Degenerations of Poisson-type algebras111 The work is supported by the Spanish
Government through the Ministry of Universities grant ‘Margarita Salas’,
funded by the European Union - NextGenerationEU; FCT UIDB/MAT/00212/2020,
UIDP/MAT/00212/2020 and 2022.02474.PTDC.

Hani Abdelwahab222Department of Mathematics, Faculty of Science, Mansoura
University, Mansoura, Egypt<EMAIL_ADDRESS>Amir Fernández
Ouaridi333Department of Mathematics, University of Cádiz, Puerto Real, España;
<EMAIL_ADDRESS>& Ivan Kaygorodov444CMA-UBI, University of
Beira Interior, Covilhã, Portugal<EMAIL_ADDRESS>

Abstract: The degenerations of Poisson-type algebras are studied in the
following varieties in dimension two: Leibniz–Poisson algebras, transposed
Leibniz–Poisson algebras, Novikov–Poisson algebras, commutative pre-Lie
algebras, anti-pre-Lie Poisson algebras and pre-Poisson algebras. For these
varieties, the algebraic and geometric classifications are given. Also, the
complete graph of degenerations is obtained, together with the description of
the orbit closures of each of its algebras and parametric families up to
isomorphism.

Keywords: Poisson-type algebras, classification, degeneration.

MSC2020: , 17A40, 17B30, 17D99, 14D06, 14L30.

###### Contents

  1. 1 Preliminaries: the algebraic and geometric classifications
    1. 1.1 The algebraic classification of algebras
      1. 1.1.1 The algebraic classification of $2$-dimensional commutative associative algebras
    2. 1.2 Degenerations and the geometric classification of algebras
  2. 2  Leibniz–Poisson algebras
    1. 2.1 The algebraic classification of $2$-dimensional Leibniz algebras
    2. 2.2 The algebraic classification of $2$-dimensional Leibniz–Poisson algebras
      1. 2.2.1 Leibniz–Poisson algebras defined on ${\rm A}_{01}$
      2. 2.2.2 Leibniz–Poisson algebras defined on ${\rm A}_{02}$
      3. 2.2.3 Leibniz–Poisson algebras defined on ${\rm A}_{03}$
      4. 2.2.4 Leibniz–Poisson algebras defined on ${\rm A}_{04}$
      5. 2.2.5 The algebraic classification of $2$-dimensional Leibniz–Poisson algebras
      6. 2.2.6 The algebraic classification of $2$-dimensional generic Poisson algebras
    3. 2.3 Degenerations of $2$-dimensional Leibniz–Poisson algebras
  3. 3 Transposed Leibniz–Poisson algebras
    1. 3.1 The algebraic classification of $2$-dimensional transposed Leibniz–Poisson algebras
      1. 3.1.1 The algebraic classification of $2$-dimensional symmertic Leibniz algebras
      2. 3.1.2 Transposed Leibniz–Poisson algebras defined on ${{\rm A}}_{01}$
      3. 3.1.3 Transposed Leibniz–Poisson algebras defined on ${{\rm A}}_{02}$
      4. 3.1.4 Transposed Leibniz–Poisson algebras defined on ${{\rm A}}_{03}$
      5. 3.1.5 Transposed Leibniz–Poisson algebras defined on ${{\rm A}}_{04}$
      6. 3.1.6 The algebraic classification of $2$-dimensional transposed Leibniz–Poisson algebras
      7. 3.1.7 The algebraic classification of $2$-dimensional transposed Poisson algebras
    2. 3.2 Degenerations of $2$-dimensional transposed Leibniz–Poisson algebras
      1. 3.2.1 The geometric classification of $2$-dimensional transposed Poisson algebras
  4. 4  Novikov–Poisson algebras
    1. 4.1 The algebraic classification of $2$-dimensional Novikov algebras
    2. 4.2 The algebraic classification of $2$-dimensional Novikov–Poisson algebras
      1. 4.2.1 Novikov–Poisson algebras defined on ${\rm A}_{01}$
      2. 4.2.2 Novikov–Poisson algebras defined on ${\rm A}_{02}$
      3. 4.2.3 Novikov–Poisson algebras defined on ${\rm A}_{03}$
      4. 4.2.4 Novikov–Poisson algebras defined on ${\rm A}_{04}$
      5. 4.2.5 The algebraic classification of $2$-dimensional Novikov-Poisson algebras
      6. 4.2.6 The algebraic classification of $2$-dimensional pre-Lie Poisson algebras
    3. 4.3 Degenerations of $2$-dimensional Novikov–Poisson algebras
      1. 4.3.1 The geometric classification of $2$-dimensional pre-Lie Poisson algebras
  5. 5  Commutative pre-Lie algebras
    1. 5.1 The algebraic classification of $2$-dimensional pre-Lie algebras
    2. 5.2 The algebraic classification of $2$-dimensional commutative pre-Lie algebras
      1. 5.2.1 Commutative pre-Lie algebras defined on ${\rm A}_{01}$
      2. 5.2.2 Commutative pre-Lie algebras defined on ${\rm A}_{02}$
      3. 5.2.3 Commutative pre-Lie algebras defined on ${\rm A}_{03}$
      4. 5.2.4 Commutative pre-Lie algebras defined on ${\rm A}_{04}$
    3. 5.3 Degenerations of $2$-dimensional commutative pre-Lie algebras
  6. 6 Anti-pre-Lie Poisson algebras
    1. 6.1 The algebraic classification of the 2-dimensional anti-pre-Lie algebras
    2. 6.2 The algebraic classification of the 2-dimensional anti-pre-Lie Poisson algebras
      1. 6.2.1 Anti-pre-Lie Poisson algebras defined on ${\rm A}_{01}$
      2. 6.2.2 Anti-pre-Lie Poisson algebras defined on ${\rm{A}}_{02}$
      3. 6.2.3 Anti-pre-Lie Poisson algebras defined on ${\rm A}_{03}$
      4. 6.2.4 Anti-pre-Lie Poisson algebras defined on ${\rm A}_{04}$
    3. 6.3 Degenerations of $2$-dimensional anti-pre-Lie Poisson algebras
  7. 7 Pre-Poisson algebras
    1. 7.1 The algebraic classification of $2$-dimensional Zinbiel algebras
    2. 7.2 The algebraic classification of $2$-dimensional pre-Poisson algebras
      1. 7.2.1 Pre-Poisson algebras defined on ${\rm Z}_{1}$
    3. 7.3 Degeneration of $2$-dimensional pre-Poisson algebras

##  Introduction

The algebraic classification (up to isomorphism) of algebras of dimension $n$
of a certain variety defined by a family of polynomial identities is a classic
problem in the theory of non-associative algebras, see [4, 5, 8, 10, 20, 21,
26, 27, 29, 39, 41, 44, 45, 51, 48, 1, 2, 3]. There are many results related
to the algebraic classification of small-dimensional algebras in different
varieties of associative and non-associative algebras. For example, algebraic
classifications of $2$-dimensional algebras [51], $3$-dimensional evolution
algebras [20], $3$-dimensional anticommutative algebras [45], $4$-dimensional
division algebras [27, 26], $6$-dimensional anticommutative nilpotent algebras
[39] and $8$-dimensional dual Mock Lie algebras [21] have been given.
Geometric properties of a variety of algebras defined by a family of
polynomial identities have been an object of study since the 1970’s, see [56,
57, 10, 8, 24, 19, 11, 9, 32, 21, 25, 34, 35, 7, 29, 4, 36, 37, 44, 54] and
references in [38, 47]. Gabriel described the irreducible components of the
variety of $4$-dimensional unital associative algebras [32]. Cibils considered
rigid associative algebras with $2$-step nilpotent radical [25]. Chouhy proved
that in the case of finite-dimensional associative algebras, the $N$-Koszul
property is preserved under the degeneration relation [24]. Burde and
Steinhoff constructed the graphs of degenerations for the varieties of
$3$-dimensional and $4$-dimensional Lie algebras [19]. Grunewald and
O’Halloran studied the degenerations for the variety of $5$-dimensional
nilpotent Lie algebras [34]. Alvarez and Hernández described the degenerations
for the variety of $5$-dimensional nilpotent Lie superalgebras [9]. Alvarez
and Vera described the degenerations for the variety of $3$-dimensional ${\rm
Hom}$-Lie algebras [11]. Fernández Ouaridi, Kaygorodov, Khrypchenko and Volkov
described all the degenerations in the variety of $4$-dimensional commutative
nilpotent algebras. Ignatyev, Kaygorodov, and Popov determined the dimensions
of the irreducible components of $n$-dimensional $2$-step nilpotent algebras
(commutative and anticommutative) [36]. On the other hand, Kaygorodov,
Khrypchenko, and Lopes studied the dimensions of the irreducible components of
$n$-dimensional (all, commutative, anticommutative) nilpotent algebras [40].
Degenerations have also been used to study the level of complexity of an
algebra [57, 56]. The notion of degenerations of binary algebras was also
extended to degenerations of $n$-ary algebras [43] and degenerations of
algebras with two multiplications [1, 2, 3, 5]. The study of degenerations of
algebras is very rich and closely related to deformation theory, in the sense
of Gerstenhaber [33].

Poisson algebras arose from the study of Poisson geometry in the 1970s and
have appeared in an extremely wide range of areas in mathematics and physics,
such as Poisson manifolds, algebraic geometry, operads, quantization theory,
quantum groups, and classical and quantum mechanics. The study of Poisson
algebras also led to other algebraic structures, such as noncommutative
Poisson algebras, generic Poisson algebras, transposed Poisson algebras,
Novikov–Poisson algebras, algebras of Jordan brackets and generalized Poisson
algebras, $F$-manifold algebras, quasi-Poisson algebras, Gerstenhaber
algebras, Poisson bialgebras, double Poisson algebras, Poisson $n$-Lie
algebras, etc. All these classes are related in some way to Poisson algebras,
for that reason we call them Poisson-type algebras.

The present paper is devoted to the algebraic, geometric, and degeneration
classification of the complex algebras from the principal varieties of
Poisson-type algebras in dimension two. In particular, we consider the
varieties of Leibniz–Poisson algebras, transposed Leibniz–Poisson algebras,
Novikov–Poisson algebras, commutative pre-Lie algebras, anti-Pre-Poisson
algebras and pre-Poisson algebras.

##  1\. Preliminaries: the algebraic and geometric classifications

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 or
can be recovered from the commutativity or from the anticommutativity are
omitted. In this section, introduce the techniques used to obtain our main
results.

###  1.1. The algebraic classification of algebras

In this paper, we work with various types of algebras with two multiplications
such that the “first” multiplication is associative and commutative (or no)
and the “second” multiplication depends on the particular type of algebras
under consideration. Let us review the method we will use to obtain the
algebraic classification for an arbitrary variety of Poisson-type algebras
(the present method, in the case of Poisson algebras, is given with more
details in [5]).

######  Definition 1.

A Poisson-type algebra is a vector space $({\rm A},\cdot,\bullet)$ equipped
with one commutative associative multiplication $\cdot$ and another
multiplication $\bullet$ satisfying a family of polynomial identities
$\Omega^{\\{\bullet\\}}_{i}(x_{1},\ldots,x_{n_{i}}),$ which includes only one
multiplication $\bullet.$ These two operations are required to satisfy a
family of Leibniz type identities
$\Theta^{\\{\cdot,\bullet\\}}_{i}(x_{1},\ldots,x_{k_{i}}),$ which includes two
of these multiplications $\cdot$ and $\bullet.$ Main examples of Poisson-type
algebras are the following: Poisson algebras, Leibniz–Poisson algebras,
transposed Poisson algebras, Novikov–Poisson algebras, commutative pre-Lie
algebras or anti-pre-Lie Poisson algebras.

######  Definition 2.

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

$\Omega^{\\{\bullet\to\theta\\}}_{i}(x_{1},\ldots,x_{n_{i}})=0$ and
$\Theta^{\\{\cdot,\bullet\to\theta\\}}_{i}(x_{1},\ldots,x_{k_{i}})=0,$

where, by $\bullet\to\theta$ we denote the changing of the multiplication
$\bullet$ to the cocycle $\theta.$ Then ${\rm Z}^{2}\left({\rm A},{\rm
A}\right)\neq\mathcal{\emptyset}$ since $\theta=0\in{\rm Z}^{2}\left({\rm
A},{\rm A}\right)$.

Now, for $\theta\in{\rm Z}^{2}\left({\rm A},{\rm A}\right)$, let us define a
multiplication $\bullet_{\theta}$ on ${\rm A}$ by
$x\bullet_{\theta}y=\theta\left(x,y\right)$ for all $x,y$ in ${\rm A}$. Then
$\left({\rm A},\cdot,\bullet_{\theta}\right)$ is a Poisson-type algebra
satisfying the two families of identities
$\Omega^{\\{\bullet_{\theta}\\}}_{i}(x_{1},\ldots,x_{n_{i}})$ and
$\Theta^{\\{\cdot,\bullet_{\theta}\\}}_{i}(x_{1},\ldots,x_{k_{i}})$.
Conversely, if $\left({\rm A},\cdot,\bullet\right)$ is a Poisson-type algebra
satisfying the two families of identities
$\Omega^{\\{\bullet\\}}_{i}(x_{1},\ldots,x_{n_{i}})$ and
$\Theta^{\\{\cdot,\bullet\\}}_{i}(x_{1},\ldots,x_{k_{i}})$, then there exists
$\theta\in{\rm Z}^{2}\left({\rm A},{\rm A}\right)$ such that $\left({\rm
A},\cdot,\bullet_{\theta}\right)\cong\left({\rm A},\cdot,\bullet\right)$. To
see this, consider the bilinear map $\theta:{\rm A}\times{\rm
A}\longrightarrow{\rm A}$ defined by $\theta\left(x,y\right)=x\bullet y$ for
all $x,y$ in ${\rm A}$. Then $\theta\in{\rm Z}^{2}\left({\rm A},{\rm
A}\right)$ and $\left({\rm A},\cdot,\bullet_{\theta}\right)=\left({\rm
A},\cdot,\bullet\right)$.

Let $\left({\rm A},\cdot\right)$ be a commutative associative algebra. The
automorphism group, ${\rm Aut}\left({\rm A}\right)$, of ${\rm A}$ acts on
${\rm Z}^{2}\left({\rm A},{\rm A}\right)$ 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 $\phi\in{\rm Aut}\left({\rm A}\right)$, and $\theta\in{\rm
Z}^{2}\left({\rm A},{\rm A}\right)$.

######  Lemma 3.

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

###### Proof.

If $\theta*\phi=\vartheta$, then
$\phi:\left(\rm{A},\cdot,\bullet_{\vartheta}\right)\longrightarrow$
$\left(\rm{A},\cdot,\bullet_{\theta}\right)$ 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
$\phi:\left(\rm{A},\cdot,\bullet_{\vartheta}\right)\longrightarrow$
$\left(\rm{A},\cdot,\bullet_{\theta}\right)$ is an isomorphism of Poisson-type
algebras, then $\phi\in{\rm Aut}\left(\rm{A}\right)$ 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 Poisson-type algebras with
given associated commutative associative algebra $\left(\rm{A},\cdot\right)$.
It consists of three steps:

  1. (1)

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

  2. (2)

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

  3. (3)

Choose a representative $\theta$ from each orbit and then construct the
Poisson-type algebra $\left(\rm{A},\cdot,\bullet_{\theta}\right)$.

####  1.1.1. The algebraic classification of $2$-dimensional commutative
associative algebras

All algebras under our consideration are constructed on the base of
$2$-dimensional commutative associative algebras. Hence, the classification of
them, which can be obtained by direct verification from [44], is given below.

######  Theorem 4.

Let $\left({\rm A},\cdot\right)$ be a $2$-dimensional commutative associative
algebra. Then ${\rm A}$ is isomorphic to one of the following algebras:

${\rm A}_{01}$ | $:$ | $e_{1}\cdot e_{1}$ | $=$ | $e_{1},$ | $e_{2}\cdot e_{2}$ | $=$ | $e_{2}$
---|---|---|---|---|---|---|---
${\rm A}_{02}$ | $:$ | $e_{1}\cdot e_{1}$ | $=$ | $e_{1},$ | $e_{1}\cdot e_{2}$ | $=$ | $e_{2}$
${\rm A}_{03}$ | $:$ | $e_{1}\cdot e_{1}$ | $=$ | $e_{1}$ |  |  |
${\rm A}_{04}$ | $:$ | $e_{1}\cdot e_{1}$ | $=$ | $e_{2}$ |  |  |

######  Lemma 5.

The description of the group of automorphisms of every $2$-dimensional
commutative associative algebra is given.

  1. (1)

If $\phi\in{\rm Aut}({\rm A}_{01})$, then $\phi\in\mathbb{S}_{2},$ i.e.
$\phi(e_{1})=e_{1},\ \phi(e_{2})=e_{2}$ or $\phi(e_{1})=e_{2},\
\phi(e_{2})=e_{1}.$

  2. (2)

If $\phi\in{\rm Aut}({\rm A}_{02})$, then $\phi(e_{1})=e_{1}$ and
$\phi(e_{2})=\xi e_{2}$, for $\xi\in\mathbb{C}^{*}$.

  3. (3)

If $\phi\in{\rm Aut}({\rm A}_{03})$, then $\phi(e_{1})=e_{1}$ and
$\phi(e_{2})=\xi e_{2}$, for $\xi\in\mathbb{C}^{*}$.

  4. (4)

If $\phi\in{\rm Aut}({\rm A}_{04})$, then $\phi(e_{1})=\xi e_{1}+\nu e_{2}$
and $\phi(e_{2})=\xi^{2}e_{2}$, for $\xi\in\mathbb{C}^{*}$ and
$\nu\in\mathbb{C}$.

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

Let us introduce the techniques used to obtain the geometric classification
for an arbitrary variety of Poisson-type 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 Poisson-type algebras, we can obtain a set of
polynomial equations in variables $c_{ij}^{k},c_{ij}^{\prime k}$. This class
of $n$-dimensional Poisson-type 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
$O((\mu,\mu^{\prime}))$, contains all the structural constants of the bilinear
pairs isomorphic to the Poisson-type algebra 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 $\overline{O((\mu,\mu^{\prime}))}$ the closure of the
orbit of $(\mu,\mu^{\prime})\in{\mathcal{T}_{n}}$.

######  Definition 6.

Let ${\rm T}$ and ${\rm T}^{\prime}$ be two $n$-dimensional Poisson-type
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
$(\lambda,\lambda^{\prime})\in\overline{O((\mu,\mu^{\prime}))}$. If ${\rm
T}\not\cong{\rm T}^{\prime}$, then we call it a proper degeneration.
Conversely, if
$(\lambda,\lambda^{\prime})\not\in\overline{O((\mu,\mu^{\prime}))}$ 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 7.

Let ${{\rm T}}(*)=\\{{{\rm T}}(\alpha):{\alpha\in I}\\}$ be a family of
$n$-dimensional Poisson-type 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
$(\lambda,\lambda^{\prime})\in\overline{\\{O((\mu(\alpha),\mu^{\prime}(\alpha)))\\}_{\alpha\in
I}}$. Conversely, if
$(\lambda,\lambda^{\prime})\not\in\overline{\\{O((\mu(\alpha),\mu^{\prime}(\alpha)))\\}_{\alpha\in
I}}$ 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 Poisson-type algebra ${\rm T}$ and ${{\rm
T}(*)}\not\to{{\rm T}}^{\prime}$ for any parametric family of $n$-dimensional
Poisson-type algebras ${\rm T}(*)$. To prove a particular algebra corresponds
to an irreducible compotent, we will use the next ideas. Firstly, since
$\mathrm{dim}\,O((\mu,\mu^{\prime}))=n^{2}-\mathrm{dim}\,\mathfrak{Der}({\rm
T})$, 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 Poisson-type 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 [5].

######  Lemma 8.

Consider two Poisson-type 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. For clarity, we assume by
convention that $c_{ij}^{k}=0$ (resp. $c_{ij}^{\prime k}=0$) if $c_{ij}^{k}$
(resp. $c_{ij}^{\prime k}$) is not explicitly mentioned on the definition of
$C$.

######  Remark 9.

Moreover, let ${{\rm T}}$ and ${{\rm T}}^{\prime}$ be two Poisson-type
algebras represented by the structures $(\mu,\mu^{\prime})$ and
$(\lambda,\lambda^{\prime})$ from ${{\rm T}}_{n}$. 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.
Fithly, to prove ${{\rm T}}(*)\not\to{{\rm T}}^{\prime}$, we may use an
analogous of Remark 9 for parametric families that follows from Lemma 10.

######  Lemma 10.

Consider the family of Poisson-type algebras ${\rm T}(*)$ and the Poisson-type
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 11.

Let $(\mu,\mu^{\prime})$ and $(\lambda,\lambda^{\prime})$ represent two
Poisson-type algebras. Suppose $(\lambda,0)\not\in\overline{O((\mu,0))}$,
(resp. $(0,\lambda^{\prime})\not\in\overline{O((0,\mu^{\prime}))}$), then
$(\lambda,\lambda^{\prime})\not\in\overline{O((\mu,\mu^{\prime}))}$. As we
construct the classification of a given class of Poisson-type algebras from a
certain class of algebras with a single multiplication which remains
unchanged, this remark becomes very useful. Namely, all the classes considered
in this paper are constructed on a $2$-dimensional commutative associative
algebra. The geometric classification of this class was given in [44].

##  2\.  Leibniz–Poisson algebras

The notion of Leibniz–Poisson algebras is a non-anticommutative generalization
of Poisson algebras and on the other hand, it is a particular case of algebras
with bracket [23] and noncommutative Leibniz–Poisson algebras [22]. Other
results related to Leibniz–Poisson algebras have appeared in [52].

###  2.1. The algebraic classification of $2$-dimensional Leibniz algebras

######  Definition 12.

An algebra is called a Leibniz algebra if it satisfies the identity

$[[x,y],z]$ | $=$ | $[[x,z],y]+[x,[y,z]].$
---|---|---

The algebraic classification of Leibniz algebras was obtained by direct
verification from [44].

######  Theorem 13.

Let ${\rm L}$ be a nonzero $2$-dimensional Leibniz algebra. Then $L$ is
isomorphic to one and only one of the following algebras:

${\rm L}_{1}$ | $:$ | $[e_{1},e_{1}]$ | $=$ | $e_{2}$ |  |  |
---|---|---|---|---|---|---|---
${\rm L}_{2}$ | $:$ | $[e_{1},e_{2}]$ | $=$ | $e_{1}$ |  |  |
${\rm L}_{3}$ | $:$ | $[e_{1},e_{2}]$ | $=$ | $e_{2},$ | $[e_{2},e_{1}]$ | $=$ | $-e_{2}$

###  2.2. The algebraic classification of $2$-dimensional Leibniz–Poisson
algebras

######  Definition 14.

A Leibniz–Poisson algebra is a vector space equipped with a commutative
associative multiplication $\cdot$ and a Leibniz multiplication $[-,-].$ These
two operations are required to satisfy the following identity:

$[x\cdot y,z]$ | $=$ | $[x,z]\cdot y+x\cdot[y,z].$
---|---|---

Note that the Leibniz–Poisson algebras $({\rm L},\cdot,[-,-])$ with zero
multiplication $\cdot$ are the Leibniz algebras. The classification of
$2$-dimensional Leibniz algebras is given in Theorem 13. Hence, we are
studying the Leibniz–Poisson algebras defined on every commutative associative
algebra from Theorem 4.

######  Definition 15.

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

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

If $\theta\in{\rm Z}_{\rm LP}^{2}\left(\rm{A},\rm{A}\right)$, then
$\left(\rm{A},\cdot,[-,-]\right)$ is a Leibniz–Poisson algebra where
$[x,y]=\theta\left(x,y\right)$ for all $x,y\in\rm{A}$.

####  2.2.1. Leibniz–Poisson algebras defined on ${\rm A}_{01}$

Since ${\rm Z}_{\rm LP}^{2}({\rm A}_{01},{\rm A}_{01})=\left\\{0\right\\}$,
then there is only the trivial structure ${\rm L}_{04}={\rm A}_{01}$.

####  2.2.2. Leibniz–Poisson algebras defined on ${\rm A}_{02}$

From the computation of ${\rm Z}_{\rm LP}^{2}({\rm A}_{02},{\rm A}_{02})$, the
Leibniz–Poisson structures defined on ${\rm A}_{02}$ are of the form:

${\rm L}_{5}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\
$[e_{2},e_{1}]$&$=$&$\alpha_{1}e_{2}$\end{tabular}\right.$

Since the action of ${\rm Aut}({\rm A}_{02})$ on this structure is the
identity, we obtain there are no isomorphic algebras in this family.

####  2.2.3. Leibniz–Poisson algebras defined on ${\rm A}_{03}$

The Leibniz–Poisson structures defined on ${\rm A}_{03}$ are:

${\rm L}_{6}^{\alpha}:=\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1}$\\\ $[e_{2},e_{1}]$&$=$&$\alpha e_{2}$\end{tabular}\right.$

Again, the action of ${\rm Aut}({\rm A}_{03})$ on this structure is the
identity, hence all algebras from this family are not isomorphic.

####  2.2.4. Leibniz–Poisson algebras defined on ${\rm A}_{04}$

The Leibniz–Poisson structures defined on ${\rm A}_{04}$ are:

${\rm L}_{7}^{\alpha}:=\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2}$\\\ $[e_{1},e_{1}]$&$=$&$\alpha e_{2}$\end{tabular}\right.$

As in the previous cases of this section, the action of ${\rm Aut}({\rm
A}_{04})$ on the structure above is the identity, hence all algebras from this
family are not isomorphic.

####  2.2.5. The algebraic classification of $2$-dimensional Leibniz–Poisson
algebras

######  Theorem 16.

Let $\left({\rm L},\cdot,[-,-]\right)$ be a nonzero $2$-dimensional
Leibniz–Poisson algebra. Then ${\rm L}$ is isomorphic to one Leibniz algebra
listed in Theorem 13 or to one algebra listed below:

${\rm L}_{4}$ | $:$ | $e_{1}\cdot e_{1}=e_{1},e_{2}\cdot e_{2}=e_{2}$
---|---|---
${\rm L}_{5}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $[e_{2},e_{1}]$&$=$&$\alpha e_{2}$\end{tabular}\right.$
${\rm L}_{6}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{1}$\\\ $[e_{2},e_{1}]$&$=$&$\alpha e_{2}$\end{tabular}\right.$
${\rm L}_{7}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{2}$\\\ $[e_{1},e_{1}]$&$=$&$\alpha e_{2}$\end{tabular}\right.$

####  2.2.6. The algebraic classification of $2$-dimensional generic Poisson
algebras

######  Definition 17.

A generic Poisson algebra (see, [42]) is a vector space equipped with a
commutative associative multiplication $\cdot$ and an anticommutative
multiplication $[-,-].$ These two operations are required to satisfy the
following identity:

$[x\cdot y,z]$ | $=$ | $[x,z]\cdot y+x\cdot[y,z].$
---|---|---

It is known [44], that each $2$-dimensional anticommutative algebra is a Lie
algebra. Hence, each $2$-dimensional generic Poisson algebra is a Poisson
algebra.

######  Corollary 18.

Let $\left({\rm G},\cdot,[-,-]\right)$ be a nonzero $2$-dimensional generic
Poisson algebra. Then ${\rm G}$ is isomorphic to one commutative associative
algebra listed in Theorem 4 or to ${\rm L}_{3}.$

###  2.3. Degenerations of $2$-dimensional Leibniz–Poisson algebras

######  Lemma 19.

The graph of primary degenerations and non-degenerations of the variety of
$2$-dimensional Leibniz–Poisson algebras is given in Figure 1, where the
numbers on the right side are the dimensions of the corresponding orbits.

$4$$3$$2$$0$${\mathbb{C}^{8}}$${\rm L}_{1}$${\rm L}_{3}$${\rm
L}_{7}^{\alpha}$${\rm L}_{2}$${\rm L}_{5}^{\beta}$${\rm L}_{6}^{\beta}$${\rm
L}_{4}$$\beta=0$$\beta=0$$\alpha=\beta$$\alpha=-\beta$

  * Legend:

    * –

Round nodes: Leibniz–Poisson algebras with trivial commutative associative
multiplication $\cdot$.

    * –

Squared nodes: Leibniz–Poisson algebras with non-trivial commutative
associative multiplication $\cdot$.

Figure 1. Graph of primary degenerations and non-degenerations.

###### Proof.

The dimensions of the orbits are deduced by computing the algebra of
derivations. The primary degenerations of the Leibniz–Poisson algebras with
trivial commutative associative multiplication follow by [44], the others are
proven using the parametric bases included in following Table:

Degeneration | Parametrized basis
---|---
${\rm L}_{4}$ | $\to$ | ${\rm L}_{5}^{0}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
${\rm L}_{4}$ | $\to$ | ${\rm L}_{6}^{0}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm L}_{5}^{\alpha}$ | $\to$ | ${\rm L}_{7}^{\alpha}$ | $E_{1}(t)=te_{1}+t^{-1}e_{2}$ | $E_{2}(t)=e_{2}$
${\rm L}_{6}^{\alpha}$ | $\to$ | ${\rm L}_{7}^{-\alpha}$ | $E_{1}(t)=te_{1}-t^{-1}e_{2}$ | $E_{2}(t)=e_{2}$

The primary non-degenerations of Leibniz–Poisson algebras with non-trivial
commutative associative multiplication are proven using the sets from the
following table:

Non-degeneration | Arguments
---|---
$\begin{array}[]{cccc}{\rm L}_{4}&\not\to&{\rm L}_{1},{\rm L}_{3},{\rm L}_{7}^{\alpha\neq 0}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{22}^{2}\in\mathbb{C},c_{12}^{2}=c_{21}^{2}\end{array}\right\\}$
$\begin{array}[]{cccc}{\rm L}_{5}^{\alpha}&\not\to&{\rm L}_{1},{\rm L}_{3},{\rm L}_{7}^{\beta\neq\alpha}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{11}^{1}=c_{12}^{2},c_{11}^{1}=c_{21}^{2},c_{11}^{\prime 2}=\alpha c_{11}^{2},c_{21}^{\prime 2}=\alpha c_{11}^{1}\end{array}\right\\}$
$\begin{array}[]{cccc}{\rm L}_{6}^{\alpha}&\not\to&{\rm L}_{1},{\rm L}_{3},{\rm L}_{7}^{\beta\neq-\alpha}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{11}^{\prime 2}=-\alpha c_{11}^{2},c_{21}^{\prime 2}=\alpha c_{11}^{1}\end{array}\right\\}$

∎

At this point, only the description of the closures of the orbits of the
parametric families is missing. Although it is not necessary to study the
closure of the orbits of each of the parametric families of the variety of
$2$-dimensional Leibniz–Poisson algebras in order to identify its irreducible
components, we will study them to give a complete description of the variety.

######  Lemma 20.

The description of the closure of the orbit of the parametric families in the
variety of $2$-dimensional Leibniz–Poisson algebras are given below:

$\overline{\\{O({\rm L}_{5}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm L}_{7}^{\alpha})\\}},\overline{\\{O({\rm L}_{2})\\}},\overline{\\{O({\rm L}_{1})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
---|---|---
$\overline{\\{O({\rm L}_{6}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm L}_{7}^{\alpha})\\}},\overline{\\{O({\rm L}_{2})\\}},\overline{\\{O({\rm L}_{1})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
$\overline{\\{O({\rm L}_{7}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm L}_{1})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$

###### Proof.

Thanks to Theorem 19 we have all necessarily degenerations. The closure of the
orbits of suitable families are given using the parametric bases and the
parametric index included in following Table:

Degeneration | Parametrized basis
---|---
${\rm L}_{5}^{t^{-1}}$ | $\to$ | ${\rm L}_{2}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=te_{1}$
${\rm L}_{6}^{t^{-1}}$ | $\to$ | ${\rm L}_{2}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=te_{1}$
${\rm L}_{7}^{t^{-1}}$ | $\to$ | ${\rm L}_{1}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
Non-degeneration | Arguments
---|---
$\begin{array}[]{cccc}{\rm L}_{5}^{*}&\not\to&{\rm L}_{3}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{11}^{1}=c_{12}^{2},c_{11}^{1}=c_{21}^{2}\end{array}\right\\}$
$\begin{array}[]{cccc}{\rm L}_{6}^{*}&\not\to&{\rm L}_{3}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C}\end{array}\right\\}$

∎

By Lemma 19 and Lemma 20, we have the following result that summarizes the
geometric classification of ${{\rm L}_{2}}$.

######  Theorem 21.

The variety of $2$-dimensional Leibniz–Poisson algebras has four irreducible
components, corresponding to the Leibniz–Poisson algebras ${\rm L}_{3}$ and
${\rm L}_{4}$ and the families of Leibniz–Poisson algebras ${\rm L}_{5}^{*}$
and ${\rm L}_{6}^{*}$.

##  3\. Transposed Leibniz–Poisson algebras

The notion of transposed Leibniz–Poisson algebras is a non-anticommutative
generalization of transposed Poisson algebras. The notion of transposed
Poisson algebras was introduced in [13]. Let us note, that they are related to
many other interesting classes of algebras, such as commutative Gelfand-
Dorfman algebras and $F$-manifold algebras [53]. The study of transposed
Leibniz–Poisson algebras is motivated by a question posed in [16].

###  3.1. The algebraic classification of $2$-dimensional transposed
Leibniz–Poisson algebras

######  Definition 22.

A transposed Leibniz–Poisson algebra is a vector space equipped with a
commutative associative multiplication, denoted by $\cdot$ and a symmetric
Leibniz multiplication $[-,-]$. These two operations are required to satisfy
the following identity:

$2[x,y]\cdot z$ | $=$ | $[x\cdot z,y]+[x,y\cdot z].$
---|---|---

######  Definition 23.

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

$\theta\left(\theta\left(x,y\right),z\right)$ | $=$ | $\theta\left(\theta\left(x,z\right),y\right)+\theta\left(x,\theta\left(y,z\right)\right),$
---|---|---
$\theta\left(x,\theta\left(y,z\right)\right)$ | $=$ | $\theta\left(\theta\left(x,y\right),z\right)+\theta\left(y,\theta\left(x,z\right)\right),$
$2\theta\left(x,y\right)\cdot z$ | $=$ | $\theta\left(x\cdot z,y\right)+\theta\left(x,y\cdot z\right),$

for all $x,y,z$ in $\rm{A}$. If $\theta\in{\rm Z}_{\rm
TLP}^{2}\left(\rm{A},\rm{A}\right)$, then $\left(\rm{A},\cdot,[-,-]\right)$ is
a transposed Leibniz–Poisson algebra where $[x,y]=\theta\left(x,y\right)$ for
all $x,y\in\rm{A}$.

####  3.1.1. The algebraic classification of $2$-dimensional symmertic Leibniz
algebras

By Theorem 13, we obtain that there are only two transposed Leibniz–Poisson
algebras with trivial multiplication $\cdot$ and non-trivial multiplication
$[-,-]:$ ${\rm T}_{1}=L_{1},{\rm T}_{2}=L_{3}$.

####  3.1.2. Transposed Leibniz–Poisson algebras defined on ${{\rm A}}_{01}$

Since ${\rm Z}_{\rm TLP}^{2}({{\rm A}}_{01},{{\rm
A}}_{01})=\left\\{0\right\\}$, the only transposed Leibniz–Poisson structure
is trivial, denote it by ${\rm T}_{3}$.

####  3.1.3. Transposed Leibniz–Poisson algebras defined on ${{\rm A}}_{02}$

From the computation of ${\rm Z}_{\rm TLP}^{2}({{\rm A}}_{02},{{\rm
A}}_{02})$, the transposed Leibniz–Poisson structures defined on ${{\rm
A}}_{02}$ are of the form:

$\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot
e_{2}$&$=$&$e_{2},$\\\
$[e_{1},e_{1}]$&$=$&$\alpha_{1}e_{2}.$\end{tabular}\right.\qquad\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2},$\\\
$[e_{1},e_{2}]$&$=$&$\alpha_{2}e_{2},$&$[e_{2},e_{1}]$&$=$&$-\alpha_{2}e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{1}\neq 0$. Then, by Lemma 3, we obtain the following
non-isomorphic transposed Leibniz–Poisson algebras:

${\rm T}_{4}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2},$\\\
$[e_{1},e_{1}]$&$=$&$e_{2}.$\end{tabular}\right.\qquad{\rm
T}_{5}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2},$\\\
$\left\\{e_{1},e_{2}\right\\}$&$=$&$\alpha
e_{2},$&$[e_{2},e_{1}]$&$=$&$-\alpha e_{2}.$\end{tabular}\right.$

####  3.1.4. Transposed Leibniz–Poisson algebras defined on ${{\rm A}}_{03}$

Since ${\rm Z}_{\rm TLP}^{2}({{\rm A}}_{03},{{\rm
A}}_{03})=\left\\{0\right\\}$, the only transposed Leibniz–Poisson structure
is trivial, denote it by ${\rm T}_{6}$.

####  3.1.5. Transposed Leibniz–Poisson algebras defined on ${{\rm A}}_{04}$

The transposed Leibniz–Poisson structures defined on ${{\rm A}}_{04}$ are of
the form:

$\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{2},$\\\
$[e_{1},e_{1}]$&$=$&$\alpha_{1}e_{2}.$\end{tabular}\right.\qquad\qquad\qquad\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2},$\\\
$[e_{1},e_{2}]$&$=$&$\alpha_{2}e_{2},$&$[e_{2},e_{1}]$&$=$&$-\alpha_{2}e_{2}.$\end{tabular}\right.$

We may assume $\alpha_{2}\neq 0$. Then, by Lemma 3, we obtain the following
non-isomorphic transposed Leibniz–Poisson algebras:

$\rm{T}_{7}^{\alpha}:=\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2},$\\\ $[e_{1},e_{1}]$&$=$&$\alpha
e_{2}.$\end{tabular}\right.\qquad\qquad\qquad{\rm
T}_{8}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2},$\\\
$[e_{1},e_{2}]$&$=$&$e_{2},$&$[e_{2},e_{1}]$&$=$&$-e_{2}.$\end{tabular}\right.$

####  3.1.6. The algebraic classification of $2$-dimensional transposed
Leibniz–Poisson algebras

######  Theorem 24.

Let $\left({\rm T},\cdot,[-,-]\right)$ be a nonzero $2$-dimensional transposed
Leibniz–Poisson algebra. Then ${\rm T}$ is isomorphic to one symmetric Leibniz
algebra listed in Subsection 3.1.1 or to one algebra listed below:

${\rm T}_{3}$ | $:$ | $e_{1}\cdot e_{1}=e_{1},e_{2}\cdot e_{2}=e_{2}$
---|---|---
${\rm T}_{4}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2},$\\\ $[e_{1},e_{1}]$&$=$&$e_{2}.$\end{tabular}\right.$
${\rm T}_{5}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2},$\\\ $[e_{1},e_{2}]$&$=$&$\alpha e_{2},$&$[e_{2},e_{1}]$&$=$&$-\alpha e_{2}.$\end{tabular}\right.$
${\rm T}_{6}$ | $:$ | $e_{1}\cdot e_{1}=e_{1}.$
${\rm T}_{7}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{2},$\\\ $[e_{1},e_{1}]$&$=$&$\alpha e_{2}.$\end{tabular}\right.$
${\rm T}_{8}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2},$\\\ $[e_{1},e_{2}]$&$=$&$e_{2},$&$[e_{2},e_{1}]$&$=$&$-e_{2}.$\end{tabular}\right.$

####  3.1.7. The algebraic classification of $2$-dimensional transposed
Poisson algebras

######  Definition 25.

A transposed Poisson algebra is a vector space equipped with a commutative
associative multiplication, denoted by $\cdot$ and a Lie multiplication
$[-,-]$. These two operations are required to satisfy the following identity:

$2[x,y]\cdot z$ | $=$ | $[x\cdot z,y]+[x,y\cdot z].$
---|---|---

######  Corollary 26.

Let $\left({\rm T},\cdot,[-,-]\right)$ be a nonzero $2$-dimensional transposed
Poisson algebra. Then ${\rm T}$ is isomorphic to one algebra listed below:

${\rm T}_{2},$ ${\rm T}_{3},$ ${\rm T}_{5}^{\alpha},$ ${\rm T}_{6},$ ${\rm
T}_{7}^{0},$ ${\rm T}_{8}.$

###  3.2. Degenerations of $2$-dimensional transposed Leibniz–Poisson algebras

######  Lemma 27.

The graph of primary degenerations and non-degenerations of the variety of
$2$-dimensional transposed Leibniz–Poisson algebras is given in Figure 2,
where the numbers on the right side are the dimensions of the corresponding
orbits.

$4$$3$$2$$0$${\mathbb{C}^{8}}$${\rm T}_{1}$${\rm T}_{2}$${\rm
T}_{7}^{\alpha}$${\rm T}_{5}^{\beta}$${\rm T}_{6}$${\rm T}_{8}$${\rm
T}_{3}$${\rm T}_{4}$$\beta=0$$\beta=0$$\alpha=0$$\alpha=0$$\alpha=0$

  * Legend:

    * –

Round nodes: transposed Leibniz–Poisson algebras with trivial commutative
associative multiplication $\cdot$.

    * –

Squared nodes: transposed Leibniz–Poisson algebras with nontrivial commutative
associative multiplication $\cdot$.

Figure 2. Graph of primary degenerations and non-degenerations.

###### Proof.

The dimensions of the orbits are deduced by computing the algebra of
derivations. The primary degenerations are proven using the parametric bases
included in following Table:

Degeneration | Parametrized basis
---|---
${\rm T}_{3}$ | $\to$ | ${\rm T}_{5}^{0}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
${\rm T}_{3}$ | $\to$ | ${\rm T}_{6}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm T}_{4}$ | $\to$ | ${\rm T}_{1}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$
${\rm T}_{4}$ | $\to$ | ${\rm T}_{5}^{0}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
${\rm T}_{4}$ | $\to$ | ${\rm T}_{7}^{\alpha\neq 0}$ | $E_{1}(t)=\alpha te_{1}+te_{2}$ | $E_{2}(t)=\alpha t^{2}e_{2}$
${\rm T}_{5}^{\alpha}$ | $\to$ | ${\rm T}_{7}^{0}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
${\rm T}_{6}$ | $\to$ | ${\rm T}_{7}^{0}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=-te_{2}$
${\rm T}_{8}$ | $\to$ | ${\rm T}_{2}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
${\rm T}_{8}$ | $\to$ | ${\rm T}_{7}^{0}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$

The primary non-degenerations are proven using the sets from the following
table:

Non-degeneration | Arguments
---|---
$\begin{array}[]{cccc}{\rm T}_{3}&\not\to&{\rm T}_{1},{\rm T}_{2},{\rm T}_{5}^{\alpha\neq 0},{\rm T}_{7}^{\alpha\neq 0}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{22}^{2}\in\mathbb{C},c_{12}^{2}=c_{21}^{2}\end{array}\right\\}$
$\begin{array}[]{cccc}{\rm T}_{4}&\not\to&{\rm T}_{2},{\rm T}_{5}^{\alpha\neq 0},{{\rm T}_{6}}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 2}\in\mathbb{C},c_{11}^{1}=c_{12}^{2},c_{11}^{1}=c_{21}^{2}\end{array}\right\\}$
$\begin{array}[]{cccc}{\rm T}_{5}^{\alpha}&\not\to&{\rm T}_{1},{\rm T}_{2},{\rm T}_{7}^{\beta\neq 0}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{11}^{1}=c_{12}^{2},c_{11}^{1}=c_{21}^{2},c_{12}^{\prime 2}=\alpha c_{11}^{1},c_{21}^{\prime 2}=-c_{12}^{\prime 2}\end{array}\right\\}$
$\begin{array}[]{cccc}{\rm T}_{8}&\not\to&{\rm T}_{1},{\rm T}_{7}^{\alpha\neq 0}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{21}^{\prime 2}=-c_{12}^{\prime 2}\end{array}\right\\}$

∎

At this point, only the description of the closures of the orbits of the
parametric families is missing. Although it is not necessary to study the
closure of the orbits of each of the parametric families of the variety of
$2$-dimensional transposed Leibniz–Poisson algebras in order to identify its
irreducible components, we will study them to give a complete description of
the variety.

######  Lemma 28.

The description of the closure of the orbit of the parametric families in the
variety of $2$-dimensional transposed Leibniz–Poisson algebras are given
below:

$\overline{\\{O({\rm T}_{5}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm T}_{2})\\}},\overline{\\{O({\rm T}_{7}^{0})\\}},\overline{\\{O({\rm T}_{8})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
---|---|---
$\overline{\\{O({\rm T}_{7}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm T}_{1})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$

###### Proof.

Thanks to Theorem 27 we have all necessarily degenerations. The closure of the
orbits of suitable families are given using the parametric bases and the
parametric index included in following Table:

Degeneration | Parametrized basis
---|---
${\rm T}_{5}^{t^{-1}}$ | $\to$ | ${\rm T}_{8}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
${\rm T}_{7}^{t^{-2}}$ | $\to$ | ${\rm T}_{1}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=e_{2}$
Non-degeneration | Arguments
---|---
$\begin{array}[]{cccc}{\rm T}_{5}^{*}&\not\to&{\rm T}_{1},{\rm T}_{6},{\rm T}_{7}^{\alpha\neq 0}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{11}^{1}=c_{12}^{2},c_{11}^{1}=c_{21}^{2},c_{21}^{\prime 2}=-c_{12}^{\prime 2}\end{array}\right\\}$
$\begin{array}[]{cccc}{\rm T}_{7}^{*}&\not\to&{\rm T}_{2}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 2}\in\mathbb{C}\end{array}\right\\}$

∎

By Lemma 27 and Lemma 28, we have the following result that summarizes the
geometric classification.

######  Theorem 29.

The variety of $2$-dimensional transposed Leibniz–Poisson algebras has three
irreducible components, corresponding to the transposed Leibniz–Poisson
algebras ${\rm T}_{3}$ and ${\rm T}_{4}$ and the family of transposed
Leibniz–Poisson algebras ${\rm T}_{5}^{*}$.

####  3.2.1. The geometric classification of $2$-dimensional transposed
Poisson algebras

######  Corollary 30.

The variety of $2$-dimensional transposed Poisson algebras has two irreducible
components, corresponding to the transposed Poisson algebra ${\rm T}_{3}$ and
the family of transposed Poisson algebras ${\rm T}_{5}^{*}$.

##  4\.  Novikov–Poisson algebras

The notion of Novikov–Poisson (also known as Gelfand-Dorfman-Novikov–Poisson
[18]) algebras is introduced in a paper of Xu in order to study Novikov
algebras [59]. He uses these new algebras to give the first examples of simple
Novikov algebras. The next paper of Xu opens the systematic study of
Novikov–Poisson algebras [58]. Novikov–Poisson algebras are related to Jordan
superalgebras and transposed Poisson algebras [13]. A first tentative of the
algebraic classification of $2$\- and $3$-dimensional Novikov–Poisson algebras
is given in [60], but the author did not separate isomorphic algebras. Our aim
in the present section is to improve and extend the results results in [60].

###  4.1. The algebraic classification of $2$-dimensional Novikov algebras

######  Definition 31.

An algebra is called Novikov if

$(x\circ y)\circ z-x\circ(y\circ z)$ | $=$ |  $(y\circ x)\circ z-y\circ(x\circ z)$,
---|---|---
$(x\circ y)\circ z$ | $=$ | $(x\circ z)\circ y.$

The algebraic classification of $2$-dimensional Novikov algebras is given by
[12].

######  Theorem 32.

Let ${\rm N}$ be a nonzero $2$-dimensional Novikov algebra. Then ${\rm N}$ is
isomorphic to one and only one of the following algebras:

${\rm N}_{01}$ | $:$ | $e_{1}\circ e_{1}$ | $=$ | $e_{2}$ |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
${\rm N}_{02}$ | $:$ | $e_{2}\circ e_{1}$ | $=$ | $-e_{1}$ |  |  |  |  |  |
${\rm N}_{03}$ | $:$ | $e_{1}\circ e_{1}$ | $=$ | $e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $e_{2}$ |  |  |
${\rm N}_{04}$ | $:$ | $e_{1}\circ e_{1}$ | $=$ | $e_{1}$ |  |  |  |  |  |
${\rm N}_{05}$ | $:$ | $e_{1}\circ e_{2}$ | $=$ | $e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $e_{1}+e_{2}$ |  |  |
${\rm N}_{06}^{\gamma}$ | $:$ | $e_{1}\circ e_{2}$ | $=$ | $e_{1},$ | $e_{2}\circ e_{1}$ | $=$ | $\gamma e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $e_{2}$

###  4.2. The algebraic classification of $2$-dimensional Novikov–Poisson
algebras

######  Definition 33.

A Novikov–Poisson algebra is a vector space equipped with a commutative
associative multiplication $\cdot$ and a Novikov multiplication $\circ.$ These
two operations are required to satisfy the following identities:

$(x\cdot y)\circ z$ | $=$ | $x\cdot(y\circ z),$
---|---|---
$(x\circ y)\cdot z-(y\circ x)\cdot z$ | $=$ | $x\circ(y\cdot z)-y\circ(x\cdot z).$

Novikov–Poisson algebras $({\rm N},\cdot,\circ)$ with zero commutative
associative multiplication are the Novikov algebras given in Theorem 32.
Hence, we are studying the Novikov–Poisson algebras defined on every
commutative associative algebra from Theorem 4.

######  Definition 34.

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

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

for all $x,y,z$ in ${\rm A}$. If $\theta\in{\rm Z}_{\rm NP}^{2}\left({\rm
A},{\rm A}\right)$, then $\left({\rm A},\cdot,\circ\right)$ is a
Novikov–Poisson algebra where $x\circ y=\theta\left(x,y\right),$ for all
$x,y\in{\rm A}$.

####  4.2.1. Novikov–Poisson algebras defined on ${\rm A}_{01}$

The Novikov–Poisson structures defined on ${\rm A}_{01}$ are:

${\rm N}_{07}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{2}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{1}\circ
e_{1}$&$=$&$\alpha e_{1},$&$e_{2}\circ e_{2}$&$=$&$\beta
e_{2}$\end{tabular}\right.$

Since ${\rm Aut}({\rm A}_{01})=\mathbb{S}_{2}$, we have ${\rm
N}_{07}^{\alpha,\beta}\cong{\rm N}_{07}^{\beta,\alpha}$.

####  4.2.2. Novikov–Poisson algebras defined on ${\rm A}_{02}$

The Novikov–Poisson structures defined on ${\rm A}_{02}$ are of the form:

$\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{1}\circ
e_{1}$&$=$&$\alpha_{1}e_{1}+\beta_{1}e_{2},$&$e_{1}\circ
e_{2}$&$=$&$\gamma_{1}e_{2},$&$e_{2}\circ
e_{1}$&$=$&$\alpha_{1}e_{2}$\end{tabular}\right.$

By Lemma 5, to study the action of ${\rm Aut}({\rm A}_{02})$, we have to
distinguish two cases:

  * •

If $\beta_{1}\neq 0$, then choose $\xi=\beta_{1}$ and obtain the parametric
family:

${\rm N}_{08}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{1}\circ
e_{1}$&$=$&$\alpha e_{1}+e_{2},$&$e_{1}\circ e_{2}$&$=$&$\beta
e_{2},$&$e_{2}\circ e_{1}$&$=$&$\alpha e_{2}$\end{tabular}\right.$

  * •

If $\beta_{1}=0$, then, we have:

${\rm N}_{09}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{lll}$e_{1}\cdot
e_{1}=e_{1},$&$e_{1}\cdot e_{2}=e_{2}$\\\ $e_{1}\circ e_{1}=\alpha
e_{1},$&$e_{1}\circ e_{2}=\beta e_{2},$&$e_{2}\circ e_{1}=\alpha
e_{2}$\end{tabular}\right.$

####  4.2.3. Novikov–Poisson algebras defined on ${\rm A}_{03}$

The Novikov–Poisson structures defined on ${\rm A}_{03}$ are:

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

The action of ${\rm Aut}({\rm A}_{03})$ produces two cases:

  * •

If $\beta_{1}\neq 0$, then set $\xi=\beta_{1}^{-1}$ and obtain the family:

${\rm N}_{10}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{2}\circ
e_{2}$&$=$&$e_{2}$\end{tabular}\right.$

  * •

If $\beta_{1}=0$, then we obtain the family:

${\rm N}_{11}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha
e_{1}$\end{tabular}\right.$

####  4.2.4. Novikov–Poisson algebras defined on ${\rm A}_{04}$

The Novikov–Poisson structures defined on ${\rm A}_{04}$ are:

$\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2}$\\\
$e_{1}\circ e_{1}$&$=$&$\alpha_{1}e_{1}+\beta_{1}e_{2},$&$e_{1}\circ
e_{2}$&$=$&$\gamma_{1}e_{2},$&$e_{2}\circ
e_{1}$&$=$&$\alpha_{1}e_{2}$\end{tabular}\right.$

The following cases arise from the action of ${\rm Aut}({\rm A}_{04})$ on this
structure:

  * •

If $\gamma_{1}\neq 0$, then choose $\xi=\gamma_{1}^{-1}$ and
$\nu=-\beta_{1}\gamma_{1}^{-2}$ obtaining the family:

${\rm N}_{12}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{1}\circ
e_{2}$&$=$&$e_{2},$&$e_{2}\circ e_{1}$&$=$&$\alpha e_{2}$\end{tabular}\right.$

  * •

If $\gamma_{1}=0$, then

    * –

If $\alpha_{1}\neq 0$, then for $\xi=\alpha_{1}^{-1}$, we obtain:

${\rm N}_{13}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$e_{1}+\alpha e_{2},$&$e_{2}\circ
e_{1}$&$=$&$e_{2}$\end{tabular}\right.$

    * –

If $\alpha_{1}=0$, then we have the algebra:

${\rm N}_{14}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha
e_{2}$\end{tabular}\right.$

####  4.2.5. The algebraic classification of $2$-dimensional Novikov-Poisson
algebras

######  Theorem 35.

Let $({\rm N},\cdot,\circ)$ be a nonzero $2$-dimensional Novikov–Poisson
algebra. Then ${\rm N}$ is isomorphic to one Novikov algebra listed in Theorem
32 or to one algebra listed below:

${\rm N}_{07}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{2}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{2}\circ e_{2}$&$=$&$\beta e_{2}$\end{tabular}\right.$
---|---|---
${\rm N}_{08}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1}+e_{2},$&$e_{1}\circ e_{2}$&$=$&$\beta e_{2},$&$e_{2}\circ e_{1}$&$=$&$\alpha e_{2}$\end{tabular}\right.$
${\rm N}_{09}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{1}\circ e_{2}$&$=$&$\beta e_{2},$&$e_{2}\circ e_{1}$&$=$&$\alpha e_{2}$\end{tabular}\right.$
${\rm N}_{10}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{2}\circ e_{2}$&$=$&$e_{2}$\end{tabular}\right.$
${\rm N}_{11}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1}$\end{tabular}\right.$
${\rm N}_{12}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{1}\circ e_{2}$&$=$&$e_{2},$&$e_{2}\circ e_{1}$&$=$&$\alpha e_{2}$\end{tabular}\right.$
${\rm N}_{13}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2},$\\\ $e_{1}\circ e_{1}$&$=$&$e_{1}+\alpha e_{2}$&$e_{2}\circ e_{1}$&$=$&$e_{2}$\end{tabular}\right.$
${\rm N}_{14}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{2}$\end{tabular}\right.$

The only non-trivial isomorphism between these families is ${\rm
N}_{07}^{\alpha,\beta}\cong{\rm N}_{07}^{\beta,\alpha}$.

####  4.2.6. The algebraic classification of $2$-dimensional pre-Lie Poisson
algebras

######  Definition 36.

A pre-Lie algebra is a vector space ${\rm P}$ equipped with a pre-Lie algebra
multiplication $\circ$ satisfying the following condition:

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

######  Definition 37.

A pre-Lie Poisson algebra is a vector space ${\rm P}$ equipped with a
commutative associative multiplication $\cdot$ and a pre-Lie algebra
multiplication $\circ.$ These two operations are required to satisfy the
following conditions:

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

######  Definition 38.

Let $\left(\rm{P},\cdot\right)$ be a commutative associative algebra. Define
${\rm Z}_{\rm PLP}^{2}\left(\rm{P},\rm{P}\right)$ to be the set of all
bilinear maps $\theta:\rm{P}\times\rm{P}\longrightarrow\mathcal{P}$ such that:

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

for all $x,y,z$ in $\rm{P}$. If $\theta\in{\rm Z}_{\rm
PLP}^{2}\left(\rm{P},\rm{P}\right)$, then $\left(\rm{P},\cdot,\circ\right)$ is
a pre-Lie Poisson algebra where $x\circ y=\theta\left(x,y\right)$ for all
$x,y\in\rm{P}$.

The main tool for the algebraic classification of $2$-dimensional pre-Lie
Poisson algebras can be obtained by a direct computations and given in next
lemma.

######  Lemma 39.

Let ${\rm P}$ be a nonzero $2$-dimensional commutative associative algebra,
then ${\rm Z}_{\rm PLP}^{2}\left(\rm{P},\rm{P}\right)={\rm Z}_{\rm
NP}^{2}\left(\rm{P},\rm{P}\right)$

It is know, that each Novikov algebra is a Pre-Lie algebra. The algebraic
classification of $2$-dimensional pre-Lie algebras can be found in [17].
Hence, we have the following statement.

######  Corollary 40.

Let $({\rm P},\cdot,\circ)$ be a nonzero $2$-dimensional pre-Lie Poisson
algebra. Then ${\rm P}$ is isomorphic to one Novikov–Poisson algebra listed in
Theorem 35 or to one pre-Lie (non-Novikov) algebra listed below:

${\rm P}_{01}^{\alpha\neq 0}$ | $:$ | $e_{2}\circ e_{1}$ | $=$ | $-e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $\alpha e_{2}$ |  |  |
---|---|---|---|---|---|---|---|---|---|---
${\rm P}_{02}$ | $:$ | $e_{1}\circ e_{1}$ | $=$ | $e_{2},$ | $e_{2}\circ e_{1}$ | $=$ | $-e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $-2e_{2}$
${\rm P}_{03}$ | $:$ | $e_{2}\circ e_{1}$ | $=$ | $-e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $e_{1}-e_{2}$ |  |  |

###  4.3. Degenerations of $2$-dimensional Novikov–Poisson algebras

######  Theorem 41.

The graph of primary degenerations and non-degenerations of the variety of
$2$-dimensional Novikov–Poisson algebras is given in Figure 3, where the
numbers on the right side are the dimensions of the corresponding orbits.

$4$$3$$2$$0$${\mathbb{C}}^{8}$${\rm N}_{14}^{\alpha}$${\rm N}_{06}^{0}$${\rm
N}_{01}$${\rm N}_{13}^{\beta}$${\rm N}_{11}^{\beta}$${\rm
N}_{09}^{\beta,\gamma}$${\rm N}_{06}^{\beta\neq 0}$${\rm N}_{05}$${\rm
N}_{04}$${\rm N}_{02}$${\rm N}_{12}^{\delta}$${\rm N}_{10}^{\delta}$${\rm
N}_{08}^{\delta,\epsilon}$${\rm N}_{07}^{\delta,\epsilon}$${\rm
N}_{03}$$\begin{array}[]{c}\delta=\beta\textrm{ or }\\\
\epsilon=\beta\end{array}$$\delta\neq\alpha$$\delta\neq\epsilon$$\delta\neq\epsilon$$\begin{array}[]{c}\delta=\beta\\\
\epsilon=\beta\\\ \gamma=\beta\end{array}$$\begin{array}[]{c}\delta=\beta\\\
\epsilon=\gamma\end{array}$$\delta=0$$\delta=\beta^{-1}$$\delta=\beta$$\alpha=\gamma$$\alpha=\beta$$\alpha=\beta$$\beta=1$

  * Legend:

    * –

Round nodes: Novikov–Poisson algebras with trivial commutative associative
multiplication $\cdot$.

    * –

Squared nodes: Novikov–Poisson algebras with non-trivial commutative
associative multiplication $\cdot$.

Figure 3. Graph of primary degenerations and non-degenerations of
$2$-dimensional Novikov–Poisson algebras.

###### Proof.

The dimensions of the orbits are deduced by computing the algebra of
derivations. The primary degenerations of the Novikov–Poisson algebras with
trivial commutative associative multiplication follow by [44], the others are
proven using the parametric bases included in following Table:

Degeneration | Parametrized basis
---|---
${\rm N}_{07}^{\alpha,\beta\neq\alpha}$ | $\to$ | ${\rm N}_{01}$ | $E_{1}(t)=te_{1}+te_{2}$ | $E_{2}(t)=t^{2}(\alpha-\beta)e_{1}$
${\rm N}_{07}^{\alpha,\alpha}$ | $\to$ | ${\rm N}_{09}^{\alpha,\alpha}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=-te_{2}$
${\rm N}_{07}^{\alpha,\beta}$ | $\to$ | ${\rm N}_{11}^{\alpha}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm N}_{07}^{\alpha,\alpha}$ | $\to$ | ${\rm N}_{14}^{\alpha}$ | $E_{1}(t)=te_{1}-(t-1)te_{2}$ | $E_{2}(t)=(t-1)t^{3}e_{2}$
${\rm N}_{08}^{\alpha,\beta}$ | $\to$ | ${\rm N}_{01}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$
${\rm N}_{08}^{\alpha,\beta}$ | $\to$ | ${\rm N}_{09}^{\alpha,\beta}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
${\rm N}_{08}^{\alpha,\beta}$ | $\to$ | ${\rm N}_{14}^{\gamma}$ | $E_{1}(t)=te_{1}+t(\gamma-\beta)^{-1}e_{2}$ | $E_{2}(t)=t^{2}(\gamma-\beta)^{-1}e_{2}$
${\rm N}_{09}^{\alpha,\beta}$ | $\to$ | ${\rm N}_{14}^{\beta}$ | $E_{1}(t)=te_{1}+t^{-1}e_{2}$ | $E_{2}(t)=e_{2}$
${\rm N}_{10}^{\alpha}$ | $\to$ | ${\rm N}_{04}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=te_{1}$
${\rm N}_{10}^{\alpha}$ | $\to$ | ${\rm N}_{11}^{\alpha}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm N}_{10}^{\alpha}$ | $\to$ | ${\rm N}_{14}^{\beta\neq\alpha}$ | $E_{1}(t)=t(\alpha-\beta)^{-1}e_{1}+te_{2}$ | $E_{2}(t)=t^{2}(\alpha-\beta)^{-2}e_{1}$
${\rm N}_{11}^{\alpha}$ | $\to$ | ${\rm N}_{14}^{\alpha}$ | $E_{1}(t)=te_{1}-t^{-1}e_{2}$ | $E_{2}(t)=e_{2}$
${\rm N}_{12}^{0}$ | $\to$ | ${\rm N}_{02}$ | $E_{1}(t)=t^{-1}e_{2}$ | $E_{2}(t)=-e_{1}$
${\rm N}_{12}^{\alpha}$ | $\to$ | ${\rm N}_{06}^{\alpha^{-1}}$ | $E_{1}(t)=t^{-1}e_{2}$ | $E_{2}(t)=\alpha^{-1}e_{1}$
${\rm N}_{12}^{\alpha}$ | $\to$ | ${\rm N}_{14}^{\beta}$ | $E_{1}(t)=te_{1}+\alpha te_{2}$ | $E_{2}(t)=t^{2}e_{2}$
${\rm N}_{13}^{\alpha}$ | $\to$ | ${\rm N}_{06}^{0}$ | $E_{1}(t)=t^{-1}e_{2}$ | $E_{2}(t)=e_{1}$
${\rm N}_{13}^{\alpha}$ | $\to$ | ${\rm N}_{14}^{\alpha}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$

The primary non-degenerations of Novikov–Poisson algebras with non-trivial
commutative associative multiplication are proven using the sets from the
following table:

Non-degeneration | Arguments
---|---
${{\rm N}_{07}^{\alpha,\beta}}$ | $\not\to$ | $\Big{\\{}\begin{array}[]{l}{\rm N}_{06}^{\gamma},{\rm N}_{02},{\rm N}_{04},\\\ {\rm N}_{11}^{\gamma\neq\alpha,\beta},{\rm N}_{09}^{\gamma,\delta}[\beta\neq\alpha]\end{array}\Big{\\}}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{22}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ c_{21}^{2}=c_{12}^{2},c_{12}^{\prime 2}=c_{21}^{\prime 2},\\\ c_{11}^{\prime 1}=\alpha c_{11}^{1},c_{12}^{\prime 2}=\beta c_{12}^{2},c_{22}^{\prime 2}=\beta c_{22}^{2}\end{array}\right\\}$
${{\rm N}_{07}^{\alpha,\alpha}}$ | $\not\to$ | $\Big{\\{}\begin{array}[]{l}{\rm N}_{06}^{0},{\rm N}_{09}^{(\beta,\gamma)\neq(\alpha,\alpha)},\\\ {\rm N}_{01},{\rm N}_{14}^{\beta\neq\alpha}\end{array}\Big{\\}}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{22}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ c_{21}^{2}=c_{12}^{2},c_{11}^{\prime 1}=\alpha c_{11}^{1},c_{11}^{\prime 2}=\alpha c_{11}^{2},\\\ c_{12}^{\prime 2}=c_{21}^{\prime 2},c_{12}^{\prime 2}=\alpha c_{12}^{2},c_{22}^{\prime 2}=\alpha c_{22}^{2}\end{array}\right\\}$
${\rm N}_{08}^{\alpha,\beta}$ | $\not\to$ | $\Big{\\{}\begin{array}[]{l}{\rm N}_{06}^{\gamma},{\rm N}_{02},{\rm N}_{04},\\\ {{\rm N}_{11}^{\gamma},}{\rm N}_{09}^{(\gamma,\delta)\neq(\alpha,\beta)}\end{array}\Big{\\}}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{12}^{2}=c_{11}^{1},c_{21}^{2}=c_{11}^{1},c_{11}^{\prime 1}=\alpha c_{11}^{1},\\\ c_{12}^{\prime 2}=\beta c_{11}^{1},c_{21}^{\prime 2}=\alpha c_{11}^{1}\end{array}\right\\}$
${\rm N}_{09}^{\alpha,\beta}$ | $\not\to$ | ${\rm N}_{01},{\rm N}_{14}^{\gamma\neq\beta}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{12}^{2}=c_{11}^{1},c_{21}^{2}=c_{11}^{1},c_{11}^{\prime 1}=\alpha c_{11}^{1},\\\ c_{21}^{\prime 2}=\alpha c_{11}^{1},c_{12}^{\prime 2}=\beta c_{11}^{1},c_{11}^{\prime 2}=\beta c_{11}^{2}\end{array}\right\\}$
${\rm N}_{10}^{\alpha}$ | $\not\to$ | ${\rm N}_{02},{\rm N}_{06}^{\beta},{{\rm N}_{09}^{\beta,\gamma},}{\rm N}_{11}^{\beta\neq\alpha}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ c_{11}^{\prime 1}=\alpha c_{11}^{1},c_{12}^{\prime 2}=c_{21}^{\prime 2},c_{11}^{1}c_{21}^{\prime 2}=-c_{11}^{2}c_{22}^{\prime 2}\end{array}\right\\}$
${\rm N}_{11}^{\alpha}$ | $\not\to$ | ${\rm N}_{01},{\rm N}_{14}^{\beta\neq\alpha}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2}\in\mathbb{C},c_{11}^{\prime 1}=\alpha c_{11}^{1},c_{11}^{\prime 2}=\alpha c_{11}^{2}\end{array}\right\\}$
${\rm N}_{12}^{\alpha}$ | $\not\to$ | $\Big{\\{}\begin{array}[]{l}{{\rm N}_{09}^{\beta,\gamma},{\rm N}_{11}^{\beta},}{\rm N}_{04},\\\ {\rm N}_{06}^{\beta\neq\alpha^{-1}},{\rm N}_{02}(\alpha\neq 0)\end{array}\Big{\\}}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{11}^{\prime 1}=\alpha c_{12}^{\prime 2},c_{21}^{\prime 2}=\alpha c_{12}^{\prime 2}\end{array}\right\\}$
${\rm N}_{13}^{\alpha}$ | $\not\to$ | ${\rm N}_{01},{\rm N}_{14}^{\beta\neq\alpha}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{11}^{\prime 1}=c_{21}^{\prime 2},c_{11}^{\prime 2}=\beta c_{11}^{2}\end{array}\right\\}$

∎

At this point, only the description of the closures of the orbits of the
parametric families is missing.

######  Lemma 42.

The description of the closure of the orbit of the parametric families in the
variety of $2$-dimensional Novikov–Poisson algebras are given below:

$\overline{\\{O({\rm N}_{06}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm N}_{05})\\}},\ \overline{\\{O({\rm N}_{02})\\}},\ \overline{\\{O({\rm N}_{01})\\}},\ \overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
---|---|---
$\overline{\\{O({\rm N}_{07}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\begin{array}[]{l}\overline{\\{O({\rm N}_{14}^{\alpha}\\}},\ \overline{\\{O({\rm N}_{12}^{1}\\}},\ \overline{\\{O({\rm N}_{11}^{\alpha}\\}},\ \overline{\\{O({\rm N}_{10}^{\alpha}\\}},\ \overline{\\{O({\rm N}_{09}^{\alpha,\alpha}\\}},\ \overline{\\{O({\rm N}_{08}^{\alpha,\alpha}\\}},\\\ \overline{\\{O({\rm N}_{06}^{1}\\}},\ \overline{\\{O({\rm N}_{04})\\}},\ \overline{\\{O({\rm N}_{03})\\}},\ \overline{\\{O({\rm N}_{01})\\}},\ \overline{\\{O({\mathbb{C}^{8}})\\}}\end{array}\Big{\\}}$
$\overline{\\{O({\rm N}_{08}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\begin{array}[]{l}\overline{\\{O({\rm N}_{14}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{13}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{12}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{09}^{\alpha,\beta})\\}},\ \overline{\\{O({\rm N}_{06}^{\alpha})\\}},\\\ \overline{\\{O({\rm N}_{05})\\}},\ \overline{\\{O({\rm N}_{02})\\}},\ \overline{\\{O({\rm N}_{01})\\}},\ \overline{\\{O({\mathbb{C}^{8}})\\}}\end{array}\Big{\\}}$
$\overline{\\{O({\rm N}_{09}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\begin{array}[]{l}\overline{\\{O({\rm N}_{14}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{13}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{06}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{05})\\}},\\\ \overline{\\{O({\rm N}_{02})\\}},\ \overline{\\{O({\rm N}_{01})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\end{array}\Big{\\}}$
$\overline{\\{O({\rm N}_{10}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\begin{array}[]{l}\overline{\\{O({\rm N}_{14}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{12}^{1})\\}},\ \overline{\\{O({\rm N}_{11}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{06}^{1})\\}},\\\ \overline{\\{O({\rm N}_{04})\\}},\ \overline{\\{O({\rm N}_{03})\\}},\ \overline{\\{O({\rm N}_{01})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\end{array}\Big{\\}}$
$\overline{\\{O({\rm N}_{11}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm N}_{14}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{04})\\}},\ \overline{\\{O({\rm N}_{01})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
$\overline{\\{O({\rm N}_{12}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\begin{array}[]{l}\overline{\\{O({\rm N}_{14}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{13}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{06}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{05})\\}},\\\ \overline{\\{O({\rm N}_{02})\\}},\ \overline{\\{O({\rm N}_{01})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\end{array}\Big{\\}}$
$\overline{\\{O({\rm N}_{13}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm N}_{14}^{\alpha})\\}},\ \overline{\\{O({\rm N}_{06}^{0})\\}},\ \overline{\\{O({\rm N}_{05})\\}},\ \overline{\\{O({\rm N}_{01})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
$\overline{\\{O({\rm N}_{14}^{*})\\}}$ | ${\supseteq}$ |  $\Big{\\{}\overline{\\{O({\rm N}_{01})\\}},\ \overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$ .

###### Proof.

Thanks to Theorem 41 we have all necessarily degenerations. The closure of the
orbits of suitable families are given using the parametric bases and the
parametric index included in following Table:

${\rm N}_{07}^{t^{-1},t^{-1}}$ | $\to$ | ${\rm N}_{03}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=te_{2}$
---|---|---|---|---
${\rm N}_{07}^{t(1+\alpha t^{-1}),\alpha}$ | $\to$ | ${\rm N}_{08}^{\alpha\alpha}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=-te_{1}$
${\rm N}_{07}^{\alpha,t^{-1}}$ | $\to$ | ${\rm N}_{10}^{\alpha}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm N}_{08}^{\alpha t^{-1},t^{-1}}$ | $\to$ | ${\rm N}_{12}^{\alpha}$ | $E_{1}(t)=te_{1}-t^{2}e_{2}$ | $E_{2}(t)=-t^{3}e_{2}$
${\rm N}_{09}^{t^{-1},\alpha t^{-1}}$ | $\to$ | ${\rm N}_{06}^{\alpha}$ | $E_{1}(t)=te_{2}$ | $E_{2}(t)=te_{1}$
${\rm N}_{09}^{t^{-1},\alpha}$ | $\to$ | ${\rm N}_{13}^{\alpha}$ | $E_{1}(t)=te_{1}+t^{-1}e_{2}$ | $E_{2}(t)=e_{2}$
${\rm N}_{10}^{t^{-1}}$ | $\to$ | ${\rm N}_{03}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=e_{2}$
${\rm N}_{10}^{t^{-1}}$ | $\to$ | ${\rm N}_{12}^{1}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=t^{2}e_{1}$
${\rm N}_{11}^{t^{-1}}$ | $\to$ | ${\rm N}_{04}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=e_{2}$
${\rm N}_{12}^{t^{-1}}$ | $\to$ | ${\rm N}_{13}^{\alpha}$ | $E_{1}(t)=te_{1}+\alpha te_{2}$ | $E_{2}(t)=t^{2}e_{2}$
${\rm N}_{13}^{t^{-1}}$ | $\to$ | ${\rm N}_{05}$ | $E_{1}(t)=t^{-1}e_{2}$ | $E_{2}(t)=e_{1}$
${\rm N}_{14}^{t^{-1}}$ | $\to$ | ${\rm N}_{01}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
Non-degeneration | Arguments
---|---
${{\rm N}_{07}^{*}}$ | $\not\to$ | ${\rm N}_{02},{\rm N}_{06}^{\alpha\neq 1},{\rm N}_{09}^{\alpha,\beta\neq\alpha}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{22}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ c_{21}^{2}=c_{12}^{2},c_{12}^{\prime 2}=c_{21}^{\prime 2},c_{22}^{2}c_{12}^{\prime 2}=c_{12}^{2}c_{22}^{\prime 2}\end{array}\right\\}$
${\rm N}_{08}^{*}$ | $\not\to$ | ${\rm N}_{04},{\rm N}_{10}^{\alpha},{\rm N}_{11}^{\alpha}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{12}^{2}=c_{11}^{1},c_{21}^{2}=c_{11}^{1},c_{11}^{\prime 1}=c_{21}^{\prime 2}\end{array}\right\\}$
${\rm N}_{09}^{*}$ | $\not\to$ | ${\rm N}_{08}^{\alpha,\beta},{\rm N}_{12}^{\alpha}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{12}^{2}=c_{11}^{1},c_{21}^{2}=c_{11}^{1},c_{11}^{\prime 1}=c_{21}^{\prime 2},c_{11}^{2}c_{12}^{\prime 2}=c_{11}^{1}c_{11}^{\prime 2}\end{array}\right\\}$
${\rm N}_{10}^{*}$ | $\not\to$ | ${\rm N}_{07}^{\alpha,\beta},{\rm N}_{09}^{\alpha,\alpha}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ c_{12}^{\prime 2}=c_{21}^{\prime 2},c_{11}^{1}c_{21}^{\prime 2}=-c_{11}^{2}c_{22}^{\prime 2}\end{array}\right\\}$
${\rm N}_{11}^{*}$ | $\not\to$ | ${\rm N}_{06}^{1}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2}\in\mathbb{C},c_{11}^{1}c_{11}^{\prime 2}=c_{11}^{2}c_{11}^{\prime 1}\end{array}\right\\}$
${\rm N}_{12}^{*}$ | $\not\to$ | ${\rm N}_{09}^{\alpha,\beta}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{11}^{\prime 1}=c_{21}^{\prime 2}\end{array}\right\\}$
${\rm N}_{13}^{*}$ | $\not\to$ | ${\rm N}_{02},{\rm N}_{06}^{\alpha\neq 0},$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{11}^{\prime 1}=c_{21}^{\prime 2}\end{array}\right\\}$

∎

By Lemma 41 and Lemma 42, we have the following result that summarizes the
geometric classification of $2$-dimensional Novikov–Poisson algebras.

######  Theorem 43.

The variety of $2$-dimensional Novikov–Poisson algebras has two irreducible
components, corresponding to the closures of the orbits of the families of
Novikov–Poisson algebras ${\rm N}_{07}^{*}$ and ${\rm N}_{08}^{*}$.

####  4.3.1. The geometric classification of $2$-dimensional pre-Lie Poisson
algebras

The geometric classification of $2$-dimensional Novikov–Poisson algebras is
given in theorem 43; a description of all degenerations of $2$-dimensional
pre-Lie algebras is given in [17]; a description of all orbit closures of
families of $2$-dimensional algebras is given in [44]. Summarizing, these
cited results and corollary 40 we have the following statement.

######  Corollary 44.

The variety of $2$-dimensional pre-Lie Poisson algebras has four irreducible
components, corresponding to the the Pre-Lie Poisson algebra ${\rm P}_{02}$
and the families of pre-Lie Poisson algebras ${\rm P}_{01}^{*},$ ${\rm
N}_{07}^{*}$ and ${\rm N}_{08}^{*}$.

##  5\.  Commutative pre-Lie algebras

Commutative pre-Lie algebras first appeared in [49] and after that, they
appeared in [30, 31, 28, 50, 46]. A description of free commutative pre-Lie
algebras with one generators in terms of partitioned trees is given in [30]
and a relation between commutative pre-Lie algebras and $F$-manifold algebras
is established in [28].

###  5.1. The algebraic classification of $2$-dimensional pre-Lie algebras

######  Definition 45.

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

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

The algebraic classification of $2$-dimensional pre-Lie algebras can be
obtained by direct verification from [44].

######  Theorem 46.

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

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

###  5.2. The algebraic classification of $2$-dimensional commutative pre-Lie
algebras

######  Definition 47.

A commutative pre-Lie algebra is a vector space equipped with a commutative
associative multiplication $\cdot$ and a pre-Lie multiplication $\circ.$ These
two operations are required to satisfy the following identities:

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

Commutative pre-Lie algebras $({\rm C},\cdot,\circ)$ with zero commutative
associative multiplication are pre-Lie algebras given in Theorem 46. Hence, we
are studying commutative pre-Lie algebras defined on every commutative
associative algebra from Theorem 4.

######  Definition 48.

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

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

If $\theta\in{\rm Z}_{\rm CPL}^{2}\left(\rm{A},\rm{A}\right)$, then
$\left(\rm{A},\cdot,\circ\right)$ is a commutative pre-Lie algebra where
$x\circ y=\theta\left(x,y\right)$ for all $x,y\in\rm{A}$.

####  5.2.1. Commutative pre-Lie algebras defined on ${\rm A}_{01}$

Since ${\rm Z}_{\rm CPL}^{2}({\rm A}_{01},{\rm A}_{01})=\left\\{0\right\\}$,
then there is only the trivial structure ${\rm C}_{09}={\rm A}_{01}$.

####  5.2.2. Commutative pre-Lie algebras defined on ${\rm A}_{02}$

From the computation of ${\rm Z}_{\rm CPL}^{2}({\rm A}_{02},{\rm A}_{02})$,
the commutative pre-Lie structures defined on ${\rm A}_{02}$ are of the form:

$\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot
e_{2}$&$=$&$e_{2}$\\\ $e_{1}\circ
e_{2}$&$=$&$\alpha_{1}e_{2}$\end{tabular}\right.\qquad\qquad\qquad\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{2}\circ
e_{2}$&$=$&$\alpha_{2}e_{2}$\end{tabular}\right.$

If $\alpha_{2}=0$, then it is included in the first family. So we may assume
$\alpha_{2}\neq 0$. Further, if we choose $\xi={\alpha_{2}}^{-1}$, we then
obtain the following non-isomorphic algebras:

${\rm C}_{10}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{1}\circ
e_{2}$&$=$&$\alpha e_{2}$\end{tabular}\right.\qquad\qquad\qquad{\rm
C}_{11}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{2}\circ
e_{2}$&$=$&$e_{2}$\end{tabular}\right.$

####  5.2.3. Commutative pre-Lie algebras defined on ${\rm A}_{03}$

The commutative pre-Lie structures defined on ${\rm A}_{03}$ are:

$\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{1}$\\\ $e_{1}\circ
e_{2}$&$=$&$\alpha_{1}e_{2}$\end{tabular}\right.\qquad\qquad\qquad\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1}$\\\ $e_{2}\circ
e_{2}$&$=$&$\alpha_{2}e_{2}$\end{tabular}\right.$

As above, we may assume $\alpha_{2}\neq 0$. Then we obtain the following non-
isomorphic algebras:

${\rm C}_{12}^{\alpha}:=\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1}$\\\ $e_{1}\circ e_{2}$&$=$&$\alpha
e_{2}$\end{tabular}\right.\qquad\qquad\qquad{\rm
C}_{13}:=\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{1}$\\\
$e_{2}\circ e_{2}$&$=$&$e_{2}$\end{tabular}\right.$

####  5.2.4. Commutative pre-Lie algebras defined on ${\rm A}_{04}$

The commutative pre-Lie structures defined on ${\rm A}_{04}$ are:

$\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2}$\\\
$e_{1}\circ e_{1}$&$=$&$\alpha_{1}e_{1}+\beta_{1}e_{2},$&$e_{1}\circ
e_{2}$&$=$&$2\alpha_{1}e_{2}$\end{tabular}\right.\qquad\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ
e_{1}$&$=$&$\alpha_{2}e_{1}+\beta_{2}e_{2},$&$e_{1}\circ
e_{2}$&$=$&$2\alpha_{2}e_{2}$\\\ $e_{2}\circ
e_{1}$&$=$&$\gamma_{2}e_{1}+\alpha_{2}e_{2},$&$e_{2}\circ
e_{2}$&$=$&$2\gamma_{2}e_{2}$\end{tabular}\right.$

Note that if $\alpha_{1}=0$, then it is included in the second one. So assume
$\alpha_{1}\neq 0$. Then, we can choose $\xi={\alpha_{1}}^{-1}$ and
$\nu=-\beta_{1}\alpha_{1}^{-2}$ to simplify the first family into:

${\rm C}_{14}:=\left\\{\begin{tabular}[]{ll}$e_{1}\cdot e_{1}=e_{2},$\\\
$e_{1}\circ e_{1}=e_{1},$&$e_{1}\circ e_{2}=2e_{2}.$\end{tabular}\right.$

Now, for the second family we have the following cases.

  * •

If $\gamma_{2}\neq 0$, then choose $\xi=-{\sqrt{\gamma_{2}^{-1}}}$ and
$\nu=\alpha_{2}\sqrt{\gamma_{2}^{-3}}-\sqrt{\gamma_{2}^{-3}\left(\alpha_{2}^{2}-\beta_{2}\gamma_{2}\right)}$
to obtain the family:

${\rm C}_{15}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{1}\circ
e_{2}$&$=$&$2\alpha e_{2},$&$e_{2}\circ e_{1}$&$=$&$e_{1}+\alpha
e_{2},$&$e_{2}\circ e_{2}$&$=$&$2e_{2}$\end{tabular}\right.$

Moreover, ${\rm C}_{15}^{\alpha}\cong{\rm C}_{15}^{-\alpha}$, setting $\xi=-1$
and $\nu=0$.

  * •

If $\gamma_{2}=0$, then if $\alpha_{2}\neq 0$, by choosing
$\xi={\alpha_{2}}^{-1}$ and $\nu=-\beta_{2}(2\alpha_{2}^{2})^{-1}$, we obtain
the algebra:

${\rm C}_{16}:=\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$e_{1},$&$e_{1}\circ
e_{2}$&$=$&$2e_{2},$&$e_{2}\circ e_{1}$&$=$&$e_{2}$\end{tabular}\right.$

Otherwise, we have the parametric family :

${\rm C}_{17}^{\alpha}:=\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha
e_{2}$\end{tabular}\right.$

######  Theorem 49.

Let $\left({\rm C},\cdot,\circ\right)$ be a nonzero $2$-dimensional
commutative pre-Lie algebra. Then ${\rm C}$ is isomorphic to one pre-Lie
algebra listed in Theorem 46 or to one algebra listed below:

${\rm C}_{09}$ | $:$ | $e_{1}\cdot e_{1}=e_{1},\ \ e_{2}\cdot e_{2}=e_{2}$
---|---|---
${\rm C}_{10}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{2}$&$=$&$\alpha e_{2}$\end{tabular}\right.$
${\rm C}_{11}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2}$\\\ $e_{2}\circ e_{2}$&$=$&$e_{2}$\end{tabular}\right.$
${\rm C}_{12}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{1}$\\\ $e_{1}\circ e_{2}$&$=$&$\alpha e_{2}$\end{tabular}\right.$
${\rm C}_{13}$ | $:$ | $\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{1}$\\\ $e_{2}\circ e_{2}$&$=$&$e_{2}$\end{tabular}\right.$
${\rm C}_{14}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$e_{1},$&$e_{1}\circ e_{2}$&$=$&$2e_{2}$\end{tabular}\right.$
${\rm C}_{15}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{1}\circ e_{2}$&$=$&$2\alpha e_{2},$&$e_{2}\circ e_{1}$&$=$&$e_{1}+\alpha e_{2},$&$e_{2}\circ e_{2}$&$=$&$2e_{2}$\end{tabular}\right.$
${\rm C}_{16}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$e_{1},$&$e_{1}\circ e_{2}$&$=$&$2e_{2},$&$e_{2}\circ e_{1}$&$=$&$e_{2}$\end{tabular}\right.$
${\rm C}_{17}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{2}$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{2}$\end{tabular}\right.$

Between these algebras, the only non-trivial isomorphism is ${\rm
C}_{15}^{\alpha}\cong{\rm C}_{15}^{-\alpha}$.

###  5.3. Degenerations of $2$-dimensional commutative pre-Lie algebras

######  Theorem 50.

The graph of primary degenerations and non-degenerations of the variety of
$2$-dimensional commutative pre-Lie algebras is given in Figure 1, where the
numbers on the right side are the dimensions of the corresponding orbits.

$4$$3$$2$$0$${\mathbb{C}^{8}}$${\rm C}_{06}^{0}$${\rm C}_{05}^{1}$${\rm
C}_{03}$${\rm C}_{17}^{\beta}$${\rm C}_{01}$${\rm C}_{02}$${\rm C}_{04}$${\rm
C}_{05}^{\alpha\neq 1}$${\rm C}_{06}^{\alpha\neq 0}$${\rm
C}_{10}^{\alpha}$${\rm C}_{12}^{\alpha}$${\rm C}_{07}$${\rm C}_{08}$${\rm
C}_{11}$${\rm C}_{13}$${\rm C}_{14}$${\rm C}_{15}^{\gamma}$${\rm C}_{16}$${\rm
C}_{09}$$\alpha=\frac{1}{2}$$\alpha=2$$\alpha=1$$\alpha=0$$\alpha=\frac{1}{2}$$\alpha=2$$\alpha=2$$\alpha=0$$\alpha=0$$\alpha=0$$\alpha=0$$\alpha=0$$\alpha=0$$\alpha=-\beta$$\alpha=\beta$

  * Legend:

    * –

Round nodes: commutative pre-Lie algebras with trivial commutative associative
multiplication $\cdot$.

    * –

Squared nodes: commutative pre-Lie algebras with non-trivial commutative
associative multiplication $\cdot$.

Figure 1. Graph of primary degenerations and non-degenerations.

###### Proof.

The dimensions of the orbits are deduced by computing the algebra of
derivations. The primary degenerations are proven using the parametric bases
included in following Table:

Degeneration | Parametrized basis
---|---
${\rm C}_{01}$ | $\to$ | ${\rm C}_{03}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$
${\rm C}_{01}$ | $\to$ | ${\rm C}_{06}^{0}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
${\rm C}_{02}$ | $\to$ | ${\rm C}_{03}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$
${\rm C}_{02}$ | $\to$ | ${\rm C}_{05}^{1}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
${\rm C}_{04}$ | $\to$ | ${\rm C}_{03}$ | $E_{1}(t)=e_{1}+te_{2}$ | $E_{2}(t)=-t^{2}e_{2}$
${\rm C}_{05}^{\alpha\neq 1}$ | $\to$ | ${\rm C}_{03}$ | $E_{1}(t)=te_{1}+t^{-1}(\alpha-1)^{-1}e_{2}$ | $E_{2}(t)=e_{2}$
${\rm C}_{06}^{\alpha\neq 0}$ | $\to$ | ${\rm C}_{03}$ | $E_{1}(t)=te_{1}+t^{-1}\alpha^{-1}e_{2}$ | $E_{2}(t)=e_{2}$
${\rm C}_{07}$ | $\to$ | ${\rm C}_{05}^{0}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm C}_{07}$ | $\to$ | ${\rm C}_{06}^{1}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
${\rm C}_{08}$ | $\to$ | ${\rm C}_{05}^{1/2}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=-\frac{t}{2}e_{1}+te_{2}$
${\rm C}_{08}$ | $\to$ | ${\rm C}_{06}^{2}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm C}_{09}$ | $\to$ | ${\rm C}_{10}^{0}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=te_{1}$
${\rm C}_{09}$ | $\to$ | ${\rm C}_{12}^{0}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm C}_{11}$ | $\to$ | ${\rm C}_{05}^{0}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=te_{1}$
${\rm C}_{11}$ | $\to$ | ${\rm C}_{10}^{0}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm C}_{13}$ | $\to$ | ${\rm C}_{05}^{0}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=te_{1}$
${\rm C}_{13}$ | $\to$ | ${\rm C}_{12}^{0}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
${\rm C}_{14}$ | $\to$ | ${\rm C}_{05}^{2}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
${\rm C}_{15}^{\alpha}$ | $\to$ | ${\rm C}_{05}^{1/2}$ | $E_{1}(t)=\frac{1}{2}e_{2}$ | $E_{2}(t)=te_{1}$
${\rm C}_{16}$ | $\to$ | ${\rm C}_{06}^{2}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
${\rm C}_{10}^{\alpha}$ | $\to$ | ${\rm C}_{17}^{\alpha}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
${\rm C}_{12}^{\alpha}$ | $\to$ | ${\rm C}_{17}^{-\alpha}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=t^{2}e_{1}$
${\rm C}_{11}$ | $\to$ | ${\rm C}_{17}^{\alpha}$ | $E_{1}(t)=te_{1}+\beta te_{2}$ | $E_{2}(t)=\beta t^{2}e_{2}$
${\rm C}_{13}$ | $\to$ | ${\rm C}_{17}^{\alpha}$ | $E_{1}(t)=-te_{1}+\beta te_{2}$ | $E_{2}(t)=\beta t^{2}e_{2}$
${\rm C}_{14}$ | $\to$ | ${\rm C}_{17}^{\alpha}$ | $E_{1}(t)=te_{1}+\alpha te_{2}$ | $E_{2}(t)=t^{2}e_{2}$
${\rm C}_{15}^{\alpha}$ | $\to$ | ${\rm C}_{17}^{\beta}$ | $E_{1}(t)=te_{1}-(\alpha t-t\sqrt{\alpha^{2}+\beta})e_{2}$ | $E_{2}(t)=t^{2}e_{2}$
${\rm C}_{16}$ | $\to$ | ${\rm C}_{17}^{\alpha}$ | $E_{1}(t)=te_{1}+\frac{\alpha t}{2}e_{2}$ | $E_{2}(t)=t^{2}e_{2}$

The primary non-degenerations are proven using the sets from the following
table:

Non-degeneration | Arguments
---|---
${\rm C}_{01}$ | $\not\to$ | ${\rm C}_{05}^{1}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{11}^{\prime 1}=c_{21}^{\prime 2}\end{array}\right\\}$
${\rm C}_{02}$ | $\not\to$ | ${\rm C}_{06}^{0}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2}\in\mathbb{C},c_{12}^{\prime 2}=c_{11}^{\prime 1}\end{array}\right\\}$
${\rm C}_{04}$ | $\not\to$ | ${\rm C}_{06}^{0},{\rm C}_{05}^{1}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{21}^{\prime 1},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{11}^{\prime 1}=-c_{21}^{\prime 2},-c_{11}^{\prime 2}c_{21}^{\prime 1}=(c_{11}^{\prime 1})^{2}\end{array}\right\\}$
${\rm C}_{05}^{\alpha\neq 1}$ | $\not\to$ | ${\rm C}_{05}^{1},{\rm C}_{06}^{0}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2}\in\mathbb{C},c_{12}^{\prime 2}=\alpha c_{11}^{\prime 1}\end{array}\right\\}$
${\rm C}_{06}^{\alpha\neq 0}$ | $\not\to$ | ${\rm C}_{05}^{1},{\rm C}_{06}^{0}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{12}^{\prime 2}=\alpha c_{11}^{\prime 1},c_{21}^{\prime 2}=c_{11}^{\prime 1}\end{array}\right\\}$
${\rm C}_{07}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{06}^{\alpha\neq 1},{\rm C}_{05}^{\alpha\neq 0},{\rm C}_{04}\\\ \end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},c_{21}^{\prime 2}=c_{12}^{\prime 2}\end{array}\right\\}$
${\rm C}_{08}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{06}^{\alpha\neq 2},{\rm C}_{05}^{\alpha\neq\frac{1}{2}},{\rm C}_{04}\\\ \end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 1},c_{21}^{\prime 2},c_{22}^{2}\in\mathbb{C},\\\ 2c_{21}^{\prime 1}=c_{22}^{\prime 2},c_{11}^{\prime 1}=c_{21}^{\prime 2},2c_{11}^{\prime 1}=c_{12}^{\prime 2}\end{array}\right\\}$
${\rm C}_{10}^{\alpha}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{17}^{\beta\neq\alpha},{\rm C}_{06}^{0},{\rm C}_{05}^{1},{\rm C}_{03}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 2},c_{12}^{\prime 2}\in\mathbb{C},c_{12}^{2}=c_{11}^{1},\\\ c_{21}^{2}=c_{11}^{1},c_{12}^{\prime 2}=\alpha c_{11}^{1},c_{11}^{\prime 2}=\alpha c_{11}^{2}\end{array}\right\\}$
${\rm C}_{12}^{\alpha}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{17}^{\beta\neq-\alpha},{\rm C}_{06}^{0},{\rm C}_{05}^{1},{\rm C}_{03}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 2},c_{12}^{\prime 2}\in\mathbb{C},c_{12}^{\prime 2}=\alpha c_{11}^{1},c_{11}^{\prime 2}=-\alpha c_{11}^{2}\end{array}\right\\}$
${\rm C}_{09}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{17}^{\alpha\neq 0},{\rm C}_{05}^{\alpha},{\rm C}_{06}^{\alpha},{\rm C}_{03}\\\ \end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{22}^{2}\in\mathbb{C},c_{12}^{2}=c_{21}^{2}\end{array}\right\\}$
${\rm C}_{11}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{04},{\rm C}_{05}^{\alpha\neq 0},{\rm C}_{06}^{\alpha},{\rm C}_{12}^{\alpha},{\rm C}_{10}^{\alpha\neq 0}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ c_{12}^{2}=c_{11}^{1},c_{21}^{2}=c_{11}^{1},c_{12}^{\prime 2}=c_{21}^{\prime 2},\\\ (c_{12}^{\prime 2})^{2}=c_{11}^{\prime 2}c_{22}^{\prime 2},c_{11}^{2}c_{22}^{\prime 2}=c_{11}^{1}c_{21}^{\prime 2}\end{array}\right\\}$
${\rm C}_{13}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{04},{\rm C}_{05}^{\alpha\neq 0},{\rm C}_{06}^{\alpha},{\rm C}_{12}^{\alpha\neq 0},{\rm C}_{10}^{\alpha}\par\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},c_{12}^{\prime 2}=c_{21}^{\prime 2},\\\ c_{11}^{\prime 2}c_{22}^{\prime 2}=(c_{12}^{\prime 2})^{2},c_{11}^{2}c_{22}^{\prime 2}=-c_{11}^{1}c_{12}^{\prime 2}\end{array}\right\\}$
${\rm C}_{14}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{05}^{\alpha\neq 2},{\rm C}_{06}^{\alpha},{\rm C}_{12}^{\alpha},{\rm C}_{10}^{\alpha},{\rm C}_{04}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2}\in\mathbb{C},c_{12}^{\prime 2}=2c_{11}^{\prime 1}\end{array}\right\\}$
${\rm C}_{15}^{\alpha}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{05}^{\beta\neq\frac{1}{2}},{\rm C}_{06}^{\beta},{\rm C}_{12}^{\beta},{\rm C}_{10}^{\beta},{\rm C}_{04}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 1},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ c_{11}^{\prime 1}=c_{21}^{\prime 2},2c_{21}^{\prime 1}=c_{22}^{\prime 2},c_{12}^{\prime 2}=2c_{11}^{\prime 1},\\\ (c_{11}^{\prime 1})^{2}=c_{11}^{\prime 2}c_{21}^{\prime 1}+\alpha^{2}c_{11}^{2}c_{21}^{\prime 1}\end{array}\right\\}$
${\rm C}_{16}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{05}^{\alpha},{\rm C}_{06}^{\alpha\neq 2},{\rm C}_{12}^{\alpha},{\rm C}_{10}^{\alpha},{\rm C}_{04}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{21}^{\prime 2}=c_{11}^{\prime 1},c_{12}^{\prime 2}=2c_{11}^{\prime 1}\end{array}\right\\}$

∎

At this point, only the description of the closures of the orbits of the
parametric families is missing. Although it is not necessary to study the
closure of the orbits of each of the parametric families of the variety of
$2$-dimensional commutative pre-Lie algebras in order to identify its
irreducible components, we will study them to give a complete description of
the variety.

######  Corollary 51.

The description of the closure of the orbit of the parametric families in the
variety of $2$-dimensional commutative pre-Lie algebras.

$\overline{\\{O({\rm C}_{05}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm C}_{04})\\}},\ \overline{\\{O({\rm C}_{03})\\}},\ \overline{\\{O({\rm C}_{02})\\}},\ \overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
---|---|---
$\overline{\\{O({\rm C}_{06}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm C}_{04})\\}},\ \overline{\\{O({\rm C}_{03})\\}},\ \overline{\\{O({\rm C}_{01})\\}},\ \overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
$\overline{\\{O({\rm C}_{10}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm C}_{17}^{\alpha})\\}},\ \overline{\\{O({\rm C}_{04})\\}},\ \overline{\\{O({\rm C}_{03})\\}},\ \overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
$\overline{\\{O({\rm C}_{12}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm C}_{17}^{\alpha})\\}},\ \overline{\\{O({\rm C}_{04})\\}},\ \overline{\\{O({\rm C}_{03})\\}},\ \overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
$\overline{\\{O({\rm C}_{15}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm C}_{17}^{\alpha})\\}},\ \overline{\\{O({\rm C}_{16})\\}},\ \overline{\\{O({\rm C}_{08})\\}},\ \overline{\\{O({\rm C}_{06}^{2})\\}},\ \overline{\\{O({\rm C}_{05}^{1/2})\\}},\ \overline{\\{O({\rm C}_{03})\\}},\ \overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$
$\overline{\\{O({\rm C}_{17}^{*})\\}}$ | ${\supseteq}$ | $\Big{\\{}\overline{\\{O({\rm C}_{03})\\}},\overline{\\{O({\mathbb{C}^{8}})\\}}\Big{\\}}$

###### Proof.

Thanks to Theorem 50 we have all necessarily degenerations. All necessarily
the closure of the orbits of suitable families are given using the parametric
bases and the parametric index included in following Table:

Degeneration | Parametrized basis
---|---
${\rm C}_{05}^{t^{-1}}$ | $\to$ | ${\rm C}_{04}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=te_{1}$
${\rm C}_{05}^{t+1}$ | $\to$ | ${\rm C}_{02}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
${\rm C}_{06}^{t^{-1}}$ | $\to$ | ${\rm C}_{04}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=te_{1}$
${\rm C}_{06}^{t}$ | $\to$ | ${\rm C}_{01}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
${\rm C}_{10}^{t^{-1}}$ | $\to$ | ${\rm C}_{04}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=te_{1}$
${\rm C}_{12}^{t^{-1}}$ | $\to$ | ${\rm C}_{04}$ | $E_{1}(t)=e_{2}$ | $E_{2}(t)=te_{1}$
${\rm C}_{15}^{t^{-1}}$ | $\to$ | ${\rm C}_{16}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$
${\rm C}_{15}^{t^{-1}}$ | $\to$ | ${\rm C}_{08}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=\frac{1}{2}e_{2}$
${\rm C}_{17}^{t^{-1}}$ | $\to$ | ${\rm C}_{03}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
Non-degeneration | Arguments
---|---
${\rm C}_{05}^{*}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{17}^{\alpha},{\rm C}_{06}^{\alpha}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2}\in\mathbb{C}\end{array}\right\\}$
${\rm C}_{06}^{*}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{17}^{\alpha},{\rm C}_{05}^{\alpha}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},c_{21}^{\prime 2}=c_{11}^{\prime 1}\end{array}\right\\}$
${\rm C}_{10}^{*}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{05}^{\alpha},{\rm C}_{06}^{\alpha},{\rm C}_{12}^{\alpha}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 2},c_{12}^{\prime 2}\in\mathbb{C},\\\ c_{12}^{2}=c_{11}^{1},c_{21}^{2}=c_{11}^{1}\end{array}\right\\}$
${\rm C}_{12}^{*}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{05}^{\alpha},{\rm C}_{06}^{\alpha},{\rm C}_{10}^{\alpha}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 2},c_{12}^{\prime 2}\in\mathbb{C},c_{11}^{1}c_{11}^{\prime 2}=-c_{11}^{2}c_{12}^{\prime 2}\end{array}\right\\}$
${\rm C}_{15}^{*}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{05}^{\alpha\neq\frac{1}{2}},{\rm C}_{06}^{\alpha\neq 2},\\\ {\rm C}_{12}^{\alpha},{\rm C}_{10}^{\alpha},{\rm C}_{04}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 1},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ c_{11}^{\prime 1}=c_{21}^{\prime 2},2c_{21}^{\prime 1}=c_{22}^{\prime 2},c_{12}^{\prime 2}=2c_{11}^{\prime 1}\end{array}\right\\}$
${\rm C}_{17}^{*}$ | $\not\to$ | $\begin{array}[]{l}{\rm C}_{05}^{\alpha},{\rm C}_{06}^{\alpha}\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ c_{12}^{\prime 2}=c_{21}^{\prime 2},c_{22}^{\prime 2}c_{11}^{\prime 2}=(c_{12}^{\prime 2})^{2}\end{array}\right\\}$

∎

The geometric classification of $2$-dimensional commutative pre-Lie algebras
follows by Theorem 50 and Corollary 51.

######  Theorem 52.

The variety of $2$-dimensional commutative pre-Lie algebras has eleven
irreducible components, corresponding to the rigid algebras ${\rm C}_{07}$,
${\rm C}_{09}$, ${\rm C}_{11}$, ${\rm C}_{13}$ and ${\rm C}_{14},$ and the
families of algebras ${\rm C}_{05}^{*}$, ${\rm C}_{06}^{*}$, ${\rm
C}_{10}^{*}$, ${\rm C}_{12}^{*}$, ${\rm C}_{15}^{*}$ and ${\rm C}_{17}^{*}$.

##  6\. Anti-pre-Lie Poisson algebras

The notion of anti-pre-Lie Poisson algebras is introduced in [14].

###  6.1. The algebraic classification of the 2-dimensional anti-pre-Lie
algebras

######  Definition 53.

An algebra $\left({\rm A},\circ\right)$ is called anti-pre-Lie algebra if the
following equations are satisfied:

$x\circ\left(y\circ z\right)-y\circ\left(x\circ z\right)-(y\circ x-x\circ y)\circ z$ | $=$ | $0,$
---|---|---
$(x\circ y-y\circ x)\circ z+(y\circ z-z\circ y)\circ x+(z\circ x-x\circ z)\circ y$ | $=$ | $0.$

It is obvious that any commutative algebra ${\rm A}$ is anti-pre-Lie algebra
if and only if ${\rm A}$ is associative. So the algebraic classification of
$2$-dimensional commutative anti-pre-Lie algebras is given in Theorem 4. The
algebraic classification of the $2$-dimensional non-commutative anti-pre-Lie
algebras is given by [14].

######  Theorem 54.

Let $\left({\rm A},\circ\right)$ be a nonzero $2$-dimensional anti-pre-Lie
algebra. Then $\left({\rm A},\circ\right)$ is isomorphic to one and only one
of the following algebras:

${\rm A}_{01}$ | $:$ | $e_{1}\circ e_{1}$ | $=$ | $e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $e_{2}$ |  |  |
---|---|---|---|---|---|---|---|---|---|---
${\rm A}_{02}$ | $:$ | $e_{1}\circ e_{1}$ | $=$ | $e_{1},$ | $e_{1}\circ e_{2}$ | $=$ | $e_{2},$ | $e_{2}\circ e_{1}$ | $=$ | $e_{2}$
${\rm A}_{03}$ | $:$ | $e_{1}\circ e_{1}$ | $=$ | $e_{1}$ |  |  |  |  |  |
${\rm A}_{04}$ | $:$ | $e_{1}\circ e_{1}$ | $=$ | $e_{2}$ |  |  |  |  |  |
${\rm A}_{05}$ | $:$ | $e_{1}\circ e_{1}$ | $=$ | $-e_{2},$ | $e_{2}\circ e_{1}$ | $=$ | $-e_{1}$ |  |  |
${\rm A}_{06}^{\lambda}$ | $:$ | $e_{2}\circ e_{1}$ | $=$ | $-e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $\lambda e_{2}$ |  |  |
${\rm A}_{07}$ | $:$ | $e_{2}\circ e_{1}$ | $=$ | $-e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $e_{1}-e_{2}$ |  |  |
${\rm A}_{08}^{\lambda\neq-1}$ | $:$ | $e_{1}\circ e_{2}$ | $=$ | $\left(\lambda+1\right)e_{1},$ | $e_{2}\circ e_{1}$ | $=$ | $\lambda e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $\left(\lambda-1\right)e_{2}$
${\rm A}_{09}$ | $:$ | $e_{1}\circ e_{2}$ | $=$ | $-e_{1},$ | $e_{2}\circ e_{1}$ | $=$ | $-2e_{1},$ | $e_{2}\circ e_{2}$ | $=$ | $e_{1}-3e_{2}.$

###  6.2. The algebraic classification of the 2-dimensional anti-pre-Lie
Poisson algebras

######  Definition 55.

An anti-pre-Lie Poisson algebra is a vector space equipped with a commutative
associative multiplication $\cdot$ and an anti-pre-Lie multiplication $\circ$.
These two operations are required to satisfy the following conditions:

$2\left(x\circ y\right)\cdot z-2\left(y\circ x\right)\cdot z$ | $=$ | $y\cdot\left(x\circ z\right)-x\cdot\left(y\circ z\right),$
---|---|---
$2\ x\circ\left(y\cdot z\right)$ | $=$ | $\left(z\cdot x\right)\circ y+z\cdot\left(x\circ y\right).$

Anti-pre-Lie algebras $\left(\rm{A},\circ\right)$ with zero commutative
associative multiplication are isomorphic to anti-pre-Lie Poisson algebras
given in Theorem 57. Hence, we only have to study the anti-pre-Lie Poisson
structures defined on every commutative associative algebra from Theorem 4 to
obtain the algebraic classification of the 2-dimensional anti-pre-Lie Poisson
algebras. To do this, we first need the following definition.

######  Definition 56.

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

$\theta\left(x,\theta\left(y,z\right)\right)-\theta\left(y,\theta\left(x,z\right)\right)-\theta\left(\theta\left(y,x\right)-\theta\left(x,y\right),z\right)$ | $=$ | $0,$
---|---|---
$\theta\left(\theta\left(x,y\right)-\theta\left(y,x\right),z\right)+\theta\left(\theta\left(y,z\right)-\theta\left(z,y\right),x\right)+\theta\left(\theta\left(z,x\right)-\theta\left(x,z\right),y\right)$ | $=$ | $0,$
$2\theta\left(x,y\right)\cdot z-2\theta\left(y,x\right)\cdot z-y\cdot\theta\left(x,z\right)+x\cdot\theta\left(y,z\right)$ | $=$ | $0,$
$2\theta\left(x,y\cdot z\right)-\theta\left(z\cdot x,y\right)-z\cdot\theta\left(x,y\right)$ | $=$ | $0.$

If $\theta\in{\rm Z}_{\rm APL}^{2}\left(\rm{A},\rm{A}\right)$, then
$\left(\rm{A},\cdot,\circ\right)$ is an anti-pre-Lie Poisson algebra where
$x\circ y=\theta\left(x,y\right)$ for all $x,y\in\rm{A}$.

####  6.2.1. Anti-pre-Lie Poisson algebras defined on ${\rm A}_{01}$

The anti-pre-Lie Poisson defined on ${\rm A}_{01}$ are:

$\mathrm{A}_{10}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{2}\cdot e_{2}$&$=$&$e_{2},$\\\ $e_{1}\circ
e_{1}$&$=$&$\alpha e_{1},$&$e_{2}\circ e_{2}$&$=$&$\beta
e_{2}.$\end{tabular}\right.$

Since $\mathrm{Aut}({\rm A}_{01})={\mathbb{S}}_{2}$, we have
$\mathrm{A}_{10}^{\alpha,\beta}\cong\mathrm{A}_{10}^{\beta,\alpha}$.

####  6.2.2. Anti-pre-Lie Poisson algebras defined on ${\rm{A}}_{02}$

The anti-pre-Lie Poisson structures defined on ${\rm A}_{02}$ are:

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

By Lemma 3, to study the action of $\mathrm{Aut}(\mathrm{A}_{02})$, we have to
distinguish two cases:

  * •

If $\alpha_{1}\neq 0$, then choose $\lambda_{22}=\alpha_{1}$ and obtain the
parametric family:

$\mathrm{A}_{11}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{lcllcl
lcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2},$&\\\
$e_{1}\circ e_{1}$&$=$&$\left(2\alpha-\beta\right)e_{1}+e_{2},$&$e_{1}\circ
e_{2}$&$=$&$\alpha e_{2},$&$e_{2}\circ e_{1}$&$=$&$\beta
e_{2}.$\end{tabular}\right.$

  * •

If $\alpha_{1}=0$, then we have:

$\mathrm{A}_{12}^{\alpha,\beta}:=\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2},$&\\\ $e_{1}\circ
e_{1}$&$=$&$\left(2\alpha-\beta\right)e_{1},$&$e_{1}\circ e_{2}$&$=$&$\alpha
e_{2},$&$e_{2}\circ e_{1}$&$=$&$\beta e_{2}.$\end{tabular}\right.$

####  6.2.3. Anti-pre-Lie Poisson algebras defined on ${\rm A}_{03}$

The anti-pre-Lie Poisson structures defined on ${\rm A}_{03}$ are:

$\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&&\\\
$e_{1}\circ e_{1}$&$=$&$\alpha_{1}e_{1},$&$e_{2}\circ
e_{2}$&$=$&$\alpha_{2}e_{2}.$&\end{tabular}\right.$

The action of $\mathrm{Aut}({\rm A}_{03})$ produces two cases:

  * •

If $\alpha_{2}\neq 0$, then set $\lambda_{22}=\alpha_{2}^{-1}$ and hence we
obtain the algebra:

$\mathrm{A}_{13}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&&\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{2}\circ
e_{2}$&$=$&$e_{2}.$&\end{tabular}\right.$

  * •

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

$\mathrm{A}_{14}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{1},$&\\\ $e_{1}\circ e_{1}$&$=$&$\alpha
e_{1}.$&\end{tabular}\right.$

####  6.2.4. Anti-pre-Lie Poisson algebras defined on ${\rm A}_{04}$

The anti-pre-Lie Poisson structures defined on ${\rm A}_{04}$ are:

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

The following cases arise from the action of $\mathrm{Aut}({\rm A}_{04})$ on
this structure:

  * •

If $\alpha_{2}\neq 0$, then

    * –

If $2\alpha_{3}-\alpha_{2}\neq 0$, choose $\lambda_{11}=\alpha_{2}^{-1}$ and
$\lambda_{21}=\frac{\alpha_{1}}{\alpha_{2}\left(\alpha_{2}-2\alpha_{3}\right)}$
obtaining the family:

$\mathrm{A}_{15}^{\alpha\neq\frac{1}{2}}:=\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2},$&&\\\ $e_{1}\circ
e_{1}$&$=$&$\left(2-\alpha\right)e_{1},$&$e_{1}\circ
e_{2}$&$=$&$e_{2},$&$e_{2}\circ e_{1}$&$=$&$\alpha
e_{2}.$\end{tabular}\right.$

Note that here $\alpha\neq\frac{1}{2}$ follows from
$2\alpha_{3}-\alpha_{2}\neq 0$.

    * –

If $2\alpha_{3}-\alpha_{2}=0$, choose $\lambda_{11}=\alpha_{2}^{-1}$ to obtain
the family:

${\mathrm{A}_{16}^{\alpha}:=\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2},$&&\\\ $e_{1}\circ e_{1}$&$=$&$\frac{3}{2}e_{1}+\alpha
e_{2},$&$e_{1}\circ e_{2}$&$=$&$e_{2},$&$e_{2}\circ
e_{1}$&$=$&$\frac{1}{2}e_{2}.$\end{tabular}\right.}$

Moreover, observe that $\mathrm{A}_{15}^{\frac{1}{2}}=\mathrm{A}_{16}^{0}$.

  * •

If $\alpha_{2}=0$, then

    * –

If $\alpha_{3}\neq 0$, choose $\lambda_{11}=\alpha_{3}^{-1}$ and
$\lambda_{21}=-\frac{\alpha_{1}}{2\alpha_{3}^{2}}$ obtaining the algebra:

$\mathrm{A}_{17}:=\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2},$\\\ $e_{1}\circ e_{1}$&$=$&$-e_{1},$&$e_{2}\circ
e_{1}$&$=$&$e_{2}.$\end{tabular}\right.$

    * –

If $\alpha_{3}=0$, then we have the family:

$\mathrm{A}_{18}^{\alpha}:=\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot
e_{1}$&$=$&$e_{2},$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha
e_{2}.$\end{tabular}\right.$

######  Theorem 57.

Let $\left({\rm A},\cdot,\circ\right)$ be a nonzero $2$-dimensional anti-pre-
Lie Poisson algebra. Then ${\rm A}$ is isomorphic to one anti-pre-Lie algebra
listed in Theorem 54 or to one algebra listed below:

${\rm A}_{10}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{2}\cdot e_{2}$&$=$&$e_{2},$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{2}\circ e_{2}$&$=$&$\beta e_{2}.$\end{tabular}\right.$
---|---|---
${\rm A}_{11}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2},$&\\\ $e_{1}\circ e_{1}$&$=$&$\left(2\alpha-\beta\right)e_{1}+e_{2},$&$e_{1}\circ e_{2}$&$=$&$\alpha e_{2},$&$e_{2}\circ e_{1}$&$=$&$\beta e_{2}.$\end{tabular}\right.$
${\rm A}_{12}^{\alpha,\beta}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&$e_{1}\cdot e_{2}$&$=$&$e_{2},$&\\\ $e_{1}\circ e_{1}$&$=$&$\left(2\alpha-\beta\right)e_{1},$&$e_{1}\circ e_{2}$&$=$&$\alpha e_{2},$&$e_{2}\circ e_{1}$&$=$&$\beta e_{2}.$\end{tabular}\right.$
${\rm A}_{13}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&&\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1},$&$e_{2}\circ e_{2}$&$=$&$e_{2}.$&\end{tabular}\right.$
${\rm A}_{14}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{1},$&\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{1}.$&\end{tabular}\right.$
${\rm A}_{15}^{\alpha\neq\frac{1}{2}}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2},$&&\\\ $e_{1}\circ e_{1}$&$=$&$\left(2-\alpha\right)e_{1},$&$e_{1}\circ e_{2}$&$=$&$e_{2},$&$e_{2}\circ e_{1}$&$=$&$\alpha e_{2}.$\end{tabular}\right.$
${\rm A}_{16}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2},$&&\\\ $e_{1}\circ e_{1}$&$=$&$\frac{3}{2}e_{1}+\alpha e_{2},$&$e_{1}\circ e_{2}$&$=$&$e_{2},$&$e_{2}\circ e_{1}$&$=$&$\frac{1}{2}e_{2}.$\end{tabular}\right.$
${\rm A}_{17}$ | $:$ | $\left\\{\begin{tabular}[]{lcllcl}$e_{1}\cdot e_{1}$&$=$&$e_{2},$\\\ $e_{1}\circ e_{1}$&$=$&$-e_{1},$&$e_{2}\circ e_{1}$&$=$&$e_{2}.$\end{tabular}\right.$
${\rm A}_{18}^{\alpha}$ | $:$ | $\left\\{\begin{tabular}[]{lcl}$e_{1}\cdot e_{1}$&$=$&$e_{2},$\\\ $e_{1}\circ e_{1}$&$=$&$\alpha e_{2}.$\end{tabular}\right.$

Between these algebras, the only non-trivial isomorphism is ${\rm
A}_{10}^{\alpha,\beta}\cong{\rm A}_{10}^{\beta,\alpha}$.

###  6.3. Degenerations of $2$-dimensional anti-pre-Lie Poisson algebras

######  Lemma 58.

The graph of primary degenerations and non-degenerations of the variety of
$2$-dimensional anti-pre-Lie Poisson algebras is given in Figure 5, where the
numbers on the right side are the dimensions of the corresponding orbits.

$4$$3$$2$$0$${\mathbb{C}^{8}}$$\mathrm{A}_{04}$$\mathrm{A}_{06}^{-1}$$\mathrm{A}_{08}^{-2}$$\mathrm{A}_{18}^{\alpha}$$\mathrm{A}_{07}$$\mathrm{A}_{09}$$\mathrm{A}_{02}$$\mathrm{A}_{06}^{\beta\neq-1}$$\mathrm{A}_{08}^{\beta\neq-1,-2}$$\mathrm{A}_{03}$$\mathrm{A}_{12}^{\beta,\gamma}$$\mathrm{A}_{14}^{\beta}$$\mathrm{A}_{16}^{\beta}$$\mathrm{A}_{01}$$\mathrm{A}_{05}$$\mathrm{A}_{10}^{\delta,\epsilon}$$\mathrm{A}_{11}^{\delta,\epsilon}$$\mathrm{A}_{13}^{\delta}$$\mathrm{A}_{15}^{\delta\neq
1/2}$$\mathrm{A}_{17}$$\begin{array}[]{c}\delta=\beta\\\ \epsilon=\beta\\\
\gamma=\beta\end{array}$$\begin{array}[]{c}\delta=\beta\\\ \textrm{ or }\\\
\epsilon=\beta\end{array}$$\delta\neq\epsilon$$\delta\neq\epsilon$$\alpha\neq
2\epsilon-\delta$$(\beta,\gamma)=(\delta,\epsilon)$$\delta\neq\alpha$$\delta=\beta$$(\delta,\beta)=(0,-2)$$\delta=1$$\beta=\frac{1}{\delta-1}$$\beta=0$$\alpha=2\gamma-\beta$$\alpha=\beta$$\alpha=\beta$$\beta=0$$\beta=0$$\delta\neq\alpha$

  * Legend:

    * –

Round nodes: anti-pre-Lie Poisson algebras with trivial commutative
associative multiplication $\cdot$.

    * –

Squared nodes: anti-pre-Lie Poisson algebras with non-trivial commutative
associative multiplication $\cdot$.

Figure 5. Graph of primary degenerations and non-degenerations.

###### Proof.

The dimensions of the orbits are deduced by computing the algebra of
derivations. The primary degenerations are proven using the parametric bases
included in following Table:

Degeneration | Parametrized basis
---|---
$\mathrm{A}_{01}$ | $\to$ | $\mathrm{A}_{02}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
$\mathrm{A}_{01}$ | $\to$ | $\mathrm{A}_{03}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
$\mathrm{A}_{02}$ | $\to$ | $\mathrm{A}_{04}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
$\mathrm{A}_{03}$ | $\to$ | $\mathrm{A}_{04}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=t^{2}e_{1}$
$\mathrm{A}_{05}$ | $\to$ | $\mathrm{A}_{06}^{0}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=e_{2}$
$\mathrm{A}_{05}$ | $\to$ | $\mathrm{A}_{08}^{0}$ | $E_{1}(t)=te_{2}$ | $E_{2}(t)=e_{1}+e_{2}$
$\mathrm{A}_{06}^{\alpha\neq-1}$ | $\to$ | $\mathrm{A}_{04}$ | $E_{1}(t)=e_{1}+te_{2}$ | $E_{2}(t)=-t(1+\alpha)e_{1}$
$\mathrm{A}_{07}$ | $\to$ | $\mathrm{A}_{04}$ | $E_{1}(t)=te_{2}$ | $E_{2}(t)=t^{2}e_{1}$
$\mathrm{A}_{07}$ | $\to$ | $\mathrm{A}_{06}^{-1}$ | $E_{1}(t)=t^{-1}e_{1}$ | $E_{2}(t)=e_{2}$
$\mathrm{A}_{08}^{\alpha\neq-2}$ | $\to$ | $\mathrm{A}_{04}$ | $E_{1}(t)=e_{1}+te_{2}$ | $E_{2}(t)=-t^{2}(2+\alpha)e_{2}$
$\mathrm{A}_{09}$ | $\to$ | $\mathrm{A}_{04}$ | $E_{1}(t)=te_{2}$ | $E_{2}(t)=t^{2}e_{1}$
$\mathrm{A}_{09}$ | $\to$ | $\mathrm{A}_{08}^{-2}$ | $E_{1}(t)=t^{-1}e_{1}$ | $E_{2}(t)=e_{2}$
$\mathrm{A}_{10}^{\alpha,\beta\neq\alpha}$ | $\to$ | $\mathrm{A}_{04}$ | $E_{1}(t)=te_{1}+te_{2}$ | $E_{2}(t)=t^{2}(\alpha-\beta)e_{1}$
$\mathrm{A}_{10}^{\alpha,\alpha}$ | $\to$ | $\mathrm{A}_{12}^{\alpha,\alpha}$ | $E_{1}(t)=e_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
$\mathrm{A}_{10}^{\alpha,\beta}$ | $\to$ | $\mathrm{A}_{14}^{\alpha}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
$\mathrm{A}_{10}^{\alpha,\beta}$ |  $\to$555Here $\alpha\neq\beta,\beta\neq\gamma$ and $\alpha\neq\gamma$. | $\mathrm{A}_{18}^{\gamma}$ | $E_{1}(t)=\frac{t(\beta-\gamma)}{\alpha-\gamma}e_{1}+te_{2}$ | $E_{2}(t)=\frac{t^{2}(\alpha-\beta)}{\alpha-\gamma}e_{2}$
$\mathrm{A}_{11}^{\alpha,\beta}$ | $\to$ | $\mathrm{A}_{04}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$
$\mathrm{A}_{11}^{\alpha,\beta}$ | $\to$ | $\mathrm{A}_{18}^{\gamma\neq 2\beta-\alpha}$ | $E_{1}(t)=t(\alpha-2\beta+\gamma)e_{1}+te_{2}$ | $E_{2}(t)=t^{2}(\alpha-2\beta+\gamma)e_{2}$
$\mathrm{A}_{11}^{\alpha,\beta}$ | $\to$ | $\mathrm{A}_{12}^{\alpha,\beta}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
$\mathrm{A}_{12}^{\alpha,\beta}$ | $\to$ | $\mathrm{A}_{18}^{2\beta-\alpha}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=te_{2}$
$\mathrm{A}_{13}^{\alpha}$ | $\to$ | $\mathrm{A}_{14}^{\alpha}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=te_{2}$
$\mathrm{A}_{13}^{\alpha}$ | $\to$ | $\mathrm{A}_{03}$ | $E_{1}(t)=t^{2}e_{1}+e_{2}$ | $E_{2}(t)=te_{1}$
$\mathrm{A}_{13}^{\alpha}$ | $\to$ | $\mathrm{A}_{18}^{\beta\neq\alpha}$ | $E_{1}(t)=te_{1}+t(\alpha-\beta)e_{2}$ | $E_{2}(t)=-t^{2}(\alpha-\beta)e_{2}$
$\mathrm{A}_{14}^{\alpha}$ | $\to$ | $\mathrm{A}_{18}^{\alpha}$ | $E_{1}(t)=te_{1}+e_{2}$ | $E_{2}(t)=-te_{2}$
$\mathrm{A}_{15}^{1}$ | $\to$ | $\mathrm{A}_{02}$ | $E_{1}(t)=e_{1}$ | $E_{2}(t)=t^{-1}e_{2}$
$\mathrm{A}_{15}^{0}$ | $\to$ | $\mathrm{A}_{06}^{-2}$ | $E_{1}(t)=t^{-1}e_{2}$ | $E_{2}(t)=-e_{1}$
$\mathrm{A}_{15}^{\alpha\neq 1}$ | $\to$ | $\mathrm{A}_{08}^{\frac{1}{\alpha-1}}$ | $E_{1}(t)=t^{-1}e_{2}$ | $E_{2}(t)=\frac{1}{\alpha-1}e_{1}+\frac{1}{\alpha-1}e_{2}$
$\mathrm{A}_{15}^{\alpha\neq\frac{1}{2}}$ | $\to$ | $\mathrm{A}_{18}^{\beta}$ | $E_{1}(t)=te_{1}+\frac{t\beta}{2\alpha-1}e_{2}$ | $E_{2}(t)=t^{2}e_{2}$
$\mathrm{A}_{16}^{\alpha}$ | $\to$ | $\mathrm{A}_{18}^{\alpha}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$
$\mathrm{A}_{16}^{\alpha}$ | $\to$ | $\mathrm{A}_{08}^{-2}$ | $E_{1}(t)=t^{-1}e_{2}$ | $E_{2}(t)=-2e_{1}$
$\mathrm{A}_{17}$ | $\to$ | $\mathrm{A}_{18}^{\alpha\neq 0}$ | $E_{1}(t)=te_{1}+\frac{t\alpha}{2}e_{2}$ | $E_{2}(t)=-\frac{t^{2}}{\alpha}e_{1}+\frac{t^{2}}{2}e_{2}$
$\mathrm{A}_{17}$ | $\to$ | $\mathrm{A}_{18}^{0}$ | $E_{1}(t)=te_{1}$ | $E_{2}(t)=t^{2}e_{2}$
$\mathrm{A}_{17}$ | $\to$ | $\mathrm{A}_{08}^{0}$ | $E_{1}(t)=t^{-1}e_{2}$ | $E_{2}(t)=e_{1}$

The primary non-degenerations are proven using the sets from the following
table:

Non-degeneration | Arguments
---|---
$\begin{array}[]{cccc}\mathrm{A}_{07}&\not\to&\mathrm{A}_{08}^{-2}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{ij}^{\prime k}\in\mathbb{C},c_{11}^{\prime 1}+2c_{21}^{\prime 2}=c_{12}^{\prime 2},2c_{12}^{\prime 1}+c_{22}^{\prime 2}=c_{21}^{\prime 1}\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{09}&\not\to&\mathrm{A}_{06}^{-1}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{ij}^{\prime k}\in\mathbb{C},c_{11}^{\prime 1}+5c_{21}^{\prime 2}=4c_{12}^{\prime 2},5c_{12}^{\prime 1}+c_{22}^{\prime 2}=4c_{21}^{\prime 1}\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{06}^{\alpha}&\not\to&\mathrm{A}_{06}^{-1},\mathrm{A}_{08}^{-2}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 1},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},-\alpha c_{11}^{\prime 1}=c_{12}^{\prime 2},\\\ -(\alpha+1)c_{11}^{\prime 1}=c_{21}^{\prime 2},-\alpha c_{21}^{\prime 1}=c_{22}^{\prime 2}\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{08}^{\alpha}&\not\to&\mathrm{A}_{06}^{-1},\mathrm{A}_{08}^{-2}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 1},c_{12}^{\prime 2},c_{21}^{\prime 1},c_{21}^{\prime 2},c_{22}^{\prime 2}\in\mathbb{C},\\\ (2\alpha+1)c_{21}^{\prime 2}+c_{11}^{\prime 1}=0,(\alpha-1)c_{21}^{\prime 1}=\alpha c_{22}^{\prime 2},\\\ c_{12}^{\prime 1}+c_{22}^{\prime 2}=2c_{21}^{\prime 1}\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{05}&\not\to&\mathrm{A}_{02},\mathrm{A}_{03},\mathrm{A}_{06}^{\alpha\neq 0},\mathrm{A}_{08}^{\alpha\neq 0}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{\prime 1},c_{11}^{\prime 2},c_{21}^{\prime 1},c_{21}^{\prime 2},\in\mathbb{C},c_{21}^{\prime 2}=-c_{11}^{\prime 1}\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{12}^{\alpha,\beta}&\not\to&\mathrm{A}_{04},\mathrm{A}_{06}^{-1},\mathrm{A}_{08}^{-2},\mathrm{A}_{18}^{\gamma\neq 2\beta-\alpha}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 1}\in\mathbb{C},\\\ c_{12}^{2}=c_{11}^{1},c_{21}^{2}=c_{11}^{1},c_{12}^{\prime 2}=\alpha c_{11}^{1},c_{21}^{\prime 2}=\beta c_{11}^{1},\\\ c_{11}^{\prime 2}=-(\alpha-2\beta)c_{11}^{2},c_{11}^{\prime 1}=c_{11}^{1}(2\alpha-\beta)\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{14}^{\alpha}&\not\to&\mathrm{A}_{04},\mathrm{A}_{06}^{-1},\mathrm{A}_{08}^{-2},\mathrm{A}_{18}^{\beta\neq\alpha}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2}\in\mathbb{C},c_{11}^{\prime 1}=\alpha c_{11}^{1},c_{11}^{\prime 2}=\alpha c_{11}^{2}\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{16}^{\alpha}&\not\to&\mathrm{A}_{04},\mathrm{A}_{06}^{-1},\mathrm{A}_{18}^{\beta\neq\alpha}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{11}^{\prime 1}=\frac{3}{2}c_{12}^{\prime 2},\\\ c_{21}^{\prime 2}=\frac{1}{2}c_{12}^{\prime 2},\alpha c_{11}^{2}=c_{11}^{\prime 2}\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{10}^{\alpha,\beta}&\not\to&\Big{\\{}\begin{array}[]{l}\mathrm{A}_{02},\mathrm{A}_{03},\mathrm{A}_{06}^{\gamma},\mathrm{A}_{08}^{\gamma},\\\ \mathrm{A}_{12}^{\gamma,\delta}[\text{Not}(\alpha=\beta=\gamma=\delta)],\\\ \mathrm{A}_{14}^{\gamma}[\alpha\neq\gamma\textrm{ and }\beta\neq\gamma]\end{array}\Big{\\}}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{ij}^{2},c_{11}^{\prime 1},c_{ij}^{\prime 2}\in\mathbb{C}\mbox{ for }1\leq i,j\leq 2\\\ \alpha c_{11}^{1}=c_{11}^{\prime 1},c_{12}^{\prime 2}=c_{21}^{\prime 2},\beta c_{22}^{2}=c_{22}^{\prime 2},\\\ c_{12}^{2}=c_{21}^{2},\beta c_{12}^{2}=c_{12}^{\prime 2}\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{10}^{\alpha,\alpha}&\not\to&\Big{\\{}\begin{array}[]{l}\mathrm{A}_{04},\mathrm{A}_{06}^{-1},\mathrm{A}_{08}^{-2},\mathrm{A}_{18}^{\beta\neq\alpha},\\\ \mathrm{A}_{12}^{\beta,\gamma}[\text{Not}(\alpha=\beta=\gamma)]\end{array}\Big{\\}}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{ij}^{2},c_{11}^{\prime 1},c_{ij}^{\prime 2}\in\mathbb{C}\mbox{ for }1\leq i,j\leq 2\\\ \alpha c_{11}^{1}=c_{11}^{\prime 1},\alpha c_{11}^{2}=c_{11}^{\prime 2},\alpha c_{12}^{2}=c_{12}^{\prime 2},\\\ \alpha c_{21}^{2}=c_{21}^{\prime 2},\alpha c_{22}^{2}=c_{22}^{\prime 2},c_{12}^{2}=c_{21}^{2}\end{array}\right\\}$
$\begin{array}[]{cccc}\mathrm{A}_{11}^{\alpha,\beta}&\not\to&\Big{\\{}\begin{array}[]{l}\mathrm{A}_{02},\mathrm{A}_{03},\mathrm{A}_{06}^{\gamma},\mathrm{A}_{14}^{\gamma},\mathrm{A}_{08}^{\gamma},\\\ \mathrm{A}_{12}^{\gamma,\delta}[\text{Not}(\alpha=\gamma\textrm{ and }\beta=\delta)]\end{array}\Big{\\}}&\end{array}$ | ${\mathcal{R}}=\left\\{\begin{array}[]{l}c_{11}^{1},c_{11}^{2},c_{12}^{2},c_{21}^{2},c_{11}^{\prime 1},c_{11}^{\prime 2},c_{12}^{\prime 2},c_{21}^{\prime 2}\in\mathbb{C},\\\ c_{11}^{1}=c_{12}^{2},c_{21}^{2}=c_{11}^{1},c_{11}^{\prime 1}=c_{11}^{1}(2\alpha-\beta),\\\ c_{12}^{\prime 2}=\alpha c_{11}^{1},c_{21}^{\prime 2}=\beta c_{11}^{1}\end{array}\right\\}$