 2.4**.**

Let $F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$ be the relatively free algebra of rank $d>1$ of the variety ${\mathfrak{N}}_{c}{\mathfrak{A}}$ of (nilpotent of class $c$)-by-abelian Lie algebras, $c>1$. Then the additive primitive length of $f\in F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$ is the same as of the image of $f$ in $F_{d}({\mathfrak{A}}^{2})$.

Proof.

There is a canonical epimorphism $\pi:F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})\to F_{d}({\mathfrak{A}}^{2})$. Since an endomorphism of $F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$ is an automorphism if and only if it induces an automorphism on $F_{d}({\mathfrak{A}}^{2})$, see [7], we obtain that $f\in F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$ is primitive if and only if its image $\pi(f)$ is primitive in $F_{d}({\mathfrak{A}}^{2})$. Hence, if $\text{plength}_{+}(\pi(f))=\infty$ in $F_{2}({\mathfrak{A}}^{2})$, then $\text{plength}_{+}(f)=\infty$ in $F_{2}({\mathfrak{N}}_{c}{\mathfrak{A}})$. Now, let $\text{plength}_{+}(\pi(f))=\ell$ in $F_{d}({\mathfrak{A}}^{2})$, $d>1$, and $\pi(f)=\pi(u_{1})+\cdots+\pi(u_{\ell})$, where $u_{1},\ldots,u_{\ell}\in F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$ and $\pi(u_{1}),\ldots,\pi(u_{\ell})$ are primitive in $F_{d}({\mathfrak{A}}^{2})$. Hence $u_{1},\ldots,u_{\ell}$ are primitive in $F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$. Clearly,

[TABLE]

and hence $u_{1}^{\prime}=u_{1}+w$ is also primitive in $F_{d}({\mathfrak{N}}_{c}{\mathfrak{A}})$. In this way $f=u_{1}^{\prime}+u_{2}+\cdots+u_{\ell}$ and $\text{plength}_{+}(f)=\ell=\text{plength}_{+}(\pi(f))$. ∎

3. Open problems

In this section we shall collect some open problems concerning the additive primitive length for different relatively free algebras.

Problem 3.1**.**

Is there an upper bound for $\text{\rm plength}_{+}(f)$, $f\in K[X_{d}]$, $d\geq 2$? In other words, does $\text{\rm pwidth}_{+}(K[X_{d}])$ exist? What happens when $\text{\rm char}(K)>0$?

By a result of Shpilrain and Yu [14] the primitive elements of the free associative algebra $K\langle X_{2}\rangle$ are palindromic, i.e., they coincide with their images under the antiautomorphism of $K\langle X_{2}\rangle$ to the opposite algebra $K\langle X_{2}\rangle^{\text{opp}}$. In other words they are the same if we read their monomials backwards, from right to the left. Hence elements of $K\langle X_{2}\rangle$ which are not palindromic cannot be presented as sums of primitive.

Problem 3.2**.**

Describe the elements of $K\langle X_{2}\rangle$ which can be presented as sums of primitive.

Since the automorphism groups of $K[X_{2}]$ and $K\langle X_{2}\rangle$ are isomorphic, one can lift in a unique way every automorphism of $K[X_{2}]$ to an automorphism of $K\langle X_{2}\rangle$, see the comments in [14]. Hence every decomposition of $f\in K[X_{2}]$ into a sum of primitive polynomials can be lifted uniquely to a similar sum in $K\langle X_{2}\rangle$. The main difficulty in the above problem is that the presentation of the elements of $K[X_{2}]$ as a sum of primitive is not unique.

Problem 3.3**.**

What happens with the presentation into a sum of primitive in $K\langle X_{d}\rangle$, $d>2$?

We may ask the same question for the free Lie algebra $L_{d}$. Since all automorphisms of $L_{2}$ are linear, the interesting case is $d>2$.

Let ${\mathfrak{T}}_{c}={\mathfrak{N}}_{c}\mathfrak{A}$ be the variety of associative algebras defined by the polynomial identity $[x_{1},x_{2}]\cdots[x_{2c-1},x_{2c}]=0$. By a theorem of Maltsev [11] in characteristic 0 and of Siderov [15] and other authors for infinite fields of positive characteristic this variety is generated by the algebra of $c\times c$ upper triangular matrices. As in the case of Lie algebras, see [7], the problem for the presentation of the elements of $F_{d}({\mathfrak{T}}_{c})$, $c>2$, is reduced to the same problem for $F_{d}({\mathfrak{T}}_{2})$.

Problem 3.4**.**

Describe the elements of the relatively free associative algebra $F_{d}({\mathfrak{T}}_{2})$ which can be presented as sums of primitive.

Applying the canonical epimorphism $\pi:F_{d}({\mathfrak{T}}_{2})\to K[X_{d}]$, when $\text{char}(K)=0$, we can present $\pi(f)$, $f\in F_{d}({\mathfrak{T}}_{2})$, as a sum $\pi(f)=\pi(u_{1})+\cdots+\pi(u_{\ell})$ of tame primitive elements and lift the presentation to $F_{d}({\mathfrak{T}}_{2})$. Hence the problem is reduced to the problem to find a presentation as a sum of primitive elements of the element $f-(u_{1}+\cdots+u_{\ell})$ in the commutator ideal of $F_{d}({\mathfrak{T}}_{2})$. This ideal has a basis

[TABLE]

see e.g., [8]. When $n\geq 3$ we can handle the elements (12) as in the case of metabelian Lie algebras and the main difficulty is to find a presentation for the elements from (12) when $n=2$.

Problem 3.5**.**

One can ask similar questions for the additive primitive length for other relatively free algebras $F_{d}({\mathfrak{V}})$, e.g., for the varieties of associative and Lie algebras generated by the $2\times 2$ matrix algebra, or for varieties of Jordan algebras.