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European Journal of Mathematics

, Volume 2, Issue 2, pp 565–577 | Cite as

Relationships between factors of canonical central series of Leibniz algebras

  • Leonid A. Kurdachenko
  • Javier OtalEmail author
  • Alexander A. Pypka
Research Article

Abstract

The aim of this paper is to show a close relationship between the factor-algebras by the terms of the upper central series \(\zeta _n(L)\) of a Leibniz algebra L and the terms of its lower central series \(\gamma _n(L)\). Specifically we show that finiteness of codimension of some \(\zeta _k(L)\) implies finiteness of dimension of \(\gamma _{k+1}(L)\) and give explicit bounds for this dimension. We also improve this in the case \(k=1\), which corresponds to the center and commutator subalgebra of the algebra, respectively. These results are analogous to the results that have been obtained for groups and Lie algebras.

Keywords

Leibniz algebra Lie algebra Center Commutator subalgebra Lower and upper central series of a Leibniz algebra 

Mathematics Subject Classification

17A32 17A60 17B30 

1 Introduction

An algebra L over a field F is said to be a (left) Leibniz algebra if it is endowed with a multiplication (or bracket) Open image in new window that satisfies the so-called Leibniz identity A right Leibniz algebra is defined similarly. Leibniz algebras arise as a generalization of Lie algebras. Indeed, if L is a Leibniz algebra such that Open image in new window for every element \(a\in L\), then it is not hard to check that L is a Lie algebra. Therefore we may think of Leibniz algebras as a non-anticommutative analog of Lie algebras.

Leibniz algebras were first introduced and investigated in the papers of A.M. Blokh [8, 9, 10], where he called them D -algebras. However, they did not receive a worthy attention immediately. Later these algebras were independently rediscovered by J.-L. Loday [17], who called them Leibniz algebras, since it was Leibniz who proved the Leibniz rule for differentiation of functions. Leibniz algebras appeared to be related in a natural way with several topics such as differential geometry, homological algebra, classic algebraic topology, algebraic K-theory, loop spaces, noncommutative geometry, etc. They also found some applications in physics (see [11, 13, 14], for example). Other papers concerning Leibniz algebras are devoted to the study of homological problems (see [12, 16, 18, 21], for example).

Structural properties of Leibniz algebras are well studied. In most cases analogs of important results from the theory of Lie algebras for them are straightforward, but some of them have a specificity. This is the case of nilpotent Leibniz algebras. The concept of nilpotency plays a very important role for Lie algebras, associative algebras, groups and other algebraic structures. The study of nilpotent Leibniz algebras started in the paper of Sh.A. Ayupov and B.A. Omirov [3] and continued in the papers [1, 2, 4, 7, 15, 20], sometimes under different approaches. The concept of nilpotency of Leibniz algebras has a serious difference from the nilpotency of Lie algebras. In all papers mentioned above it relies on the concept of lower central series. To define it as well as other concepts we recall some standard definitions and notation.

Let L be a Leibniz algebra over a field F. If A and B are subspaces of L, then [AB] is the subspace generated by Open image in new window . A subspace A of L is called a subalgebra of L if \([A,A]\leqslant A\). A subalgebra A is called a left ideal of L (respectively, a right ideal of L) if \([L,A]\leqslant A\) (respectively, \([A,L]\leqslant A\)), and it is called an (two-sided) ideal of L if it is both a left ideal and a right ideal. If A is an ideal of L, then the factor subspace L / A is in fact a Leibniz algebra. We denote by Open image in new window is the subspace generated by Open image in new window . Open image in new window is an ideal of L (called the Leibniz kernel of L); if H is an ideal of L such that L / H is a Lie algebra, then Open image in new window . Note the following important property:If A and B are ideals of a Leibniz algebra L, then [AB] need not be an ideal of L; a specific example can be found in [6]. However, if H is an ideal, then [HH] is an ideal, which is important in what follows.
The lower central series of L
$$\begin{aligned} L = \gamma _1(L)\geqslant \gamma _2(L)\geqslant \cdots \geqslant \gamma _\alpha (L)\geqslant \gamma _{\alpha +1}(L)\geqslant \cdots \geqslant \gamma _\delta (L) \end{aligned}$$
is defined inductively by the rules: \(\gamma _1(L) = L\), \(\gamma _{\alpha +1}(L) = [L,\gamma _\alpha (L)]\) for every ordinal \(\alpha \) and \(\gamma _\lambda (L) = \bigcap _{\mu <\lambda }\gamma _\mu (L)\) for every limit ordinal \(\lambda \). It is rather easy to show that every term of this series is an ideal of L. The last term \(\gamma _\delta (L)\) of the series is called the lower hypercenter of L. We note that \(\gamma _\delta (L) = [L,\gamma _\delta (L)]\). Also if \(\alpha = k\) is a positive integer, then \(\gamma _k(L) = [L,[L,[L,\dots ]\dots ]]\) is a left normed product of k copies of L.

The Leibniz algebra L is called nilpotent if there exists a positive integer k such that \(\gamma _k(L) = \{0\}\). More precisely, L is said to be nilpotent of nilpotency class c if \(\gamma _{c+1}(L) = \{0\}\) but \(\gamma _c(L)\ne \{0\}\). We denote the nilpotency class of L by Open image in new window . The concept of lower central series is specific not only for Leibniz algebras but also for other algebraic structures such as Lie algebras and groups. However, for the latter structures there exists another canonical series, which is dual to the lower central series, the upper central series. The case of Leibniz algebras has essential differences that are worth to study, for example, the notion of the center.

By definition, the left center \(\zeta ^\mathrm{l}(L)\) and the right center \(\zeta ^\mathrm{r}(L)\) of L areClearly, \(\zeta ^\mathrm{l}(L)\) is an ideal and Open image in new window and so \(L/\zeta ^\mathrm{l}(L)\) is a Lie algebra. On the other hand, the right center is a subalgebra of L but it need not be an ideal. In general, the left and right centers are different. In Sect. 2, we will present the corresponding example. The center \(\zeta (L)\) of L is the intersection of the left and right centers, that isThe center \(\zeta (L)\) is an ideal of L, and thus the factor-space \(L/\zeta (L)\) is an algebra.
The upper central series of L,
$$\begin{aligned} \{0\} = \zeta _0(L)\leqslant \zeta _1(L)\leqslant \cdots \leqslant \zeta _\alpha (L)\leqslant \zeta _{\alpha +1}(L)\leqslant \cdots \leqslant \zeta _\eta (L)=\zeta _\infty (L), \end{aligned}$$
is defined inductively by the rules \(\zeta _1(L) = \zeta (L)\), \(\zeta _{\alpha +1}(L)/\zeta _\alpha (L) = \zeta (L/\zeta _\alpha (L))\) for every ordinal \(\alpha \) and \(\zeta _\lambda (L) = \bigcup _{\mu <\lambda }\zeta _\mu (L)\) for every limit ordinal \(\lambda \). Note that, by construction, every term of this series is an ideal of L. The last term \(\zeta _\infty (L)\) of the series is called the upper hypercenter of L. We set Open image in new window to denote the length of this series.

For Lie algebras L, the fact that \(\gamma _{c+1}(L) = \{0\}\) is equivalent to \(\zeta _c(L)=L\), that is for a nilpotent Lie algebra the lower and upper central series have the same length (similar facts hold for groups). In this paper we will show that this remains true for Leibniz algebras. The case of finite upper or lower central series naturally raises the question of investigation of relationships between \(L/\zeta _k(L)\) and \(\gamma _{k+1}(L)\). If L is a Lie algebra such that \(L/\zeta _k(L)\) is finitely dimensional, I.N. Stewart [22, Theorem 5.2] showed that \(\gamma _{k+1}(L)\) is also finitely dimensional. For groups, the corresponding result was obtained earlier by R. Baer [5]. Our first main result is an analog of these theorems.

Theorem A

Let L be a Leibniz algebra over a field F. If \(\mathrm{codim}_F\zeta _k(L) = d\) is finite for some \(k\geqslant 1\), then \(\mathrm{dim}_F\gamma _{k+1}(L)\leqslant 2^{k-1}d^{k+1}\).

As a corollary we may improve the bound for the dimension of \(\gamma _{k+1}(L)\) for Lie algebras, thus completing the result of Stewart mentioned above.

Corollary A1

Let L be a Lie algebra over a field F. If \(\mathrm{codim}_F\zeta _k(L) = d\) is finite for some \(k\geqslant 1\), then Open image in new window .

An important specific case in these results is the case \(k=1\) when the center of the Leibniz algebra L has finite codimension. For Lie algebras the following result is well-known (see [23], for example): if L is a Lie algebra over a field F and the factor-algebra \(L/\zeta (L)\) has finite dimension d, then Open image in new window .

For groups the corresponding result was proved much earlier and makes up one of the most outstanding results of the theory: if C is a central subgroup of a group G such that G / C is finite, then the derived subgroup [GG] is finite. In this formulation, it appears for the first time in the paper of B.H. Neumann [19] but the result was also obtained by R. Baer [5]. For Leibniz algebras we obtain the following analogous results.

Theorem B

Let L be a Leibniz algebra over a field F. If \(\mathrm{codim}_F\zeta ^\mathrm{l}(L) = d\) and \(\mathrm{codim}_F\zeta ^\mathrm{r}(L) = r\) are finite, then Open image in new window .

In this setting we can raise the question of studying the above result when only one of the above codimensions is finite. In Sect. 3, we construct an example that shows that the answer to the question “If \(\mathrm{codim}_F\zeta ^\mathrm{l}(L)\) is finite, is \(\dim _F[L,L]\) finite?” is negative. Despite this, other cases are more favourable.

Corollary B1

Let L be a Leibniz algebra over a field F. If \(\mathrm{codim}_F\zeta (L) = d\) is finite, then \(\dim _F[L,L]\leqslant d^2\).

Corollary B2

Let L be a Leibniz algebra over a field F. If \(\mathrm{codim}_F\zeta (L) = d\) is finite, then Open image in new window .

2 Auxiliary results

First we construct an example of a Leibniz algebra in which left and right centers are different. Actually they will have different dimensions.

Example 2.1

Let F be a field. Consider a vector space L over F of dimension 4, say Open image in new window , and define an operation Open image in new window as follows:It is not hard to check that this operation defines a Leibniz algebra. We can also see that \(\zeta ^\mathrm{r}(L) = Fe_4\) and so \(\zeta ^\mathrm{r}(L)\) is not an ideal. Furthermore, Open image in new window , and then \(\zeta ^\mathrm{r}(L)\cap \zeta ^\mathrm{l}(L) = \{0\}\), \(\mathrm{dim}_F\zeta ^\mathrm{r}(L) = 1\) and \(\mathrm{dim}_F\zeta ^\mathrm{l}(L) = 2\). We also note that Open image in new window .

Now we collect some technical results needed in the sequel.

Proposition 2.2

Let L be a Leibniz algebra over a field F and H an ideal of L. Then
  1. (i)

    [HH] is an ideal of L;

     
  2. (ii)

    [LH] is a subalgebra of L;

     
  3. (iii)

    [HL] is a subalgebra of L;

     
  4. (iv)

    \([L,H] + [H,L]\) is an ideal of L;

     
  5. (v)

    Open image in new window for all positive integers jk;

     
  6. (vi)

    \(\gamma _j(H)\) is an ideal of L for every positive integer j, in particular, \(\gamma _j(L)\) is an ideal of L for every positive integer j; and

     
  7. (vii)

    \(\gamma _j(\gamma _k(H))\leqslant \gamma _{jk}(H)\) for all positive integers jk.

     

Proof

The proof of (i)–(iv) is not complicated and we omit it.

(v) We proceed by induction on j. If \(j = 1\), the result follows from the definition. Suppose now that \(j > 1\) and we have already proved the inclusionWe have \(\gamma _j(H) = [H,\gamma _{j-1}(H)]\). Pick \(x\in H\), \(y\in \gamma _{j-1}(H)\) and \(z\in \gamma _k(H)\). We have
$$\begin{aligned}{}[[x,y],z] = [x,[y,z]]-[y,[x,z]]. \end{aligned}$$
Since Open image in new window , by induction, \([y,z]\in \gamma _{j-1+k}(H)\) whence Open image in new window . FurtherApplying induction,Hence Open image in new window , which gives Open image in new window .
(vi) Again we proceed by induction on j. If \(j = 1\), the result follows from definitions. Suppose now that \(j > 1\) and we have already proved that \(\gamma _m(H)\) is an ideal of L for all \(m < j\). We have \(\gamma _j(H) = [H,\gamma _{j-1}(H)]\). Pick \(x\in H\), \(y\in \gamma _{j-1}(H)\) and \(z\in L\). We have Open image in new window . By induction, \([y,z]\in \gamma _{j-1}(H)\) and so Open image in new window . Since H is an ideal of L, Open image in new window and then Open image in new window . By induction,Hence Open image in new window . Similarly, Open image in new window .
(vii) One more time we proceed by induction on j. Since \(\gamma _1(\gamma _k(H)) = \gamma _k(H)\), the case \(j=1\) is trivial. Suppose now that \(j > 1\) and we have already proved the inclusion \(\gamma _m(\gamma _k(H))\leqslant \gamma _{mk}(H)\) for all \(m < j\). We haveApplying (iii), we obtain the inclusion
$$\begin{aligned} \gamma _j(\gamma _k(H))\leqslant \gamma _{jk-k+k}(H) = \gamma _{jk}(H), \end{aligned}$$
and we are done. \(\square \)
An ascending series of ideals of a Leibniz algebra L
$$\begin{aligned} \{0\} = C_0\leqslant C_1\leqslant \cdots \leqslant C_\alpha \leqslant C_{\alpha +1}\leqslant \cdots \leqslant C_\rho = L \end{aligned}$$
is said to be central if \(C_{\alpha +1}/C_\alpha \leqslant \zeta (L/C_\alpha )\) for every ordinal \(\alpha <\rho \). In other words, \([C_{\alpha +1},L], [L,C_{\alpha +1}]\leqslant C_\alpha \) for every ordinal \(\alpha <\rho \).

As in other structures, for example in groups, central series of finite length characterizes nilpotency as the following result shows.

Proposition 2.3

Let L be a Leibniz algebra over a field F and
$$\begin{aligned} \{0\} = C_0\leqslant C_1\leqslant \cdots \leqslant C_n = L \end{aligned}$$
be a central series of L of finite length. Then we have
  1. (i)

    \(\gamma _j(L)\leqslant C_{n-j+1}\) for every \(1\leqslant j\leqslant n+1\). In particular, \(\gamma _{n+1}(L) = \{0\}\); and

     
  2. (ii)

    \(C_j\leqslant \zeta _j(L)\) for every \(0\leqslant j\leqslant n\). In particular, \(\zeta _n(L) = L\).

     

Proof

(i) We proceed by induction on j. If \(j = 2\), we have
$$\begin{aligned} \gamma _2(L) = [L,L] = [L,C_n]\leqslant C_{n-1}. \end{aligned}$$
Suppose now that \(j > 2\) and we have already proved the inclusion \(\gamma _m(L)\leqslant C_{n-m+1}\) for all \(m < j\). Then \(\gamma _j(L) = [L,\gamma _{j-1}(L)]\leqslant [L,C_{n-j+1+1}]\leqslant C_{n-j+1}\) and we are done.
(ii) We proceed again by induction on j. If \(j = 1\), we have \([C_1,L] = [L,C_1] = \{0\}\), which shows that \(C_1\leqslant \zeta _1(L)\). Suppose now that \(j > 1\) and we have already proved the inclusion \(C_m\leqslant \zeta _m(L)\) for all \(m < j\). It follows that
$$\begin{aligned}{}[C_j,L], [L,C_j]\leqslant C_{j-1}\leqslant \zeta _{j-1}(L). \end{aligned}$$
Then
$$\begin{aligned}{}[C_j + \zeta _{j-1}(L),L], [L,C_j + \zeta _{j-1}(L)]\leqslant \zeta _{j-1}(L), \end{aligned}$$
which gives the inclusions
$$\begin{aligned} (C_j + \zeta _{j-1}(L))/\zeta _{j-1}(L)\leqslant \zeta L/\zeta _{j-1}(L) = \zeta _j(L)/\zeta _{j-1}(L) \end{aligned}$$
and we are done.\(\square \)

This result has the following consequences.

Corollary 2.4

Let L be a Leibniz algebra over a field F and suppose that L has a finite central series
$$\begin{aligned} \{0\} = C_0\leqslant C_1\leqslant \cdots \leqslant C_n = L. \end{aligned}$$
Then L is nilpotent and Open image in new window . Furthermore, the upper central series of L has finite length Open image in new window . Moreover, Open image in new window .

The above corollary shows that the Leibniz algebra L is nilpotent if and only if there is a positive integer k such that \(L = \zeta _k(L)\). The least positive integer with this property coincides with the nilpotency class of L.

Corollary 2.5

Let L be a Leibniz algebra over a field F and H an ideal of L. If the factor-algebra L / H is nilpotent of nilpotency class at most n, then \(\gamma _{n+1}(L)\leqslant H\).

With some extra work we may prove something more.

Corollary 2.6

Let L be a Leibniz algebra over a field F and H an ideal of L. Then Open image in new window for every positive integer n.

Proof

We proceed by induction on n. If \(n = 1\), we have \(\gamma _1(L/H) = L/H = \gamma _1(L)/H\). Suppose now that \(n > 1\) and we have already proved Open image in new window for all \(m < n\). Thenas required.\(\square \)

3 Proof of Theorem B

We start by describing an example mentioned in Introduction.

Example 3.1

Let F be a field. We consider an F-vector space with the formwhere the subspace Z has a countable basis, say Open image in new window . We define Open image in new window for every \(x\in L\) andThen we haveTaking into account the equalitieswe obtain Open image in new window for every \(z\in Z\). Now we putFrom these definitions we deducefor all \(j, k\in \{1,2\}\) and \(z\in Z\). As we have seen above,for all \(j, k\in \{1,2\}\) and \(z\in Z\). Hence L is a Leibniz algebra. By construction, Z is the left center of L, the right center coincides with the center and this is equal to \(Fz_1\). It follows that the left center has finite codimension (and then infinite dimension) and the right center and the center have finite dimension.
Also, by construction, \([L,L] = Z\). Furthermore we havefor all \(j >1\). It follows that Open image in new window .

Example 3.1 shows that we may have a Leibniz algebra L whose left center has finite codimension and whose derived subalgebra has infinite dimension. However, as we shall see, when the center of L itself has finite codimension, the conclusion is rather different.

For the proof of our further results the following notion is fundamental. Let L be a Leibniz algebra over a field F, M a non-empty subset of L and H a subalgebra of L. We putThese subsets are called the left and right annihilators of M in the subalgebra H, respectively, and the intersectionis called the annihilator or centralizer of M in the subalgebra H. It is not hard to see that all these subsets are subalgebras of L. Moreover, if M is a left ideal of L then \(\mathrm{Ann}_L^\mathrm{l}(M)\) is an ideal of L, and if M is an ideal then \(\mathrm{Ann}_L(M)\) is also an ideal of L. For example, the center of L is the intersection of the annihilators of all elements of L.

Proposition 3.2

Let L be a Leibniz algebra over a field F and H an ideal of L. Then the factor-algebra \(L/\mathrm{Ann}_L^\mathrm{l}(H)\) is isomorphic to a subalgebra of the algebra of derivations of H.

Proof

If \(a\in L\) we define a mapping \(\iota _a:H\rightarrow H\) by the rule Open image in new window , \(x\in H\). It is rather easy to show that \(\iota _a\) is a derivation of H that satisfies \(\beta \iota _a = \iota _{\beta a}\), \(\iota _a+\iota _b = \iota _{a+b}\) and Open image in new window , where \(\beta \in F\), \(a,b\in L\).

We consider now the mapping Open image in new window defined by the rule \(\delta (a) = \iota _a\), \(a\in L\). Applying the expressions proved in the above paragraphs, we haveThese equations show that the mapping \(\delta \) is a homomorphism from the Leibniz algebra L to the Lie algebra Open image in new window . Then Open image in new window is a subalgebra of Open image in new window . Since Open image in new window and Open image in new window , the result follows.\(\square \)

Proof of Theorem B

We have Open image in new window for some subspace E. Choose in E a basis, say \(\{e_1,\dots , e_d\}\). If \(x, y\in L\), then we may write
$$\begin{aligned} x =\alpha _1e_1 + \cdots + \alpha _de_d + z_1,\qquad y=\beta _1e_1+\cdots +\beta _de_d+z_2 \end{aligned}$$
for suitable \(\alpha _1,\dots ,\alpha _d,\beta _1,\dots ,\beta _d\in F\) and \(z_1,z_2\in \zeta ^\mathrm{l}(L)\). Then we haveIt follows that the subspace S generated by the elements \([e_j,e_m]\), \(1\leqslant j,m\leqslant d\), and the subspaces Open image in new window , \(1\leqslant j\leqslant d\), include [LL].
Put \(Z = \zeta ^\mathrm{l}(L)\) and let a be an arbitrary element of L. We define a mapping \(\iota _a:Z\rightarrow Z\) by the rule Open image in new window , \(z\in Z\). By Proposition 3.2, this mapping is linear, Open image in new window and Open image in new window . HenceSince \(\zeta ^\mathrm{r}(L)\leqslant \mathrm{Ann}_L^\mathrm{r}(a)\), we have \(\mathrm{codim}_F\mathrm{Ann}_L^\mathrm{r}(a)\leqslant r\) and soIn particular, Open image in new window for every \(1\leqslant j\leqslant d\). It follows that the subspace S has dimension at most Open image in new window , as required.\(\square \)

Proof of Corollary B1

We have \(\zeta (L) = \zeta ^\mathrm{l}(L)\cap \zeta ^\mathrm{r}(L)\). Since \(\mathrm{codim}_F\zeta ^\mathrm{l}(L)\) and \(\mathrm{codim}_F\zeta ^\mathrm{r}(L)\) are finite, \(\mathrm{codim}_F\zeta (L)\) is finite. Then it suffices to take the proof of Theorem B into account to obtain Corollary B1.\(\square \)

Proof of Corollary B2

Put Open image in new window and \(Z = \zeta (L)\). Clearly
$$\begin{aligned}{}[L/K,L/K] = [L,L]/K,\qquad Z/K\leqslant \zeta (L/K), \end{aligned}$$
therefore \(\mathrm{codim}_F\zeta (L/K)\leqslant d\). The factor-algebra L / K is in fact a Lie algebra, and Open image in new window . Henceas required.\(\square \)

4 Proof of Theorem A

To carry out the proof we need some auxiliary results.

Lemma 4.1

Let L be a Leibniz algebra over a field F. For every pair \(k\geqslant j\) of positive integers we have Open image in new window .

Proof

We proceed by induction on j. If \(j = 1\), we have Open image in new window and Open image in new window . Suppose that \(j > 1\) and we have already proved that Open image in new window for all \(m < j\). We have \(\gamma _j(L) = [L,\gamma _{j-1}(L)]\). Pick \(x\in L\), \(y\in \gamma _{j-1}(L)\) and \(z\in \zeta _k(L)\). We have Open image in new window . Since \([y,z]\in [\gamma _{j-1}(L),\zeta _k(L)]\), by induction, \([y,z]\in \zeta _{k-j+1}(L)\), and so Open image in new window . Further,Since \(k-1 > j-1\), by induction, we obtainWe have Open image in new window . Since Open image in new window ,and so, by induction, Open image in new window . Again, by induction, Open image in new window . Then Open image in new window and we are done.\(\square \)

Lemma 4.2

Let L be a Leibniz algebra over a field F and H an ideal of L. Suppose that Open image in new window , where E is a suitable subspace such that \(\mathrm{dim}_FE = d\) is finite. If \(\mathrm{dim}_F H/\mathrm{Ann}_H^\mathrm{r}(E) = t\) is finite, then \(\mathrm{dim}_F[L,H]\leqslant dt\).

Proof

Let \(\{e_1,\dots , e_d\}\) be a basis of E. If x is an arbitrary element of L, then
$$\begin{aligned}x = \alpha _1e_1+\cdots +\alpha _de_d+c,\end{aligned}$$
for suitable \(\alpha _1,\dots ,\alpha _d\in F\) and \(c\in \mathrm{Ann}_L^\mathrm{l}(H)\). If \(h\in H\), thenIt follows that Open image in new window . Pick \(a\in L\) and define the mapping \(\iota _a:H\rightarrow H\) by the rule Open image in new window , \(h\in H\). By Proposition 3.2, this mapping is linear, Open image in new window and Open image in new window . HenceLet \(1\leqslant j\leqslant d\). Since \(\mathrm{Ann}_H^\mathrm{r}(E)\leqslant \mathrm{Ann}_H^\mathrm{r}(e_j)\),
$$\begin{aligned} \mathrm{dim}_FH/\mathrm{Ann}_H^\mathrm{r}(e_j)\leqslant \mathrm{dim}_FH/\mathrm{Ann}_H^\mathrm{r}(E) = t. \end{aligned}$$
Thus Open image in new window . This holds for every j so Open image in new window and therefore \(\mathrm{dim}_F[L,H]\leqslant td\), as required.\(\square \)

Similarly we can prove the next result.

Lemma 4.3

Let L be a Leibniz algebra over a field F and H an ideal of L. Suppose that Open image in new window , where E is a suitable subspace such that \(\mathrm{dim}_FE = d\) is finite. If \(\mathrm{dim}_FH/\mathrm{Ann}_H^\mathrm{l}(E) = t\) is finite, then [HL] has finite dimension and \(\mathrm{dim}_F[H,L]\leqslant dt\).

Lemmas 4.2 and 4.3 immediately imply the following conclusion.

Corollary 4.4

Let L be a Leibniz algebra over a field F and H an ideal of L. Suppose that Open image in new window , where E is a suitable subspace such that \(\mathrm{dim}_FE = d\) is finite. If \(\mathrm{dim}_FH/\mathrm{Ann}_H(E) = t\) is finite, then \(\mathrm{dim}_F[L,H], \mathrm{dim}_F[H,L]\leqslant dt\).

Corollary 4.5

Let L be a Leibniz algebra over a field F and H an ideal of L. Suppose that \(\mathrm{codim}_F\mathrm{Ann}_L(H) = d\) is finite. If \(\mathrm{dim}_FH/(H\cap \zeta (L)) = t\) is finite, then \(\mathrm{dim}_F[L,H],\mathrm{dim}_F[H,L]\leqslant dt\).

Indeed, \(\mathrm{Ann}_H(E)\geqslant H\cap \zeta (L)\), so that \(\mathrm{dim}_FH/\mathrm{Ann}_H(E)\leqslant t\).

Proof of Theorem A

Let
$$\begin{aligned} \{0\} = Z_0\leqslant Z_1\leqslant \cdots \leqslant Z_{k-1}\leqslant Z_k = Z \end{aligned}$$
be the upper central series of L. We proceed by induction on k. If \(k = 1\), then the first center \(Z_1\) has finite codimension d and it suffices to apply Corollary B1 to obtain that \(\gamma _2(L) = [L,L]\) has dimension at most \(d^2\).
Assume that \(k > 1\) and we have already proved that
$$\begin{aligned} \mathrm{dim}_F\gamma _k(L/Z_1)\leqslant 2^{k-2}d^k = t. \end{aligned}$$
By Corollary 2.5, Open image in new window . Put \(K/Z_1 = \gamma _k(L/Z_1)\) and \(T = \gamma _k(L)\) so that \(T\leqslant K\) by Corollary 2.5. Further,
$$\begin{aligned} \dim _FT/(T\cap Z_1)\leqslant \mathrm{dim}_F K/Z_1\leqslant t. \end{aligned}$$
By Lemma 4.1, \(Z_k\leqslant \mathrm{Ann}_L(T)\) so that \(\mathrm{codim}_F\mathrm{Ann}_L(T)\leqslant d\). By Corollary 4.5, \(\mathrm{dim}_F[L,T],\mathrm{dim}_F[T,L]\leqslant dt\).
By Proposition 2.2, \(D = [L,T] + [T,L]\) is an ideal of L. The factor K / D is abelian hence Open image in new window is abelian. We have Open image in new window . By construction of D, we have \([L/D,T/D]\leqslant T/D\) and \([T/D,L/D]\leqslant T/D\). These inclusions show that \(T/D\leqslant \zeta (L/D)\). Since L / T is nilpotent of nilpotency class \(k-1\), the latter inclusion shows that L / D is nilpotent of nilpotency class at most k. By Corollary 2.5, \(\gamma _{k+1}(L)\leqslant D\), which impliesas required.\(\square \)

Using similar arguments we can obtain Corollary A1 for Lie algebras.

References

  1. 1.
    Albeverio, S., Ayupov, S.A., Omirov, B.A.: On nilpotent and simple Leibniz algebras. Commun. Algebr. 33(1), 159–172 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Albeverio, S., Omirov, B.A., Rakhimov, I.S.: Varieties of nilpotent complex Leibniz algebras of dimension less than five. Commun. Algebr. 33(5), 1575–1585 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ayupov, Sh.A., Omirov, B.A.: On Leibniz algebras. In: Khakimdjanov, Yu., Goze, M., Ayupov, Sh.A. (eds.) Algebra and Operators Theory, pp. 1–12. Kluwer, Dordrecht (1998)Google Scholar
  4. 4.
    Ayupov, Sh.A., Omirov, B.A.: On some classes of nilpotent Leibniz algebras. Sib. Math. J. 42(1), 15–24 (2001)Google Scholar
  5. 5.
    Baer, R.: Endlichkeitskriterien für Kommutatorgruppen. Math. Ann. 124, 161–177 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barnes, D.W.: Schunck classes of soluble Leibniz algebras. Commun. Algebr. 41(11), 4046–4065 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Batten Ray, C., Combs, A., Gin, N., Hedges, A., Hird, J.T., Zack, L.: Nilpotent Lie and Leibniz algebras. Commun. Algebr. 42(6), 2404–2410 (2014)Google Scholar
  8. 8.
    Blokh, A.: A generalization of the concept of a Lie algebra. Soviet Math. Dokl. 6, 1450–1452 (1965)zbMATHGoogle Scholar
  9. 9.
    Blokh, A.: Cartan–Eilenberg homology theory for a generalized class of Lie algebras. Soviet Math. Dokl. 8, 824–826 (1967)Google Scholar
  10. 10.
    Blokh, A.: A certain generalization of the concept of Lie algebra. Algebra and Number Theory, Moskov. Gos. Ped. Inst. Učen. Zap 375, 9–20 (1971) (in Russian)Google Scholar
  11. 11.
    Butterfield, J., Pagonis, C. (eds.): From Physics to Philosophy. Cambridge University Press, Cambridge (1999)Google Scholar
  12. 12.
    Casas, J.M., Pirashvili, T.: Ten-term exact sequence of Leibniz homology. J. Algebr. 231(1), 258–264 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dobrev, V. (ed.): Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol. 36. Springer, Tokyo (2013)Google Scholar
  14. 14.
    Duplij, S., Wess, J. (eds.): Noncommutative Structures in Mathematics and Physics. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 22. Kluwer, Dordrecht (2001)Google Scholar
  15. 15.
    Fialowski, A., Khudoyberdiyev, A.Kh., Omirov, B.A.: A characterization of nilpotent Leibniz algebras. Algebr. Represent. Theory 16(5), 1489–1505 (2013)Google Scholar
  16. 16.
    Frabetti, A.: Leibniz homology of dialgebras of matrices. J. Pure Appl. Algebra 129(2), 123–141 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Loday, J.-L.: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math. 39(3–4), 269–293 (1993)MathSciNetGoogle Scholar
  18. 18.
    Loday, J.-L., Pirashvili, T.: Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296(1), 139–158 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Neumann, B.H.: Groups with finite classes of conjugate elements. Proc. London Math. Soc. s3–1, 178–187 (1951)Google Scholar
  20. 20.
    Patsourakos, A.: On nilpotent properties of Leibniz algebras. Commun. Algebr. 35(12), 3828–3834 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pirashvili, T.: On Leibniz homology. Ann. Inst. Fourier (Grenoble) 44(2), 401–411 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Stewart, I.N.: Verbal and marginal properties of non-associative algebras. Proc. London Math. Soc. s3–28, 129–140 (1974)Google Scholar
  23. 23.
    Vaughan-Lee, M.R.: Metabelian BFC \(p\)-groups. J. London Math. Soc. s2–5(4), 673–680 (1972)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Leonid A. Kurdachenko
    • 1
  • Javier Otal
    • 2
    Email author
  • Alexander A. Pypka
    • 1
  1. 1.Faculty of Mechanics and MathematicsNational University of DnepropetrovskDnepropetrovskUkraine
  2. 2.Department of Mathematics-IUMAUniversity of ZaragozaZaragozaSpain

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