The geometric classification of Leibniz algebras

TL;DR

This paper classifies all rigid four-dimensional Leibniz algebras over complex numbers, identifies irreducible components in their algebraic variety, and examines conjectures related to their structure, providing a comprehensive geometric understanding.

Contribution

It provides a complete geometric classification of four-dimensional Leibniz algebras, including the validation and refutation of key conjectures in the field.

Findings

Identified all rigid algebras in $\

,

,

Abstract

We describe all rigid algebras and all irreducible components in the variety of four dimensional Leibniz algebras $\mathfrak{Leib}_4$ over $\mathbb{C}.$ In particular, we prove that the Grunewald--O'Halloran conjecture is not valid and the Vergne conjecture is valid for $\mathfrak{Leib}_4.$

Equations

$$(xy)z=(xz)y+x(yz).$$

$$(g\ast \mu )(x\otimes y)=g\mu (g^{-1}x\otimes g^{-1}y)$$

$$\begin{array}[]{llllll}\mathfrak{R}_{2}&e_{1}e_{2}=-e_{2}e_{1}=e_{2},&e_{3}e_{4}=-e_{4}e_{3}=e_{4};\\ \mathfrak{sl}_{2}\oplus\mathbb{C}&e_{1}e_{2}=-e_{2}e_{1}=e_{2},&e_{1}e_{3}=-e_{3}e_{1}=-e_{3},&e_{2}e_{3}=-e_{3}e_{2}=e_{1};\\ g_{4}(a,b)&e_{1}e_{2}=-e_{2}e_{1}=e_{2},&e_{1}e_{3}=-e_{3}e_{1}=e_{2}+ae_{3},&e_{1}e_{4}=-e_{4}e_{1}=e_{3}+be_{4};\\ g_{5}(a)&e_{1}e_{2}=-e_{2}e_{1}=e_{2},&e_{1}e_{3}=-e_{3}e_{1}=e_{2}+ae_{3},\\ &e_{1}e_{4}=-e_{4}e_{1}=(a+1)e_{4},&e_{2}e_{3}=-e_{3}e_{2}=e_{4}.\end{array}$$

$$\overline{\{O(g_4(a,b)){\}}_{a,b\in \mathbb{C}}},\overline{\{O(g_5(a)){\}}_{a\in \mathbb{C}}},\overline{\{O({\mathfrak{L}}_4^a){\}}_{a\in \mathbb{C}}},\overline{\{O({\mathfrak{L}}_8^a){\}}_{a\in \mathbb{C}}},\overline{\{O({\mathfrak{L}}_9^a){\}}_{a\in \mathbb{C}}},\overline{\{O({\mathfrak{L}}_{10}^a){\}}_{a\in \mathbb{C}}},$$

$$\overline{\{O({\mathfrak{L}}_{15}^a){\}}_{a\in \mathbb{C}}},\overline{\{O({\mathfrak{L}}_{18}^a){\}}_{a\in \mathbb{C}}},\overline{\{O({\mathfrak{L}}_{21}^{a,b}){\}}_{a,b\in \mathbb{C}}},\overline{\{O({\mathfrak{L}}_{22}^{a,b}){\}}_{a,b\in \mathbb{C}}},\overline{\{O({\mathfrak{L}}_{23}^{a,b}){\}}_{a,b\in \mathbb{C}}},$$

$$\overline{O({\mathfrak{sl}}_2\oplus \mathbb{C})},\overline{O({\mathfrak{R}}_1)},\overline{O({\mathfrak{R}}_2)},\overline{O({\mathfrak{R}}_3)},\overline{O({\mathfrak{L}}_{44})}.$$

$${\mathfrak{sl}}_2\oplus \mathbb{C},{\mathfrak{R}}_1,{\mathfrak{R}}_2,{\mathfrak{R}}_3,{\mathfrak{L}}_{44}.$$

$$\begin{array}[]{lcllllll}\mathfrak{N}_{3}^{a}&e_{1}e_{1}=e_{4},&e_{1}e_{2}=ae_{4},&e_{2}e_{1}=-ae_{4},&e_{2}e_{2}=e_{4},&e_{3}e_{3}=e_{4};\\ \mathfrak{L}^{n}_{2}&e_{1}e_{1}=e_{2},&e_{2}e_{1}=e_{3},&e_{3}e_{1}=e_{4};\\ \mathfrak{L}^{n}_{5}&e_{1}e_{1}=e_{3},&e_{2}e_{1}=e_{3},&e_{2}e_{2}=e_{4},&e_{3}e_{1}=e_{4};\\ \mathfrak{L}^{n}_{11}&e_{1}e_{1}=e_{4},&e_{1}e_{2}=-e_{3},&e_{1}e_{3}=-e_{4},&e_{2}e_{1}=e_{3},&e_{2}e_{2}=e_{4},&e_{3}e_{1}=e_{4}.\\ \end{array}$$

$$E_1^t{E}_1^t=t^2{e}_2+t^2{e}_4=E_2^t,E_2^t{E}_1^t=t^3{e}_3+t^3{e}_4=E_3^t,E_3^t{E}_1^t=t^4{e}_4=t{E}_3^t+E_4^t.$$

$$\mathcal{R}=\left\{\mu\in\mathfrak{Leib}_{4}\left|\begin{array}[]{c}S_{1}S_{3}+S_{4}S_{2}=0,S_{3}S_{2}+S_{4}S_{1}\subset S_{4},S_{2}S_{2}+S_{3}S_{1}\subset S_{3},S_{2}S_{1}+S_{1}S_{2}\subset S_{2},\\ c_{12}^{2}+c_{21}^{2}=0,c_{31}^{3}=2c_{21}^{2},c_{12}^{3}=c_{21}^{3}\end{array}\right.\right\}.$$

Full text

The geometric classification of Leibniz algebras 111The work was supported by RFBR 17-01-00258 and by the President’s Program ”Support of Young Russian Scientists” (grant MK-1378.2017.1).

Nurlan Ismailova, Ivan Kaygorodovb, Yury Volkovc

a Suleyman Demirel University, Almaty, Kazakhstan

b Universidade Federal do ABC, CMCC, Santo André, Brazil.

c Saint Petersburg State University, Saint Petersburg, Russia.

E-mail addresses:

Nurlan Ismailov (nurlan.ismail@gmail.com),

Ivan Kaygorodov (kaygorodov.ivan@gmail.com),

Yury Volkov (wolf86_666@list.ru).

Abstract. We describe all rigid algebras and all irreducible components in the variety of four dimensional Leibniz algebras $\mathfrak{Leib}_{4}$ over $\mathbb{C}.$ In particular, we prove that the Grunewald–O’Halloran conjecture is not valid and the Vergne conjecture is valid for $\mathfrak{Leib}_{4}.$

Keywords: Leibniz algebra, Grunewald–O’Halloran conjecture, Vergne conjecture, orbit closure, degeneration, rigid algebra

2010 MSC: 14D06, 14L30.

1. Introduction

Degenerations of algebras is an interesting subject, which was studied in various papers (see, for example, [M79, CKLO13, BC99, S90, GRH, GRH2, BB14, kpv16, kppv, kv17]). In particular, there are many results concerning degenerations of algebras of low dimensions in a variety defined by a set of identities. One of important problems in this direction is the description of so-called rigid algebras. These algebras are of big interest, since the closures of their orbits under the action of generalized linear group form irreducible components of a variety under consideration (with respect to the Zariski topology). For example, the rigid algebras were classified in the varieties of two dimensional bicommutative algebras in [kv17], three dimensional Novikov algebras in [BB14], three dimensional Leibniz in [ra12], four dimensional Lie algebras in [BC99], four dimensional Jordan algebras in [KE14], four dimensional Zinbiel algebras and nilpotent four dimensional Leibniz algebras in [kppv], unital five dimensional associative algebras in [M79], nilpotent five- and six-dimensional Lie algebras in [S90, GRH], nilpotent five- and six-dimensional Malcev algebras in [kpv16], and some other.

The Leibniz algebras were introduced as a generalization of Lie algebras. The study of the structure theory and other properties of Leibniz algebras was initiated by Loday in [lodaypir]. Leibniz algebras were also studied in [leib2, leib4, yau]. An algebra $A$ is called a Leibniz algebra if it satisfies the identity

[TABLE]

It is easy to see that any Lie algebra is a Leibniz algebra. At this moment, the algebraic classification of $n$ dimensional Leibniz algebras over $\mathbb{C}$ is known only for $n\leq 4$.

In this paper we describe all rigid algebras and all irreducible components in variety of $\mathfrak{Leib}_{4}$. As a result, we show that $\mathfrak{Leib}_{4}$ has $5$ rigid algebras and $16$ irreducible components.

2. Definitions and notation

All spaces in this paper are considered over $\mathbb{C}$, and we write simply $dim$, $Hom$ and $\otimes$ instead of $dim_{\mathbb{C}}$, $Hom_{\mathbb{C}}$ and $\otimes_{\mathbb{C}}$. An algebra $A$ is a set with a structure of a vector space and a binary operation that induces a bilinear map from $A\times A$ to $A$.

Given an $n$-dimensional vector space $V$, the set $Hom(V\otimes V,V)\cong V^{*}\otimes V^{*}\otimes V$ is a vector space of dimension $n^{3}$. This space has a structure of the affine variety $\mathbb{C}^{n^{3}}.$ Indeed, let us fix a basis $e_{1},\dots,e_{n}$ of $V$. Then any $\mu\in Hom(V\otimes V,V)$ is determined by $n^{3}$ structure constants $c_{i,j}^{k}\in\mathbb{C}$ such that $\mu(e_{i}\otimes e_{j})=\sum\limits_{k=1}^{n}c_{i,j}^{k}e_{k}$. A subset of $Hom(V\otimes V,V)$ is Zariski-closed if it can be defined by a set of polynomial equations in the variables $c_{i,j}^{k}$ ($1\leq i,j,k\leq n$).

All algebra structures on $V$ satisfying Leibniz identity form a Zariski-closed subset of the variety $Hom(V\otimes V,V)$. We denote this subset by $\mathfrak{Leib}_{n}$. The general linear group $GL(V)$ acts on $\mathfrak{Leib}_{n}$ by conjugations:

[TABLE]

for $x,y\in V$, $\mu\in\mathfrak{Leib}_{n}\subset Hom(V\otimes V,V)$ and $g\in GL(V)$. Thus, $\mathfrak{Leib}_{n}$ is decomposed into $GL(V)$-orbits that correspond to the isomorphism classes of algebras. Let $O(\mu)$ denote the orbit of $\mu\in\mathfrak{Leib}_{n}$ under the action of $GL(V)$ and $\overline{O(\mu)}$ denote the Zariski closure of $O(\mu)$.

Let $A$ and $B$ be two $n$-dimensional Leibniz algebras and $\mu,\lambda\in\mathfrak{Leib}_{n}$ represent $A$ and $B$ respectively. We say that $A$ degenerates to $B$ and write $A\to B$ if $\lambda\in\overline{O(\mu)}$. Note that in this case we have $\overline{O(\lambda)}\subset\overline{O(\mu)}$. Hence, the definition of a degeneration does not depend on the choice of $\mu$ and $\lambda$. We write $A\not\to B$ if $\lambda\not\in\overline{O(\mu)}$.

Let $A$ be represented by $\mu\in\mathfrak{Leib}_{n}$. Then $A$ is rigid in $\mathfrak{Leib}_{n}$ if $O(\mu)$ is an open subset of $\mathfrak{Leib}_{n}.$ Recall that a subset of a variety is called irreducible if it cannot be represented as a union of two non-trivial closed subsets. A maximal irreducible closed subset of a variety is called irreducible component. It is well known that any affine variety can be represented as a finite union of its irreducible components in a unique way. In particular, $A$ is rigid in $\mathfrak{Leib}_{n}$ iff $\overline{O(\mu)}$ is an irreducible component of $\mathfrak{Leib}_{n}.$ This is a general fact about algebraic varieties whose proof can be found, for example, in [kppv].

We use the following notation:

1. (1)

$Ann_{L}(A)=\{a\in A\mid xa=0\mbox{ for all }x\in A\}$ is the left annihilator of $A;$ 2. (2)

$Ann_{R}(A)=\{a\in A\mid ax=0\mbox{ for all }x\in A\}$ is the right annihilator of $A;$ 3. (3)

$Ann(A)=Ann_{R}(A)\cap Ann_{L}(A)$ is the annihilator of $A;$ 4. (4)

$A^{(+2)}$ is the space $\{xy+yx\mid x,y\in A\}.$

Given spaces $U$ and $W$, we write simply $U>W$ instead of $dim\,U>dim\,W$. We write $UW$ ($U,W\subset V$) for the product of subspaces of $V$ with respect to the multiplication $\mu$. We use the notation $S_{i}=\langle e_{i},\dots,e_{4}\rangle,\ i=1,\ldots,4$.

3. Methods

In the present work we use the methods applied in our previous works (see [kv17, kpv16, kppv]).

To prove degenerations, we will construct families of matrices parametrized by $t$. Namely, let $A$ and $B$ be two Leibniz algebras represented by the structures $\mu$ and $\lambda$ from $\mathfrak{Leib}_{n}$ respectively. Let $e_{1},\dots,e_{n}$ be a basis of $V$ and $c_{i,j}^{k}$ ($1\leq i,j,k\leq n$) be the structure constants of $\lambda$ in this basis. If there exist $a_{i}^{j}(t)\in\mathbb{C}$ ($1\leq i,j\leq n$, $t\in\mathbb{C}^{*}$) such that $E_{i}^{t}=\sum\limits_{j=1}^{n}a_{i}^{j}(t)e_{j}$ ($1\leq i\leq n$) form a basis of $V$ for any $t\in\mathbb{C}^{*}$, and the structure constants of $\mu$ in the basis $E_{1}^{t},\dots,E_{n}^{t}$ are such polynomials $c_{i,j}^{k}(t)\in\mathbb{C}[t]$ that $c_{i,j}^{k}(0)=c_{i,j}^{k}$, then $A\to B$. In this case $E_{1}^{t},\dots,E_{n}^{t}$ is called a parametrized basis for $A\to B$. It is easy to see that any algebra degenerates to the algebra with zero multiplication.

Let now $A(*):=\{A(\alpha)\}_{\alpha\in I}$ be a set of algebras, and let $B$ be another algebra. Suppose that, for $\alpha\in I$, $A(\alpha)$ is represented by the structure $\mu(\alpha)\in\mathfrak{Leib}_{n}$ and $B\in\mathfrak{Leib}_{n}$ is represented by the structure $\lambda$. Then $A(*)\to B$ means $\lambda\in\overline{\{O(\mu(\alpha))\}_{\alpha\in I}}$, and $A(*)\not\to B$ means $\lambda\not\in\overline{\{O(\mu(\alpha))\}_{\alpha\in I}}$.

Let $A(*)$, $B$, $\mu(\alpha)$, and $\lambda$ be as above. To prove $A(*)\to B$ it is enough to construct a family of pairs $(g(t),f(t))$ parametrized by $t\in\mathbb{C}^{*}$, where $f(t)\in I$ and $g(t)\in GL(V)$, with the following properties: Let $e_{1},\dots,e_{n}$ be a basis of $V$ and $c_{i,j}^{k}$ ($1\leq i,j,k\leq n$) be the structure constants of $\lambda$ in this basis. If we construct $a_{i}^{j}:\mathbb{C}^{*}\to\mathbb{C}$ ($1\leq i,j\leq n$) and $f:\mathbb{C}^{*}\to I$ such that $E_{i}^{t}=\sum\limits_{j=1}^{n}a_{i}^{j}(t)e_{j}$ ($1\leq i\leq n$) form a basis of $V$ for any $t\in\mathbb{C}^{*}$, and the structure constants of $\mu_{f(t)}$ in the basis $E_{1}^{t},\dots,E_{n}^{t}$ are such polynomials $c_{i,j}^{k}(t)\in\mathbb{C}[t]$ that $c_{i,j}^{k}(0)=c_{i,j}^{k}$, then $A(*)\to B$. In this case $E_{1}^{t},\dots,E_{n}^{t}$ and $f(t)$ are called a parametrized basis and a parametrized index for $A(*)\to B$ respectively.

Note also the following fact. Suppose that, for $\alpha\in\mathbb{C}$, the structure $\mu(\alpha)\in\mathfrak{Leib}_{n}$ has structure constants $c_{i,j}^{k}(\alpha)\in\mathbb{C}$ in the basis $e_{1},\dots,e_{n}$, where $c_{i,j}^{k}(t)\in\mathbb{C}[t]$ for all $1\leq i,j,k\leq n$. Let $X$ be some subset of $\mathfrak{Leib}_{n}$ such that $\mu(\alpha)\in\overline{X}$ for $\alpha\in\mathbb{C}\setminus S$, where $S$ is a finite subset of $\mathbb{C}$. Then $\mu(\alpha)\in\overline{X}$ for all $\alpha\in\mathbb{C}$. Indeed, $\mu(\alpha)\in\overline{\{\mu(\beta)\}_{\beta\in\mathbb{C}\setminus S}}\subset\overline{X}$ for any $\alpha\in\mathbb{C}$. Thus, to prove that $\mu(\alpha)\in\overline{X}$ for all $\alpha\in\mathbb{C}$ we will prove that $\mu(\alpha)\in\overline{X}$ for all but finitely many $\alpha$.

Note that $A(*)\not\to B$ if $dim\,Der(A(\alpha))>dim\,Der(B)$ for all $\alpha\in I$, where $Der(A)$ is the Lie algebra of derivations of $A$. Moreover, in the case of one algebra, if $A\to B$ and $A\not\cong B$, then $Der(A)<Der(B)$.

In other cases, the main tool for proving assertions of the form $A(*)\not\to B$ will be the following lemma.

Lemma 1** (Lemma 2 of [kppv]).**

Let $\mathcal{B}$ be a Borel subgroup of $GL(V)$ and $\mathcal{R}\subset\mathfrak{Leib}_{n}$ be a $\mathcal{B}$-stable closed subset. If $A(*)\to B$ and for any $\alpha\in I$ the algebra $A(\alpha)$ can be represented by a structure $\mu(\alpha)\in\mathcal{R}$, then there is $\lambda\in\mathcal{R}$ representing $B$.

In all our applications of Lemma 1 we will use the group of lower triangular matrices as a Borel subgroup. This lemma can be applied in the case $|I|=1$ to prove non-degenerations. In particular, if $Ann_{L}(A)>Ann_{L}(B)$, $A^{2}<B^{2}$ or $A^{(+2)}<B^{(+2)}$, then $A\not\to B$ by Lemma 1. We will use also the following two criteria that follow from Lemma 1. If $A\to B$, then the dimension of a maximal trivial subalgebra of $B$ is greater or equal to the dimension of a maximal trivial subalgebra of $A$ and the dimension of a maximal anticommutative subalgebra $D$ of $B$ such that $dim(BD)\leq 1$ is greater or equal to the dimension of a maximal anticommutative subalgebra of $A$ satisfying the same condition.

In some cases we will construct the set $\mathcal{R}$ for the assertion $A(*)\not\to B$ explicitly. In this case we will define $\mathcal{R}$ as a set of structures $\mu$ satisfying some polynomial equations. In such a description we always denote by $c_{ij}^{k}$ ($1\leq i,j,k\leq n$) the structure constants of $\mu$. Note also that in this case a condition of the form $S_{i}S_{j}\subset S_{k}$ is equivalent to a set of polynomial equations. Moreover, the set defined by such a condition is stable under the action of the group of lower triangular matrices.

4. Main theorem

The goal of this section is to describe irreducible components in $\mathfrak{Leib}_{4}$. The algebraic classification of four dimensional Lebniz algebras is based on the papers [ort13, Abror13, alb06]. Let us give a brief introduction to this classification. The analogue of Levi-Malcev’s theorem about the splitting of the solvable radical for Leibniz algebras was proved in [barnes]. Note that any semisimple (here this means that the solvable radical is zero) Leibniz algebra is a Lie algebra, and hence a Leibniz algebra is always the hemisemidirect product of a semisimple Lie algebra and a solvable ideal. The first step in the classification of four dimensional Leibniz algebras was done in [alb06], where all nilpotent four dimensional Leibniz algebras were classified. Then the description of all four dimensional solvable non-nilpotent Lebniz algebras was obtained in [Abror13]. Finally, it was proved in [ort13] that there is only one non-solvable indecomposable Leibniz algebra whose dimension is less or equal to four, namely, the simple Lie algebra $\mathfrak{sl}_{2}$.

Based on the just mentioned works, we have constructed Table 1 that describes all four dimensional non-nilpotent non-Lie Leibniz algebras. Let us also introduce the following four dimensional Lie algebra structures:

[TABLE]

Due to the results of [BC99], the variety of four dimensional Lie algebras contains $4$ irreducible components, namely, $\overline{O(\mathfrak{sl}_{2}\oplus\mathbb{C})}$, $\overline{O(\mathfrak{R}_{2})}$, $\overline{\{O(g_{5}(a))\}_{a\in\mathbb{C}}}$, and $\overline{\{O(g_{4}(a,b))\}_{a,b\in\mathbb{C}}}$. In particular, there are $2$ rigid four dimensional Lie algebras, namely, $\mathfrak{sl}_{2}\oplus\mathbb{C}$ and $\mathfrak{R}_{2}$.

The main result of the present paper is the following theorem.

Theorem 2**.**

The variety $\mathfrak{Leib}_{4}$ has $16$ irreducible components:

[TABLE]

[TABLE]

[TABLE]

In particular, there are $5$ rigid four dimensional Leibniz algebras:

[TABLE]

Non-Lie algebras mentioned in the theorem are described in Table 1. The rest of the paper is devoted to the proof of Theorem 2.

4.1. Nilpotent Leibniz algebras and conjectures about them.

Several conjectures state that nilpotent Lie algebras form a very small subvariety in the variety of Lie algebras. Grunewald and O’Halloran conjectured in [GRH3] that for any $n$-dimensional nilpotent Lie algebra $A$ there exists an $n$-dimensional non-nilpotent Lie algebra $B$ such that $B\to A$. At the same time, Vergne conjectured in [V70] that a nilpotent Lie algebra cannot be rigid in the variety of all Lie algebras. Analogous assertions can be conjectured for other varieties. We will say that the variety $\mathfrak{A}$ of algebras has Grunewald–O’Halloran Property if for any nilpotent algebra $A\in\mathfrak{A}$ there is a non-nilpotent algebra $B\in\mathfrak{A}$ such that $B\to A$. We will say that $\mathfrak{A}$ has Vergne Property if there are no nilpotent rigid algebras in $\mathfrak{A}$. We will also say that $\mathfrak{A}$ has Vergne–Grunewald–O’Halloran Property if any irreducible component of $\mathfrak{A}$ contains a non-nilpotent algebra. Grunewald–O’Halloran Property was proved for four dimensional Lie and Jordan algebras in [BC99, KE14] and for three dimensional Novikov and Leibniz algebras in [BB14, CKLO13]. Also, some results concerning Grunewald–O’Halloran Conjecture for Lie algebras were obtained in [ht14, ht16].

It is clear that Vergne–Grunewald–O’Halloran Property follows from Grunewald–O’Halloran Property and Vergne Property follows from Vergne–Grunewald–O’Halloran Property. As a part of the proof of Theorem 2, we will show that $\mathfrak{Leib}_{4}$ has Vergne–Grunewald–O’Halloran Property. On the other hand, we will show that the same variety does not have Grunewald–O’Halloran Property.

Let us also introduce the following four dimensional nilpotent Lebniz algebra structures:

[TABLE]

It was proved in [kppv] that the variety of four dimensional nilpotent Leibniz algebras is formed by four irreducible components, namely, $\overline{O(\mathfrak{L}_{2}^{n})}$, $\overline{O(\mathfrak{L}_{5}^{n})}$, $\overline{O(\mathfrak{L}_{11}^{n})}$ and $\overline{\{O(\mathfrak{N}_{3}^{a})\}_{a\in\mathbb{C}}}$.

Lemma 3**.**

Any irreducible component in $\mathfrak{Leib}_{4}$ contains a non-nilpotent algebra.

Proof.

It is enough to prove that $\mathfrak{L}^{n}_{2}$, $\mathfrak{L}^{n}_{5}$, $\mathfrak{L}^{n}_{11}$ and $\mathfrak{N}_{3}^{a}$ belong to the closure of the union of orbits of non-nilpotent Lebniz algebras for any $a\in\mathbb{C}$. Let us prove this assertion case by case.

• •

The parametrized basis $E_{1}^{t}=te_{1}+te_{4}$, $E_{2}^{t}=t^{2}e_{2}+t^{2}e_{4}$, $E_{3}^{t}=t^{3}e_{3}+t^{3}e_{4}$, $E_{4}^{t}=-t^{4}e_{3}$ gives the degeneration $\mathfrak{L}_{40}\to\mathfrak{L}_{2}^{n}$.

• •

The parametrized basis $E_{1}^{t}=te_{1}-te_{2}+e_{3}$, $E_{2}^{t}=t^{2}e_{2}+e_{3}$, $E_{3}^{t}=te_{3}-t^{3}e_{4}$, $E_{4}^{t}=t^{4}e_{4}$ with parametrized index $\epsilon(t)=-t$ give the assertion $\mathfrak{L}_{18}^{*}\to\mathfrak{L}_{5}^{n}$.

• •

The parametrized basis $E_{1}^{t}=te_{1}-\frac{t}{2}e_{2}$, $E_{2}^{t}=\frac{t^{2}}{2}e_{2}+t^{2}e_{3}$, $E_{3}^{t}=t^{3}e_{3}-\frac{t^{3}}{4}e_{4}$, $E_{4}^{t}=\frac{t^{4}}{4}e_{4}$ with parametrized index $\epsilon(t)=\frac{t^{2}+1}{4}$ give the assertion $\mathfrak{L}_{15}^{*}\to\mathfrak{L}_{11}^{n}$.

• •

The parametrized basis $E_{1}^{t}=te_{1}+\frac{t}{2}e_{4}$, $E_{2}^{t}=ite_{1}-ie_{3}-\frac{it}{2}e_{4}$, $E_{3}^{t}=te_{2}$, $E_{4}^{t}=t^{2}e_{4}$ gives the degeneration $\mathfrak{L}_{9}^{1+ia}\to\mathfrak{N}_{3}^{a}$ for any $a\in\mathbb{C}$.

Let us consider the degeneration $\mathfrak{L}_{40}\to\mathfrak{L}_{2}^{n}$ to clarify our proof. Writing nonzero products of $\mathfrak{L}_{40}$ in the basis $E_{i}^{t}$, we get

[TABLE]

It is easy to see now that for $t=0$ we obtain the multiplication table of $\mathfrak{L}_{2}^{n}.$ The remaining assertions can be considered in the same way. ∎

Though we have proved that $\overline{O(\mathfrak{L}_{5}^{n})}$ and $\overline{O(\mathfrak{L}_{11}^{n})}$ are not irreducible components of $\mathfrak{Leib}_{4}$, we have not found an algebra that degenerates either to $\mathfrak{L}_{5}^{n}$ or to $\mathfrak{L}_{11}^{n}$ in the proof of Lemma 3. In the proof of the next result we show that in fact there is no algebra that degenerates to $\mathfrak{L}_{5}^{n}$.

Theorem 4**.**

$\mathfrak{Leib}_{4}$* does not have Grunewald–O’Halloran Property.*

Proof.

To prove the theorem, we will show that there is no four dimensional Leibniz algebra that degenerates to $\mathfrak{L}_{5}^{n}.$ Since $\mathfrak{L}_{5}^{n}$ is not a Lie algebra, only algebras from Table 1 can degenerate to it. Direct calculations show that the dimension of the Lie algebra of derivations is greater or equal to $3$ for all algebras from Table 1 except $\mathfrak{R}_{1}$ and $\mathfrak{L}_{44}$. Since $dim\,Der(\mathfrak{L}_{5}^{n})=3$ (see, for example [kppv]) it suffices to prove that $\mathfrak{R}_{1}\not\to\mathfrak{L}_{5}^{n}$ and $\mathfrak{L}_{44}\not\to\mathfrak{L}_{5}^{n}$.

• •

$\mathfrak{R}_{1}\not\to\mathfrak{L}_{5}^{n}$ follows from the fact that $Ann_{L}(\mathfrak{R}_{1})>Ann_{L}(\mathfrak{L}_{5}^{n})$.

• •

To prove the assertion $\mathfrak{L}_{44}\not\to\mathfrak{L}_{5}^{n}$ let us consider the set

[TABLE]

It is not difficult to show that $\mathcal{R}$ is a closed subset of $\mathfrak{Leib}_{4}$ that is stable under the action of the subgroup of lower triangular matrices and contains the structure $\mathfrak{L}_{44}$. It is also not difficult to show that $\mathcal{R}\cap O(\mathfrak{L}_{5}^{n})=\varnothing$.

∎

4.2. The proof of the main theorem.

Now we are ready to prove Theorem 2. As a first step, we are going to prove that the irreducible components of the variety of four dimensional Lie algebras remain irreducible in $\mathfrak{Leib}_{4}$. This fact follows from the next general lemma.

Lemma 5**.**

Suppose that $A(*)=\{A(\alpha)\}_{\alpha\in T}$ is a set of $n$-dimensional non-Lie Leibniz algebras. If $B$ is an $n$-dimensional Lie algebra such that $Ann(B)=0$, then $A(*)\not\to B$.

• Proof.

Since $A(\alpha)$ is non-Lie, the ideal $I(\alpha)=\{xy+yx\mid x,y\in A(\alpha)\}$ is nonzero for any $\alpha\in T$. Since I(\alpha)\subset Ann_{L}\big{(}A(\alpha)\big{)}, we have Ann_{L}\big{(}A(\alpha)\big{)}>0=Ann(B)=Ann_{L}(B) for any $\alpha\in T$, and hence $A(*)\not\to B$.

$\Box$

It easily follows from Lemma 5 that if $\mathcal{C}$ is an irreducible component in the variety of $n$-dimensional Lie algebras containing an algebra with zero annihilator, then $\mathcal{C}$ is an irreducible component in $\mathfrak{Leib}_{n}$. Indeed, suppose that $\mathcal{C}_{0}$ is an irreducible component of $\mathfrak{Leib}_{n}$ containing $\mathcal{C}$. Let $\mathcal{L}$ be a set of all Lie algebras in $\mathcal{C}_{0}$. We have $\mathcal{C}\not\subset\overline{\mathcal{C}_{0}\setminus\mathcal{L}}$ by Lemma 5. Since $\mathcal{C}_{0}=\overline{\mathcal{C}_{0}\setminus\mathcal{L}}\cup\mathcal{L}$ and $\mathcal{C}_{0}$ is irreducible, we have $\mathcal{C}_{0}=\mathcal{L}$, and hence $\mathcal{C}_{0}=\mathcal{C}$.

Corollary 6**.**

$\overline{O(\mathfrak{sl}_{2}\oplus\mathbb{C})}$, $\overline{O(\mathfrak{R}_{2})}$, $\overline{\{O(g_{5}(a))\}_{a\in\mathbb{C}}}$ and $\overline{\{O(g_{4}(a,b))\}_{a,b\in\mathbb{C}}}$ are irreducible components of $\mathfrak{Leib}_{4}$.

• Proof.

The structure $\mathfrak{sl}_{2}\oplus\mathbb{C}$ is rigid in $\mathfrak{Leib}_{4}$ as a unique non-solvable structure in the variety. The remaining part of the corollary follows from the fact that $Ann(\mathfrak{R}_{2})=0$, Ann\big{(}g_{5}(a)\big{)}=0 for $a\not=-1$, and Ann\big{(}g_{4}(a,b)\big{)}=0 for all $a,b\in\mathbb{C}^{*}$.