The variety of $2$-dimensional algebras over an algebraically closed field

TL;DR

This paper classifies 2-dimensional algebras over an algebraically closed field, analyzes their degenerations, and describes the structure of subvarieties including flexible and bicommutative algebras, identifying rigid algebras and irreducible components.

Contribution

It provides a comprehensive classification and structural analysis of 2-dimensional algebras and their subvarieties, including degenerations and rigid algebras.

Findings

Classification of 2-dimensional algebras up to isomorphism.

Description of degenerations and closures of algebra series.

Identification of rigid algebras and irreducible components.

Abstract

The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the variety under consideration. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible, and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

Equations

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

$$\begin{gathered} {\tilde{c}}_{11}^1=c_{11}^1+\alpha (c_{12}^1+c_{21}^1)+{\alpha }^2{c}_{22}^1,{\tilde{c}}_{21}^1=c_{21}^1+\alpha c_{22}^1,{\tilde{c}}_{12}^1=c_{12}^1+\alpha c_{22}^1,{\tilde{c}}_{22}^1=c_{22}^1, \\ {\tilde{c}}_{11}^2=c_{11}^2+\alpha (c_{12}^2+c_{21}^2-c_{11}^1)+{\alpha }^2(c_{22}^2-c_{12}^1-c_{21}^1)-{\alpha }^3{c}_{22}^1, \\ {\tilde{c}}_{21}^2=c_{21}^2+\alpha (c_{22}^2-c_{21}^1)-{\alpha }^2{c}_{22}^1,{\tilde{c}}_{12}^2=c_{12}^2+\alpha (c_{22}^2-c_{12}^1)-{\alpha }^2{c}_{22}^1,{\tilde{c}}_{22}^2=c_{22}^2-\alpha c_{22}^1. \end{gathered}$$

Full text

The variety of $2$-dimensional algebras over an algebraically closed field 111 The work was supported by FAPESP 14/24519-8; RFBR 17-51-04004; the President’s Program ”Support of Young Russian Scientists” (grant MK-1378.2017.1).

Ivan Kaygorodova, Yury Volkovb

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

b Saint Petersburg state university, Saint Petersburg, Russia.

E-mail addresses:

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

Yury Volkov (wolf86_666@list.ru).

Abstract. The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of $2$-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible, and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

Keywords: $2$-dimensional algebras, orbit closure, degeneration, rigid algebra

1. Introduction

In this paper, an algebra is simply a vector space over a field with a bilinear binary operation that doesn’t have to be associative. Algebras of a fixed dimension form a variety with a natural action of a general linear group. Orbits under this action correspond to isomorphism classes of algebras. There are many classifications up to isomorphism for varieties of algebras of some fixed dimension satisfying some polynomial identities. For example, there exist such classifications of $3$-dimensional Novikov algebras [bai], $4$-dimensional Leibniz algebras [Abror13], $6$-dimensional Lie algebras [SW] and many others.

In this paper we classify all $2$-dimensional algebras over an algebraically closed field up to isomorphism. It is not the first work devoted to this problem, classifications of different types were made in [goze, GR11, mir00], but all of them are not convenient for our main goal, the geometric description of the algebraic variety of $2$-dimensional algebras. One of the advantages of our paper is that our approach deals uniformly with all possible characteristics while the authors of [mir00] don’t consider the characteristics $2$ and $3$ and the authors of [goze] consider only the two elements field in the characteristic $2$. The authors of [mir00] in fact don’t give an explicit classification of $2$-dimensional algebras up to isomorphism because they have other purposes. They describe the moduli space by proving that $2$-dimensional algebras can be divided into parts that can be naturally included into projective spaces of different dimensions. The authors pretend that the classification up to isomorphism is easy and could be extracted from their proofs. The classification is really not very difficult and we believe that one can extract it from [mir00] after reading the paper, taking parts of the classification from different places and taking in account carefully all the details while for us it was easier to produce this classification from scratch. The authors of [goze] have produced a full classification. One of the problems is that this classification is outstretched through the whole paper and is mixed with other formulas. To collect all the parts of the classification from [goze] in one place and find all the additional conditions for these parts one has to fulfill a tedious work. At the same time, [goze] contains some inaccuracies. For example, the series $\mu_{10}$ parametrized by two scalars has to be divided into two series parametrized by one scalar, the series $\mu_{11}$ admits nontrivial isomorphisms, and in the case of a commutative $2$-dimensional algebra with one idempotent $e$ it may be impossible to find $f$ linear independent with $e$ such that $f^{2}$ and $e$ are linearly dependent. The paper [GR11] is very nice and gives the full classification of $2$-dimensional algebras over any field. Unfortunately, the answer is not given in terms of multiplication tables. The translation of this answer to the language of multiplication tables as well as its direct usage for the description of orbit closures is very difficult and it seems to be easier to produce a new appropriate classification. Also the consideration of arbitrary fields complicates the result and the extraction of the answer for an algebraically closed field becomes tedious. For these reasons, we give a classification that is valid over an algebraically closed field of arbitrary characteristic. In the same part of the paper, we also describe the automorphism groups for all algebras under consideration.

In the main part of our paper we develop the geometry of the variety of $2$-dimensional algebras. Namely, we describe the closures of orbits of some sets with respect the Zariski topology. Firstly, we describe all possible degenerations, i.e. closures of orbits of one point sets. Degenerations are an interesting subject, which was studied in various papers (see, for example, [B99, B05, M79, M80, BC99, S90, GRH, GRH2, GRH3, BB09, ikv17, BB14, laur03, kppv, kpv, gorb91, gorb93, gorb98]). One of the problems in this direction is to describe all degenerations in a variety of algebras of some fixed dimension satisfying some set of identities. For example, this problem was solved for $2$-dimensional pre-Lie algebras in [BB09], for $3$-dimensional Novikov algebras in [BB14], for $4$-dimensional Lie algebras in [BC99], for $4$-dimensional Zinbiel and nilpotent Leibniz algebras in [kppv], for nilpotent $5$- and $6$-dimensional Lie algebras in [S90, GRH], and for nilpotent $5$- and $6$-dimensional Malcev algebras in [kpv]. As an application of our results, one can easily recover the results of [BB09].

Another interesting notion concerning degenerations is the so-called level of an algebra defined in the end of Section 5. The algebras of the first level and the associative, Lie and Jordan algebras of the second level are classified in [khud13, khud15]. In the papers [gorb91, gorb93, gorb98], the author defined the notion of an infinite level and described all anticommutative algebras that have an infinite level not greater than $3$. This notion is much easier in the sense that the infinite level of an algebra can be easily expressed in terms of the usual level. Algebras of low dimension play a special role in problems of such type, because they have small levels. The complete description of degenerations obtained in this work allows to compute the level for all $2$-dimensional algebras.

The next result of this paper is the description of orbit closures of certain series that appear in the classification up to isomorphism. Let $T$ be some subvariety of the variety of $n$-dimensional algebra closed under the action of the general linear group. An $n$-dimensional algebra from $T$ is called rigid if its orbit is an open subset of $T$. Another important characteristic of a variety is its partition into irreducible components. The notion of a rigid algebra is closely related to this characteristic, because orbit closures of such algebras form irreducible components. For example, irreducible components and rigid algebras were classified for low dimensional associative (see [M79, M80]) and Jordan (see [KE14]) algebras. Since the variety of $2$-dimensional algebras is simply ${\bf k}^{8}$, it is clear that there is only one irreducible component and there are no rigid algebras in it. Thus, this problem is not relevant for the variety of all $2$-dimensional algebras itself. Nevertheless, it is relevant for subvarieties. In the last part we apply our results about the variety of all $2$-dimensional algebras to its subvarieties consisting of flexible and bicommutative algebras. We describe all degenerations and closures of orbits in these varieties. In particular, we classify the irreducible components and rigid algebras. Our results allow to get such descriptions and classifications for varieties of $2$-dimensional algebras defined by any identities without any problems.

Let us give a resume of our motivations. The problems considered in this paper are classical and their solution is interesting itself. In general the classification of all $n$-dimensional algebras is a wild problem and it is interesting to get the solution in particular cases where it is still possible. Our main motivation was the classification of all the algebras of the second level that we produce in [kayvo] using the results of this paper. In fact, there are reasons to guess that our classification will allow to classify also algebras of the third and the fourth levels. Thus, our results are important for the classification of algebras of small levels and constitute a necessary part of it. Another application that we have in mind is the geometric description of subvarieties of the variety of $2$-dimensional algebras. There are some works (for example, [BB09]) devoted to this problem and our work gives a powerful tool to solve it in all certain cases. Our results can be applied whenever the natural action of $GL({\bf k}^{2})$ on $({\bf k}^{2})^{*}\otimes({\bf k}^{2})^{*}\otimes{\bf k}^{2}$ appears and we expect that they will have other applications, for example, in the theory of algebras with polynomial identities or in the geometric representation theory. Even when one deals with $n$-dimensional algebras for $n>2$ it may be useful to consider $2$-dimensional subalgebras, and our results could be applied in this case. For example, the classification of $n$-dimensional algebras with an $(n-2)$-dimensional annihilator is fulfilled using our classification in [bi2].

2. Definitions and notation

Throughout the paper we fix an algebraically closed field ${\bf k}$, a $2$-dimensional ${\bf k}$-linear vector space $V$ and a basis $\{e_{1},e_{2}\}$ of $V$. All spaces in this paper are considered over ${\bf k}$, and we write simply $dim$, $Hom$ and $\otimes$ instead of $dim_{{\bf k}}$, $Hom_{{\bf k}}$ and $\otimes_{{\bf k}}$. 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 $A$.

Since this paper is devoted to $2$-dimensional algebras, we give all definitions and notation only for this case, though everything in this section can be rewritten for any dimension.

The set $\mathcal{A}_{2}:=Hom(V\otimes V,V)\cong V^{*}\otimes V^{*}\otimes V$ is a vector space of dimension $8$. This space has a structure of the affine variety ${\bf k}^{8}.$ Indeed, any $\mu\in\mathcal{A}_{2}$ is determined by $8$ structure constants $c_{ij}^{k}\in{\bf k}$ ($i,j,k=1,2$) such that $\mu(e_{i}\otimes e_{j})=c_{ij}^{1}e_{1}+c_{ij}^{2}e_{2}$. A subset of $\mathcal{A}_{2}$ is Zariski-closed if it can be defined by a set of polynomial equations in the variables $c_{ij}^{k}$.

The general linear group $GL(V)$ acts on $\mathcal{A}_{2}$ by conjugations:

[TABLE]

for $x,y\in V$, $\mu\in\mathcal{A}_{2}$ and $g\in GL(V)$. Thus, $\mathcal{A}_{2}$ is decomposed into $GL(V)$-orbits that correspond to the isomorphism classes of $2$-dimensional algebras. The classification of $2$-dimensional algebras up to isomorphism is equivalent to the classification of $GL(V)$-orbits.

Let $O(\mu)$ denote the orbit of $\mu\in\mathcal{A}_{2}$ under the action of $GL(V)$ and $\overline{O(\mu)}$ denote the Zariski closure of $O(\mu)$. Let $A$ and $B$ be two $2$-dimensional algebras and $\mu,\lambda\in\mathcal{A}_{2}$ 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 doesn’t depend on the choice of $\mu$ and $\lambda$. If $A\not\cong B$, then the assertion $A\to B$ is called a proper degeneration. We write $A\not\to B$ if $\lambda\not\in\overline{O(\mu)}$. Let now $A(*):=\{A(\alpha)\}_{\alpha\in I}$ be a set of $2$-dimensional algebras and $\mu_{\alpha}\in\mathcal{A}_{2}$ represent $A(\alpha)$ for $\alpha\in I$. If $\lambda\in\overline{\{O(\mu_{\alpha})\}_{\alpha\in I}}$, then we write $A(*)\to B$ and say that $A(*)$ degenerates to $B$. In the opposite case we write $A(*)\not\to B$.

Let $A(*)$, $B$, $\mu_{\alpha}$ ($\alpha\in I$) and $\lambda$ be as above. Let $c_{ij}^{k}$ ($i,j,k=1,2$) be the structure constants of $\lambda$ in the basis $e_{1},e_{2}$. If we construct maps $a_{i}^{j}:{\bf k}^{*}\to{\bf k}$ ($i,j=1,2$) and $f:{\bf k}^{*}\to I$ such that $a_{1}^{1}(t)e_{1}+a_{1}^{2}(t)e_{2}$ and $a_{2}^{1}(t)e_{1}+a_{2}^{2}(t)e_{2}$ form a basis of $V$ for any $t\in{\bf k}^{*}$, and the structure constants of $\mu_{f(t)}$ in this basis are polynomials $c_{ij}^{k}(t)\in{\bf k}[t]$ such that $c_{ij}^{k}(0)=c_{ij}^{k}$, then $A(*)\to B$. Indeed, if there is some closed subset $\mathcal{R}$ containing $O(\mu_{\alpha})$ for all $\alpha\in I$, then it contains, in particular, $O(\mu_{f(t)})$ for all $t\in{\bf k}^{*}$, and hence the element $\lambda_{t}$ of $\mathcal{A}_{2}$ with structure constants $c_{ij}^{k}(t)$ belongs to $\mathcal{R}$ for any $t\in{\bf k}^{*}$. Note that the assertion $\lambda_{t}\in\mathcal{R}$ is equivalent to the annihilation of some set polynomials in one variable in the point $t$. But if this set of polynomials vanishes for all $t\in{\bf k}^{*}$, then each of these polinomials has infinitely many roots, and hence it equals zero. Thus, $t=0$ annihilates all the required polynomials too, i.e. $\lambda=\lambda_{0}\in\mathcal{R}$. We will call $(a_{1}^{1}(t)e_{1}+a_{1}^{2}(t)e_{2},a_{2}^{1}(t)e_{1}+a_{2}^{2}(t)e_{2})$ and $f(t)$ a parametrized basis and a parametrized index for $A(*)\to B$ respectively. The case of degeneration between two algebras corresponds to the case $|I|=1$. In this case we need only a parametrized basis, because $f(t)$ is the unique element of $I$ for any $t\in{\bf k}^{*}$.

We take the ideas for proving non-degenerations from [S90]. Let $Q$ be a set of polynomial the equations in the variables $x_{i,j}^{k}$ ($i,j,k=1,2$). Suppose that $Q$ satisfies the following property: if $x_{i,j}^{k}=c_{ij}^{k}$ is a solution to all equations in $Q$, then also $x_{i,j}^{k}=\tilde{c}_{ij}^{k}$ is a solution to all equations in $Q$ too in the following cases:

1. (1)

there are $\alpha_{1},\alpha_{2}\in{\bf k}^{*}$ such that $\tilde{c}_{ij}^{k}=\frac{\alpha_{i}\alpha_{j}}{\alpha_{k}}c_{ij}^{k}$; 2. (2)

there is $\alpha\in{\bf k}$ such that

[TABLE]

Let $\mathcal{R}\subset\mathcal{A}_{2}$ be a set of all algebra structures whose structure constants satisfy all equations in $Q$. We will call such a set $\mathcal{R}$ a closed upper invariant set. Let $\{A(\alpha)\}_{\alpha\in I}$ be a set of $2$-dimensional algebras such that $A(\alpha)$ can be represented by a structure from $\mathcal{R}$ for any $\alpha\in I$. Let $B$ be a $2$-dimensional algebra represented by the structure $\lambda\in\mathcal{A}_{2}$. If $O(\lambda)\cap\mathcal{R}=\varnothing$, then $A(*)\not\to B$. In this case we call $\mathcal{R}$ a separating set for $A(*)\not\to B$.

Let us recall two more tools for proving degenerations and non-degenerations. Firstly, if $A\to B$, then $dim\,Aut(A)<dim\,Aut(B)$. Note that if $A(*)\to B$, then either $dim\,Aut(A(\alpha))=dim\,Aut(B)$ for infinitely many $\alpha\in I$ or $dim\,Aut(A(\alpha))<dim\,Aut(B)$ for some $\alpha\in I$, but it is possible that $dim\,Aut(A(\alpha))\geq dim\,Aut(B)$ for all $\alpha\in I$. Note also that $dim\,Aut(A)=dim\,Der(A)$. Secondly, if $A\to C$ and $C\to B$ then $A\to B$. If there is no $C$ such that $A\to C$ and $C\to B$ are proper degenerations, then the assertion $A\to B$ is called a primary degeneration. If there are no $C$ and $D$ such that $C\to A$, $B\to D$, $C\not\to D$ and one of the assertions $C\to A$ and $B\to D$ is a proper degeneration, then the assertion $A\not\to B$ is called a primary non-degeneration. It suffices to prove only primary degenerations and non-degenerations to describe degenerations in the variety under consideration. Note also that any algebra degenerates to the algebra with zero multiplication.

3. Algebraic classification

The first of our aims is to classify all $2$-dimensional algebras over ${\bf k}$ modulo isomorphism. Our classification is based on the following lemma.

Lemma 1**.**

Let $A$ be a $2$-dimensional algebra. Then there exists a non-zero element $x\in A$ such that $x$ and $x^{2}$ are linearly dependent.

• Proof.

The required assertion is equivalent to the existence of a $1$-dimensional subalgebra in $A$. Then the lemma follows from the discussion right after [mir00, Proposition 1].

$\Box$

Note that if $x\in A$ and $x^{2}$ are linearly dependent, then either $x^{2}=0$ or $x=\alpha e$ for some $\alpha\in{\bf k}^{*}$ and some $e\in A$ such that $e^{2}=e$. If $x^{2}=0$, then $x$ is called a $2$-nil element. An element $e$ such that $e^{2}=e$ is called an idempotent.

Corollary 2**.**

Any $2$-dimensional ${\bf k}$-algebra belongs to one of the following disjoint classes:

1. ${\bf A}.$

algebras that don’t have nonzero idempotents and have a unique $1$-dimensional subspace of $2$-nil elements; 2. ${\bf B}.$

algebras that don’t have nonzero idempotents and have two linearly independent $2$-nil elements; 3. ${\bf C}.$

algebras that have a unique nonzero idempotent and don’t have nonzero $2$-nil elements; 4. ${\bf D}.$

algebras that have a unique nonzero idempotent and a nonzero $2$-nil element; 5. ${\bf E}.$

algebras that have two different nonzero idempotents.

• Proof.

The fact that the classes are disjoint is obvious. The fact that any $2$-dimensional algebra belongs to one of the classes follows easily from Lemma 1 and the remark after it.