The Nowicki Conjecture for free metabelian Lie algebras

TL;DR

This paper extends the Nowicki conjecture to free metabelian Lie algebras, providing explicit generators for the module of constants under Weitzenb"ock derivations, revealing new algebraic structures and properties.

Contribution

It introduces an analogue of the Nowicki conjecture for free metabelian Lie algebras and explicitly constructs generators for the module of constants.

Findings

The module of constants in the Lie algebra setting is finitely generated.

Explicit generators for the module of constants are provided.

The algebra of constants in the Lie algebra case differs from the polynomial case.

Abstract

Let $K[X_d]=K[x_1,\ldots,x_d]$ be the polynomial algebra in $d$ variables over a field $K$ of characteristic 0. The classical theorem of Weitzenb\"ock from 1932 states that for linear locally nilpotent derivations $\delta$ (known as Weitzenb\"ock derivations) the algebra of constants $K[X_{d}]^{\delta}$ is finitely generated. When the Weitzenb\"ock derivation $\delta$ acts on the polynomial algebra $K[X_d,Y_d]$ in $2d$ variables by $\delta(y_i)=x_i$, $\delta(x_i)=0$, $i=1,\ldots,d$, Nowicki conjectured that $K[X_d,Y_d]^{\delta}$ is generated by $X_d$ and $x_iy_j-y_ix_j$ for all $1\leq i<j\leq d$. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenb\"ock derivations of the free $d$-generated metabelian Lie algebra $F_d$, with few trivial exceptions, the algebra $F_d^{\delta}$ is not finitely generated. However, the vector subspace $(F_d')^{\delta}$ of the commutator ideal $F_d'$ of $F_d$ is finitely generated as a $K[X_d]^{\delta}$-module. In this paper we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the $K[X_d,Y_d]^{\delta}$-module $(F_{2d}')^{\delta}$.

Equations

$$\delta (uv)=\delta (u)v+u\delta (v)\text{ for all }u,v\in R.$$

$$K[X_d{]}^{\delta }=\ker \delta =\{u\in K[X_d]\mid \delta (u)=0\}$$

$$\exp (\alpha \delta )=1+\frac{\alpha \delta }{1!}+\frac{{\alpha }^2{\delta }^2}{2!}+\cdots$$

$$U{T}_2(K)=\left\{\left(\begin{array}{cc}1& \alpha \\ 0& 1\end{array}\right)\mid \alpha \in K\right\}.$$

$$\alpha :x_i\to x_i,y_i\to y_i+\alpha x_i,i=1,\dots ,d,\alpha \in K.$$

$$u_{pq}=\left|\begin{array}{cc}{x}_p& y_p\\ x_q& y_q\end{array}\right|,1\le p<q\le d.$$

$$F_d=F_d({\mathfrak{A}}^2)=L_d/L_d^{\prime \prime }$$

$$[z_1,z_2,z_3]=[[z_1,z_2],z_3]=[z_1,z_2]\text{ad}{z}_3$$

$$[z_{j_1},z_{j_2},z_{j_{\sigma (3)}},\dots ,z_{j_{\sigma (k)}}]=[z_{j_1},z_{j_2},z_{j_3},\dots ,z_{j_k}],$$

$$uf(x_1,\dots ,x_d,y_1,\dots ,y_d)=uf(\text{ad}{x}_1,\dots ,\text{ad}{x}_d,\text{ad}{y}_1,\dots ,\text{ad}{y}_d),$$

$$A_d\cup B_d=\{a_1,\dots ,a_d\}\cup \{b_1,\dots ,b_d\}\text{ and }{P}_d\cup Q_d=\{p_1,\dots ,p_d\}\cup \{q_1,\dots ,q_d\},$$

$$\sum _{i=1}^d{a}_i{f}_i(X_d,Y_d)+\sum _{i=1}^d{b}_i{g}_i(X_d,Y_d)+\sum _{i=1}^d{\alpha }_i{p}_i+\sum _{i=1}^d{\beta }_i{q}_i,$$

$$[C_{2d},C_{2d}]=[{\Gamma }_{2d},{\Gamma }_{2d}]=0,$$

$$[a_i{f}_i(X_d,Y_d),p_j]=a_i{f}_i(X_d,Y_d)x_j,[b_i{f}_i(X_d,Y_d),p_j]=b_i{f}_i(X_d,Y_d)x_j,$$

$$[a_i{f}_i(X_d,Y_d),q_j]=a_i{f}_i(X_d,Y_d)y_j,[b_i{f}_i(X_d,Y_d),q_j]=b_i{f}_i(X_d,Y_d)y_j,$$

$$\{x_1,\dots ,x_n,y_1,\dots ,y_n\}\to W_{2d}$$

$$\epsilon (x_i)=a_i+p_i,\epsilon (y_i)=b_i+q_i,i=1,\dots ,d,$$

$$C_{2d}=a_{1}K[X_d,Y_d]\oplus \cdots \oplus a_{d}K[X_d,Y_d]\oplus b_{1}K[X_d,Y_d]\oplus \cdots \oplus b_{d}K[X_d,Y_d]$$

$$\epsilon ([x_i,x_j])=a_i{x}_j-a_j{x}_i,\epsilon ([y_i,y_j])=b_i{y}_j-b_j{y}_i,\epsilon ([x_i,y_j])=a_i{y}_j-b_j{x}_i,$$

$$\epsilon ([w,x_j])=\epsilon (w)x_j,\epsilon ([w,y_j])=\epsilon (w)y_j,j=1,\dots ,d.$$

$$\sum _{i=1}^d{a}_i{f}_i(X_d,Y_d)+\sum _{i=1}^d{b}_i{g}_i(X_d,Y_d)$$

$$\sum _{i=1}^d{x}_i{f}_i(X_d,Y_d)+\sum _{i=1}^d{y}_i{g}_i(X_d,Y_d)=0.$$

$$\delta (y_i)=x_i,\delta (x_i)=0,i=1,\dots ,d,$$

$$J(\delta )=\left(\begin{array}{ccccc}0& 1& \cdots & 0& 0\\ 0& 0& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 0& 1\\ 0& 0& \cdots & 0& 0\end{array}\right).$$

$$K[X_d,Y_d{]}^{\delta }=K[X_d,U]=K[X_d,u_{pq}\mid 1\le p<q\le d]$$

$$x_i{u}_{jk}-x_j{u}_{ik}+x_k{u}_{ij}=0,1\le i<j<k\le d.$$

$$u_{ij}{u}_{kl}-u_{ik}{u}_{jl}+u_{il}{u}_{jk}=0,1\le i<j<k<l\le d.$$

$$x_{i_1}\cdots x_{i_m}{u}_{k_1{l}_1}\cdots u_{k_s{l}_s}$$

$$\left|\begin{array}{cc}{a}_p& b_p\\ a_q& b_q\end{array}\right|,u_{pq}=\left|\begin{array}{cc}{x}_p& y_p\\ x_q& y_q\end{array}\right|,1\le p<q\le d,$$

$$w_{pq}=a_p{y}_q-b_p{x}_q=\left|\begin{array}{cc}{a}_p& b_p\\ x_q& y_q\end{array}\right|,p,q=1,\dots ,d.$$

Full text

The research of both authors was partially supported by Grant I02/18 of the Bulgarian Science Fund.

The research of the second named author was partially supported by the Council of Higher Education (YÖK) in Turkey.

1. Introduction

A linear operator $\delta$ of a (not necessarily commutative or associative) algebra $R$ over a field $K$ is called a derivation if

[TABLE]

The kernel $R^{\delta}$ of $\delta$ is called the algebra of constants of $\delta$.

In the sequel $K$ will be a field of characteristic 0. Let $K[X_{d}]=K[x_{1},\ldots,x_{d}]$, $d\geq 2$, be the polynomial algebra in $d$ variables over $K$. A derivation $\delta$ of $K[X_{d}]$ acting as a nonzero nilpotent linear operator of the vector space $KX_{d}$ with basis $X_{d}$ is called a Weitzenböck derivation. The Jordan normal form $J(\delta)=(J_{1},\ldots,J_{s})$ of the matrix of $\delta$ considered as a linear operator acting on $KX_{d}$ consists of Jordan cells $J_{i}$, $i=1,\ldots,s$, with zero diagonals. In 1932 Weitzenböck [17] proved that the algebra of constants

[TABLE]

is finitely generated. For more information on Weitzenböck derivations one can see the books by Nowicki [14], Derksen and Kemper [4], and Sturmfels [16]. The algebra of constants $K[X_{d}]^{\delta}$ can be considered also from the point of view of classical invariant theory. The linear operator $\alpha\delta$ of $KX_{d}$ is nilpotent for all $\alpha\in K$. The exponent

[TABLE]

is a well defined invertible linear operator of $KX_{d}$ and this defines a $d$-dimensional representation of the unitriangular group

[TABLE]

This action can be extended diagonally on the whole algebra $K[X_{d}]$ and $K[X_{d}]^{\delta}$ is equal to the algebra of invariants $K[X_{d}]^{UT_{2}(K)}$.

Let $K[X_{d},Y_{d}]=K[x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}]$, be the polynomial algebra in $2d$ variables and let ${\delta}$ be the Weitzenböck derivation defined by ${\delta}(y_{i})=x_{i}$, ${\delta}(x_{i})=0$, $i=1,\ldots,d$. In the language of invariant theory $K[X_{d},Y_{d}]^{\delta}$ is the algebra of invariants for the action of the additive group $(K,+)$ on $K[X_{d},Y_{d}]$ by

[TABLE]

In 1994 Nowicki conjectured [14, p. 76, Conjecture 6.9.10] that the algebra of constants $K[X_{d},Y_{d}]^{\delta}$ is generated by $x_{1},\ldots,x_{d}$ and the determinants

[TABLE]

This conjecture attracted many mathematicians and was verified by several authors with proofs based on different ideas: In his Ph.D. thesis in 2004 Khoury [9, 10] gave a computational proof using Gröbner basis techniques. The unpublished proofs of Derksen and Panyushev applied ideas of classical invariant theory. Several proofs appeared in 2009. Drensky and Makar-Limanov [7] gave an elementary proof using easy arguments from undergraduate algebra and a simple induction only, without involving any invariant theory. In his proof Bedratyuk [2] reduced the Nowicki conjecture to a well known problem of classical invariant theory. Kuroda [12] gave a short proof based on the ideas of Kurano [11] in his study on the analogue in positive characteristic of the Roberts’ counterexample to the Hilbert fourteenth problem. As Kuroda mentioned Hashimoto informed him that Goto, Hayasaka, Kurano, and Nakamura [8] and Miyazaki [13] determined sets of generators for certain invariant rings where $K[X_{d},Y_{d}]^{\delta}$ is included, and this gives one more proof of the Nowicki conjecture.

Let $K\langle X_{d}\rangle$, $d\geq 2$, be the free (unitary or nonunitary) associative algebra freely generated by $X_{d}$ and let, as above, $\delta$ be a nilpotent linear operator acting on the vector space $KX_{d}$. Then the action of $\delta$ on $KX_{d}$ can be extended to an action as a derivation on the whole algebra $K\langle X_{d}\rangle$. If $V$ is a T-ideal (or a verbal ideal) of $K\langle X_{d}\rangle$, i.e., an ideal which is invariant under all endomorphisms of $K\langle X_{d}\rangle$, then it is well known that $\delta(V)\subseteq V$ and $\delta$ induces a derivation on the factor algebra $K\langle X_{d}\rangle/V$. We shall use the same notation $\delta$ for this derivation of $K\langle X_{d}\rangle/V$ and again shall call it Weitzenböck. The factor algebra $F_{d}({\mathfrak{V}})=K\langle X_{d}\rangle/V$ is a relatively free algebra in the variety $\mathfrak{V}$ of associative algebras defined by the polynomial identities from $V$. As in the case of polynomial algebras, the kernel $F_{d}({\mathfrak{V}})^{\delta}$ of $\delta$ is the algebra of constants of $\delta$. Similarly, if $L_{d}=L(X_{d})$, $d\geq 2$, is the free Lie algebra freely generated by $X_{d}$ and $W$ is a T-ideal (or a verbal ideal) of $L_{d}$, then the action of $\delta$ on $KX_{d}$ defines a Weitzenböck derivation on the relatively free algebra $F_{d}({\mathfrak{W}})=L_{d}/W$ in the variety of Lie algebras $\mathfrak{W}$ defined by the polynomial identities from $W$. See e.g., the book by Bahturin [1] for a background on varieties of Lie algebras, the book [5] by one of the authors for associative PI-algebras, and his paper [6] with Gupta for Weitzenböck derivations acting on free and relatively free algebras, and for the properties of their algebras of constants.

In the sequel, let

[TABLE]

be the factor algebra of $L_{d}$ modulo the second term $L_{2d}^{\prime\prime}$ of the derived series of $L_{d}$. This is the free metabelian Lie algebra generated by $X_{d}$. It is a relatively free algebra in the variety ${\mathfrak{A}}^{2}$ of the metabelian (solvable of class 2) Lie algebras defined by the identity $[[x_{1},x_{2}],[x_{3},x_{4}]]=0$. The variety ${\mathfrak{A}}^{2}$ has a key position in the theory of varieties of Lie algebras. By the well-known dichotomy a variety $\mathfrak{W}$ of Lie algebras either satisfies the Engel condition and by the theorem of Zelmanov [18] is nilpotent or contains the metabelian variety ${\mathfrak{A}}^{2}$. Since finitely generated nilpotent algebras are finite dimensional, the algebra $F_{d}=F_{d}({\mathfrak{A}}^{2})$ is the minimal relatively free algebra which is not finite dimensional. If $\delta$ is a Weitzenböck derivation of $F_{d}$, then Drensky and Gupta [6] showed that $F_{d}^{\delta}$ is finitely generated only in the trivial case when the Jordan normal form of $\delta$ consists of one Jordan cell of size $2\times 2$ and $d-2$ Jordan cells of size $1\times 1$, i.e., when the rank of the matrix of $\delta$ is equal to 1. The commutator ideal $F_{d}^{\prime}$ has a natural structure of a $K[X_{d}]$-module. Recently Dangovski and the authors [3] established that the vector space $(F_{d}^{\prime})^{\delta}$ of the constants of $\delta$ in the commutator ideal $F_{d}^{\prime}$ of $F_{d}$ is a finitely generated $K[X_{d}]^{\delta}$-module. Freely speaking, this means that the algebra of constants $F_{d}^{\delta}$ is very close to be finitely generated.

In the present paper we consider the free metabelian Lie algebra $F_{2d}$ of rank $2d$ generated by the set $X_{d}\cup Y_{d}$. We assume that ${\delta}$ is its Weitzenböck derivation acting similarly as in the Nowicki conjecture. We give a complete set of generators of the $K[X_{d},Y_{d}]^{\delta}$-module $(F_{2d}^{\prime})^{\delta}$. This gives also an infinite set of generators of the Lie algebra $(F_{2d})^{\delta}$.

2. Preliminaries

Till the end of the paper we fix the notation $F_{2d}=L_{2d}/L_{2d}^{\prime\prime}$ for the free metabelian Lie algebra of rank $2d$ freely generated by $X_{d}\cup Y_{d}=\{x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}\}$. We assume that all Lie commutators are left normed, e.g.,

[TABLE]

for all $z_{1},z_{2},z_{3}\in F_{2d}$. The metabelian identity implies, see, e.g., [1], that

[TABLE]

where $\sigma$ is an arbitrary permutation of $3,\ldots,k$. Thus the polynomial algebra $K[X_{d},Y_{d}]$ acts on $F_{2d}^{\prime}$ by the rule

[TABLE]

where $u\in F_{2d}^{\prime}$, $f(X_{d},Y_{d})=f(x_{1},\ldots,x_{d},y_{1},\ldots,y_{d})\in K[X_{d},Y_{d}]$.

We construct the abelian wreath product due to Shmel’kin [15]. Let $\Lambda_{2d}=K(A_{d}\cup B_{d})$ and $\Gamma_{2d}=K(P_{d}\cup Q_{d})$ denote the abelian Lie algebras with linear bases

[TABLE]

respectively, and let $C_{2d}$ be the free right $K[X_{d},Y_{d}]$-module with free generators $A_{d}\cup B_{d}$. Equipping $C_{2d}$ with trivial multiplication we give it the structure of an abelian Lie algebra. The abelian wreath product $W_{2d}=\Lambda_{2d}\text{\rm wr}\Gamma_{2d}$ is equal to the semidirect sum $C_{2d}\leftthreetimes\Gamma_{2d}$. The elements of $W_{2d}$ are of the form

[TABLE]

where $\alpha_{i},\beta_{i}\in K$. The multiplication in $W_{2d}$ is defined by

[TABLE]

[TABLE]

[TABLE]

$j=1,\ldots,d$. Hence $W_{2d}$ is a metabelian Lie algebra and every mapping

[TABLE]

can be extended to a homomorphism $F_{2d}\to W_{2d}$. As a special case of the embedding theorem of Shmel’kin, the homomorphism $\varepsilon:F_{2d}\to W_{2d}$ defined by

[TABLE]

is a monomorphism. By this action of $\varepsilon$, the commutator ideal $F_{2d}^{\prime}$ is embedded into the free right $K[X_{d},Y_{d}]$-module

[TABLE]

as follows:

[TABLE]

and then, by induction, if $w\in F_{2d}^{\prime}$, then

[TABLE]

As a consequence of this construction, we have the following result.

Lemma 2.1**.**

[15, Theorem 2]** An element

[TABLE]

from $C_{2d}$ is an image of an element from the commutator ideal $F_{2d}^{\prime}$ if and only if

[TABLE]

It follows immediately from Lemma 2.1 that the image $\varepsilon(F_{2d}^{\prime})$ of $F_{2d}^{\prime}$ in $C_{2d}$ is a submodule of the $K[X_{d},Y_{d}]$-module $C_{2d}$. In the sequel we shall identify the elements of $F_{2d}^{\prime}$ with their images in $C_{2d}$.

If $\delta$ is a Weitzenböck derivation of $K[X_{d},Y_{d}]$ we shall assume that it acts on $K[X_{d},Y_{d}]$ by the rule

[TABLE]

and shall use without reference that the algebra of constants $K[X_{d},Y_{d}]^{\delta}$ is generated by $x_{1},\ldots,x_{d}$ and the determinants (1) as conjectured by Nowicki [14] and proved in [9, 10, 7, 2, 12]. In this special case the Jordan normal form $J(\delta)$ of $\delta$ consist of $2\times 2$ Jordan cells only, i.e.,

[TABLE]

The action of $\delta$ on $\{a_{i},b_{i}\mid i=1,\ldots,d\}$ will be defined in the same way as on $\{x_{i},y_{i}\mid i=1,\ldots,d\}$. Thus $\delta$ is extended to a derivation of $F_{2d}$ and $W_{2d}$. The vector space $C_{2d}^{\delta}$ of the constants of $\delta$ in the free $K[X_{d},Y_{d}]$-module $C_{2d}$ is a $K[X_{d},Y_{d}]^{\delta}$-module. The following theorem is a partial case of a result of [3].

Theorem 2.2**.**

Let $\delta$ be a Weitzenböck derivation of the free metabelian Lie algebra $F_{2d}$. Then the vector space $(F_{2d}^{\prime})^{\delta}$ of the constants of $\delta$ in the commutator ideal $F_{2d}^{\prime}$ of $F_{2d}$ is a finitely generated $K[X_{d},Y_{d}]^{\delta}$-module.

Since it will not cause misunderstanding with the notation for the polynomial algebra we shall use the notation

[TABLE]

for the algebra generated by $X_{d}$ and the elements $U=\{u_{pq}\mid 1\leq i<j<k\leq d\}$ defined in (1). Drensky and Makar-Limanov [7] showed that the algebra $K[X_{d},Y_{d}]^{\delta}$ has the following defining relations

[TABLE]

[TABLE]

and gave a canonical linear basis consisting of the elements of the form

[TABLE]

such that the generators $u_{k_{\alpha}l_{\alpha}}$ and $u_{k_{\beta}l_{\beta}}$ do not intersect each other and $u_{k_{\alpha}l_{\alpha}}$ does not cover $x_{i_{\gamma}}$ for any $\alpha,\beta,\gamma$. Here each $u_{k_{\alpha}l_{\alpha}}$ is identified with the open interval $(k_{\alpha},l_{\alpha})$ on the real line. The generators $u_{k_{\alpha}l_{\alpha}}$ and $u_{k_{\beta}l_{\beta}}$ intersect each other if the intervals $(k_{\alpha},l_{\alpha})$ and $(k_{\beta},l_{\beta})$ have a nonempty intersection and are not contained in each other. We say also that $u_{k_{\alpha}l_{\alpha}}$ covers $x_{i_{\gamma}}$ if $i_{\gamma}$ belongs to the open interval $(k_{\alpha},l_{\alpha})$. The order of generators in this basis is assumed to be as follows: $p_{1}\leq\cdots\leq p_{s}$ and if $p_{n}=p_{n+1}$, then $q_{n}\leq q_{n+1}$; and the order among $x_{i_{\gamma}}$ is such that $i_{1}\leq\cdots\leq i_{m}$.

As a direct consequence of the affirmative answer to the Nowicki Conjecture the algebra of constants $K[A_{d},B_{d},X_{d},Y_{d}]^{\delta}$ of the derivation $\delta$ acting on the polynomial algebra $K[A_{d},B_{d},X_{d},Y_{d}]$ is generated by $A_{d},X_{d}$ and the determinants

[TABLE]

[TABLE]

Hence the $K[X_{d},U]$-module $C_{2d}^{\delta}$ is generated by the elements $a_{1},\dots,a_{d}$ and the determinants (5), and as a vector space $C_{2d}^{\delta}$ is spanned by the elements of the form

[TABLE]

[TABLE]

for $i_{0},p_{0},q_{0}=1,\ldots,d$. Ordering the elements $A_{d}\cup B_{d}\cup X_{d}\cup Y_{d}$ and assuming that the elements from $A_{d}$ and $B_{d}$ precede, respectively, the elements from $X_{d}$ and $Y_{d}$, we obtain as an application of (2) and (3) that the $K[X_{d},U]$-module $C_{2d}^{\delta}$ has the following defining relations

[TABLE]

[TABLE]

In order to fix a basis of $C_{2d}^{\delta}$ as a vector space, the factors $x_{i_{1}}\cdots x_{i_{m}}u_{k_{1}l_{1}}\cdots u_{k_{s}l_{s}}$ and $x_{j_{1}}\cdots x_{j_{n}}u_{p_{1}q_{1}}\cdots u_{p_{r}q_{r}}$ of the elements (6) and (7) have to satisfy the restrictions in (4). Additionally, for the elements in (7) we require $q_{0}\leq p_{1}$.

3. Main Results

In this section we give the generators of the $K[X_{d},Y_{d}]^{\delta}$-module of constants $(F_{2d}^{\prime})^{\delta}$ in the commutator ideal $F_{2d}^{\prime}$ of the free metabelian Lie algebra $F_{2d}$. Since $(F_{2d}^{\prime})^{\delta}$ is canonically embedded in $C_{2d}^{\delta}$ we shall work in $C_{2d}^{\delta}$ instead of directly in $(F_{2d}^{\prime})^{\delta}$.

Definition 3.1**.**

We define the $K[X_{d},U]$-submodule $L$ of $C_{2d}^{\delta}$ generated by the elements

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By Lemma 2.1, one can easily observe that the generating elements (10)–(15) of $L$ are Lie elements, i.e., images of elements in the commutator ideal $F_{2d}^{\prime}$ of $F_{2d}$.

Lemma 3.2**.**

The following elements span the quotient space $C_{2d}^{\delta}/L$.

[TABLE]

[TABLE]

where $u_{p_{0}q_{0}}u_{p_{1}q_{1}}\cdots u_{p_{r}q_{r}}$ and $x_{i_{0}}x_{i_{1}}\cdots x_{i_{m}}u_{k_{1}l_{1}}\cdots u_{k_{s}l_{s}}$ are elements of the form (4); i.e., they are canonical basis elements of the algebra $K[X_{d},U]$.

Proof.

We shall work in the vector space $C_{2d}^{\delta}$ modulo the subspace $L$. It is sufficient to handle the basis elements of $C_{2d}^{\delta}$ of the form (6) and (7). Starting with the element $w_{p_{0}q_{0}}x_{j_{1}}\cdots x_{j_{n}}u_{p_{1}q_{1}}\cdots u_{p_{r}q_{r}}$ in (7) we apply the relation $w_{pq}x_{i}\equiv a_{i}u_{pq}$ (mod $L$) from (13) and bring the element from (7) to an element from (6). If the element from (7) is of the form $w_{p_{0}q_{0}}u_{p_{1}q_{1}}\cdots u_{p_{r}q_{r}}$, then (10) and (11) imply that we may assume that $p_{0}<q_{0}$. Then using the relation (9) we obtain that the interval $(p_{0},q_{0})$ does not intersect with $(p_{1},q_{1}),\ldots,(p_{r},q_{r})$, and the generator (15) fixes the order among $u_{p_{0}q_{0}},u_{p_{1}q_{1}}\ldots,u_{p_{r}q_{r}}$. This closes the case (7). Now we consider the element $a_{i_{0}}x_{i_{1}}\cdots x_{i_{m}}u_{k_{1}l_{1}}\cdots u_{k_{s}l_{s}}$ in (6). By assumption, the integers $i_{1},\ldots,i_{m}$ do not belong to the open intervals $(p_{l},q_{l})$. If $i_{0}\in(p_{l},q_{l})$ for some $l=1,\ldots,s$, then the relation $a_{j}u_{ik}\equiv a_{i}u_{jk}+a_{k}u_{ij}$ (mod $L$) from (14) replaces $a_{i_{0}}u_{p_{l}q_{l}}$ with $a_{p_{l}}u_{i_{0}q_{l}}$ and $a_{q_{l}}u_{p_{l}i_{0}}$. Since the intervals $(i_{0},q_{l})$ and $(p_{l},i_{0})$ are shorter than the interval $(p_{l},q_{l})$, the integers $i_{1},\ldots,i_{m}$ are not covered by the intervals $(p_{1},q_{1}),\ldots,(p_{s},q_{s})$. In finite number of steps the same holds for the integer $i_{0}$. Finally the generator (12) fixes the order among $x_{i_{t}}$, $t=0,1,\ldots,m$. ∎

Theorem 3.3**.**

The $K[X_{d},U]$-module $L$ consists of all Lie elements in $C_{2d}^{\delta}$.

Proof.

Let

[TABLE]

be a Lie element in the vector space $C_{2d}^{\delta}/L$. Then by Lemma 2.1 and Lemma 3.2 we have that

[TABLE]

where $x_{i_{0}}x_{i_{1}}\cdots x_{i_{m}}u_{k_{1}l_{1}}\cdots u_{k_{s}l_{s}}$ and $u_{p_{0}q_{0}}u_{p_{1}q_{1}}\cdots u_{p_{r}q_{r}}$ are basis elements of the algebra $K[X_{d},U]$. Clearly each element from the first sum is linearly independent from the elements of the second sum, since there is at least one multiplier of the form $x_{i_{0}}$ in each summand of the first sum which does not appear in the second sum. This implies that

[TABLE]

Thus $\xi_{ikl}=0=\psi_{pq}$ for all $i,k,l,p,q$, because $x_{i_{0}}x_{i_{1}}\cdots x_{i_{m}}u_{k_{1}l_{1}}\cdots u_{k_{s}l_{s}}$ and $u_{p_{0}q_{0}}u_{p_{1}q_{1}}\cdots u_{p_{r}q_{r}}$ uniquely determine the monomials $a_{i_{0}}x_{i_{1}}\cdots x_{i_{m}}u_{k_{1}l_{1}}\cdots u_{k_{s}l_{s}}$ and $w_{p_{0}q_{0}}u_{p_{1}q_{1}}\cdots u_{p_{r}q_{r}}$, respectively. ∎

Finally the generators of $(F_{2d}^{\prime})^{\delta}$ are obtained by computing the inverse images of generators of $L$.

Corollary 3.4**.**

The $K[X_{d},U]$-module $(F_{2d}^{\prime})^{\delta}$ is generated by the following elements

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where $1\leq i<j\leq d$, $1\leq p<q\leq d$.

Acknowledgements

The second named author is very thankful to the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences for the creative atmosphere and the warm hospitality during his visit as a post-doctoral fellow when this project was carried out.