# Conservative algebras of $2$-dimensional algebras, IV

Amir Fernández Ouaridi  A. Fernández Ouaridi: CMUC, Universidade de Coimbra,
Coimbra, Portugal; Universidad de Cádiz, Cádiz, Spain.
<EMAIL_ADDRESS>,  Ivan Kaygorodov  I. Kaygorodov: CMA-UBI,
Universidade da Beira Interior, Covilhã, Portugal; Moscow Center for
Fundamental and Applied Mathematics, Moscow, Russia; Saint Petersburg State
University, Russia<EMAIL_ADDRESS>and  Cándido Martín González
C. Martín: Departamento de Álgebra Geometría y Topología, Facultad de
Ciencias, Universidad de Málaga, Campus de Teatinos s/n. 29071 Málaga. Spain.
<EMAIL_ADDRESS>
###### Abstract.

The notion of conservative algebras appeared in a paper of Kantor in 1972.
Later, he defined the conservative algebra $W(n)$ of all algebras (i.e.
bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra
$W(n)$ does not belong to any well-known class of algebras (such as
associative, Lie, Jordan, or Leibniz algebras). It looks like that $W(n)$ in
the theory of conservative algebras plays a similar role with the role of
$\mathfrak{gl}_{n}$ in the theory of Lie algebras. Namely, an arbitrary
conservative algebra can be obtained from a universal algebra $W(n)$ for some
$n\in\mathbb{N}.$ The present paper is a part of a series of papers, which
dedicated to the study of the algebra $W(2)$ and its principal subalgebras.

Keywords: bilinear maps, conservative algebra, contraction, identities.

MSC2020:  17A30111Corresponding author: Ivan Kaygorodov
<EMAIL_ADDRESS>
## Introduction

A multiplication on a vector space $W$ is a bilinear mapping $W\times W\to W$.
We denote by $(W,P)$ the algebra with underlining space $W$ and multiplication
$P$. Given a vector space $W$, a linear mapping ${\bf A}:W\rightarrow W$, and
a bilinear mapping ${\bf B}:W\times W\to W$, we can define a multiplication
$[{\bf A},{\bf B}]:W\times W\to W$ by the formula

$[{\bf A},{\bf B}](x,y)={\bf A}({\bf B}(x,y))-{\bf B}({\bf A}(x),y)-{\bf
B}(x,{\bf A}(y))$

for $x,y\in W$. For an algebra ${\bf A}$ with a multiplication $P$ and
$x\in{\bf A}$ we denote by $L_{x}^{P}$ the operator of left multiplication by
$x$. If the multiplication $P$ is fixed, we write $L_{x}$ instead of
$L_{x}^{P}$.

In 1990 Kantor [14] defined the multiplication $\cdot$ on the set of all
algebras (i.e. all multiplications) on the $n$-dimensional vector space
$V_{n}$ as follows:

${\bf A}\cdot{\bf B}=[L_{e}^{\bf A},{\bf B}],$

where ${\bf A}$ and ${\bf B}$ are multiplications and $e\in V_{n}$ is some
fixed vector. Let $W(n)$ denote the algebra of all algebra structures on
$V_{n}$ with multiplication defined above. If $n>1$, then the algebra $W(n)$
does not belong to any well-known class of algebras (such as associative, Lie,
Jordan, or Leibniz algebras). The algebra $W(n)$ turns out to be a
conservative algebra (see below).

In 1972 Kantor [11] introduced conservative algebras as a generalization of
Jordan algebras (also, see a good written survey about the study of
conservative algebras and superalgebras [25]). Namely, an algebra ${\bf
A}=(W,P)$ is called a conservative algebra if there is a new multiplication
$F:W\times W\rightarrow W$ such that

(1) $\displaystyle[L_{b}^{P},[L_{a}^{P},P]]=-[L_{F(a,b)}^{P},P]$

for all $a,b\in W$. In other words, the following identity holds for all
$a,b,x,y\in W$:

(2) $b(a(xy)-(ax)y-x(ay))-a((bx)y)+(a(bx))y+(bx)(ay)\\\
-a(x(by))+(ax)(by)+x(a(by))=-F(a,b)(xy)+(F(a,b)x)y+x(F(a,b)y).$

The algebra $(W,F)$ is called an algebra associated to ${\bf A}$. The main
subclass of conservative algebras is the variety of terminal algebras, which
defined by the identity (2) with $F(a,b)=\frac{1}{3}(2ab+ba).$ It includes the
varieties of Leibniz and Jordan algebras as subvarieties.

Let us recall some well-known results about conservative algebras. In [11]
Kantor classified all simple conservative algebras and triple systems of
second-order and defined the class of terminal algebras as algebras satisfying
some certain identity. He proved that every terminal algebra is a conservative
algebra and classified all simple finite-dimensional terminal algebras with
left quasi-unit over an algebraically closed field of characteristic zero
[12]. Terminal trilinear operations were studied in [13]. After that,
Cantarini and Kac classified simple finite-dimensional (and linearly compact)
super-commutative and super-anticommutative conservative superalgebras and
some generalization of these algebras (also known as “rigid” or quasi-
conservative superalgebras) over an algebraically closed field of
characteristic zero [3]. The classification of all $2$-dimensional
conservative and rigid (in sense of Kac-Cantarini) algebras is given in [2];
and also, the algebraic and geometric classification of nilpotent low
dimensional terminal algebras is given in [16, 17].

The algebra $W(n)$ plays a similar role in the theory of conservative algebras
as the Lie algebra of all $n\times n$ matrices $\mathfrak{gl}_{n}$ plays in
the theory of Lie algebras. Namely, in [14] Kantor considered the category
$\mathscr{S}_{n}$ whose objects are conservative algebras of non-Jacobi
dimension $n$. It was proven that the algebra $W(n)$ is the universal
attracting object in this category, i.e., for every $M\in\mathscr{S}_{n}$
there exists a canonical homomorphism from $M$ into the algebra $W(n)$. In
particular, all Jordan algebras of dimension $n$ with unity are contained in
the algebra $W(n)$. The same statement also holds for all noncommutative
Jordan algebras of dimension $n$ with unity. Some properties of the product in
the algebra $W(n)$ were studied in [5, 15]. The universal conservative
superalgebra was constructed in [20]. The study of low dimensional
conservative algebras was started in [18]. The study of properties of
$2$-dimensional algebras is also one of popular topic in non-associative
algebras (see, for example, [6, 22, 23, 4, 24]) and as we can see the study of
properties of the algebra $W(2)$ could give some applications on the theory of
$2$-dimensional algebras. So, from the description of idempotents of the
algebra $W(2)$ it was received an algebraic classification of all
$2$-dimensional algebras with left quasi-unit [21]. Derivations and
subalgebras of codimension 1 of the algebra $W(2)$ and of its principal
subalgebras $W_{2}$ and $S_{2}$ were described [18]. Later, the automorphisms,
one-sided ideals, idempotents, local (and $2$-local) derivations and
automorphisms of $W(2)$ and its principal subalgebras were described in [1,
21]. Note that $W_{2}$ and $S_{2}$ are simple terminal algebras with left
quasi-unit from the classification of Kantor [12]. The present paper is
devoted to continuing the study of properties of $W(2)$ and its principal
subalgebras. Throughout this paper, unless stated otherwise, $\mathbb{F}$
denotes a field of characteristic zero. All algebras are defined over
$\mathbb{F}$.

The multiplication table of $W(2)$ is given by the following table.

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $2e_{1}$ | $e_{3}$ | $0$ | $-e_{5}$ | $e_{8}$ | $0$
$e_{3}$ | $-2e_{3}$ | $-e_{1}$ | $-3e_{4}$ | $0$ | $e_{6}$ | $0$ | $0$ | $-e_{7}$
$e_{4}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{5}$ | $-2e_{1}$ | $-3e_{2}$ | $-e_{3}$ | $0$ | $-2e_{5}$ | $-e_{6}$ | $-e_{7}$ | $-2e_{8}$
$e_{6}$ | $2e_{3}$ | $e_{1}$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{7}$ | $2e_{3}$ | $e_{1}$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{8}$ | $0$ | $e_{2}$ | $-e_{3}$ | $-2e_{4}$ | $0$ | $-e_{6}$ | $-e_{7}$ | $0$

##  1\. Inönü-Wigner contractions of $W(2)$ and its subalgebras

The class of conservative algebras includes the variety of terminal algebras,
which includes all Leibniz and Jordan algebras. On the other hand, the variety
of terminal algebras is ”dual” to the variety of commutative algebras (in the
sense of generalized TTK-functor). The algebra $W(2)$ is not terminal, but its
principal subalgebras $W_{2}$ and $S_{2}$ are terminal. Our main aim is to try
to understand how the algebra $W(2)$ ”far” from terminal algebras. For a
particular answer for our question, we will consider contractions to some
certain subalgebras of $W(2)$ and study its relations with the variety of
terminal algebras. The standard Inönü-Wigner contraction was introduced in
[8]. We will call it IW contraction for short.

######  Definition 1.

Let $\mu,\chi$ represent algebras ${\bf A}$ and ${\bf B}$ respectively defined
on a vector space $V$. Suppose that there are some elements $E_{i}^{t}\in V$
$(1\leq i\leq n$, $t\in{\mathbb{F}}^{*})$ such that
$E^{t}=(E_{1}^{t},\dots,E_{n}^{t})$ is a basis of $V$ for any
$t\in{\mathbb{F}}^{*}$ and the structure constants of $\mu$ in this basis are
$\mu_{i,j}^{k}(t)$ for some polynomials $\mu_{i,j}^{k}(t)\in{\mathbb{F}}[t]$.
If $\mu_{i,j}^{k}(0)=\chi_{i,j}^{k}$ for all $1\leq i,j,k\leq n$, then ${\bf
A}\to{\bf B}$. To emphasize that the parametrized basis
$E^{t}=(E_{1}^{t},\dots,E_{n}^{t})$ $(t\in{\mathbb{F}}^{*})$ gives a
degeneration between the algebras represented by the structures $\mu$ and
$\chi$, we will write $\mu\xrightarrow{E^{t}}\chi$. Suppose that ${\bf A}_{0}$
is an $(n-m)$-dimensional subalgebra of the $n$-dimensional algebra ${\bf A}$
and $\mu$ is a structure representing ${\bf A}$ such that ${\bf A}_{0}$
corresponds to the subspace $\langle e_{m+1},\dots,e_{n}\rangle$ of $V$. Then
$\mu\xrightarrow{(te_{1},\dots,te_{m},e_{m+1},\dots,e_{n})}\chi$ for some
$\chi$ and the algebra ${\bf B}$ represented by $\chi$ is called the IW
contraction of ${\bf A}$ with respect to ${\bf A}_{0}$.

###  1.1. IW contraction of $W(2)$

####  1.1.1. The algebra $\overline{W(2)}$

The description of all subalgebras of codimension $1$ for the algebra $W(2)$
is given in [18]. Namely, $W(2)$ has only one $7$-dimensional subalgebra. It
is generated by elements $e_{1},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8},$ and it
is terminal. Let us consider the IW contraction
$W(2)\xrightarrow{(e_{1},te_{2},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8})}\overline{W(2)}.$
It is easy to see, that the multiplication table of $\overline{W(2)}$ is given
by the following table.

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{3}$ | $-2e_{3}$ | $0$ | $-3e_{4}$ | $0$ | $e_{6}$ | $0$ | $0$ | $-e_{7}$
$e_{4}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{5}$ | $-2e_{1}$ | $-3e_{2}$ | $-e_{3}$ | $0$ | $-2e_{5}$ | $-e_{6}$ | $-e_{7}$ | $-2e_{8}$
$e_{6}$ | $2e_{3}$ | $0$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{7}$ | $2e_{3}$ | $0$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{8}$ | $0$ | $e_{2}$ | $-e_{3}$ | $-2e_{4}$ | $0$ | $-e_{6}$ | $-e_{7}$ | $0$

After a carefully checking of the dimension of the algebra of derivation of
$\overline{W(2)},$ we have $\dim\mathfrak{Der}(\overline{W(2)})=3.$ Since
$\dim\mathfrak{Der}(W(2))=2,$ it follows that the degeneration
$W(2)\to\overline{W(2)}$ is primary, that is, there is no algebra ${\bf A}$
such that ${W(2)}\to{\bf A}$ and ${\bf A}\to\overline{W(2)}$, where ${\bf A}$
is neither isomorphic to $W(2)$ or $\overline{W(2)}$ (see [9]).

######  Lemma 2.

The algebra $\overline{W(2)}$ is a non-terminal conservative non-simple
algebra.

###### Proof.

The subspace $\langle e_{2},e_{3},e_{4},e_{6},e_{7},e_{8}\rangle$ gives a
$6$-dimensional ideal, it gives that $\overline{W(2)}$ is non-simple. The non-
terminal property is following from the direct verification of the terminal
identity (for example, using a modification of the Wolfram code presented in
[10]). The conservative property is following from the direct verification of
the conservative identity with the additional multiplication $*:$

$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{1}*e_{5}=-2e_{1}$ | $e_{2}*e_{1}=e_{2}$ | $e_{2}*e_{5}=-e_{2}$
---|---|---|---|---
$e_{2}*e_{8}=-e_{2}$ | $e_{5}*e_{1}=-2e_{1}$ | $e_{5}*e_{2}=-2e_{2}$ | $e_{5}*e_{5}=-4e_{1}.$ |

∎

######  Lemma 3.

Let $\mathscr{S}$ be a subalgebra of $\overline{W(2)}$ of codimension $1,$ the
$\mathscr{S}$ is one of the following conservative subalgebras

${\mathscr{S}}_{1}=\langle e_{2},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8}\rangle$ | ${\mathscr{S}}_{2}=\langle e_{1},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8}\rangle$
---|---
${\mathscr{S}}_{5}=\langle e_{1},e_{2},e_{3},e_{4},e_{6},e_{7},e_{8}\rangle$ | ${\mathscr{S}}_{\alpha,\beta}=\langle e_{1}+\alpha e_{8},e_{2},e_{3},e_{4},e_{5}+\beta e_{8},e_{6},e_{7}\rangle_{\alpha,\beta\in{\mathbb{F}}},$

where only ${\mathscr{S}}_{1},$ ${\mathscr{S}}_{2}$, ${\mathscr{S}}_{0,0}$ and
${\mathscr{S}}_{-1,1}$ are terminal.

###### Proof.

Let $\mathscr{S}$ be generated by the following set
$\\{e_{1},\ldots,\widehat{e_{i}},\ldots,e_{8}\\}.$ By some easy verification
of $8$ possibilities, we have that there are only $4$ subalgebras of this
type: for $i=1,2,5,8.$

Let us consider the situation when $\mathscr{S}$ is generated by seven vectors
of the following type:
$\\{\sum\alpha_{i1}e_{i},\ldots,\sum\alpha_{i7}e_{i}\\}.$ By some linear
combinations, we can reduce this basis to a basis considered above, or a basis
of the following type:
$\\{e_{1}+\alpha_{1}e_{8},\ldots,e_{7}+\alpha_{1}e_{8}\\}.$ It is easy to see,
that

$(e_{2}+\alpha_{2}e_{8})^{2}=\alpha_{2}e_{2}\in{\mathscr{S}}$ | $(e_{3}+\alpha_{3}e_{8})^{2}=-3e_{4}-\alpha_{3}(e_{3}+e_{7})\in{\mathscr{S}}$
---|---
$(e_{4}+\alpha_{4}e_{8})^{2}=-2\alpha_{4}e_{4}\in{\mathscr{S}}$ | $(e_{6}+\alpha_{6}e_{8})^{2}=-\alpha_{6}(e_{6}-e_{7})\in{\mathscr{S}},$

which gives that $e_{2},e_{4}\in{\mathscr{S}}$ and there are four cases:

I. $e_{3},e_{6}\in{\mathscr{S}}$ | II. $e_{3},e_{6}-e_{7}\in{\mathscr{S}}$
---|---
III. $e_{3}+e_{7},e_{6}\in{\mathscr{S}}$ | IV. $e_{3}+e_{7},e_{6}-e_{7}\in{\mathscr{S}}$

Analysing all these cases, we have that ${\mathscr{S}}$ is a subalgebra
considered above, or it has the following basis $\langle e_{1}+\alpha
e_{8},e_{2},e_{3},e_{4},e_{5}+\beta
e_{8},e_{6},e_{7}\rangle_{\alpha,\beta\in{\mathbb{F}}}.$

The conservative property of the subalgebra ${\mathscr{S}}_{5}$ is following
from the direct verification of the conservative identity with the additional
multiplication $*:$

$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{2}*e_{1}=e_{2}$ | $e_{2}*e_{8}=-e_{2}$
---|---|---|---

Let us give the multiplication table of $\overline{W(2)}$ in more useful way
(here the subalgebra $\langle e_{1},\ldots,e_{7}\rangle$ gives
${\mathscr{S}}_{\alpha,\beta}$):

$e_{1}e_{1}=-e_{1}$ | $e_{1}e_{2}=(-3+\alpha)e_{2}$ | $e_{1}e_{3}=(1-\alpha)e_{3}$ | $e_{1}e_{4}=(3-2\alpha)e_{4}$
---|---|---|---
$e_{1}e_{5}=-e_{5}$ | $e_{1}e_{6}=(1-\alpha)e_{6}$ | $e_{1}e_{7}=(1-\alpha)e_{7}$ | $e_{1}e_{8}=-e_{8}$
$e_{2}e_{1}=3e_{2}$ | $e_{3}e_{1}=-2e_{3}-\alpha e_{7}$ | $e_{3}e_{3}=-3e_{4}$ | $e_{3}e_{5}=e_{6}-\beta e_{7}$
$e_{3}e_{8}=-e_{7}$ | $e_{5}e_{1}=-2e_{1}$ | $e_{5}e_{2}=(-3+\beta)e_{2}$ | $e_{5}e_{3}=(-1-\beta)e_{3}$
$e_{5}e_{4}=-2\beta e_{4}$ | $e_{5}e_{5}=-2e_{5}$ | $e_{5}e_{6}=(-1-\beta)e_{6}$ | $e_{5}e_{7}=(-1-\beta)e_{7}$
$e_{5}e_{8}=-2e_{8}$ | $e_{6}e_{1}=2e_{3}+\alpha e_{7}$ | $e_{6}e_{3}=3e_{4}$ | $e_{6}e_{5}=-e_{6}+\beta e_{7}$
$e_{6}e_{8}=e_{7}$ | $e_{7}e_{1}=2e_{3}+\alpha e_{7}$ | $e_{7}e_{3}=3e_{4}$ | $e_{7}e_{5}=-e_{6}+\beta e_{7}$
$e_{7}e_{8}=e_{7}$ | $e_{8}e_{2}=e_{2}$ | $e_{8}e_{3}=-e_{3}$ | $e_{8}e_{4}=-2e_{4}$
| $e_{8}e_{6}=-e_{6}$ | $e_{8}e_{7}=-e_{7}$ |

The conservative property of the subalgebra ${\mathscr{S}}_{\alpha,\beta}$ is
following from the direct verification of the conservative identity with the
additional multiplication $*:$

$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{1}*e_{5}=-2e_{1}$ | $e_{2}*e_{1}=(1-\alpha)e_{2}$
---|---|---|---
$\par e_{2}*e_{5}=(-1-\beta)e_{2}$ | $e_{5}*e_{1}=-2e_{1}$ | $e_{5}*e_{2}=-2e_{2}$ | $e_{5}*e_{5}=-4e_{1}$

∎

####  1.1.2. Algebras $\overline{{\mathscr{S}}}$

In the present subsection, we have to talk about contractions of the algebra
$\overline{W(2)}$ to its subalgebra of codimension $1.$

$\bullet$
$\overline{W(2)}\xrightarrow{(te_{1},{\mathscr{S}}_{1})}\overline{{\mathscr{S}}_{1}}.$
It is easy to see that the multiplication of $\overline{{\mathscr{S}}_{1}}$ is
given by the following table.

$e_{3}e_{3}=-3e_{4}$ | $e_{3}e_{5}=e_{6}$ | $e_{3}e_{8}=-e_{7}$ | $e_{5}e_{1}=-2e_{1}$ | $e_{5}e_{2}=-3e_{2}$ | $e_{5}e_{3}=-e_{3}$ | $e_{5}e_{5}=-2e_{5}$
---|---|---|---|---|---|---
$e_{5}e_{6}=-e_{6}$ | $e_{5}e_{7}=-e_{7}$ | $e_{5}e_{8}=-2e_{8}$ | $e_{6}e_{3}=3e_{4}$ | $e_{6}e_{5}=-e_{6}$ | $e_{6}e_{8}=e_{7}$ | $e_{7}e_{3}=3e_{4}$
$e_{7}e_{5}=-e_{6}$ | $e_{7}e_{8}=e_{7}$ | $e_{8}e_{2}=e_{2}$ | $e_{8}e_{3}=-e_{3}$ | $e_{8}e_{4}=-2e_{4}$ | $e_{8}e_{6}=-e_{6}$ | $e_{8}e_{7}=-e_{7}$

######  Lemma 4.

The algebra $\overline{{\mathscr{S}}_{1}}$ is terminal.

$\bullet$
$\overline{W(2)}\xrightarrow{(te_{5},{\mathscr{S}}_{5})}\overline{{\mathscr{S}}_{5}}.$
It is easy to see that the multiplication of $\overline{{\mathscr{S}}_{5}}$ is
given by the following table.

$e_{1}e_{1}=-e_{1}$ | $e_{1}e_{2}=-3e_{2}$ | $e_{1}e_{3}=e_{3}$ | $e_{1}e_{4}=3e_{4}$ | $e_{1}e_{5}=-e_{5}$ | $e_{1}e_{6}=e_{6}$
---|---|---|---|---|---
$e_{1}e_{7}=e_{7}$ | $e_{1}e_{8}=-e_{8}$ | $e_{2}e_{1}=3e_{2}$ | $e_{3}e_{1}=-2e_{3}$ | $e_{3}e_{3}=-3e_{4}$ | $e_{3}e_{8}=-e_{7}$
$\par e_{6}e_{1}=2e_{3}$ | $e_{6}e_{3}=3e_{4}$ | $e_{6}e_{8}=e_{7}$ | $e_{7}e_{1}=2e_{3}$ | $e_{7}e_{3}=3e_{4}$ | $e_{7}e_{8}=e_{7}$
$\par e_{8}e_{2}=e_{2}$ | $e_{8}e_{3}=-e_{3}$ | $e_{8}e_{4}=-2e_{4}$ | $e_{8}e_{6}=-e_{6}$ | $e_{8}e_{7}=-e_{7}$ |

######  Lemma 5.

The algebra $\overline{{\mathscr{S}}_{5}}$ is a non-terminal conservative
algebra.

###### Proof.

The conservative property of the algebra $\overline{{\mathscr{S}}_{5}}$ is
following from the direct verification of the conservative identity with the
additional multiplication $*:$

$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{2}*e_{1}=e_{2}$ | $e_{2}*e_{8}=-e_{2}$
---|---|---|---

∎

$\bullet$
$\overline{W(2)}\xrightarrow{(te_{8},{\mathscr{S}}_{\alpha,\beta})}\overline{{\mathscr{S}}_{\alpha,\beta}}.$
It is easy to see that the multiplication of
$\overline{{\mathscr{S}}_{\alpha,\beta}}$ is given by the following table.

$e_{1}e_{1}=-e_{1}$ | $e_{1}e_{2}=(-3+\alpha)e_{2}$ | $e_{1}e_{3}=(1-\alpha)e_{3}$ | $e_{1}e_{4}=(3-2\alpha)e_{4}$
---|---|---|---
$\par e_{1}e_{5}=-e_{5}$ | $e_{1}e_{6}=(1-\alpha)e_{6}$ | $e_{1}e_{7}=(1-\alpha)e_{7}$ | $e_{1}e_{8}=-e_{8}$
$\par e_{2}e_{1}=3e_{2}$ | $e_{3}e_{1}=-2e_{3}-\alpha e_{7}$ | $e_{3}e_{3}=-3e_{4}$ | $e_{3}e_{5}=e_{6}-\beta e_{7}$
$\par e_{5}e_{1}=-2e_{1}$ | $e_{5}e_{2}=(-3+\beta)e_{2}$ | $e_{5}e_{3}=(-1-\beta)e_{3}$ | $e_{5}e_{4}=-2\beta e_{4}$
$\par e_{5}e_{5}=-2e_{5}$ | $e_{5}e_{6}=(-1-\beta)e_{6}$ | $e_{5}e_{7}=(-1-\beta)e_{7}$ | $e_{5}e_{8}=-2e_{8}$
$\par e_{6}e_{1}=2e_{3}+\alpha e_{7}$ | $e_{6}e_{3}=3e_{4}$ | $e_{6}e_{5}=-e_{6}+\beta e_{7}$ | $e_{7}e_{1}=2e_{3}+\alpha e_{7}$
| $\par e_{7}e_{3}=3e_{4}$ | $e_{7}e_{5}=-e_{6}+\beta e_{7}$ |

######  Lemma 6.

The algebra $\overline{{\mathscr{S}}_{\alpha,\beta}}$ is a conservative
algebra; and it is a terminal algebra if and only if $(\alpha,\beta)=(-1,1)$
or $(\alpha,\beta)=(0,0)$.

###### Proof.

The conservative property of the algebra
$\overline{{\mathscr{S}}_{\alpha,\beta}}$ is following from the direct
verification of the conservative identity with the additional multiplication
$*:$

$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{1}*e_{5}=-2e_{1}$ | $e_{2}*e_{1}=(1-\alpha)e_{2}$
---|---|---|---
$\par e_{2}*e_{5}=(-1-\beta)e_{2}$ | $e_{5}*e_{1}=-2e_{1}$ | $e_{5}*e_{2}=-2e_{2}$ | $e_{5}*e_{5}=-4e_{1}$

∎

####  1.1.3. The algebra $\widehat{W(2)}$

The second interesting ”big” subalgebra of $W(2)$ is $W_{2},$ which is
generated by $e_{1},\ldots,e_{6}.$ Let us consider the IW contraction
$W(2)\xrightarrow{(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},te_{7},te_{8})}{\widehat{W(2)}}.$
It is easy to see, that the multiplication table (for nonzero products) of
${\widehat{W(2)}}$ is given by the following table.

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $2e_{1}$ | $e_{3}$ | $0$ | $-e_{5}$ | $e_{8}$ | $0$
$e_{3}$ | $-2e_{3}$ | $-e_{1}$ | $-3e_{4}$ | $0$ | $e_{6}$ | $0$ | $0$ | $-e_{7}$
$e_{4}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{5}$ | $-2e_{1}$ | $-3e_{2}$ | $-e_{3}$ | $0$ | $-2e_{5}$ | $-e_{6}$ | $-e_{7}$ | $-2e_{8}$
$e_{6}$ | $2e_{3}$ | $e_{1}$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$

######  Lemma 7.

The algebra ${\widehat{W(2)}}$ is terminal.

####  1.1.4. The algebra $\widehat{\widehat{W(2)}}$

The next interesting subalgebra of $W(2)$ is $S_{2},$ which is generated by
$e_{1},\ldots,e_{4}.$ Let us consider the IW contraction
$W(2)\xrightarrow{(e_{1},e_{2},e_{3},e_{4},te_{5},te_{6},te_{7},te_{8})}{\widehat{\widehat{W(2)}}}.$
It is easy to see, that the multiplication table (for nonzero products) of
${\widehat{\widehat{W(2)}}}$ is given by the following table.

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $2e_{1}$ | $e_{3}$ | $0$ | $-e_{5}$ | $e_{8}$ | $0$
$e_{3}$ | $-2e_{3}$ | $-e_{1}$ | $-3e_{4}$ | $0$ | $e_{6}$ | $0$ | $0$ | $-e_{7}$

######  Corollary 8.

The algebra ${\widehat{\widehat{W(2)}}}$ is terminal.

####  1.1.5. The algebra $\widetilde{W(2)}$

The next interesting subalgebra of $W(2)$ is generated by $e_{1}$ and $e_{2}.$
Let us consider the IW contraction
$W(2)\xrightarrow{(e_{1},e_{2},te_{3},te_{4},te_{5},te_{6},te_{7},te_{8})}{\widetilde{W(2)}}.$
It is easy to see, that the multiplication table (for nonzero products) of
${\widetilde{W(2)}}$ is given by the following table.

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $0$ | $e_{3}$ | $0$ | $-e_{5}$ | $e_{8}$ | $0$
$e_{3}$ | $-2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{6}$ | $2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{7}$ | $2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$

######  Lemma 9.

The algebra ${\widetilde{W(2)}}$ is a non-Leibniz, non-Jordan terminal
algebra.

####  1.1.6. The algebra $\widetilde{\widetilde{W(2)}}$

The last interesting subalgebra of $W(2)$ is generated by $e_{1}.$ Let us
consider the IW contraction
$W(2)\xrightarrow{(e_{1},te_{2},te_{3},te_{4},te_{5},te_{6},te_{7},te_{8})}\widetilde{\widetilde{W(2)}}.$
It is easy to see, that the multiplication table (for nonzero products) of
$\widetilde{\widetilde{W(2)}}$ is given by the following table.

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{3}$ | $-2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{6}$ | $2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{7}$ | $2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$

######  Lemma 10.

The algebra $\widetilde{\widetilde{W(2)}}$ is a non-Leibniz, non-Jordan
terminal algebra.

###  1.2. IW contraction of $S_{2}$ and $W_{2}$

Thanks to [18], algebras $S_{2}$ and $W_{2}$ have also only one subalgebra of
codimension $1,$ which are $\langle e_{1},e_{3},e_{4}\rangle$ and $\langle
e_{1},e_{3},e_{4},e_{5},e_{6}\rangle.$ Other important subalgebras of $S_{2}$
and $W_{2}$ are $\langle e_{1}\rangle,$ $\langle e_{1},e_{2}\rangle$ and
$\langle e_{1},e_{2},e_{3},e_{4}\rangle.$ All contractions of $S_{2}$ (and
$W_{2}$) with respected to all cited subalgebras can be obtained as
$4$-dimensional subalgebras $\langle e_{1},e_{2},e_{3},e_{4}\rangle$
($6$-dimensional subalgebras $\langle e_{1},\ldots,e_{6}\rangle$) of the
following algebras $\overline{W(2)},$ ${\widehat{\widehat{W(2)}}},$
${\widetilde{W(2)}}$ and $\widetilde{\widetilde{W(2)}}.$ All these subalgebras
are non-Leibniz, non-Jordan terminal algebras.

##  2\. Varieties related to $W(2)$ and its subalgebras

###  2.1. Identities

Let ${\mathbb{F}}$ be a field of characteristic zero and ${\mathbb{F}}\langle
x_{1},x_{2},\ldots,x_{n}\rangle$ the free nonassociative
${\mathbb{F}}$-algebra in $n$ indeterminates. Let ${\bf A}$ be any algebra and
${\mathfrak{S}}^{n}_{\bf A}$ the subspace of ${\mathbb{F}}\langle
x_{1},x_{2},\ldots,x_{n}\rangle$ of all $n$-linear
$w(x_{1},x_{2},\ldots,x_{n})$ which vanish on ${\bf A}$ (so
$w(x_{1},x_{2},\ldots,x_{n})$ is of degree one in each variable). We have
studied by direct verification the subspace ${\mathfrak{S}}^{n}_{\bf A}$ for
$n=3,4$ of $W(2)$ and of its IW contractions mentioned in this paper.

######  Proposition 11.

In the following table, we summarize the dimension of the subspaces
${\mathfrak{S}}^{n}_{\bf A}$ for ${\bf
A}\in\\{W(2),\overline{W(2)},\overline{\mathscr{S}_{1}},\overline{\mathscr{S}_{5}},\overline{\mathscr{S}_{-1,1}},\overline{\mathscr{S}_{0,0}},\widehat{W(2)},\widehat{\widehat{W(2)}},{\widetilde{W(2)}}\\}$
and $n=3,4$.

Algebra (${\bf A}$) $\textrm{dim}({\mathfrak{S}}^{3}_{\bf A})$
$\textrm{dim}({\mathfrak{S}}^{4}_{\bf A})$ Comments $W(2)$ 0 0 non-terminal
conservative $\overline{W(2)}$ 0 20 non-terminal, conservative
$\overline{\mathscr{S}_{1}}$ 0 64 terminal $\overline{\mathscr{S}_{5}}$ 0 40
non-terminal, conservative $\overline{\mathscr{S}_{-1,1}}$ 0 64 terminal
$\overline{\mathscr{S}_{0,0}}$ 0 44 terminal $\widehat{W(2)}$ 0 24 terminal
$\widehat{\widehat{W(2)}}$ 0 47 terminal ${\widetilde{W(2)}}$ 0 82 terminal
${\widetilde{\widetilde{W(2)}}}$ 2 101 terminal

Moreover, if ${\bf A}=\overline{\mathscr{S}_{\alpha,\beta}}$ then
$\dim({\mathfrak{S}}^{3}_{\bf A})=0$ for $(\alpha,\beta)\neq(2,1)$ and
$(\alpha,\beta)\neq(0,-3)$.

###### Proof.

We have determined the spaces ${\mathfrak{S}}^{n}_{\bf A}$ for $n=3,4$ by
constructing an arbitrary $n$-linear map $w(x_{1},\ldots,x_{n})$ and solving
$w(x_{1},\ldots,x_{n})=0$ in ${\bf A}$ using Wolfram.

∎

######  Proposition 12.

If ${\bf A}=\overline{\mathscr{S}_{2,1}}$ then $\dim({\mathfrak{S}}^{3}_{\bf
A})=3$ and a basis of the ${\mathbb{F}}$-vector space ${\mathfrak{S}}^{3}_{\bf
A}$ is the set of identities:

  1. (1)

$x_{1}(x_{2}x_{3})-x_{2}(x_{1}x_{3}),$

  2. (2)

$x_{2}(x_{3}x_{1})-x_{3}(x_{2}x_{1}),$

  3. (3)

$x_{3}(x_{1}x_{2})-x_{1}(x_{3}x_{2}).$

Now, consider the following identities:

$\mathfrak{st}^{n}_{1}=\sum\limits_{\sigma\in\mathbb{S}_{n}}(-1)^{\sigma}(\ldots(x_{\sigma(1)}x_{\sigma(2)})x_{\sigma(3)}\ldots)x_{\sigma(n)}\mbox{
and
}\mathfrak{st}^{n}_{2}=\sum\limits_{\sigma\in\mathbb{S}_{n}}(-1)^{\sigma}x_{\sigma(n)}(\ldots
x_{\sigma(3)}(x_{\sigma(2)}x_{\sigma(1)})\ldots).$

It is clear that if an algebra satisfies $\mathfrak{st}^{n}_{1}$ (resp.
$\mathfrak{st}^{n}_{2}$) then it also satisfies $\mathfrak{st}^{n+1}_{1}$
(resp. $\mathfrak{st}^{n+1}_{2}$).

######  Proposition 13.

If ${\bf A}=\overline{\mathscr{S}_{0,-3}}$ then $\dim({\mathfrak{S}}^{3}_{\bf
A})=1$. This ${\mathbb{F}}$-vector space is generated by

$2\mathfrak{st}^{3}_{1}-3\mathfrak{st}^{3}_{2}.$

######  Proposition 14.

If ${\bf A}=\widetilde{\widetilde{W(2)}}$ then $\dim({\mathfrak{S}}^{3}_{\bf
A})=2$ and a basis of the ${\mathbb{F}}$-vector space ${\mathfrak{S}}^{3}_{\bf
A}$ is the set of identities $\mathfrak{st}^{3}_{1},\mathfrak{st}^{3}_{2}$.

We have studied the space ${\mathfrak{S}}^{n}_{\bf A}$ for the subalgebras of
$W(2)$ mentioned in this paper. We have also studied the identities
$\mathfrak{st}^{n}_{1}$ and $\mathfrak{st}^{n}_{2}$ for these subalgebras for
$n=3,4,5$.

######  Proposition 15.

In the following table, we summarize the dimension of the subspaces
${\mathfrak{S}}^{n}_{\bf A}$ for ${\bf A}$ a subalgebra of $W(2)$ and $n=3,4$.

Subalgebra $({\bf A})$ $\textrm{dim}({\mathfrak{S}}^{3}_{\bf A})$
$\textrm{dim}({\mathfrak{S}}^{4}_{\bf A})$ Comments $B_{2}:=\langle
e_{1},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8}\rangle$ 0 64 no
$\mathfrak{st}^{4}_{1}$, no $\mathfrak{st}^{4}_{2}$,
$\mathfrak{st}^{5}_{1},\mathfrak{st}^{5}_{2}$ $W_{2}=\langle
e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\rangle$ 0 24 no $\mathfrak{st}^{5}_{1}$,
no $\mathfrak{st}^{4}_{2},$ $\mathfrak{st}^{5}_{2}$ $C_{2}:=\langle
e_{1},e_{3},e_{4},e_{5},e_{6}\rangle$ 0 64 no $\mathfrak{st}^{4}_{1}$, no
$\mathfrak{st}^{4}_{2}$, $\mathfrak{st}^{5}_{1},$ $\mathfrak{st}^{5}_{2}$
$S_{2}=\langle e_{1},e_{2},e_{3},e_{4}\rangle$ 3 86 no
$\mathfrak{st}^{4}_{1}$, no $\mathfrak{st}^{4}_{2}$, $\mathfrak{st}^{5}_{1},$
$\mathfrak{st}^{5}_{2}$ $D_{2}:=\langle e_{1},e_{3},e_{4}\rangle$ 6 110
$\mathfrak{st}^{3}_{1},\mathfrak{st}^{3}_{2}$ $E_{2}:=\langle
e_{1},e_{2}\rangle$ 8 115 $\mathfrak{st}^{3}_{1},\mathfrak{st}^{3}_{2}$

Moreover, ${\mathfrak{S}}^{4}_{B_{2}}={\mathfrak{S}}^{4}_{C_{2}};$ all present
algebras are terminal, non-Leibniz and non-Jordan.

######  Proposition 16.

If ${\bf A}=S_{2}$, the subalgebra of $W(2)$ generated by
$e_{1},\ldots,e_{4}$, then a basis of the ${\mathbb{F}}$-vector space
${\mathfrak{S}}^{3}_{\bf A}$ is the set of identities:

  1. (1)

$30x_{1}(x_{2}x_{3})-42x_{1}(x_{3}x_{2})-25x_{2}(x_{1}x_{3})+7x_{2}(x_{3}x_{1})+39x_{3}(x_{1}x_{2})-9x_{3}(x_{2}x_{1})+10(x_{1}x_{3})x_{2}+15(x_{2}x_{1})x_{3}-19(x_{3}x_{1})x_{2}-6(x_{3}x_{2})x_{1}$,

  2. (2)

$-5x_{1}(x_{2}x_{3})+11x_{1}(x_{3}x_{2})+5x_{2}(x_{1}x_{3})-11x_{2}(x_{3}x_{1})-12x_{3}(x_{1}x_{2})+12x_{3}(x_{2}x_{1})-5(x_{1}x_{3})x_{2}+5(x_{2}x_{3})x_{1}+2(x_{3}x_{1})x_{2}-2(x_{3}x_{2})x_{1}$,

  3. (3)

$-3x_{1}(x_{2}x_{3})-3x_{1}(x_{3}x_{2})+4x_{2}(x_{1}x_{3})-4x_{2}(x_{3}x_{1})+3x_{3}(x_{1}x_{2})+3x_{3}(x_{2}x_{1})+3(x_{1}x_{2})x_{3}+2(x_{1}x_{3})x_{2}-2(x_{3}x_{1})x_{2}-3(x_{3}x_{2})x_{1}$.

Finally, we have studied the family of identities $\mathfrak{st}^{n}_{1}$ and
$\mathfrak{st}^{n}_{2}$ for $W(2)$ and its contractions.

######  Proposition 17.

In the following table, we summarize which identities from the families
$\mathfrak{st}^{n}_{1}$ and $\mathfrak{st}^{n}_{1}$ are satisfies for every
contraction of $W(2)$, for $n=3,4,5$.

Algebra $\mathfrak{st}^{3}_{1}$ $\mathfrak{st}^{4}_{1}$
$\mathfrak{st}^{5}_{1}$ $\mathfrak{st}^{3}_{2}$ $\mathfrak{st}^{4}_{2}$
$\mathfrak{st}^{5}_{2}$ $W(2)$ ✗ ✗ ✗ ✗ ✗ ✓ $\overline{W(2)}$ ✗ ✗ ✓ ✗ ✗ ✓
$\overline{\mathscr{S}_{1}}$ ✗ ✗ ✓ ✗ ✗ ✓ $\overline{\mathscr{S}_{5}}$ ✗ ✗ ✓ ✗
✗ ✓ $\overline{\mathscr{S}_{\alpha,\beta}}$ ✗ $\text{\char
51}_{\alpha=\frac{3+\beta}{2}}$ ✓ $\text{\char 51}_{(\alpha,\beta)=(2,1)}$
$\text{\char 51}_{\alpha=\frac{3+\beta}{2}}$ ✓ $\widehat{W(2)}$ ✗ ✗ ✗ ✗ ✗ ✓
$\widehat{\widehat{W(2)}}$ ✗ ✗ ✓ ✗ ✗ ✓ ${\widetilde{W(2)}}$ ✗ ✓ ✓ ✗ ✓ ✓

######  Corollary 18.

${\mathfrak{S}}^{4}_{W(n)}=0$ and ${\mathfrak{S}}^{5}_{W(2)}\neq 0$.

The present corollary gives the following question.

######  Open question.

Find minimal $k,$ such that ${\mathfrak{S}}^{k}_{W(n)}\neq 0$. In this case,
is $W(n)$ satisfying $\mathfrak{st}^{k}_{2}$?

###  2.2. Other degree five identities for $W(2)$

In this subsection we are interested in finding other degree five identities
for $W(2)$. Consider the set of free monomials
$w(x_{1},x_{2},x_{3},x_{4},x_{5})$ of degree five up to permutations of the
variables. There are exactly fourteen monomials:

$w_{1}(x_{1},x_{2},x_{3},x_{4},x_{5})=(((x_{1}x_{2})x_{3})x_{4})x_{5}$ | $w_{2}(x_{1},x_{2},x_{3},x_{4},x_{5})=((x_{1}x_{2})x_{3})(x_{4}x_{5})$
---|---
$w_{3}(x_{1},x_{2},x_{3},x_{4},x_{5})=((x_{1}x_{2})(x_{3}x_{4}))x_{5}$ | $w_{4}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}x_{2})((x_{3}x_{4})x_{5})$
$w_{5}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}x_{2})(x_{3}(x_{4}x_{5}))$ | $w_{6}(x_{1},x_{2},x_{3},x_{4},x_{5})=((x_{1}(x_{2}x_{3}))x_{4})x_{5}$
$w_{7}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}(x_{2}x_{3}))(x_{4}x_{5})$ | $w_{8}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}((x_{2}x_{3})x_{4}))x_{5}$
$w_{9}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}(((x_{2}x_{3})x_{4})x_{5})$ | $w_{10}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}((x_{2}x_{3})(x_{4}x_{5}))$
$w_{11}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}(x_{2}(x_{3}x_{4})))x_{5}$ | $w_{12}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}((x_{2}(x_{3}x_{4}))x_{5})$
$w_{13}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}(x_{2}((x_{3}x_{4})x_{5}))$ | $w_{14}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}(x_{2}(x_{3}(x_{4}x_{5})))$

Now, consider the $\mathbb{F}$-vector spaces ${\mathfrak{Z}}^{i}$ generated by
the set:

$\left\\{w_{i}(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)},x_{\sigma(4)},x_{\sigma(5)}):\sigma\in\mathbb{S}_{5}\right\\}.$

Denote by ${\mathfrak{Z}}^{i}_{\bf A}$ the subspace of ${\mathfrak{Z}}^{i}$ of
all $5$-linear polynomials vanishing on $\bf A$. Then we have the following
result regarding the dimension of these subspaces:

######  Proposition 19.

If ${\bf A}=W(2)$, then $\textrm{dim}({\mathfrak{Z}}^{i}_{\bf A})=0$ for
$1\leq i\leq 13$ and $\textrm{dim}({\mathfrak{Z}}^{14}_{\bf A})=5$. A basis of
the ${\mathbb{F}}$-vector space ${\mathfrak{Z}}^{14}_{\bf A}$ is the set of
identities:

  1. (1)

$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(2)}(x_{\sigma(3)}(x_{\sigma(4)}(x_{\sigma(5)}x_{1}))).$

  2. (2)

$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(1)}(x_{\sigma(3)}(x_{\sigma(4)}(x_{\sigma(5)}x_{2}))).$

  3. (3)

$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(1)}(x_{\sigma(2)}(x_{\sigma(4)}(x_{\sigma(5)}x_{3}))).$

  4. (4)

$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(1)}(x_{\sigma(2)}(x_{\sigma(3)}(x_{\sigma(5)}x_{4}))).$

  5. (5)

$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(1)}(x_{\sigma(2)}(x_{\sigma(3)}(x_{\sigma(4)}x_{5}))).$

Moreover, the linear combination with parameters $(1,-1,1,-1,1)$ is
$\mathfrak{st}^{5}_{2}$.

###  2.3. Central extensions

The notion of central extensions appeared in the study of Lie algebras, but it
can be considered in an arbitrary variety of algebras (see, for example,
[19]). The calculation of central extensions of an algebra ${\bf A}$ of
dimension $n$ from a certain variety of algebras gives the classification of
all algebras with $(k-n)$-dimensional annihilator, such that its factor
algebra by the annihilator is isomorphic to ${\bf A}$ (see, for example, [7]).
These calculations are carried out by studying the cohomology, with respect to
a polynomial identity, of the algebra ${\bf A}$. In this section, we are
interested in the central extensions of the contractions and subalgebras of
$W(2)$ considered in this paper. Some of these contractions and subalgebras
have turned out to be terminal, i.e., they satisfy the terminal identity
(degree four). The following result is about these particular algebras.

######  Proposition 20.

There are no terminal central extensions of the terminal contractions of
$W(2)$: $\overline{\mathscr{S}_{1}}$, $\overline{\mathscr{S}_{-1,1}}$,
$\overline{\mathscr{S}_{0,0}}$, $\widehat{W(2)}$, $\widehat{\widehat{W(2)}}$,
$\widetilde{W(2)}$ and $\widetilde{\widetilde{W(2)}}.$

###### Proof.

Recall that if $Z^{2}_{P}\left({\bf A},{\mathbb{F}}\right)$ denotes the space
of cocycles with respect to the polynomial identity $P$ of the algebra ${\bf
A}$, $B^{2}\left({\bf A},{\mathbb{F}}\right)$ denotes the space of coborders
of the algebra ${\bf A}$ and $H^{2}_{P}\left({\bf
A},{\mathbb{F}}\right):=Z^{2}_{P}\left({\bf
A},{\mathbb{F}}\right)/B^{2}\left({\bf A},{\mathbb{F}}\right)$ denotes the
cohomology space with respect to the polynomial identity $P$ of the algebra
${\bf A}$, then if $H^{2}_{P}\left({\bf A},{\mathbb{F}}\right)$ is trivial, we
have that ${\bf A}$ has no central extensions for the identity $P$. Now, fix
$P=T$ the terminal identity. Thus, the result is proven by direct calculation
of the cohomology space, obtaining that $H^{2}_{T}\left({\bf
A},{\mathbb{F}}\right)$ is trivial for any of the terminal contractions ${\bf
A}$ considered. ∎

######  Proposition 21.

There are no terminal central extensions of the terminal subalgebras of
$W(2)$: $B_{2},W_{2},C_{2},S_{2},D_{2},E_{2}$.

###### Proof.

The result follows by the direct calculation of the cohomology space,
obtaining that $H^{2}_{T}\left({\bf A},{\mathbb{F}}\right)$ is trivial for any
of the terminal subalgebras ${\bf A}$ considered. ∎

Similarly, we can determine if there are central extensions for the rest of
identities mentioned in the previous section.

######  Proposition 22.

By calculating the correspoding cohomology space, we conclude the following.

  1. (1)

There are no central extensions of $\overline{\mathscr{S}_{2,1}}$ in the
variety defined by one identity from the proposition 12.

  2. (2)

$\textrm{dim}\,Z^{2}_{P}\left(\overline{\mathscr{S}_{0,-3}},{\mathbb{F}}\right)=31$
and
$\textrm{dim}\,H^{2}_{P}\left(\overline{\mathscr{S}_{0,-3}},{\mathbb{F}}\right)=23$,
where $P$ is the identity $2\mathfrak{st}^{3}_{1}-3\mathfrak{st}^{3}_{2}$ from
proposition 13.

  3. (3)

There are no central extensions of $S_{2}$ in the variety defined by one
identity from the proposition 16.

Regarding the central extensions with respect to the identities
$\mathfrak{st}^{n}_{1}$ and $\mathfrak{st}^{n}_{2}$ for $n=3,4,5$ (see
Proposition 15 and Proposition 17), we have the following result.

######  Proposition 23.

The dimensions of the spaces of cocycles and coborders of the subalgebras of
$W(2)$ are given.

Algebra $\textrm{dim}\,B^{2}$ $\textrm{dim}\,Z^{2}_{\mathfrak{st}^{3}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{4}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{5}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{3}_{2}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{4}_{2}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{5}_{2}}$ $B_{2}$ 7 - - 44 - - 44 $W_{2}$
6 - - - - - 30 $C_{2}$ 5 - - 24 - - 24 $S_{2}$ 4 - - 16 - - 16 $D_{2}$ 3 8 9 9
8 9 9 $E_{2}$ 2 4 4 4 4 4 4

######  Proposition 24.

The dimensions of the spaces of cocycles and coborders of the subalgebras of
$W(2)$ are given.

Algebra $\textrm{dim}\,B^{2}$ $\textrm{dim}\,Z^{2}_{\mathfrak{st}^{3}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{4}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{5}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{3}_{2}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{4}_{2}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{5}_{2}}$ $W(2)$ 8 - - - - - 38
$\overline{W(2)}$ 8 - - 54 - - 54 $\overline{\mathscr{S}_{1}}$ 8 - - 60 - - 60
$\overline{\mathscr{S}_{5}}$ 8 - - 60 - - 60 $\overline{\mathscr{S}_{-1,1}}$ 8
- - 60 - - 60 $\overline{\mathscr{S}_{0,0}}$ 8 - - 60 - - 60 $\widehat{W(2)}$
8 - - - - - 40 $\widehat{\widehat{W(2)}}$ 8 - - 50 - - 54 ${\widetilde{W(2)}}$
8 - 52 64 - 52 64 $\widetilde{\widetilde{W(2)}}$ 8 43 64 64 43 64 64

By Proposition 23 and Proposition 24, we can conclude that for any of the
subalgebras and contractions considered there are central extensions with
respect to the identities $\mathfrak{st}^{n}_{1}$ and $\mathfrak{st}^{n}_{2}$
for $n=3,4,5$.

## Acknowledgements

Funding The first part of this work is supported by the Spanish Government
through the Ministry of Universities grant ‘Margarita Salas’, funded by the
European Union - NextGenerationEU; by the Junta de Andalucía through projects
UMA18-FEDERJA-119 and FQM-336 and by the Spanish Ministerio de Ciencia e
Innovación through project PID 2019-104236GB-I00, all of them with FEDER
funds; FCT UIDB/00212/2020 and UIDP/00212/2020. The second part of this work
is supported by the Russian Science Foundation under grant 22-11-00081.

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