The algebraic and geometric classification of
compatible Pre-Lie algebras111The authors thank Amir Fernández Ouaridi for
sharing some useful software programs.222 The work is supported by FCT
UIDB/00212/2020, UIDP/00212/2020 and 2022.14950.CBM (Programa PESSOA).

Hani Abdelwahab333Department of Mathematics, Faculty of Science, Mansoura
University, Mansoura, Egypt<EMAIL_ADDRESS>Ivan Kaygorodov 444CMA-
UBI, Universidade da Beira Interior, Covilhã, Portugal;
<EMAIL_ADDRESS>& Abdenacer Makhlouf 555IRIMAS - Département de
Mathématiques, University of Haute Alsace, Mulhouse, France;
<EMAIL_ADDRESS>
Abstract: In this paper, we develop a method to obtain the algebraic
classification of compatible pre-Lie algebras from the classification of pre-
Lie algebras of the same dimension. We use this method to obtain the algebraic
classification of complex $2$-dimensional compatible pre-Lie algebras. As a
byproduct, we obtain the classification of complex $2$-dimensional compatible
commutative associative, compatible associative and compatible Novikov
algebras. In addition, we consider the geometric classification of varieties
of cited algebras, that is the description of its irreducible components.

Keywords: compatible algebra, compatible associative algebra, compatible pre-
Lie algebra, algebraic classification, geometric classification.

MSC2020: 17A30 (primary); 17D25, 14L30 (secondary).

###### Contents

  1. 1 The algebraic classification of compatible algebras
    1. 1.1 The algebraic classification of algebras
    2. 1.2 $2$-dimensional pre-Lie algebras
    3. 1.3 The algebraic classification of compatible pre-Lie algebras
      1. 1.3.1 Compatible pre-Lie algebras defined on $\mathcal{C}_{01}$
      2. 1.3.2 Compatible pre-Lie algebras defined on $\mathcal{C}_{02}$
      3. 1.3.3 Compatible pre-Lie algebras defined on $\mathcal{C}_{03}$
      4. 1.3.4 Compatible pre-Lie algebras defined on $\mathcal{C}_{04}$
      5. 1.3.5 Compatible pre-Lie algebras defined on $\mathcal{C}_{05}^{\alpha\neq 0,\frac{1}{2},1}$
      6. 1.3.6 Compatible pre-Lie algebras defined on $\mathcal{C}_{05}^{\frac{1}{2}}$
      7. 1.3.7 Compatible pre-Lie algebras defined on $\mathcal{C}_{05}^{1}$
      8. 1.3.8 Compatible pre-Lie algebras defined on $\mathcal{C}_{05}^{0}$
      9. 1.3.9 Compatible pre-Lie algebras defined on $\mathcal{C}_{06}^{\alpha\neq 0,1,2}$
      10. 1.3.10 Compatible pre-Lie algebras defined on $\mathcal{C}_{06}^{0}$
      11. 1.3.11 Compatible pre-Lie algebras defined on $\mathcal{C}_{06}^{1}$
      12. 1.3.12 Compatible pre-Lie algebras defined on $\mathcal{C}_{06}^{2}$
      13. 1.3.13 Compatible pre-Lie algebras defined on $\mathcal{C}_{07}$
      14. 1.3.14 Compatible pre-Lie algebras defined on $\mathcal{C}_{08}$
    4. 1.4 The algebraic classification of compatible commutative associative algebras
    5. 1.5 The algebraic classification of compatible associative algebras
    6. 1.6 The algebraic classification of compatible Novikov algebras
  2. 2 The geometric classification of compatible algebras
    1. 2.1 Degenerations and the geometric classification of algebras
    2. 2.2 The geometric classification of compatible commutative associative algebras
    3. 2.3 The geometric classification of compatible associative algebras
    4. 2.4 The geometric classification of compatible Novikov algebras
    5. 2.5 The geometric classification of compatible pre-Lie algebras

##  Introduction

The algebraic classification (up to isomorphism) of algebras of small
dimensions from a certain variety defined by a family of polynomial identities
is a classic problem in the theory of non-associative algebras. Another
interesting approach to studying algebras of a fixed dimension is to study
them from a geometric point of view (that is, to study the degenerations and
deformations of these algebras). The results in which the complete information
about degenerations of a certain variety is obtained are generally referred to
as the geometric classification of the algebras of these varieties. There are
many results related to the algebraic and geometric classification of Jordan,
Lie, Leibniz, Zinbiel, and other algebras (see, [7, 6, 28, 1, 4, 5, 18, 3, 25,
27, 16, 17, 23, 2] and references in [24, 32, 26]).

Pre-Lie algebras, also known as right symmetric algebras, appeared in some
papers by Gerstenhaber, Koszul, and Vinberg in the 1960s. It is a
generalization of associative algebras and the most popular non-associative
subvariety of Lie-admissible algebras. They have various applications in
geometry and physics [8]; recently some connections between them and trees,
braces and $F$-manifold algebras were established in [38, 37, 9, 30]. Novikov
algebras are pre-Lie algebras with an additional identity. They were
introduced in papers by Gel’fand and Dorfman, Balinskiĭ and Novikov (about
Novikov algebras see [15] and references therein). Let $\Omega$ be a variety
of algebras. We say that an algebra $(\bf A,\cdot,\ast)$ is a compatible
$\Omega$-algebra, if and only if $(\bf A,\cdot),$ $(\bf A,\ast)$ and $(\bf
A,\cdot+\ast)$ are $\Omega$-algebras. Compatible Lie algebras are considered
in the study of the classical Yang-Baxter equation [21], integrable equations
of the principal chiral model type [20], elliptic theta functions [33], and
other areas. The study of non-Lie compatible algebras is also very popular.
So, cohomology and deformations were studied for compatible Lie algebras [31],
compatible associative algebras [10], compatible ${\rm Hom}$-Lie algebras
[12], compatible $L_{\infty}$-algebras [11], compatible $3$-Lie algebras [19],
compatible dendriform algebras [13], and so on. Free compatible algebras were
studied in the associative algebra case in [14] and in the Lie algebra case in
[22, 29]. Compatible associative structures on matrix algebras were studied in
[34, 35, 36]. General compatible structures have been studied from an operadic
point of view in [39, 40]. At this moment there is only one paper about the
classification of small dimensional compatible algebras. Namely, all
$4$-dimensional nilpotent compatible Lie algebras were classified in [28].

The main goal of the present paper is to obtain the algebraic and geometric
description of the variety of complex $2$-dimensional compatible pre-Lie
algebras. To do so, we first determine all such $2$-dimensional algebra
structures, up to isomorphism (what we call the algebraic classification), and
then proceed to determine the geometric properties of the corresponding
variety, namely its dimension and description of the irreducible components
(the geometric classification). As some corollaries, we have the algebraic and
geometric classification of complex $2$-dimensional compatible commutative
associative, compatible associative and compatible Novikov algebras.

Our main results regarding the algebraic classification are summarized below.

######  Theorem A1.

There are infinitely many isomorphism classes of complex $2$-dimensional
compatible pre-Lie algebras, described explicitly in Theorem 7 in terms of $6$
three-parameter families, $14$ two-parameter families, $13$ one-parameter
families, and $8$ additional isomorphism classes.

######  Theorem A2.

There are infinitely many isomorphism classes of complex $2$-dimensional
compatible commutative associative algebras, described explicitly in Theorem 8
in terms of $1$ three-parameter family, $3$ two-parameter families, $8$ one-
parameter families, and $6$ additional isomorphism classes.

######  Theorem A3.

There are infinitely many isomorphism classes of complex $2$-dimensional
compatible associative algebras, described explicitly in Theorem 10 in terms
of $1$ three-parameter family, $3$ two-parameter families, $10$ one-parameter
families, and $10$ additional isomorphism classes.

######  Theorem A4.

There are infinitely many isomorphism classes of complex $2$-dimensional
compatible Novikov algebras, described explicitly in Theorem 12 in terms of
$3$ three-parameter family, $7$ two-parameter families, $11$ one-parameter
families, and $7$ additional isomorphism classes.

The geometric part of our work aims to generalize previously obtained results
about the geometric classification of $2$-dimensional Novikov [7] and pre-Lie
[6] algebras. Our main results regarding the geometric classification are
summarized below.

######  Theorem G1.

The variety of complex $2$-dimensional compatible pre-Lie algebras has
dimension $7$. It is defined by $2$ rigid algebras, $1$ one-parametric family
of algebras, $7$ two-parametric family of algebras, and $4$ three-parametric
families of algebras and can be described as the closure of the union of
$\mathrm{GL}_{2}(\mathbb{C})$-orbits of the algebras given in Theorem 22.

######  Theorem G2.

The variety of complex $2$-dimensional compatible commutative associative
algebras has dimension $7$. It is defined by $1$ two-parametric family of
algebras and $1$ three-parametric families of algebras and can be described as
the closure of the union of $\mathrm{GL}_{2}(\mathbb{C})$-orbits of the
algebras given in Theorem 19.

######  Theorem G3.

The variety of complex $2$-dimensional compatible associative algebras has
dimension $7$. It is defined by $2$ rigid algebras, $2$ two-parametric family
of algebras, and $1$ three-parametric families of algebras and can be
described as the closure of the union of $\mathrm{GL}_{2}(\mathbb{C})$-orbits
of the algebras given in Theorem 20.

######  Theorem G4.

The variety of complex $2$-dimensional compatible Novikov algebras has
dimension $7$. It is defined by $1$ rigid algebra, $3$ two-parametric family
of algebras, and $1$ three-parametric families of algebras and can be
described as the closure of the union of $\mathrm{GL}_{2}(\mathbb{C})$-orbits
of the algebras given in Theorem 21.

##  1\. The algebraic classification of compatible algebras

All the algebras below will be over $\mathbb{C}$ and all the linear maps will
be $\mathbb{C}$-linear. For simplicity, every time we write the multiplication
table of an algebra the products of basic elements whose values are zero are
omitted.

###  1.1. The algebraic classification of algebras

In this paper, we work with compatible pre-Lie algebras with two
multiplications. Let us review the method we will use to obtain the algebraic
classification for the variety of compatible pre-Lie algebras (the present
method, in the case of Poisson algebras, is given with more details in [3,
2]).

######  Definition 1.

An algebra is called a pre-Lie algebra if it satisfies the identity

$\left(x\cdot y\right)\cdot z-x\cdot\left(y\cdot z\right)$ | $=$ | $\left(y\cdot x\right)\cdot z-y\cdot\left(x\cdot z\right).$
---|---|---

######  Definition 2.

A compatible pre-Lie algebra is a vector space ${\bf A}$ equipped with two
multiplication $\cdot$ and another multiplication $\ast,$ such that, $({\bf
A},\cdot),$ $({\bf A},\ast)$ and $({\bf A},\cdot+\ast)$ are pre-Lie algebras.
These two operations are required to satisfy the following identities.

$\left(x\cdot y\right)\cdot z-x\cdot\left(y\cdot z\right)$ | $=$ | $\left(y\cdot x\right)\cdot z-y\cdot\left(x\cdot z\right),$
---|---|---
$\left(x\ast y\right)\ast z-x\ast\left(y\ast z\right)$ | $=$ | $\left(y\ast x\right)\ast z-y\ast\left(x\ast z\right),$
$(x\ast y)\cdot z-x\ast(y\cdot z)+(x\cdot y)\ast z-x\cdot(y\ast z)$ | $=$ | $(y\ast x)\cdot z-y\ast(x\cdot z)+(y\cdot x)\ast z-y\cdot(x\ast z).$

The main examples of compatible pre-Lie algebras are the following: compatible
commutative associative, compatible associative and compatible Novikov
algebras.

######  Definition 3.

Let $\left({\bf A},\cdot\right)$ be a pre-Lie algebra. Define ${\rm
Z}^{2}\left({\bf A},{\bf A}\right)$ to be the set of all bilinear maps
$\theta:{\bf A}\times{\bf A}\longrightarrow{\bf A}$ such that:

$\theta(\theta\left(x,y\right),z)-\theta(x,\theta\left(y,z\right))$ | $=$ | $\theta(\theta\left(y,x\right),z)-\theta(y,\theta\left(x,z\right)),$
---|---|---
$\theta(x,y)\cdot z-\theta(x,y\cdot z)+\theta(x\cdot y,z)-x\cdot\theta(y,z)$ | $=$ | $\theta(y,x)\cdot z-\theta(y,x\cdot z)+\theta(y\cdot x,z)-y\cdot\theta(x,z).$

Then ${\rm Z}^{2}\left({\bf A},{\bf A}\right)\neq{\varnothing}$ since
$\theta=0\in{\rm Z}^{2}\left({\bf A},{\bf A}\right)$.

Now, for $\theta\in{\rm Z}^{2}\left({\bf A},{\bf A}\right)$, let us define a
multiplication $\bullet_{\theta}$ on ${\bf A}$ by
$x\bullet_{\theta}y=\theta\left(x,y\right)$ for all $x,y\in{\bf A}$. Then
$\left({\bf A},\cdot,\bullet_{\theta}\right)$ is a compatible pre-Lie algebra.
Conversely, if $\left({\bf A},\cdot,\bullet\right)$ is a compatible pre-Lie
algebra, then there exists $\theta\in{\rm Z}^{2}\left({\bf A},{\bf A}\right)$
such that $\left({\bf A},\cdot,\bullet_{\theta}\right)\cong\left({\bf
A},\cdot,\bullet\right)$. To see this, consider the bilinear map $\theta:{\bf
A}\times{\bf A}\longrightarrow{\bf A}$ defined by
$\theta\left(x,y\right)=x\bullet y$ for all $x,y$ in ${\bf A}$. Then
$\theta\in{\rm Z}^{2}\left({\bf A},{\bf A}\right)$ and $\left({\bf
A},\cdot,\bullet_{\theta}\right)=\left({\bf A},\cdot,\bullet\right)$.

Let $\left({\bf A},\cdot\right)$ be a pre-Lie algebra. The automorphism group,
${\rm Aut}\left({\bf A}\right)$, of ${\bf A}$ acts on ${\rm Z}^{2}\left({\bf
A},{\bf A}\right)$ by

$\left(\theta*\phi\right)\left(x,y\right)$ | $=$ | $\phi^{-1}\left(\theta\left(\phi\left(x\right),\phi\left(y\right)\right)\right),$
---|---|---

for $\phi\in{\rm Aut}\left({\bf A}\right)$, and $\theta\in{\rm
Z}^{2}\left({\bf A},{\bf A}\right)$.

######  Lemma 4.

Let $\left({\bf A},\cdot\right)$ be a pre-Lie algebra and
$\theta,\vartheta\in{\rm Z}^{2}\left({\bf A},{\bf A}\right)$. Then $\left({\bf
A},\cdot,\bullet_{\theta}\right)$ and $\left({\bf
A},\cdot,\bullet_{\vartheta}\right)$ are isomorphic if and only if there is a
$\phi\in{\rm Aut}\left({\bf A}\right)$ with $\theta*\phi=\vartheta$.

###### Proof.

If $\theta*\phi=\vartheta$, then $\phi:\left({\bf
A},\cdot,\bullet_{\vartheta}\right)\longrightarrow$ $\left({\bf
A},\cdot,\bullet_{\theta}\right)$ is an isomorphism since
$\phi\left(\vartheta\left(x,y\right)\right)=\theta\left(\phi\left(x\right),\phi\left(y\right)\right)$.
On the other hand, if $\phi:\left({\bf
A},\cdot,\bullet_{\vartheta}\right)\longrightarrow$ $\left({\bf
A},\cdot,\bullet_{\theta}\right)$ is an isomorphism of compatible pre-Lie
algebras, then $\phi\in{\rm Aut}\left({\bf A}\right)$ and
$\phi\left(x\bullet_{\vartheta}y\right)=\phi\left(x\right)\bullet_{\theta}\phi\left(y\right)$.
Hence

$\vartheta\left(x,y\right)$ | $=$ | $\phi^{-1}\left(\theta\left(\phi\left(x\right),\phi\left(y\right)\right)\right)$ | $=$ | $\left(\theta*\phi\right)\left(x,y\right)$
---|---|---|---|---

and therefore $\theta*\phi=\vartheta$. ∎

Consequently, we have a procedure to classify the compatible pre-Lie algebras
with given associated pre-Lie algebra $\left({\bf A},\cdot\right)$. It
consists of three steps:

  1. (1)

Compute ${\rm Z}^{2}\left({\bf A},{\bf A}\right)$.

  2. (2)

Find the orbits of ${\rm Aut}\left({\bf A}\right)$ on ${\rm Z}^{2}\left({\bf
A},{\bf A}\right)$.

  3. (3)

Choose a representative $\theta$ from each orbit and then construct the
compatible pre-Lie algebra $\left({\bf A},\cdot,\bullet_{\theta}\right)$.

###  1.2. $2$-dimensional pre-Lie algebras

######  Lemma 5.

Let $\mathcal{C}$ be a nonzero $2$-dimensional pre-Lie algebra. Then
$\mathcal{C}$ is isomorphic to one and only one of the following algebras:

$\mathcal{C}_{01}$ | $:$ | $e_{1}\cdot e_{1}=e_{1}+e_{2},$ | $e_{2}\cdot e_{1}=e_{2}.$ |  |
---|---|---|---|---|---
$\mathcal{C}_{02}$ | $:$ | $e_{1}\cdot e_{1}=e_{1}+e_{2},$ | $e_{1}\cdot e_{2}=e_{2}.$ |  |
$\mathcal{C}_{03}$ | $:$ | $e_{1}\cdot e_{1}=e_{2}.$ |  |  |
$\mathcal{C}_{04}$ | $:$ | $e_{2}\cdot e_{1}=e_{1}.$ |  |  |
$\mathcal{C}_{05}^{\alpha}$ | $:$ | $e_{1}\cdot e_{1}=e_{1},$ | $e_{1}\cdot e_{2}=\alpha e_{2}.$ |  |
$\mathcal{C}_{06}^{\alpha}$ | $:$ | $e_{1}\cdot e_{1}=e_{1},$ | $e_{1}\cdot e_{2}=\alpha e_{2},$ | $e_{2}\cdot e_{1}=e_{2}.$ |
$\mathcal{C}_{07}$ | $:$ | $e_{1}\cdot e_{1}=e_{1},$ | $e_{2}\cdot e_{2}=e_{2}.$ |  |
$\mathcal{C}_{08}$ | $:$ | $e_{1}\cdot e_{1}=e_{1},$ | $e_{1}\cdot e_{2}=2e_{2},$ | $e_{2}\cdot e_{1}=\frac{1}{2}e_{1}+e_{2},$ | $e_{2}\cdot e_{2}=e_{2}.$

######  Lemma 6.

The description of the group of automorphisms of every $2$-dimensional pre-Lie
algebra is given below.

  1. (1)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{01})$, then $\phi(e_{1})=e_{1}+\nu e_{2}$
and $\phi(e_{2})=e_{2}$, for $\nu\in\mathbb{C}$.

  2. (2)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{02})$, then $\phi(e_{1})=e_{1}+\nu e_{2}$
and $\phi(e_{2})=e_{2}$, for $\nu\in\mathbb{C}$.

  3. (3)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{03})$, then $\phi(e_{1})=\xi e_{1}+\nu
e_{2}$ and $\phi(e_{2})=\xi^{2}e_{2}$, for $\xi\in\mathbb{C}^{\ast}$ and
$\nu\in\mathbb{C}$.

  4. (4)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{04})$, then $\phi(e_{1})=\xi e_{1}$ and
$\phi(e_{2})=e_{2}$, for $\xi\in\mathbb{C}^{\ast}$.

  5. (5)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{05}^{\alpha\neq 1})$, then
$\phi(e_{1})=e_{1}$ and $\phi(e_{2})=\xi e_{2}$, for
$\xi\in\mathbb{C}^{\ast}$.

  6. (6)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{05}^{1})$, then $\phi(e_{1})=e_{1}+\nu
e_{2}$ and $\phi(e_{2})=\xi e_{2}$, for $\xi\in\mathbb{C}^{\ast}$ and
$\nu\in\mathbb{C}$.

  7. (7)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{06}^{\alpha\neq 0})$, then
$\phi(e_{1})=e_{1}$ and $\phi(e_{2})=\xi e_{2}$, for
$\xi\in\mathbb{C}^{\ast}$.

  8. (8)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{06}^{0})$, then $\phi(e_{1})=e_{1}+\nu
e_{2}$ and $\phi(e_{2})=\xi e_{2}$, for $\xi\in\mathbb{C}^{\ast}$ and
$\nu\in\mathbb{C}$.

  9. (9)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{07})$, then $\phi\in\mathbb{S}_{2},$ i.e.
$\phi(e_{1})=e_{1},\ \phi(e_{2})=e_{2}$ or $\phi(e_{1})=e_{2},\
\phi(e_{2})=e_{1}.$

  10. (10)

If $\phi\in\mathrm{Aut}(\mathcal{C}_{08})$, then $\phi(e_{1})=e_{1},\
\phi(e_{2})=e_{2}$ or $\phi(e_{1})=-e_{1}+4e_{2},\ \phi(e_{2})=e_{2}.$

###  1.3. The algebraic classification of compatible pre-Lie algebras

The main aim of the present section is to prove the following results.

######  Theorem 7.

Let $\mathcal{C}$ be a nonzero $2$-dimensional compatible pre-Lie algebra.
Then $\mathcal{C}$ is isomorphic to one and only one of the following
algebras:

$\mathcal{C}_{01}$ | $:$ | $e_{1}\ast e_{1}=e_{1}+e_{2},\ e_{2}\ast e_{1}=e_{2}.$
---|---|---
$\mathcal{C}_{02}$ | $:$ | $e_{1}\ast e_{1}=e_{1}+e_{2},\ e_{1}\ast e_{2}=e_{2}.$
$\mathcal{C}_{03}$ | $:$ | $e_{1}\ast e_{1}=e_{2}.$
$\mathcal{C}_{04}$ | $:$ | $e_{2}\ast e_{1}=e_{1}.$
$\mathcal{C}_{05}^{\alpha}$ | $:$ | $e_{1}\ast e_{1}=e_{1},\ e_{1}\ast e_{2}=\alpha e_{2}.$
$\mathcal{C}_{06}^{\alpha}$ | $:$ | $e_{1}\ast e_{1}=e_{1},\ e_{1}\ast e_{2}=\alpha e_{2},\ e_{2}\ast e_{1}=e_{2}.$
$\mathcal{C}_{07}$ | $:$ | $e_{1}\ast e_{1}=e_{1},\ e_{2}\ast e_{2}=e_{2}.$
$\mathcal{C}_{08}$ | $:$ | $e_{1}\ast e_{1}=e_{1},\ e_{1}\ast e_{2}=2e_{2},\ e_{2}\ast e_{1}=\frac{1}{2}e_{1}+e_{2},\ e_{2}\ast e_{2}=e_{2}.$
$\mathcal{C}_{09}^{\alpha,\beta\neq 0}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1}+e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\beta e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2}.$\\\ \end{tabular}\right.$
$\mathcal{C}_{10}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1}+e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{11}^{\alpha,\beta\neq\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1}+e_{2},$&$e_{1}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{12}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1}+e_{2},$&$e_{1}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{1}\ast e_{2}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{13}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{2},$&&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{14}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=e_{1}+\alpha e_{2},$&$e_{2}\ast e_{1}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{15}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{l}$e_{1}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{16}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=e_{1},$&$e_{1}\ast e_{2}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{17}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=e_{1}+\alpha e_{2},$&$e_{1}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{18}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{2},$&&&\\\ $e_{1}\ast e_{1}=\alpha e_{2},$&$e_{1}\ast e_{2}=e_{1},$&$e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{19}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{2},$&&\\\ $e_{1}\ast e_{1}=\alpha e_{2},$&$e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast e_{2}=2e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{20}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{2}\cdot e_{1}=e_{1},$&\\\ $e_{2}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast e_{2}=e_{1}+\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{21}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{2}\cdot e_{1}=e_{1},$&\\\ $e_{2}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast e_{2}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{22}^{\alpha\neq 0,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{2}\cdot e_{1}=e_{1},$&&\\\ $e_{1}\ast e_{2}=\alpha e_{1},$&$e_{2}\ast e_{1}=\beta e_{1},$&$e_{2}\ast e_{2}=e_{1}+\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{23}^{\alpha\neq 0,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{2}\cdot e_{1}=e_{1},$&&\\\ $e_{1}\ast e_{2}=\alpha e_{1},$&$e_{2}\ast e_{1}=\beta e_{1},$&$e_{2}\ast e_{2}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{24}^{\alpha,\beta,\gamma}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1}+e_{2},$&$e_{1}\ast e_{2}=\gamma e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{25}^{\alpha,\beta,\gamma}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1},$&$e_{1}\ast e_{2}=\gamma e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{26}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\frac{1}{2}e_{2},$&\\\ $e_{1}\ast e_{1}=2\alpha e_{1},$&$e_{1}\ast e_{2}=\alpha e_{2},$&$e_{2}\ast e_{2}=e_{1}.$\end{tabular}\right.$
$\mathcal{C}_{27}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\frac{1}{2}e_{2},$&&\\\ $e_{1}\ast e_{1}=2\alpha e_{1},$&$e_{1}\ast e_{2}=e_{1}+\alpha e_{2},$&$e_{2}\ast e_{1}=2e_{1}$&$e_{2}\ast e_{2}=\beta e_{1}+e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{28}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$\\\ $e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{29}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{30}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&&&\\\ $e_{1}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{1}\ast e_{2}=e_{1},$&$e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{31}^{\alpha,\beta,\gamma}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1}+e_{2},$&$e_{1}\ast e_{2}=\gamma e_{2},$&$e_{2}\ast e_{1}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{32}^{\alpha,\beta,\gamma}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1},$&$e_{1}\ast e_{2}=\gamma e_{2},$&$e_{2}\ast e_{1}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{33}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{2}=e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{34}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\alpha e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2},$&$e_{2}\ast e_{2}=e_{1}+\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{35}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\alpha e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{36}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=2e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{1}\ast e_{2}=2\alpha e_{2},$&$e_{2}\ast e_{1}=e_{1}+\alpha e_{2},$&$e_{2}\ast e_{2}=2e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{37}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=2e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=e_{1}+2\alpha e_{2},$&$e_{2}\ast e_{1}=2e_{1}+\alpha e_{2},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{38}^{\alpha,\beta,\gamma}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot e_{2}=e_{2},$&&\\\ $e_{1}\ast e_{1}=\left(\gamma+\beta-\alpha\right)e_{1}-\beta e_{2},$&$e_{1}\ast e_{2}=\alpha e_{1}+\beta e_{2},$\\\ $e_{2}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{2}\ast e_{2}=-\alpha e_{1}+\gamma e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{39}^{\alpha,\beta\neq\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast e_{2}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{40}^{\alpha,\beta,\gamma}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=2e_{2},$&$e_{2}\cdot e_{1}=\frac{1}{2}e_{1}+e_{2},$&$e_{2}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{1}\ast e_{2}=2\alpha e_{2},$&$e_{2}\ast e_{1}=\gamma e_{1}+\alpha e_{2},$&$e_{2}\ast e_{2}=2\gamma e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{41}^{\alpha,\beta\neq 0}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=2e_{2},$&\\\ $e_{2}\cdot e_{1}=\frac{1}{2}e_{1}+e_{2},$&$e_{2}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\beta e_{1}+2\alpha e_{2},$&\\\ $e_{2}\ast e_{1}=\left(2\beta+\frac{1}{2}\alpha\right)e_{1}+\alpha e_{2},$&$e_{2}\ast e_{2}=\frac{1}{2}\beta e_{1}+\left(\alpha+\beta\right)e_{2}.$\end{tabular}\right.$

All algebras are non-isomorphic, except

$\mathcal{C}_{24}^{1,\beta,\gamma}$ | $\cong$ | $\mathcal{C}_{25}^{1,\beta,\gamma},$ | $\mathcal{C}_{31}^{0,\beta,\gamma}$ | $\cong$ | $\mathcal{C}_{32}^{0,\beta,\gamma},$ | $\mathcal{C}_{38}^{\alpha,\beta,\gamma}$ | $\cong$ | $\mathcal{C}_{38}^{\beta,\alpha,-\alpha+\beta+\gamma},$
---|---|---|---|---|---|---|---|---
$\mathcal{C}_{39}^{\alpha,\beta}$ | $\cong$ | $\mathcal{C}_{39}^{\beta,\alpha},$ | $\mathcal{C}_{40}^{\alpha,\beta,\gamma}$ | $\cong$ | $\mathcal{C}_{40}^{4\gamma-\alpha,\beta-8\alpha+16\gamma,\gamma},$ | $\mathcal{C}_{41}^{\alpha,\beta}$ | $\cong$ | $\mathcal{C}_{41}^{\alpha+4\beta,-\beta}.$

####  1.3.1. Compatible pre-Lie algebras defined on $\mathcal{C}_{01}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{01},\mathcal{C}_{01})$, the
compatible pre-Lie algebra structures defined on $\mathcal{C}_{01}$ are of the
form:

$\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1}+e_{2},$&$e_{2}\cdot
e_{1}=e_{2},$&\\\ $e_{1}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{1}e_{2}.$\end{tabular}\right.\qquad$

Then we have the following cases:

  * •

If $\alpha_{3}\neq 0$, then choose $\nu=-\frac{\alpha_{2}}{\alpha_{3}}$ and
obtain the parametric family:

$\mathcal{C}_{09}^{\alpha,\beta\neq
0}:=\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1}+e_{2},$&$e_{2}\cdot
e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\beta
e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.\qquad$

The algebras $\mathcal{C}_{09}^{\alpha,\beta}$ and
$\mathcal{C}_{09}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

  * •

If $\alpha_{3}=0$, we obtain the parametric family:

$\mathcal{C}_{10}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{1}+e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha
e_{1}+\beta e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.\qquad$

The algebras $\mathcal{C}_{10}^{\alpha,\beta}$ and
$\mathcal{C}_{10}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

####  1.3.2. Compatible pre-Lie algebras defined on $\mathcal{C}_{02}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{02},\mathcal{C}_{02})$, the
compatible pre-Lie algebra structures defined on $\mathcal{C}_{02}$ are of the
form:

$\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1}+e_{2},$&$e_{1}\cdot
e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2}.$\end{tabular}\right.\qquad$

Then we have the following cases:

  * •

If $\alpha_{1}\neq\alpha_{3}$, then choose
$\nu=\frac{\alpha_{2}}{\alpha_{1}-\alpha_{3}}$ and obtain the parametric
family:

$\mathcal{C}_{11}^{\alpha,\beta\neq\alpha}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{1}+e_{2},$&$e_{1}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha
e_{1},$&$e_{1}\ast e_{2}=\beta e_{2}.$\end{tabular}\right.\qquad$

The algebras $\mathcal{C}_{11}^{\alpha,\beta}$and
$\mathcal{C}_{11}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

  * •

If $\alpha_{1}=\alpha_{3}$, we obtain the parametric family:

$\mathcal{C}_{12}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{1}+e_{2},$&$e_{1}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha
e_{1}+\beta e_{2},$&$e_{1}\ast e_{2}=\alpha e_{2}.$\end{tabular}\right.\qquad$

The algebras $\mathcal{C}_{12}^{\alpha,\beta}$ and
$\mathcal{C}_{12}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

####  1.3.3. Compatible pre-Lie algebras defined on $\mathcal{C}_{03}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{03},\mathcal{C}_{03})$, the
compatible pre-Lie algebra structures defined on $\mathcal{C}_{03}$ are of the
forms:

  1. (1)

$\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{2},$&&\\\ $e_{1}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{1}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast
e_{1}=\alpha_{4}e_{1}+\alpha_{5}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{6}e_{2}.$\end{tabular}\right.$

  3. (3)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{2},$&&&\\\ $e_{1}\ast
e_{1}=\alpha_{7}e_{1}+\alpha_{8}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{9}e_{1},$&$e_{2}\ast e_{1}=\alpha_{9}e_{1},$&$e_{2}\ast
e_{2}=\alpha_{9}e_{2}.$\end{tabular}\right.$

  4. (4)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{2},$&&&\\\ $e_{1}\ast
e_{1}=\alpha_{10}e_{1}+\alpha_{11}e_{2},$&$e_{1}\ast
e_{2}=2\alpha_{10}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{12}e_{1}+\alpha_{10}e_{2},$&$e_{2}\ast
e_{2}=2\alpha_{12}e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{4}\alpha_{9}\alpha_{12}\neq 0$.

First, we consider the first form. Then we have the following cases:

  * •

If $\alpha_{3}\neq 0$, then choose
$\xi=\alpha^{-1}_{3},\nu=-{\alpha_{2}}{\alpha_{3}^{-2}}$ and obtain the
parametric family:

$\mathcal{C}_{13}^{\alpha}:=\left\\{\begin{tabular}[]{lll}$e_{1}\cdot
e_{1}=e_{2},$&&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast
e_{2}=e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.\qquad$

The algebras $\mathcal{C}_{13}^{\alpha}$ and
$\mathcal{C}_{13}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

  * •

If $\alpha_{3}=0,\alpha_{1}\neq 0$, then choose $\xi={\alpha^{-1}_{1}}$ and
obtain the parametric family:

$\mathcal{C}_{14}^{\alpha}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=e_{1}+\alpha e_{2},$&$e_{2}\ast
e_{1}=e_{2}.$\end{tabular}\right.\qquad$

The algebras $\mathcal{C}_{14}^{\alpha}$ and
$\mathcal{C}_{14}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

  * •

If $\alpha_{1}=\alpha_{3}=0$, then we obtain the parametric family:

$\mathcal{C}_{15}^{\alpha}:=\left\\{\begin{tabular}[]{l}$e_{1}\cdot
e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.\qquad$

The algebras $\mathcal{C}_{15}^{\alpha}$ and
$\mathcal{C}_{15}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

Second, we consider the second form. Then we have the following cases:

  * •

If $\alpha_{4}\neq\alpha_{6}$, then choose
$\xi={\alpha^{-1}_{4}},\nu=\frac{\alpha_{5}}{\alpha_{4}-\alpha_{6}}$ and
obtain the parametric family:

$\mathcal{C}_{16}^{\alpha}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=e_{1},$&$e_{1}\ast e_{2}=\alpha
e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{16}^{\alpha}$ and
$\mathcal{C}_{16}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

  * •

If $\alpha_{4}=\alpha_{6}$, then choose $\xi={\alpha^{-1}_{4}}$ and obtain the
parametric family:

$\mathcal{C}_{17}^{\alpha}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=e_{1}+\alpha e_{2},$&$e_{1}\ast
e_{2}=e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{17}^{\alpha}$ and
$\mathcal{C}_{17}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

Third, we consider the third form. Then, we choose
$\xi={\alpha^{-\frac{1}{2}}_{9}},\nu=-\frac{\alpha_{7}\alpha_{9}^{-\frac{3}{2}}}{2}$
and obtain the parametric family:

$\mathcal{C}_{18}^{\alpha}:=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{2},$&&&\\\ $e_{1}\ast e_{1}=\alpha e_{2},$&$e_{1}\ast
e_{2}=e_{1},$&$e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast
e_{2}=e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{18}^{\alpha}$ and
$\mathcal{C}_{18}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

Finally, we consider the fourth form. Then, we choose
$\xi=\alpha^{-\frac{1}{2}}_{12},$
$\nu=-{\alpha_{10}}{\alpha_{12}^{-\frac{3}{2}}}$ and obtain the parametric
family:

$\mathcal{C}_{19}^{\alpha}:=\left\\{\begin{tabular}[]{lll}$e_{1}\cdot
e_{1}=e_{2},$&&\\\ $e_{1}\ast e_{1}=\alpha e_{2},$&$e_{2}\ast
e_{1}=e_{1},$&$e_{2}\ast e_{2}=2e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{19}^{\alpha}$ and
$\mathcal{C}_{19}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

####  1.3.4. Compatible pre-Lie algebras defined on $\mathcal{C}_{04}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{04},\mathcal{C}_{04})$, the
compatible pre-Lie algebra structures defined on $\mathcal{C}_{04}$ are of the
form:

  1. (1)

$\left\\{\begin{tabular}[]{ll}$e_{2}\cdot e_{1}=e_{1},$&\\\ $e_{2}\ast
e_{1}=\alpha_{1}e_{1},$&$e_{2}\ast
e_{2}=\alpha_{2}e_{1}+\alpha_{3}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{lll}$e_{2}\cdot e_{1}=e_{1},$&&\\\ $e_{1}\ast
e_{2}=\alpha_{4}e_{1},$&$e_{2}\ast e_{1}=\alpha_{5}e_{1},$&$e_{2}\ast
e_{2}=\alpha_{6}e_{1}+\alpha_{4}e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{4}\neq 0$.

First, we study the first form. Then we have the following cases:

  * •

If $\alpha_{2}\neq 0$, then choose $\xi=\alpha_{2}$ and obtain the parametric
family:

$\mathcal{C}_{20}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{ll}$e_{2}\cdot
e_{1}=e_{1},$&\\\ $e_{2}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast e_{2}=e_{1}+\beta
e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{20}^{\alpha,\beta}$ and
$\mathcal{C}_{20}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

  * •

If $\alpha_{2}=0$, we obtain the parametric family:

$\mathcal{C}_{21}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{ll}$e_{2}\cdot
e_{1}=e_{1},$&\\\ $e_{2}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast e_{2}=\beta
e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{21}^{\alpha,\beta}$ and
$\mathcal{C}_{21}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

Second, we consider the second form. Then we have the following cases:

  * •

If $\alpha_{6}\neq 0$, then choose $\xi=\alpha_{6}$ and obtain the parametric
family:

$\mathcal{C}_{22}^{\alpha\neq
0,\beta}:=\left\\{\begin{tabular}[]{lll}$e_{2}\cdot e_{1}=e_{1},$&&\\\
$e_{1}\ast e_{2}=\alpha e_{1},$&$e_{2}\ast e_{1}=\beta e_{1},$&$e_{2}\ast
e_{2}=e_{1}+\alpha e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{22}^{\alpha,\beta}$ and
$\mathcal{C}_{22}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

  * •

If $\alpha_{6}=0$, we obtain the parametric family:

$\mathcal{C}_{23}^{\alpha\neq
0,\beta}:=\left\\{\begin{tabular}[]{lll}$e_{2}\cdot e_{1}=e_{1},$&&\\\
$e_{1}\ast e_{2}=\alpha e_{1},$&$e_{2}\ast e_{1}=\beta e_{1},$&$e_{2}\ast
e_{2}=\alpha e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{23}^{\alpha,\beta}$ and
$\mathcal{C}_{23}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

####  1.3.5. Compatible pre-Lie algebras defined on
$\mathcal{C}_{05}^{\alpha\neq 0,\frac{1}{2},1}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{05}^{\alpha},\mathcal{C}_{05}^{\alpha})$,
the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{05}^{\alpha}$ are of the form:

$\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=\alpha e_{2},$\\\ $e_{1}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2}.$\end{tabular}\right.$

Then we have the following cases:

  * •

If $\alpha_{2}\neq 0$, then choose $\xi=\alpha_{2}$ and obtain the parametric
family:

$\mathcal{C}_{24}^{\alpha,\beta,\gamma}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$\\\ $e_{1}\ast e_{1}=\beta
e_{1}+e_{2},$&$e_{1}\ast e_{2}=\gamma e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{24}^{\alpha,\beta,\gamma}$ and
$\mathcal{C}_{24}^{\alpha^{\prime}\beta^{\prime},\gamma^{\prime}}$ are
isomorphic if and only if
$(\alpha,\beta,\gamma)=(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$.

  * •

If $\alpha_{2}=0$, we obtain the parametric family:

$\mathcal{C}_{25}^{\alpha,\beta,\gamma}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$\\\ $e_{1}\ast e_{1}=\beta
e_{1},$&$e_{1}\ast e_{2}=\gamma e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{25}^{\alpha,\beta,\gamma}$ and
$\mathcal{C}_{25}^{\alpha^{\prime},\beta^{\prime},\gamma^{\prime}}$ are
isomorphic if and only if
$(\alpha,\beta,\gamma)=(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$.

####  1.3.6. Compatible pre-Lie algebras defined on
$\mathcal{C}_{05}^{\frac{1}{2}}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{05}^{\frac{1}{2}},\mathcal{C}_{05}^{\frac{1}{2}})$,
the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{05}^{\frac{1}{2}}$ are of the form:

  1. (1)

$\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=\frac{1}{2}e_{2},$\\\ $e_{1}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=\frac{1}{2}e_{2},$&\\\ $e_{1}\ast e_{1}=2\alpha_{4}e_{1},$&$e_{1}\ast
e_{2}=\alpha_{4}e_{2},$&$e_{2}\ast
e_{2}=\alpha_{5}e_{1}.$\end{tabular}\right.$

  3. (3)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=\frac{1}{2}e_{2},$&&\\\ $e_{1}\ast e_{1}=2\alpha_{6}e_{1},$&$e_{1}\ast
e_{2}=\alpha_{7}e_{1}+\alpha_{6}e_{2},$&$e_{2}\ast
e_{1}=2\alpha_{7}e_{1}$&$e_{2}\ast
e_{2}=\alpha_{8}e_{1}+\alpha_{7}e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{5}\alpha_{7}\neq 0$.

If the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{05}^{\frac{1}{2}}$ is of the first form, then we obtain the
algebras $\mathcal{C}_{24}^{\frac{1}{2},\beta,\gamma}$ and
$\mathcal{C}_{25}^{\frac{1}{2},\beta,\gamma}$.

Assume now that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{05}^{\frac{1}{2}}$ is of the second form. Then, choose
$\xi={\alpha^{-\frac{1}{2}}_{5}}$ and obtain the parametric family:

$\mathcal{C}_{26}^{\alpha}:=\left\\{\begin{tabular}[]{lll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\frac{1}{2}e_{2},$&\\\ $e_{1}\ast
e_{1}=2\alpha e_{1},$&$e_{1}\ast e_{2}=\alpha e_{2},$&$e_{2}\ast
e_{2}=e_{1}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{26}^{\alpha}$ and
$\mathcal{C}_{26}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

Finally, assume that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{05}^{\frac{1}{2}}$ is of the third form. Then, choose
$\xi={\alpha^{-1}_{7}}$ and obtain the parametric family:

$\mathcal{C}_{27}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\frac{1}{2}e_{2},$&&\\\ $e_{1}\ast
e_{1}=2\alpha e_{1},$&$e_{1}\ast e_{2}=e_{1}+\alpha e_{2},$&$e_{2}\ast
e_{1}=2e_{1}$&$e_{2}\ast e_{2}=\beta e_{1}+e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{27}^{\alpha,\beta}$ and
$\mathcal{C}_{27}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

####  1.3.7. Compatible pre-Lie algebras defined on $\mathcal{C}_{05}^{1}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{05}^{1},\mathcal{C}_{05}^{1})$,
the compatible pre-Lie algebra structures defined on $\mathcal{C}_{05}^{1}$
are of the form:

  1. (1)

$\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=e_{2},$&&\\\ $e_{1}\ast e_{1}=\alpha_{4}e_{1},$&$e_{1}\ast
e_{2}=\alpha_{4}e_{2},$&$e_{2}\ast e_{1}=\alpha_{5}e_{1},$&$e_{2}\ast
e_{2}=\alpha_{5}e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{5}\neq 0$.

First, suppose that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{05}^{1}$ is of the first form. Then we have the following cases:

  * •

If $\alpha_{1}\neq\alpha_{3}$, then choose
$\nu=\frac{\alpha_{2}}{\alpha_{1}-\alpha_{3}}$ and obtain the parametric
family $\mathcal{C}_{25}^{1,\beta,\gamma\neq\beta}$.

  * •

If $\alpha_{1}=\alpha_{3}$, then we will consider the following two cases.

    * –

If $\alpha_{2}\neq 0$, then choose $\xi=\alpha_{2}$ and obtain the parametric
family $\mathcal{C}_{24}^{1,\beta,\beta}$.

    * –

If $\alpha_{2}=0$, then we obtain the parametric family
$\mathcal{C}_{25}^{1,\beta,\beta}$.

Assume now that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{05}^{1}$ is of the second form. Then, choose
$\xi={\alpha^{-1}_{5}},\nu=-{\alpha_{4}}{\alpha^{-1}_{5}}$ and obtain the
algebra:

$\mathcal{C}_{28}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$\\\ $e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast
e_{2}=e_{2}.$\end{tabular}\right.$

####  1.3.8. Compatible pre-Lie algebras defined on $\mathcal{C}_{05}^{0}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{05}^{0},\mathcal{C}_{05}^{0})$,
the compatible pre-Lie algebra structures defined on $\mathcal{C}_{05}^{0}$
are of the form:

  1. (1)

$\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&\\\ $e_{1}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&\\\ $e_{1}\ast
e_{1}=\alpha_{4}e_{1},$&$e_{2}\ast
e_{2}=\alpha_{5}e_{2}.$\end{tabular}\right.$

  3. (3)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&&&\\\ $e_{1}\ast
e_{1}=\alpha_{6}e_{1}+\alpha_{7}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{8}e_{1},$&$e_{2}\ast e_{1}=\alpha_{8}e_{1},$&$e_{2}\ast
e_{2}=\alpha_{8}e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{5}\alpha_{8}\neq 0$.

If the compatible pre-Lie algebra structures defined on $\mathcal{C}_{05}^{0}$
is of the first form, then we obtain the algebras
$\mathcal{C}_{24}^{0,\beta,\gamma}$ and $\mathcal{C}_{25}^{0,\beta,\gamma}$.

Assume now that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{05}^{0}$ are of the second form. Then, choose
$\xi={\alpha^{-1}_{5}}$ and obtain the parametric family:

$\mathcal{C}_{29}^{\alpha}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{1},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast
e_{2}=e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{29}^{\alpha}$ are
$\mathcal{C}_{29}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

Finally, assume that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{05}^{0}$ is of the third form. Then, choose
$\xi={\alpha^{-1}_{8}}$ and obtain the parametric family:

$\mathcal{C}_{30}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{1},$&&&\\\ $e_{1}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{1}\ast
e_{2}=e_{1},$&$e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast
e_{2}=e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{30}^{\alpha,\beta}$ and
$\mathcal{C}_{30}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

####  1.3.9. Compatible pre-Lie algebras defined on
$\mathcal{C}_{06}^{\alpha\neq 0,1,2}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{06}^{\alpha},\mathcal{C}_{06}^{\alpha})$,
the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{06}^{\alpha}$ are of the form:

$\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=\alpha e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{1}e_{2}.$\end{tabular}\right.$

Then we have the following cases:

  * •

If $\alpha_{2}\neq 0$, then choose $\xi=\alpha_{2}$ and obtain the parametric
family:

$\mathcal{C}_{31}^{\alpha,\beta,\gamma}:=\left\\{\begin{tabular}[]{lll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\
$e_{1}\ast e_{1}=\beta e_{1}+e_{2},$&$e_{1}\ast e_{2}=\gamma
e_{2},$&$e_{2}\ast e_{1}=\beta e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{31}^{\alpha,\beta,\gamma}$ and
$\mathcal{C}_{31}^{\alpha^{\prime},\beta^{\prime},\gamma^{\prime}}$ are
isomorphic if and only if
$(\alpha,\beta,\gamma)=(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$.

  * •

If $\alpha_{2}=0$, we obtain the parametric family:

$\mathcal{C}_{32}^{\alpha,\beta,\gamma}:=\left\\{\begin{tabular}[]{lll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\
$e_{1}\ast e_{1}=\beta e_{1},$&$e_{1}\ast e_{2}=\gamma e_{2},$&$e_{2}\ast
e_{1}=\beta e_{2}.$\end{tabular}\right.$

The algebras
$\mathcal{C}_{32}^{\alpha,\beta,\gamma},\mathcal{C}_{32}^{\alpha^{\prime},\beta^{\prime},\gamma^{\prime}}$
are isomorphic if and only if
$(\alpha,\beta,\gamma)=(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$.

####  1.3.10. Compatible pre-Lie algebras defined on $\mathcal{C}_{06}^{0}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{06}^{0},\mathcal{C}_{06}^{0})$,
the compatible pre-Lie algebra structures defined on $\mathcal{C}_{06}^{0}$
are of the forms:

  1. (1)

$\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot
e_{1}=e_{2},$&\\\ $e_{1}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{1}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot
e_{1}=e_{2},$&&\\\ $e_{1}\ast e_{1}=\alpha_{4}e_{1},$&$e_{1}\ast
e_{2}=\alpha_{5}e_{1},$&$e_{2}\ast e_{1}=\alpha_{4}e_{2},$&$e_{2}\ast
e_{2}=\alpha_{5}e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{5}\neq 0$.

Let us first consider the first form. Then we have the following cases:

  * •

If $\alpha_{3}\neq 0$, then $\nu=-\frac{\alpha_{2}}{\alpha_{3}}$ choose and
obtain the parametric family $\mathcal{C}_{32}^{0,\beta,\gamma\neq 0}$.

  * •

If $\alpha_{3}=0$, then we consider the following two cases.

    * –

If $\alpha_{2}\neq 0$, then choose $\xi=\alpha_{2}$ and obtain the parametric
family ${\mathcal{C}_{31}^{0,\beta,0}}$.

    * –

If $\alpha_{2}=0$, then we obtain the parametric family
$\mathcal{C}_{32}^{0,\beta,0}$

Now, we consider the second form. Then choose
$\xi={\alpha^{-1}_{5}},\nu=-{\alpha_{4}}{\alpha^{-1}_{5}}$ and obtain the
algebra:

${\mathcal{C}_{33}}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{2}=e_{1},$&$e_{2}\ast
e_{2}=e_{2}.$\end{tabular}\right.$

####  1.3.11. Compatible pre-Lie algebras defined on $\mathcal{C}_{06}^{1}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{06}^{1},\mathcal{C}_{06}^{1})$,
the compatible pre-Lie algebra structures defined on $\mathcal{C}_{06}^{1}$
are of the forms:

  1. (1)

$\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{1}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast
e_{1}=\alpha_{4}e_{1},$&$e_{1}\ast e_{2}=\alpha_{4}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{4}e_{2},$&$e_{2}\ast
e_{2}=\alpha_{5}e_{1}+\alpha_{6}e_{2}.$\end{tabular}\right.$

We may assume $\left(\alpha_{5},\alpha_{6}\right)\neq\left(0,0\right)$.

If the compatible pre-Lie algebra structures defined on $\mathcal{C}_{06}^{1}$
is of the first form, then we obtain the algebras
$\mathcal{C}_{31}^{1,\beta,\gamma}$ and $\mathcal{C}_{32}^{1,\beta,\gamma}$.

Assume now that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{06}^{1}$ is of the second form. Then we have the following
cases:

  * •

If $\alpha_{5}\neq 0$, then choose $\xi={\alpha^{-\frac{1}{2}}_{5}}$ and
obtain the parametric family:

$\mathcal{C}_{34}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\
$e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\alpha e_{2},$&$e_{2}\ast
e_{1}=\alpha e_{2},$&$e_{2}\ast e_{2}=e_{1}+\beta e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{34}^{\alpha,\beta}$ and
$\mathcal{C}_{34}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

  * •

If $\alpha_{5}=0$, then choose $\xi={\alpha^{-1}_{6}}$ and obtain the
parametric family:

$\mathcal{C}_{35}^{\alpha}:=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\
$e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\alpha e_{2},$&$e_{2}\ast
e_{1}=\alpha e_{2},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{35}^{\alpha}$ and
$\mathcal{C}_{35}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

####  1.3.12. Compatible pre-Lie algebras defined on $\mathcal{C}_{06}^{2}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{06}^{2},\mathcal{C}_{06}^{2})$,
the compatible pre-Lie algebra structures defined on $\mathcal{C}_{06}^{2}$
are of the forms:

  1. (1)

$\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=2e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{3}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{1}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=2e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast
e_{1}=\alpha_{4}e_{1}+\alpha_{5}e_{2},$&$e_{1}\ast
e_{2}=2\alpha_{4}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{6}e_{1}+\alpha_{4}e_{2},$&$e_{2}\ast
e_{2}=2\alpha_{6}e_{2}.$\end{tabular}\right.$

  3. (3)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=2e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast
e_{1}=\alpha_{7}e_{1},$&$e_{1}\ast
e_{2}=\alpha_{8}e_{1}+2\alpha_{7}e_{2},$&$e_{2}\ast
e_{1}=2\alpha_{8}e_{1}+\alpha_{7}e_{2},$&$e_{2}\ast
e_{2}=\alpha_{8}e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{6}\alpha_{8}\neq 0$.

If the compatible pre-Lie algebra structures defined on $\mathcal{C}_{06}^{2}$
is of the first form, then we obtain the algebras
$\mathcal{C}_{31}^{2,\beta,\gamma}$ and $\mathcal{C}_{32}^{2,\beta,\gamma}$.

Assume now that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{06}^{2}$ are of the second form. Then, choose
$\xi={\alpha^{-1}_{6}}$ and obtain the parametric family:

$\mathcal{C}_{36}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=2e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\
$e_{1}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{1}\ast e_{2}=2\alpha
e_{2},$&$e_{2}\ast e_{1}=e_{1}+\alpha e_{2},$&$e_{2}\ast
e_{2}=2e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{36}^{\alpha,\beta}$ and
$\mathcal{C}_{36}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$.

Finally, assume that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{06}^{2}$ are of the third form. Then, choose
$\xi={\alpha^{-1}_{8}}$ and obtain the parametric family:

$\mathcal{C}_{37}^{\alpha}:=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=2e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\
$e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=e_{1}+2\alpha
e_{2},$&$e_{2}\ast e_{1}=2e_{1}+\alpha e_{2},$&$e_{2}\ast
e_{2}=e_{2}.$\end{tabular}\right.$

The algebras $\mathcal{C}_{37}^{\alpha}$ and
$\mathcal{C}_{37}^{\alpha^{\prime}}$ are isomorphic if and only if
$\alpha=\alpha^{\prime}$.

####  1.3.13. Compatible pre-Lie algebras defined on $\mathcal{C}_{07}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{07},\mathcal{C}_{07})$, the
compatible pre-Lie algebra structures defined on $\mathcal{C}_{07}$ are of the
forms:

  1. (1)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot
e_{2}=e_{2},$&&\\\ $e_{1}\ast
e_{1}=\left(\alpha_{3}+\alpha_{2}-\alpha_{1}\right)e_{1}-\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$\\\ $e_{2}\ast
e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{2}\ast
e_{2}=-\alpha_{1}e_{1}+\alpha_{3}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot
e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha_{4}e_{1},$&$e_{2}\ast
e_{2}=\alpha_{5}e_{2}.$\end{tabular}\right.$

We may assume that $\alpha_{4}\neq\alpha_{5}$.

Suppose first that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{07}$ has the first form. Then we obtain the algebras:

$\mathcal{C}_{38}^{\alpha,\beta,\gamma}:=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{2}\cdot e_{2}=e_{2},$&&\\\ $e_{1}\ast
e_{1}=\left(\gamma+\beta-\alpha\right)e_{1}-\beta e_{2},$&$e_{1}\ast
e_{2}=\alpha e_{1}+\beta e_{2},$&\\\ $e_{2}\ast e_{1}=\alpha e_{1}+\beta
e_{2},$&$e_{2}\ast e_{2}=-\alpha e_{1}+\gamma
e_{2}.$\end{tabular}\right.\qquad$

The algebras $\mathcal{C}_{38}^{\alpha,\beta,\gamma}$ and
$\mathcal{C}_{38}^{\alpha^{\prime},\beta^{\prime},\gamma^{\prime}}$ are
isomorphic if and only if

$(\alpha,\beta,\gamma)=(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$ or
$(\alpha,\beta,\gamma)=(\beta^{\prime},\alpha^{\prime},-\alpha^{\prime}+\beta^{\prime}+\gamma^{\prime})$.

Now, assume that the compatible pre-Lie algebra structures defined on
$\mathcal{C}_{07}$ has the second form. Then we obtain the algebras:

$\mathcal{C}_{39}^{\alpha,\beta\neq\alpha}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{2}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha
e_{1},$&$e_{2}\ast e_{2}=\beta e_{2}.$\end{tabular}\right.\qquad$

The algebras $\mathcal{C}_{39}^{\alpha,\beta}$ and
$\mathcal{C}_{39}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$ or
$(\alpha,\beta)=(\beta^{\prime},\alpha^{\prime}).$

####  1.3.14. Compatible pre-Lie algebras defined on $\mathcal{C}_{08}$

From the computation of
$\mathrm{Z}_{\mathrm{CPL}}^{2}(\mathcal{C}_{08},\mathcal{C}_{08})$, the
compatible pre-Lie algebra structures defined on $\mathcal{C}_{08}$ are of the
forms:

  1. (1)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=2e_{2},$&$e_{2}\cdot e_{1}=\frac{1}{2}e_{1}+e_{2},$&$e_{2}\cdot
e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha_{1}e_{1}+\alpha_{2}e_{2},$&$e_{1}\ast
e_{2}=2\alpha_{1}e_{2},$&$e_{2}\ast
e_{1}=\alpha_{3}e_{1}+\alpha_{1}e_{2},$&$e_{2}\ast
e_{2}=2\alpha_{3}e_{2}.$\end{tabular}\right.$

  2. (2)

$\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot
e_{2}=2e_{2},$\\\ $e_{2}\cdot e_{1}=\frac{1}{2}e_{1}+e_{2},$&$e_{2}\cdot
e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha_{4}e_{1},$&$e_{1}\ast
e_{2}=\alpha_{5}e_{1}+2\alpha_{4}e_{2},$\\\ $e_{2}\ast
e_{1}=\left(2\alpha_{5}+\frac{1}{2}\alpha_{4}\right)e_{1}+\alpha_{4}e_{2},$&$e_{2}\ast
e_{2}=\frac{1}{2}\alpha_{5}e_{1}+\left(\alpha_{4}+\alpha_{5}\right)e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{5}\neq 0$. Then we obtain the following compatible pre-
Lie algebras:

$\displaystyle\mathcal{C}_{40}^{\alpha,\beta,\gamma}$ $\displaystyle:$
$\displaystyle=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=2e_{2},$&$e_{2}\cdot
e_{1}=\frac{1}{2}e_{1}+e_{2},$&$e_{2}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast
e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{1}\ast e_{2}=2\alpha e_{2},$&$e_{2}\ast
e_{1}=\gamma e_{1}+\alpha e_{2},$&$e_{2}\ast e_{2}=2\gamma
e_{2}.$\end{tabular}\right.$ $\displaystyle\mathcal{C}_{41}^{\alpha,\beta\neq
0}$ $\displaystyle:$ $\displaystyle=\left\\{\begin{tabular}[]{llll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=2e_{2},$&$e_{2}\cdot
e_{1}=\frac{1}{2}e_{1}+e_{2},$&$e_{2}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast
e_{1}=\alpha e_{1},$&\lx@intercol$e_{1}\ast e_{2}=\beta e_{1}+2\alpha
e_{2},$\hfil\lx@intercol \\\ $e_{2}\ast
e_{1}=\left(2\beta+\frac{1}{2}\alpha\right)e_{1}+\alpha
e_{2},$&\lx@intercol$e_{2}\ast e_{2}=\frac{1}{2}\beta
<EMAIL_ADDRESS>
Moreover, the algebras $\mathcal{C}_{40}^{\alpha,\beta,\gamma}$ and
$\mathcal{C}_{40}^{\alpha^{\prime},\beta^{\prime},\gamma^{\prime}}$ are
isomorphic if and only if

$\left(\alpha^{\prime},\beta^{\prime},\gamma^{\prime}\right)=\left(\alpha,\beta,\gamma\right)$
or
$\left(\alpha^{\prime},\beta^{\prime},\gamma^{\prime}\right)=\left(4\gamma-\alpha,\beta-8\alpha+16\gamma,\gamma\right)$.

Also, the algebras $\mathcal{C}_{41}^{\alpha,\beta}$ and
$\mathcal{C}_{41}^{\alpha^{\prime},\beta^{\prime}}$ are isomorphic if and only
if

$\left(\alpha^{\prime},\beta^{\prime}\right)=\left(\alpha,\beta\right)$ or
$\left(\alpha^{\prime},\beta^{\prime}\right)=\left(\alpha+4\beta,-\beta\right)$.

###  1.4. The algebraic classification of compatible commutative associative
algebras

It is easy to see that each commutative pre-Lie algebra is associative. Hence,
we can choose only commutative compatible pre-Lie algebras from the Theorem 7.

######  Theorem 8.

Let $\mathcal{C}$ be a nonzero $2$-dimensional compatible commutative
associative algebra. Then $\mathcal{C}$ is isomorphic to one and only one and
only one of the following algebras:

$\mathcal{C}_{03}$ | $:$ | $e_{1}\ast e_{1}=e_{2}.$
---|---|---
$\mathcal{C}_{05}^{0}$ | $:$ | $e_{1}\ast e_{1}=e_{1}.$
$\mathcal{C}_{06}^{1}$ | $:$ | $e_{1}\ast e_{1}=e_{1},\ e_{1}\ast e_{2}=e_{2},\ e_{2}\ast e_{1}=e_{2}.$
$\mathcal{C}_{07}$ | $:$ | $e_{1}\ast e_{1}=e_{1},\ e_{2}\ast e_{2}=e_{2}.$
$\mathcal{C}_{13}^{1}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{2},$&&\\\ $e_{1}\ast e_{1}=e_{1},$&$e_{1}\ast e_{2}=e_{2},$&$e_{2}\ast e_{1}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{15}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{l}$e_{1}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{16}^{0}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=e_{1}.$\end{tabular}\right.$
$\mathcal{C}_{18}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{2},$&&&\\\ $e_{1}\ast e_{1}=\alpha e_{2},$&$e_{1}\ast e_{2}=e_{1},$&$e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{24}^{0,\beta,0}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$\\\ $e_{1}\ast e_{1}=\beta e_{1}+e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{25}^{0,\beta,0}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$\\\ $e_{1}\ast e_{1}=\beta e_{1}.$\end{tabular}\right.$
$\mathcal{C}_{29}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{30}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&&&\\\ $e_{1}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{1}\ast e_{2}=e_{1},$&$e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{31}^{1,\beta,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1}+e_{2},$&$e_{1}\ast e_{2}=\beta e_{2},$&$e_{2}\ast e_{1}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{32}^{1,\beta,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1},$&$e_{1}\ast e_{2}=\beta e_{2},$&$e_{2}\ast e_{1}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{34}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\alpha e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2},$&$e_{2}\ast e_{2}=e_{1}+\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{35}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\alpha e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{38}^{\alpha,\beta,\gamma}$ | $:$ | $\left\\{\begin{tabular}[]{llll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot e_{2}=e_{2},$&&\\\ $e_{1}\ast e_{1}=\left(\gamma+\beta-\alpha\right)e_{1}-\beta e_{2},$&$e_{1}\ast e_{2}=\alpha e_{1}+\beta e_{2},$\\\ $e_{2}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{2}\ast e_{2}=-\alpha e_{1}+\gamma e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{39}^{\alpha,\beta\neq\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast e_{2}=\beta e_{2}.$\end{tabular}\right.$

All algebras are non-isomorphic, except
$\mathcal{C}_{38}^{\alpha,\beta,\gamma}\cong\mathcal{C}_{38}^{\beta,\alpha,-\alpha+\beta+\gamma},$
$\mathcal{C}_{39}^{\alpha,\beta}\cong\mathcal{C}_{39}^{\beta,\alpha}.$

###  1.5. The algebraic classification of compatible associative algebras

######  Definition 9.

A compatible associative algebra is a vector space ${\bf A}$ equipped with two
multiplication $\cdot$ and another multiplication $\ast,$ such that, $({\bf
A},\cdot),$ $({\bf A},\ast)$ and $({\bf A},\cdot+\ast)$ are associative
algebras. These two operations are required to satisfy the following
identities.

$\left(x\cdot y\right)\cdot z$ | $=$ | $x\cdot\left(y\cdot z\right),$
---|---|---
$\left(x\ast y\right)\ast z$ | $=$ | $x\ast\left(y\ast z\right),$
$(x\ast y)\cdot z+(x\cdot y)\ast z$ | $=$ | $x\ast(y\cdot z)+x\cdot(y\ast z).$

######  Theorem 10.

Let $\mathcal{C}$ be a nonzero $2$-dimensional compatible associative algebra.
Then $\mathcal{C}$ is isomorphic to one and only one compatible commutative
associative algebra listed in Theorem 8 or one of the following algebras:

$\mathcal{C}_{05}^{1}$ | $:$ | $e_{1}\ast e_{1}=e_{1},\ e_{1}\ast e_{2}=e_{2}.$
---|---|---
$\mathcal{C}_{06}^{0}$ | $:$ | $e_{1}\ast e_{1}=e_{1},\ \ e_{2}\ast e_{1}=e_{2}.$
$\mathcal{C}_{25}^{1,\beta,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1},$&$e_{1}\ast e_{2}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{28}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2},$\\\ $e_{2}\ast e_{1}=e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{32}^{0,\beta,0}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1},$&$e_{2}\ast e_{1}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{33}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{2}=e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$

###  1.6. The algebraic classification of compatible Novikov algebras

######  Definition 11.

A compatible Novikov algebra is a vector space ${\bf A}$ equipped with two
multiplication $\cdot$ and another multiplication $\ast,$ such that, $({\bf
A},\cdot),$ $({\bf A},\ast)$ and $({\bf A},\cdot+\ast)$ are Novikov algebras.
These two operations are required to satisfy the following identities.

$\left(x\cdot y\right)\cdot z-x\cdot\left(y\cdot z\right)$ | $=$ | $\left(y\cdot x\right)\cdot z-y\cdot\left(x\cdot z\right),$
---|---|---
$(x\cdot y)\cdot z$ | $=$ | $(x\cdot z)\cdot y,$
$\left(x\ast y\right)\ast z-x\ast\left(y\ast z\right)$ | $=$ | $\left(y\ast x\right)\ast z-y\ast\left(x\ast z\right),$
$(x\ast y)\ast z$ | $=$ | $(x\ast z)\ast y,$
$(x\ast y)\cdot z-x\ast(y\cdot z)+(x\cdot y)\ast z-x\cdot(y\ast z)$ | $=$ | $(y\ast x)\cdot z-y\ast(x\cdot z)+(y\cdot x)\ast z-y\cdot(x\ast z),$
$(x\ast y)\cdot z+(x\cdot y)\ast z$ | $=$ | $(x\ast z)\cdot y+(x\cdot z)\ast y.$

######  Theorem 12.

Let $\mathcal{C}$ be a nonzero $2$-dimensional compatible Novikov algebra.
Then $\mathcal{C}$ is isomorphic to one and only one compatible commutative
associative algebra listed in Theorem 8 or one of the following algebras:

$\mathcal{C}_{01}$ | $:$ | $e_{1}\ast e_{1}=e_{1}+e_{2},\ e_{2}\ast e_{1}=e_{2}.$
---|---|---
$\mathcal{C}_{04}$ | $:$ | $e_{2}\ast e_{1}=e_{1}.$
$\mathcal{C}_{06}^{\alpha\neq 1}$ | $:$ | $e_{1}\ast e_{1}=e_{1},\ e_{1}\ast e_{2}=\alpha e_{2},\ e_{2}\ast e_{1}=e_{2}.$
$\mathcal{C}_{09}^{\alpha,\beta\neq 0}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1}+e_{2},$&$e_{2}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=\beta e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2}.$\\\ \end{tabular}\right.$
$\mathcal{C}_{10}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1}+e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\alpha e_{1}+\beta e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{13}^{\alpha\neq 1}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{2},$&&\\\ $e_{1}\ast e_{1}=\alpha e_{1},$&$e_{1}\ast e_{2}=e_{2},$&$e_{2}\ast e_{1}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{14}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{2},$&\\\ $e_{1}\ast e_{1}=e_{1}+\alpha e_{2},$&$e_{2}\ast e_{1}=e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{20}^{\alpha,0}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{2}\cdot e_{1}=e_{1},$&\\\ $e_{2}\ast e_{1}=\alpha e_{1},$&$e_{2}\ast e_{2}=e_{1}.$\end{tabular}\right.$
$\mathcal{C}_{21}^{\alpha,0}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{2}\cdot e_{1}=e_{1},$&\\\ $e_{2}\ast e_{1}=\alpha e_{1}.$\end{tabular}\right.$
$\mathcal{C}_{22}^{\alpha\neq 0,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{2}\cdot e_{1}=e_{1},$&&\\\ $e_{1}\ast e_{2}=\alpha e_{1},$&$e_{2}\ast e_{1}=\beta e_{1},$&$e_{2}\ast e_{2}=e_{1}+\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{23}^{\alpha\neq 0,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{2}\cdot e_{1}=e_{1},$&&\\\ $e_{1}\ast e_{2}=\alpha e_{1},$&$e_{2}\ast e_{1}=\beta e_{1},$&$e_{2}\ast e_{2}=\alpha e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{31}^{(\alpha,\beta,\gamma)\neq(1,\beta,\beta)}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1}+e_{2},$&$e_{1}\ast e_{2}=\gamma e_{2},$&$e_{2}\ast e_{1}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{32}^{(\alpha,\beta,\gamma)\neq(1,\beta,\beta)}$ | $:$ | $\left\\{\begin{tabular}[]{lll}$e_{1}\cdot e_{1}=e_{1},$&$e_{1}\cdot e_{2}=\alpha e_{2},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{1}=\beta e_{1},$&$e_{1}\ast e_{2}=\gamma e_{2},$&$e_{2}\ast e_{1}=\beta e_{2}.$\end{tabular}\right.$
$\mathcal{C}_{33}$ | $:$ | $\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{1},$&$e_{2}\cdot e_{1}=e_{2},$\\\ $e_{1}\ast e_{2}=e_{1},$&$e_{2}\ast e_{2}=e_{2}.$\end{tabular}\right.$

All algebras are non-isomorphic, except
$\mathcal{C}_{31}^{0,\beta,\gamma}\cong\mathcal{C}_{32}^{0,\beta,\gamma}.$

##  2\. The geometric classification of compatible algebras

###  2.1. Degenerations and the geometric classification of algebras

Let us introduce the techniques used to obtain the geometric classification
for an arbitrary variety of compatible $\Omega$-algebras. Given a complex
vector space ${\mathbb{V}}$ of dimension $n$, the set of bilinear maps

$\textrm{Bil}({\mathbb{V}}\times{\mathbb{V}},{\mathbb{V}})\cong\textrm{Hom}({\mathbb{V}}^{\otimes
2},{\mathbb{V}})\cong({\mathbb{V}}^{*})^{\otimes 2}\otimes{\mathbb{V}}$

is a vector space of dimension $n^{3}$. The set of pairs of bilinear maps

$\textrm{Bil}({\mathbb{V}}\times{\mathbb{V}},{\mathbb{V}})\oplus\textrm{Bil}({\mathbb{V}}\times{\mathbb{V}},{\mathbb{V}})\cong({\mathbb{V}}^{*})^{\otimes
2}\otimes{\mathbb{V}}\oplus({\mathbb{V}}^{*})^{\otimes 2}\otimes{\mathbb{V}}$

which is a vector space of dimension $2n^{3}$. This vector space has the
structure of the affine space $\mathbb{C}^{2n^{3}}$ in the following sense:
fixed a basis $e_{1},\ldots,e_{n}$ of ${\mathbb{V}}$, then any pair with
multiplication $(\mu,\mu^{\prime})$, is determined by some parameters
$c_{ij}^{k},c_{ij}^{\prime k}\in\mathbb{C}$, called structural constants, such
that

$\mu(e_{i},e_{j})=\sum_{p=1}^{n}c_{ij}^{k}e_{k}\textrm{ and
}\mu^{\prime}(e_{i},e_{j})=\sum_{p=1}^{n}c_{ij}^{\prime k}e_{k},$

which corresponds to a point in the affine space $\mathbb{C}^{2n^{3}}$. Then a
set of bilinear pairs $\mathcal{S}$ corresponds to an algebraic variety, i.e.,
a Zariski closed set, if there are some polynomial equations in variables
$c_{ij}^{k},c_{ij}^{\prime k}$ with zero locus equal to the set of structural
constants of the bilinear pairs in $\mathcal{S}$. Given the identities
defining a particular class of compatible $\Omega$-algebras, we can obtain a
set of polynomial equations in variables $c_{ij}^{k},c_{ij}^{\prime k}$. This
class of $n$-dimensional compatible $\Omega$-algebras is a variety. Denote it
by $\mathcal{T}_{n}$.Now, consider the following action of
$\rm{GL}({\mathbb{V}})$ on ${\mathcal{T}}_{n}$:

$(g*(\mu,\mu^{\prime}))(x,y):=(g\mu(g^{-1}x,g^{-1}y),g\mu^{\prime}(g^{-1}x,g^{-1}y))$

for $g\in\rm{GL}({\mathbb{V}})$, $(\mu,\mu^{\prime})\in\mathcal{T}_{n}$ and
for any $x,y\in{\mathbb{V}}$. Observe that the
$\textrm{GL}({\mathbb{V}})$-orbit of $(\mu,\mu^{\prime})$, denoted
${\mathcal{O}}((\mu,\mu^{\prime}))$, contains all the structural constants of
the bilinear pairs isomorphic to the compatible $\Omega$-algebras with
structural constants $(\mu,\mu^{\prime})$.

A geometric classification of a variety of algebras consists on describing the
irreducible components of the variety. Recall that any affine variety can be
represented as a finite union of its irreducible components uniquely. Note
that describing the irreducible components of ${\mathcal{T}_{n}}$ gives us the
rigid algebras of the variety, which are those bilinear pairs with an open
$\textrm{GL}(\mathbb{V})$-orbit. This is due to the fact that a bilinear pair
is rigid in a variety if and only if the closure of its orbit is an
irreducible component of the variety. For this reason, the following notion is
convenient. Denote by $\overline{{\mathcal{O}}((\mu,\mu^{\prime}))}$ the
closure of the orbit of $(\mu,\mu^{\prime})\in{\mathcal{T}_{n}}$.

######  Definition 13.

Let ${\rm T}$ and ${\rm T}^{\prime}$ be two $n$-dimensional compatible
$\Omega$-algebras of a fixed class corresponding to the variety
$\mathcal{T}_{n}$ and
$(\mu,\mu^{\prime}),(\lambda,\lambda^{\prime})\in\mathcal{T}_{n}$ be their
representatives in the affine space, respectively. The algebra ${\rm T}$ is
said to degenerate to ${\rm T}^{\prime}$, and we write ${\rm T}\to{\rm
T}^{\prime}$, if
$(\lambda,\lambda^{\prime})\in\overline{{\mathcal{O}}((\mu,\mu^{\prime}))}$.
If ${\rm T}\not\cong{\rm T}^{\prime}$, then we call it a proper degeneration.
Conversely, if
$(\lambda,\lambda^{\prime})\not\in\overline{{\mathcal{O}}((\mu,\mu^{\prime}))}$
then we say that ${{\rm T}}$ does not degenerate to ${{\rm T}}^{\prime}$ and
we write ${{\rm T}}\not\to{{\rm T}}^{\prime}$.

Furthermore, for a parametric family of algebras we have the following notion.

######  Definition 14.

Let ${{\rm T}}(*)=\\{{{\rm T}}(\alpha):{\alpha\in I}\\}$ be a family of
$n$-dimensional compatible $\Omega$-algebras of a fixed class corresponding to
${\mathcal{T}}_{n}$ and let ${{\rm T}}^{\prime}$ be another algebra. Suppose
that ${{\rm T}}(\alpha)$ is represented by the structure
$(\mu(\alpha),\mu^{\prime}(\alpha))\in{\mathcal{T}}_{n}$ for $\alpha\in I$ and
${{\rm T}}^{\prime}$ is represented by the structure
$(\lambda,\lambda^{\prime})\in{\mathcal{T}}_{n}$. We say that the family
${{\rm T}}(*)$ degenerates to ${{\rm T}}^{\prime}$, and write ${{\rm
T}}(*)\to{{\rm T}}^{\prime}$, if
$(\lambda,\lambda^{\prime})\in\overline{\\{{\mathcal{O}}((\mu(\alpha),\mu^{\prime}(\alpha)))\\}_{\alpha\in
I}}$.Conversely, if
$(\lambda,\lambda^{\prime})\not\in\overline{\\{{\mathcal{O}}((\mu(\alpha),\mu^{\prime}(\alpha)))\\}_{\alpha\in
I}}$ then we call it a non-degeneration, and we write ${{\rm
T}}(*)\not\to{{\rm T}}^{\prime}$.

Observe that ${\rm T}^{\prime}$ corresponds to an irreducible component of
$\mathcal{T}_{n}$ (more precisely, $\overline{{\rm T}^{\prime}}$ is an
irreducible component) if and only if ${{\rm T}}\not\to{{\rm T}}^{\prime}$ for
any $n$-dimensional compatible $\Omega$-algebra ${\rm T}$ and ${{\rm
T}(*)}\not\to{{\rm T}}^{\prime}$ for any parametric family of $n$-dimensional
compatible $\Omega$-algebras ${\rm T}(*)$. To prove a particular algebra
corresponds to an irreducible compotent, we will use the next ideas.Firstly,
since
$\mathrm{dim}\,{\mathcal{O}}((\mu,\mu^{\prime}))=n^{2}-\mathrm{dim}\,\mathfrak{Der}({\rm
T})$, then if ${\rm T}\to{\rm T}^{\prime}$ and ${\rm T}\not\cong{\rm
T}^{\prime}$, we have that $\mathrm{dim}\,\mathfrak{Der}({\rm
T})<\mathrm{dim}\,\mathfrak{Der}({\rm T}^{\prime})$, where
$\mathfrak{Der}({\rm T})$ denotes the Lie algebra of derivations of ${\rm T}$.
Secondly, to prove degenerations, let ${{\rm T}}$ and ${{\rm T}}^{\prime}$ be
two compatible $\Omega$-algebras represented by the structures
$(\mu,\mu^{\prime})$ and $(\lambda,\lambda^{\prime})$ from
${{\mathcal{T}}}_{n}$, respectively. Let $c_{ij}^{k},$ $c_{ij}^{\prime k}$ be
the structure constants of $(\lambda,\lambda^{\prime})$ in a basis
$e_{1},\dots,e_{n}$ of ${\mathbb{V}}$. If there exist $n^{2}$ maps
$a_{i}^{j}(t):\mathbb{C}^{*}\to\mathbb{C}$ such that
$E_{i}(t)=\sum_{j=1}^{n}a_{i}^{j}(t)e_{j}$ ($1\leq i\leq n$) form a basis of
${\mathbb{V}}$ for any $t\in\mathbb{C}^{*}$ and the structure constants
$c_{ij}^{k}(t),c_{ij}^{\prime k}(t)$ of $(\mu,\mu^{\prime})$ in the basis
$E_{1}(t),\dots,E_{n}(t)$ satisfy $\lim\limits_{t\to
0}c_{ij}^{k}(t)=c_{ij}^{k}$ and $\lim\limits_{t\to 0}c_{ij}^{\prime
k}(t)=c_{ij}^{\prime k}$, then ${{\rm T}}\to{{\rm T}}^{\prime}$. In this case,
$E_{1}(t),\dots,E_{n}(t)$ is called a parametrized basis for ${{\rm
T}}\to{{\rm T}}^{\prime}$. Thirdly, to prove non-degenerations we may use a
remark that follows from this lemma, see [3].

######  Lemma 15.

Consider two compatible $\Omega$-algebras ${\rm T}$ and ${\rm T}^{\prime}$.
Suppose ${\rm T}\to{\rm T}^{\prime}$. Let C be a Zariski closed in
${\mathcal{T}}_{n}$ that is stable by the action of the invertible upper
(lower) triangular matrices. Then if there is a representation
$(\mu,\mu^{\prime})$ of ${\rm T}$ in C, then there is a representation
$(\lambda,\lambda^{\prime})$ of ${\rm T}^{\prime}$ in C.

In order to apply this lemma, we will give the explicit definition of the
appropriate stable Zariski closed $C$ in terms of the variables
$c_{ij}^{k},c_{ij}^{\prime k}$ in each case.

######  Remark 16.

Moreover, let ${{\rm T}}$ and ${{\rm T}}^{\prime}$ be two compatible
$\Omega$-algebras represented by the structures $(\mu,\mu^{\prime})$ and
$(\lambda,\lambda^{\prime})$ from ${\mathcal{T}_{n}}$. Suppose ${\rm T}\to{\rm
T}^{\prime}$. Then if $\mu,\mu^{\prime},\lambda,\lambda^{\prime}$ represents
algebras ${\rm T}_{0},{\rm T}_{1},{\rm T}^{\prime}_{0},{\rm T}^{\prime}_{1}$
in the affine space $\mathbb{C}^{n^{3}}$ of algebras with a single
multiplication, respectively, we have ${\rm T}_{0}\to{\rm T}^{\prime}_{0}$ and
${\rm T}_{1}\to{\rm T}^{\prime}_{1}$. So for example, $(0,\mu)$ can not
degenerate in $(\lambda,0)$ unless $\lambda=0$.

Fourthly, to prove ${{\rm T}}(*)\to{{\rm T}}^{\prime}$, suppose that ${{\rm
T}}(\alpha)$ is represented by the structure
$(\mu(\alpha),\mu^{\prime}(\alpha))\in{\mathcal{T}}_{n}$ for $\alpha\in I$ and
${{\rm T}}^{\prime}$ is represented by the structure
$(\lambda,\lambda^{\prime})\in{\mathcal{T}}_{n}$. Let
$c_{ij}^{k},c_{ij}^{\prime k}$ be the structure constants of
$(\lambda,\lambda^{\prime})$ in a basis $e_{1},\dots,e_{n}$ of ${\mathbb{V}}$.
If there is a pair of maps $(f,(a_{i}^{j}))$, where $f:\mathbb{C}^{*}\to I$
and $a_{i}^{j}:\mathbb{C}^{*}\to\mathbb{C}$ are such that
$E_{i}(t)=\sum_{j=1}^{n}a_{i}^{j}(t)e_{j}$ ($1\leq i\leq n$) form a basis of
${\mathbb{V}}$ for any $t\in\mathbb{C}^{*}$ and the structure constants
$c_{ij}^{k}(t),c_{ij}^{\prime k}(t)$ of
$(\mu\big{(}f(t)\big{)},\mu^{\prime}\big{(}f(t)\big{)})$ in the basis
$E_{1}(t),\dots,E_{n}(t)$ satisfy $\lim\limits_{t\to
0}c_{ij}^{k}(t)=c_{ij}^{k}$ and $\lim\limits_{t\to 0}c_{ij}^{\prime
k}(t)=c_{ij}^{\prime k}$, then ${{\rm T}}(*)\to{{\rm T}}^{\prime}$. In this
case $E_{1}(t),\dots,E_{n}(t)$ and $f(t)$ are called a parametrized basis and
a parametrized index for ${{\rm T}}(*)\to{{\rm T}}^{\prime}$, respectively.
Fifthly, to prove ${{\rm T}}(*)\not\to{{\rm T}}^{\prime}$, we may use an
analogous of Remark 16 for parametric families that follows from Lemma 17.

######  Lemma 17.

Consider the family of compatible $\Omega$-algebras ${\rm T}(*)$ and the
compatible $\Omega$-algebra ${\rm T}^{\prime}$. Suppose ${\rm T}(*)\to{\rm
T}^{\prime}$. Let C be a Zariski closed in $\mathcal{T}_{n}$ that is stable by
the action of the invertible upper (lower) triangular matrices. Then if there
is a representation $(\mu(\alpha),\mu^{\prime}(\alpha))$ of ${\rm T}(\alpha)$
in C for every $\alpha\in I$, then there is a representation
$(\lambda,\lambda^{\prime})$ of ${\rm T}^{\prime}$ in C.

Finally, the following remark simplifies the geometric problem.

######  Remark 18.

Let $(\mu,\mu^{\prime})$ and $(\lambda,\lambda^{\prime})$ represent two
compatible $\Omega$-algebras. Suppose
$(\lambda,0)\not\in\overline{{\mathcal{O}}((\mu,0))}$, (resp.
$(0,\lambda^{\prime})\not\in\overline{{\mathcal{O}}((0,\mu^{\prime}))}$), then
$(\lambda,\lambda^{\prime})\not\in\overline{{\mathcal{O}}((\mu,\mu^{\prime}))}$.
As we construct the classification of a given class of compatible
$\Omega$-algebras from a certain class of algebras with a single
multiplication that remains unchanged, this remark becomes very useful.

###  2.2. The geometric classification of compatible commutative associative
algebras

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

######  Theorem 19.

The variety of complex $2$-dimensional compatible commutative associative
algebras has dimension $7$ and it has $2$ irreducible components defined by
$\overline{\mathcal{O}(\mathcal{C}_{39}^{\alpha,\beta})}$ and
$\overline{\mathcal{O}(\mathcal{C}_{38}^{\alpha,\beta,\gamma})}.$ In
particular, there are no rigid algebras in this variety.

###### Proof.

Thanks to Theorem 8, we have the algebraic classification of $2$-dimensional
compatible commutative associative algebras. After carefully checking the
dimensions of orbit closures of the more important for us algebras, we have

$\dim\mathcal{O}(\mathcal{C}_{38}^{\alpha,\beta,\gamma})$ | $=$ | $7$ |  and | $\dim\mathcal{O}(\mathcal{C}_{39}^{\alpha,\beta})$ | $=$ | $6.$
---|---|---|---|---|---|---

$\mathcal{C}_{38}^{*}\not\to\mathcal{C}_{39}^{\alpha,\beta}$ due to the
following relation

$\mathcal{R}=\left\\{c_{22}^{1}=c_{12}^{1}=0,\ c_{22}^{2}c_{12}^{\prime
2}+c_{12}^{2}c_{21}^{\prime 1}+c_{11}^{1}c_{22}^{\prime 2}=c_{11}^{\prime
1}c_{22}^{2}+c_{11}^{1}c_{21}^{\prime 1}+c_{12}^{2}c_{22}^{\prime
2}\right\\}.$

Thanks to [27], we have
$\mathcal{C}_{07}\to\\{\mathcal{C}_{03},\mathcal{C}_{05}^{0},\mathcal{C}_{06}^{1}\\}.$
All necessary degenerations are given below

$\mathcal{C}_{38}^{0,0,t^{-1}}\xrightarrow{(te_{1},\
te_{2})}\mathcal{C}_{07},$
$\mathcal{C}_{39}^{-it^{-1},it^{-1}}\xrightarrow{(ite_{1}-ite_{2},\
-t^{2}e_{1}-t^{2}e_{2})}\mathcal{C}_{13}^{1},$
$\mathcal{C}_{38}^{0,0,\alpha}\xrightarrow{(ite_{1}-ite_{2},\
-t^{2}e_{1}-t^{2}e_{2})}\mathcal{C}_{15}^{\alpha},$
$\mathcal{C}_{38}^{\frac{i}{2t},0,\frac{i}{t}}\xrightarrow{(ite_{1}-ite_{2},\
-t^{2}e_{1}-t^{2}e_{2})}\mathcal{C}_{16}^{0},$
$\mathcal{C}_{38}^{-\frac{1+\alpha t^{2}}{4t^{2}},-\frac{1+\alpha
t^{2}}{4t^{2}},\frac{\alpha t^{2}-3}{4t^{2}}}\xrightarrow{(ite_{1}-ite_{2},\
-t^{2}e_{1}-t^{2}e_{2})}\mathcal{C}_{18}^{\alpha},$
$\mathcal{C}_{38}^{0,-t,\beta+t}\xrightarrow{(e_{1},\
te_{2})}\mathcal{C}_{24}^{0,\beta,0},$
$\mathcal{C}_{38}^{0,0,\beta}\xrightarrow{(e_{1},\
te_{2})}\mathcal{C}_{25}^{0,\beta,0},$
$\mathcal{C}_{39}^{\alpha,t^{-1}}\xrightarrow{(e_{1},\
te_{2})}\mathcal{C}_{29}^{\alpha},$
$\mathcal{C}_{39}^{-t+\beta,\beta}\xrightarrow{(e_{1}+e_{2},\
te_{2})}\mathcal{C}_{31}^{1,\beta,\beta},$ $\mathcal{C}_{38}^{-\alpha-\beta
t+t^{-1},-\beta t,t^{-1}}\xrightarrow{(e_{1}+e_{2},\
te_{2})}\mathcal{C}_{30}^{\alpha,\beta},$
$\mathcal{C}_{38}^{0,\beta,0}\xrightarrow{(e_{1}+e_{2},\
te_{2})}\mathcal{C}_{32}^{1,\beta,\beta},$
$\mathcal{C}_{38}^{-t^{-2},\alpha-(1+\beta t)t^{-2},(1+\beta
t)t^{-2}}\xrightarrow{(e_{1}+e_{2},\ te_{2})}\mathcal{C}_{34}^{\alpha,\beta},$
$\mathcal{C}_{38}^{\frac{1+2\alpha t}{2t},\alpha,0}\xrightarrow{(e_{1}+e_{2},\
-te_{1}+te_{2})}\mathcal{C}_{35}^{\alpha}.$

∎

###  2.3. The geometric classification of compatible associative algebras

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

######  Theorem 20.

The variety of complex $2$-dimensional compatible associative algebras has
dimension $7$ and it has $4$ irreducible components defined by
$\overline{\mathcal{O}(\mathcal{C}_{28}),}$
$\overline{\mathcal{O}(\mathcal{C}_{33})},$
$\overline{\mathcal{O}(\mathcal{C}_{39}^{\alpha,\beta})}$ and
$\overline{\mathcal{O}(\mathcal{C}_{38}^{\alpha,\beta,\gamma})}.$ In
particular, there are two rigid algebras in this variety.

###### Proof.

Thanks to Theorem 10, we have the algebraic classification of $2$-dimensional
compatible associative algebras. After carefully checking the dimensions of
orbit closures of the more important for us algebras, we have

$\dim\mathcal{O}(\mathcal{C}_{38}^{\alpha,\beta,\gamma})$ | $=$ | $7,$ | $\dim\mathcal{O}(\mathcal{C}_{39}^{\alpha,\beta})$ | $=$ | $6,$ | $\dim\mathcal{O}(\mathcal{C}_{28})$ | $=$ | $\dim\mathcal{O}(\mathcal{C}_{34})$ | $=$ | $4.$
---|---|---|---|---|---|---|---|---|---|---

Thanks to Theorem 20, we have that $\mathcal{C}_{38}^{\alpha,\beta,\gamma}$
and $\mathcal{C}_{39}^{\alpha,\beta}$ give irreducible components in the
variety of $2$-dimensional compatible commutative associative algebras. The
rest of the necessary degenerations are given below.

$\mathcal{C}_{28}\xrightarrow{(e_{2},\ te_{1})}\mathcal{C}_{05}^{1},$
$\mathcal{C}_{33}\xrightarrow{(e_{2},\ te_{1})}\mathcal{C}_{06}^{0},$
$\mathcal{C}_{28}\xrightarrow{(e_{1}+\beta e_{2},\
te_{2})}\mathcal{C}_{25}^{1,\beta,\beta},$
$\mathcal{C}_{33}\xrightarrow{(e_{1}+\beta e_{2},\
te_{2})}\mathcal{C}_{32}^{0,\beta,0}.$

∎

###  2.4. The geometric classification of compatible Novikov algebras

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

######  Theorem 21.

The variety of complex $2$-dimensional compatible Novikov algebras has
dimension $7$ and it has $6$ irreducible components defined by
$\overline{\mathcal{O}(\mathcal{C}_{33})},$
$\overline{\mathcal{O}(\mathcal{C}_{09}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{22}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{39}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{31}^{\alpha,\beta,\gamma})}$ and
$\overline{\mathcal{O}(\mathcal{C}_{38}^{\alpha,\beta,\gamma})}.$ In
particular, there is one rigid algebra in this variety.

###### Proof.

Thanks to Theorem 12, we have the algebraic classification of $2$-dimensional
compatible Novikov algebras. After carefully checking the dimensions of orbit
closures of the more important for us algebras, we have

$\dim\mathcal{O}(\mathcal{C}_{38}^{\alpha,\beta,\gamma})$ | $=$ | $7,$
---|---|---
$\dim\mathcal{O}(\mathcal{C}_{39}^{\alpha,\beta})=\dim\mathcal{O}(\mathcal{C}_{09}^{\alpha,\beta})=\dim\mathcal{O}(\mathcal{C}_{22}^{\alpha,\beta})=\dim\mathcal{O}(\mathcal{C}_{31}^{\alpha,\beta,\gamma})$ | $=$ | $6,$
$\dim\mathcal{O}(\mathcal{C}_{33})$ | $=$ | $4.$

Thanks to Theorem 21, we have that $\mathcal{C}_{38}^{\alpha,\beta,\gamma}$
and $\mathcal{C}_{39}^{\alpha,\beta}$ give irreducible components in the
variety of $2$-dimensional compatible commutative associative algebras.
Algebras $\mathcal{C}_{09}^{\alpha,\beta},$ $\mathcal{C}_{22}^{\alpha,\beta},$
and $\mathcal{C}_{31}^{\alpha,\beta,\gamma}$ have a one-dimensional subalgebra
with zero multiplication (concerning both multiplications), but
$\mathcal{C}_{33}$ has not. Hence, $\mathcal{C}_{33}$ is not in the orbit
closure of $\mathcal{C}_{09}^{\alpha,\beta},$
$\mathcal{C}_{22}^{\alpha,\beta},$ and $\mathcal{C}_{31}.$ The rest of the
necessary degenerations are given below.

$\mathcal{C}_{31}^{t,\frac{t+2}{2t},-\frac{1}{2}}\xrightarrow{(te_{1}+\frac{2t}{3}e_{2},\
\frac{t^{2}}{2}e_{1}+t^{2}e_{2})}\mathcal{C}_{01},$
$\mathcal{C}_{31}^{0,0,t^{-1}}\xrightarrow{(t^{2}e_{1}+e_{2},\
te_{1}-t^{2}e_{2})}\mathcal{C}_{04},$ $\mathcal{C}_{31}^{0,t^{-1},\alpha
t^{-1}}\xrightarrow{(te_{1}-\alpha^{-1}t^{2}e_{2},\
te_{2})}\mathcal{C}_{06}^{\alpha\neq 0},$
$\mathcal{C}_{09}^{\frac{(2\alpha+t)\beta}{2\beta+t},-\frac{\beta
t}{2\beta+t}}\xrightarrow{(\frac{2\beta+t}{2\beta}e_{1}-\frac{(2\beta+t)^{2}}{2\beta
t}e_{2},\
\frac{(2\beta+t)t}{4\beta^{2}}te_{1})}\mathcal{C}_{10}^{\alpha,\beta\neq 0},$
$\mathcal{C}_{31}^{1,\alpha t^{-1},t^{-1}}\xrightarrow{(te_{1}-t^{2}e_{2},\
-t^{3}e_{2})}\mathcal{C}_{13}^{\alpha},$
$\mathcal{C}_{31}^{1,t^{-1},0}\xrightarrow{(te_{1}+\alpha^{-1}te_{2},\alpha^{-1}t^{2}e_{2})}\mathcal{C}_{14}^{\alpha\neq
0},$ $\mathcal{C}_{31}^{t^{-1}-t,0,-1+\alpha
t^{-1}}\xrightarrow{(t^{2}e_{1}+\frac{t^{2}}{1-\alpha t}e_{2},\
te_{1}+\frac{t^{3}}{1-\alpha t}e_{2})}\mathcal{C}_{20}^{\alpha,0},$
$\mathcal{C}_{31}^{t^{-1}-t,0,-\alpha
t^{-1}}\xrightarrow{(t^{2}e_{1}-\alpha^{-1}te_{2},\
te_{1}-\alpha^{-1}te_{2})}\mathcal{C}_{21}^{\alpha,0},$
$\mathcal{C}_{31}^{t^{-1}-t,\alpha t^{-1},\beta
t^{-1}}\xrightarrow{(t^{2}e_{1}-\beta^{-1}te_{2},\
te_{1}-\beta^{-1}t^{2}e_{2})}\mathcal{C}_{23}^{\alpha,\beta},$
$\mathcal{C}_{31}^{\alpha,\beta,\gamma}\xrightarrow{(e_{1},\
t^{-1}e_{2})}\mathcal{C}_{32}^{\alpha,\beta,\gamma}.$

∎

###  2.5. The geometric classification of compatible pre-Lie algebras

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

######  Theorem 22.

The variety of complex $2$-dimensional compatible pre-Lie algebras has
dimension $7$ and it has $14$ irreducible components defined by

$\overline{\mathcal{O}(\mathcal{C}_{28})},$
$\overline{\mathcal{O}(\mathcal{C}_{33})},$
$\overline{\mathcal{O}(\mathcal{C}_{37}^{\alpha})},$
$\overline{\mathcal{O}(\mathcal{C}_{09}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{11}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{22}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{27}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{36}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{39}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{41}^{\alpha,\beta})},$
$\overline{\mathcal{O}(\mathcal{C}_{24}^{\alpha,\beta,\gamma})},$
$\overline{\mathcal{O}(\mathcal{C}_{31}^{\alpha,\beta,\gamma})},$
$\overline{\mathcal{O}(\mathcal{C}_{38}^{\alpha,\beta,\gamma})}$ and
$\overline{\mathcal{O}(\mathcal{C}_{40}^{\alpha,\beta,\gamma})}.$

In particular, there are two rigid algebras in this variety.

###### Proof.

Thanks to Theorem 7, we have the algebraic classification of $2$-dimensional
compatible pre-Lie algebras. After carefully checking the dimensions of orbit
closures of the more important for us algebras, we have

$\dim\mathcal{O}(\mathcal{C}_{24}^{\alpha,\beta,\gamma})=\dim\mathcal{O}(\mathcal{C}_{31}^{\alpha,\beta,\gamma})=\dim\mathcal{O}(\mathcal{C}_{38}^{\alpha,\beta,\gamma})=\dim\mathcal{O}(\mathcal{C}_{40}^{\alpha,\beta,\gamma})$ | $=$ | $7,$
---|---|---
$\dim\mathcal{O}(\mathcal{C}_{09}^{\alpha,\beta})=\dim\mathcal{O}(\mathcal{C}_{11}^{\alpha,\beta})=\dim\mathcal{O}(\mathcal{C}_{22}^{\alpha,\beta})=\dim\mathcal{O}(\mathcal{C}_{27}^{\alpha,\beta})$ | $=$ |
$\dim\mathcal{O}(\mathcal{C}_{36}^{\alpha,\beta})=\dim\mathcal{O}(\mathcal{C}_{39}^{\alpha,\beta})=\dim\mathcal{O}(\mathcal{C}_{41}^{\alpha,\beta})$ | $=$ | $6,$
$\dim\mathcal{O}(\mathcal{C}_{37}^{\alpha})$ | $=$ | $5.$
$\dim\mathcal{O}(\mathcal{C}_{28})=\dim\mathcal{O}(\mathcal{C}_{33})$ | $=$ | $4.$

All necessary reasons for non-degenerations are listed below.

Non-degenerations reasons
---
$\begin{array}[]{llll}\mathcal{C}_{24}^{*}\end{array}$ | $\not\rightarrow$ | $\begin{array}[]{llll}\mathcal{C}_{22}^{\alpha,\beta},\ \mathcal{C}_{26}^{\alpha},\ \mathcal{C}_{27}^{\alpha,\beta},\ \mathcal{C}_{28}\end{array}$ | $\mathcal{R}=\left\\{\begin{array}[]{lllll}c_{21}^{\prime 1}=c_{21}^{\prime 2}=c_{22}^{1}=c_{22}^{2}=c_{22}^{\prime 1}=c_{22}^{\prime 2}=0\par\end{array}\right\\}$
$\begin{array}[]{llll}\mathcal{C}_{31}^{*}\end{array}$ | $\not\rightarrow$ | $\begin{array}[]{llll}\mathcal{C}_{33},\ \mathcal{C}_{36}^{\alpha,\beta},\ \mathcal{C}_{37}^{\alpha}\end{array}$ | $\mathcal{R}=\left\\{\begin{array}[]{lllll}c_{22}^{\prime 1}=c_{23}^{\prime 2}=c_{22}^{1}=c_{22}^{2}=0\par\end{array}\right\\}$
$\begin{array}[]{llll}\mathcal{C}_{36}^{*}\end{array}$ | $\not\rightarrow$ | $\begin{array}[]{llll}\mathcal{C}_{37}^{\alpha}\end{array}$ | $\mathcal{R}=\left\\{\begin{array}[]{lllll}c_{22}^{1}=c_{12}^{1}=c_{12}^{\prime 1}=0\par\end{array}\right\\}$
$\begin{array}[]{llll}\mathcal{C}_{40}^{*}\end{array}$ | $\not\rightarrow$ | $\begin{array}[]{llll}\mathcal{C}_{41}^{\alpha,\beta}\end{array}$ | $\mathcal{R}=\left\\{\begin{array}[]{lllll}c_{22}^{1}=c_{12}^{1}=c_{12}^{\prime 1}=0,\ 2c_{11}^{\prime 1}=c_{12}^{\prime 1}\par\end{array}\right\\}$

The rest of the non-degenerations between indicated algebras follows from
Remark 18.

Thanks to Theorem 21, we have that $\mathcal{C}_{33},$
$\mathcal{C}_{09}^{\alpha,\beta},$ $\mathcal{C}_{22}^{\alpha,\beta},$
$\mathcal{C}_{39}^{\alpha,\beta},$ $\mathcal{C}_{31}^{\alpha,\beta,\gamma},$
and $\mathcal{C}_{38}^{\alpha,\beta,\gamma}$ give irreducible components in
the variety of $2$-dimensional compatible Novikov algebras. Hence, each
$2$-dimensional compatible Novikov algebras is on the orbit closure of these
algebras. The rest of the necessary degenerations are given below.

$\mathcal{C}_{24}^{1,1+t^{-1},-1+t^{-1}}\xrightarrow{(te_{1}+(-1+t)e_{2},\
t^{2}e_{1}+te_{2})}\mathcal{C}_{02},$ $\mathcal{C}_{24}^{1,t^{-1},\alpha
t^{-1}}\xrightarrow{(te_{1}+\frac{t^{2}}{1-\alpha}e_{2},\
t^{2}e_{1}+e_{2})}\mathcal{C}_{05}^{\alpha\neq 1},$
$\mathcal{C}_{40}^{t^{-1},0,\frac{1}{2t}}\xrightarrow{(te_{1},\
te_{2})}\mathcal{C}_{08},$
$\mathcal{C}_{11}^{\alpha,\alpha+t,}\xrightarrow{(e_{1}+\beta t^{-1}e_{2},\
e_{2})}\mathcal{C}_{12}^{\alpha,\beta},$
$\mathcal{C}_{24}^{-1+\alpha,t^{-1},\alpha
t^{-1}}\xrightarrow{(te_{1}-\frac{t^{2}}{\alpha-1}e_{2},\
t^{2}e_{1}-t^{3}e_{2})}\mathcal{C}_{16}^{\alpha\neq 1},$
$\mathcal{C}_{24}^{-2,\alpha+t^{-1},-\alpha+t^{-1}}\xrightarrow{(te_{1}-\alpha^{-1}te_{2},\
t^{2}e_{1}+2\alpha^{-1}t^{2}e_{2})}\mathcal{C}_{17}^{\alpha\neq 0},$
$\mathcal{C}_{40}^{-\alpha-2t^{-2},-6\alpha-6t^{-2},-\frac{4+\alpha
t^{2}}{6t^{2}}}\xrightarrow{(te_{1}-3te_{2},\frac{t^{2}}{2}e_{1}-3t^{2}e_{2})}\mathcal{C}_{19}^{\alpha},$
$\mathcal{C}_{24}^{-1-t^{-2},-(\beta+t)t^{-2},(t-\alpha)t^{-2}}\xrightarrow{(-t^{3}e_{1}+\frac{t^{4}+t^{6}}{1+\alpha
t-\beta t}e_{2},\ -t^{2}e_{1}+\frac{t^{5}}{\beta t-\alpha
t-1}e_{2})}\mathcal{C}_{20}^{\alpha,\beta},$
$\mathcal{C}_{24}^{-1-t^{-2},-\beta t^{-2},-\alpha
t^{-2}}\xrightarrow{(-t^{3}e_{1}+\frac{t^{3}+t^{5}}{\alpha-\beta}e_{2},\
-t^{2}e_{1}-\frac{t^{4}}{\alpha-\beta}e_{2})}\mathcal{C}_{21}^{\alpha,\beta},$
$\mathcal{C}_{27}^{\alpha-5t^{2},t^{-2}}\xrightarrow{(e_{1}+2t^{2}e_{2},\
te_{2})}\mathcal{C}_{26}^{\alpha},$
$\mathcal{C}_{24}^{\alpha,\beta,\gamma}\xrightarrow{(e_{1},\
t^{-1}e_{2})}\mathcal{C}_{25}^{\alpha,\beta,\gamma}.$

∎

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