# Non-associative algebraic structures: classification and structure111 The
work was supported by FCT UIDB/MAT/00212/2020, UIDP/MAT/00212/2020, and
2022.02474.PTDC.

Ivan Kaygorodov

###### Abstract

These are detailed notes for a lecture on “Non-associative Algebraic
Structures: Classification and Structure” which I presented as part of my
Agregação em Matemática e Applicações (University of Beira Interior, Covilhã,
Portugal, 13-14/03/2023).

###### keywords:

Non-associative algebra, Poisson algebra, $n$-ary algebra, superalgebra.

[I. Kaygorodov ]CMA-UBI, Universidade da Beira Interior, Covilhã,
<EMAIL_ADDRESS>Primary 17A30; secondary 17A36, 17A42,
17A70, 17B63.

32 2024 3 https://doi.org/10.46298/cm.11419

###### Contents

  1. 1 Introduction
    1. 1.1 Non-associative algebras
    2. 1.2 Poisson algebras
    3. 1.3 $n$-Ary algebras
  2. 2 Classifications of non-associative algebras
    1. 2.1 The algebraic classification of algebras
    2. 2.2 The geometric classification of algebras
    3. 2.3 Degenerations of algebras
  3. 3 Conservative algebras and superalgebras
    1. 3.1 Kantor product
    2. 3.2 Universal conservative superalgebras
    3. 3.3 Conservative algebras of $2$-dimensional algebras
  4. 4 Non-associative algebras and superalgebras
    1. 4.1 Leibniz algebras
    2. 4.2 Anticommutative $\mathfrak{CD}$-algebras
    3. 4.3 Noncommutative Jordan superalgebras
  5. 5 Maps on associative and non-associative algebras
    1. 5.1 Alternative and Jordan algebras with invertible derivations
    2. 5.2 Maps on alternative algebras
    3. 5.3 Higher derivations of incidence algebras
  6. 6 Generalized derivations of non-associative algebras
    1. 6.1 Local derivations of $n$-ary algebras
    2. 6.2 Leibniz-derivations of non-associative algebras
    3. 6.3 Generalized derivations of non-associative algebras
  7. 7 Poisson type algebras and superalgebras
    1. 7.1 Poisson structures on finitary incidence algebras
    2. 7.2 Generalized and generic Poisson algebras and superalgebras
    3. 7.3 Transposed Poisson algebra structures
  8. 8 $n$-Ary algebras
    1. 8.1 Ternary Jordan algebras
    2. 8.2 Split regular ${\rm Hom}$-Leibniz color $3$-algebras
    3. 8.3 $n$-Ary algebras with a multiplicative type basis

##  1 Introduction

In this lecture, we aim to summarize our research interests and achievements
as well as motivate what drives our work. The present work is based on various
papers written together with Abror Khudoyberdiyev, Alexandre Pozhidaev,
Aloberdi Sattarov, Amir Fernández Ouaridi, Antonio Jesús Calderón, Artem
Lopatin, Bakhrom Omirov, Bruno Ferreira, Crislaine Kuster, Doston Jumaniyozov,
Elisabete Barreiro, Farukh Mashurov, Feng Wei, Henrique Guzzo, Gulkhayo
Solijanova, Iqboljon Karimjanov, Ilya Gorshkov, Inomjon Yuldashev, Isabel
Hernández, Isamiddin Rakhimov, Ivan Shestakov, Jobir Adashev, José María
Sánchez, Karimbergan Kudaybergenov, Luisa Camacho, Manat Mustafa, Manuel
Ladra, María Alejandra Alvarez, Mikhail Ignatyev, Mohamed Salim, Mykola
Khrypchenko, Nurlan Ismailov, Pasha Zusmanovich, Patrícia Beites, Paulo
Saraiva, Pilar Páez-Guillán, Renato Fehlberg Júnior, Samuel Lopes, Sharifah
Kartini Said Husain, Thiago Castilho de Mello, Ualbai Umirbaev, Vasily
Voronin, Viktor Lopatkin, Yury Popov and Yury Volkov [119, 164, 46, 156, 7,
88, 121, 144, 146, 162, 57, 155, 173, 131, 9, 3, 4, 130, 129, 150, 151, 176,
177, 171, 170, 157, 163, 153, 137, 128, 123, 120, 122, 110, 109, 108, 89, 47,
8, 1, 152, 48, 178, 160, 142, 87, 169, 175, 159, 180, 59, 44, 158, 154, 91,
90, 165, 139, 140, 28, 167, 166, 92, 93, 145, 56, 138, 161, 174, 58, 141, 24,
22, 50, 168, 172].

The main direction of our research is concentrated on non-associative
algebraic structures, their classifications, and the study of their
properties. Namely, we work with many types of non-associative algebras and
superalgebras, $n$-ary algebras and also algebraic structures with two binary
multiplications (known as algebras of Poisson type). We consider small-
dimensional algebras, simple and semisimple algebras, PI-algebras, and free
algebras.

###  1.1 Non-associative algebras

Unless otherwise noted, all vector spaces and algebras are considered over the
complex field. In our work, “algebra” will mean “non-associative algebra”,
i.e., a vector space $\mathsf{A}$ equipped with a bilinear product which need
not be associative or unital. In the case of associative algebras, we work
with algebras defined by the following identity: $(xy)z=x(yz).$ On the other
hand, in non-associative cases, there are algebras defined by some special
conditions (not identities!), such as axial algebras, evolution algebras,
composition algebras, conservative algebras, etc., and also algebras defined
by polynomial identities. It seems that the first example of non-associative
algebras defined by some family of polynomial identities are Lie algebras. All
these algebras are closely related to associative algebras. For example, each
associative algebra under the commutator product $[x,y]:=xy-yx$ gives a Lie
algebra (the variety of algebras satisfying this property is called Lie-
admissible algebras).

Commutative associative algebras and Lie algebras admit many generalizations,
which have triggered some interest. So, the varieties of Jordan algebras,
alternative algebras, right alternative algebras, assosymmetric algebras,
bicommutative algebras, Novikov algebras, left-symmetric algebras, right
commutative algebras, $\mathfrak{CD}$-algebras, weakly associative algebras,
noncommutative Jordan algebras or terminal algebras, are generalizations of
commutative associative algebras. On the other hand, the varieties of Malcev,
symmetric Leibniz, Leibniz, binary Lie, $\mathfrak{CD}$-algebras, and also
noncommutative Jordan algebras and terminal algebras can be seen as
generalizations of Lie algebras. Some types of algebras appeared in another
way: Zinbiel algebras are Koszul dual to Leibniz algebras, Tortkara algebras
are defined by identities that are found in Zinbiel algebras under the
commutator multiplication; mock Lie algebras are commutative analogs of Lie
algebras; and dual mock Lie algebras are Koszul dual to mock Lie algebras.
There are many other relations between cited varieties, such as the following.
Lie algebras and Jordan algebras are related by Kantor–Koecher–Tits
construction. Alternative and noncommutative Jordan algebras under the
symmetric product $x\bullet y=xy+yx$ give Jordan algebras, assosymetric
algebras under the symmetric product give commutative
$\mathfrak{CD}$-algebras; Novikov, bicommutative, assosymmetric, and left-
symmetric algebras are Lie-admissible algebras (i.e., under the commutator
product they give Lie algebras) and so on.

Below, we provide the list of identities that defined the varieties of
algebras that are under consideration in this manuscript.

$(-1,1)$- | : | $(xy)y=xy^{2},\ (x,y,z)+(z,x,y)+(y,z,x)=0$
---|---|---
Alternative | : | $(x,y,z)=-(y,x,z),\ (x,y,z)=-(x,z,y)$
Antiassociative | : | $(xy)z=-x(yz)$
Assosymmetric | : | $(x,y,z)=(y,x,z),\ (x,y,z)=(x,z,y)$
Bicommutative | : | $x(yz)=y(xz),\ (xy)z=(xz)y$
Binary Lie | : | $xy=-yx,((xy)y)x=((xy)x)y$
$\mathfrak{CD}$- | : |  $((xy)a)b-((xy)b)a=((xa)b-(xb)a)y+x((ya)b-(yb)a)$,
|  |  $(a(xy))b-a((xy)b)=((ax)b-a(xb))y+x((ay)b-a(yb))$,
|  | $a(b(xy))-b(a(xy))=(a(bx)-b(ax))y+x(a(by)-b(ay))$
Dual mock-Lie | : | $xy=-yx,(xy)z=-x(yz)$
Jordan | : | $xy=yx,\ (x^{2}y)x=x^{2}(yx)$
Left-symmetric | : | $(x,y,z)=(y,x,z)$
Leibniz | : | $(xy)z=(xz)y+x(yz)$
Malcev | : | $xy=-yx,\ (xy)(xz)+(y(xz))x+$
|  | $+((xz)x)y=((xy)z+(yz)x+(zx)y)x$
Noncommutative Jordan | : | $(xy)x=x(yx),\ (x^{2}y)x=x^{2}(yx)$
Novikov | : | $(xy)z=(xz)y,\ (x,y,z)=(y,x,z)$
Right alternative | : | $(x,y,z)=-(x,z,y)$
Right commutative | : | $(xy)z=(xz)y$
Symmetric Leibniz | : | $x(yz)=(xy)z+y(xz),\ (xy)z=(xz)y+x(yz)$
Terminal | : | $3(b(a(xy)-(ax)y-x(ay))-a((bx)y)+(a(bx))y+$
|  | $(bx)(ay)-a(x(by))+(ax)(by)+x(a(by)))=$
|  | $=-(2ab+ba)(xy)+((2ab+ba)x)y+x((2ab+ba)y)$
Tortkara | : | $xy=-yx,\ (xy)(zy)=((xy)z+(yz)x+(zx)y)y$
Weakly associative | : | $(xy)z-x(yz)+(yz)x-y(zx)=(yx)z-y(xz)$
Zinbiel | : | $(xy)z=x(yz+zy)$

Superalgebras. Let $\mathbb{F}$ be a field and let $G$ be the Grassmann
algebra over $\mathbb{F}$ given by the generators $1$, $\xi_{1}$, $\ldots$,
$\xi_{n}$, $\ldots$ and the defining relations $\xi_{i}^{2}=0$ and
$\xi_{i}\xi_{j}=-\xi_{j}\xi_{i}$. The elements $1$,
$\xi_{i_{1}}\xi_{i_{2}}\ldots\xi_{i_{k}}$, $i_{1}<i_{2}<\ldots<i_{k}$, form a
basis of the algebra $G$ over $\mathbb{F}$. Denote by $G_{0}$ and $G_{1}$ the
subspaces spanned by the products of even and odd lengths, respectively; then
$G$ can be represented as the direct sum of these subspaces, $G=G_{0}\oplus
G_{1}$. Here the relations

$G_{i}G_{j}\subseteq G_{i+j(mod\ 2)},\ i,j=0,1,$

hold. In other words, $G$ is a $\mathbb{Z}_{2}$-graded algebra (or a
superalgebra) over $\mathbb{F}$. Suppose now that
$\mathsf{A}=\mathsf{A}_{0}\oplus\mathsf{A}_{1}$ is an arbitrary superalgebra
over $\mathbb{F}$. Consider the tensor product $G\otimes\mathsf{A}$ of
$\mathbb{F}$-algebras. The subalgebra

$G(\mathsf{A})=G_{0}\otimes\mathsf{A}_{0}+G_{1}\otimes\mathsf{A}_{1}$

of $G\otimes\mathsf{A}$ is referred to as the Grassmann envelope of the
superalgebra $\mathsf{A}$. Let $\Omega$ be a variety of algebras over
$\mathbb{F}$. A superalgebra $\mathsf{A}=\mathsf{A}_{0}\oplus\mathsf{A}_{1}$
is referred to as an $\Omega$-superalgebra if its Grassmann envelope
$G(\mathsf{A})$ is an algebra in $\Omega$. In particular,
$\mathsf{A}=\mathsf{A}_{0}\oplus\mathsf{A}_{1}$ is referred to as a
(noncommutative) Jordan superalgebra if its Grassmann envelope $G(\mathsf{A})$
is a (noncommutative) Jordan algebra.

###  1.2 Poisson algebras

The notion of a Poisson bracket has its origin in the works of S.D. Poisson on
celestial mechanics at the beginning of the XIX century. Since then, Poisson
algebras have appeared in several areas of mathematics, such as: Poisson
manifolds [198], algebraic geometry [29, 105], noncommutative geometry [239],
operads [204], quantization theory [118, 191], quantum groups [78], and
classical and quantum mechanics. The study of Poisson algebras also led to
other algebraic structures, such as generic Poisson algebras [174, 190],
algebras of Jordan brackets and generalized Poisson algebras [60, 179, 206,
188], Gerstenhaber algebras [192], Novikov-Poisson algebras [248], Malcev-
Poisson-Jordan algebras, transposed Poisson algebras [20, 93], $n$-ary Poisson
algebras [62], etc.

######  Definition 1.1.

A Poisson algebra is a vector space $\mathsf{A}$ with two bilinear operations
$\cdot$ and $\\{\cdot,\cdot\\}$, such that $(\mathsf{A},\cdot)$ is a
commutative associative algebra, $(\mathsf{A},\\{\cdot,\cdot\\})$ is a Lie
algebra and

$\\{x,y\cdot z\\}=\\{x,y\\}\cdot z+y\cdot\\{x,z\\}.$

Poisson algebras are at the intersection of Lie and commutative associative
algebras. They inherit many common properties of these algebras, and they can
be used to solve some well-known problems in the theory of Lie or associative
algebras. So, Poisson algebras have been used to prove the Nagata conjecture
about wild automorphisms of the polynomial ring with three generators [234]
and to describe simple Jordan [207] and noncommutative Jordan [225, 226]
algebras and superalgebras. Namely, from each simple Poisson algebra, it is
possible to obtain simple Jordan superalgebra by a process named as “Kantor
double”; but on the other hand, each noncommutative Jordan superalgebra under
the commutator product gives a structure of Poisson bracket. The systematic
study of free Poisson algebras began in the paper by Shestakov [233], where he
constructed a basis for the free Poisson algebra. Later, in a series papers by
Makar-Limanov and Umirbaev (see [201, 200] and references therein), many
analogs of classical results for free Poisson algebras were obtained. For
example, they proved that there are no wild automorphisms in the free Poisson
algebra with two generators, proved the freedom theorem (The Freiheitssatz),
and described universal multiplicative envelopings of the free Poisson fields.
The systematic study of noncommutative Poisson algebra structures began in the
paper by Kubo [195]. He obtained a description of all the Poisson structures
on the full and upper triangular matrix algebras, which was later generalized
to prime associative algebras in [86]. Namely, it was proved in [86] that any
Poisson bracket on a prime noncommutative associative algebra is the
commutator bracket multiplied by an element of the extended centroid of the
algebra. On the other hand, in his next paper, Kubo studied noncommutative
Poisson algebra structures on affine Kac-Moody algebras. The investigation of
Poisson structures on associative algebras continued in some papers of Yao, Ye
and Zhang [246]; Mroczyńska, Jaworska-Pastuszak, and Pogorzały [125, 215],
where Poisson structures on finite-dimensional path algebras and on canonical
algebras were studied. Crawley-Boevey introduced a noncommutative Poisson
structure, called an $H_{0}$-Poisson structure, on the $0$-th cyclic homology
of a noncommutative associative algebra [69] and showed that an
$H_{0}$-Poisson structure can be induced on the affine moduli space of
(semisimple) representations of an associative algebra from a suitable Lie
algebra structure on the $0$-th Hochschild homology of the algebra [69]. The
derived noncommutative Poisson bracket on Koszul Calabi-Yau algebras has been
studied in [65]. Van den Bergh introduced double Poisson algebras in [239],
and Van de Weyer described all the double Poisson structures on finite-
dimensional semisimple algebras. Recently, the notion of a noncommutative
Poisson bialgebra appeared in [199].

###  1.3 $n$-Ary algebras

Following Kurosh, an $\Omega$-algebra over a field $\mathbb{F}$ is a linear
space over $\mathbb{F}$ equipped with a system of multilinear algebraic
operations $\Omega=\\{\omega_{i}:|\omega_{i}|=n_{i},n_{i}\in\mathbb{N}\\}$,
where $|\omega_{i}|$ denotes the arity of $\omega_{i}$. The present notion of
$\Omega$-algebras is very wide. It includes associative and non-associative
algebras (with one binary multiplication); dialgebras and Poisson algebras and
their generalizations (with two binary multiplications); trialgebras (with
three binary multiplications); quadri-algebras (with four binary
multiplications), ennea-algebras (with nine binary multiplications);
pluriassociative algebras (with $2\gamma$ binary multiplications); Jordan
triple disystems and comtrans algebras (with two ternary multiplications);
Akivis algebras (with one binary and one ternary multiplications); Bol
algebras, Lie-Yamaguti algebras, Poisson $n$-Lie algebras, Sabinin algebras
and so on. A particular case of $\Omega$-algebras, that will be interesting
for us is $n$-ary algebras.

Let $\mathsf{A}$ be an $\Omega$-algebra equipped with one $n$-ary operation
$[\cdot,\ldots,\cdot]:\mathsf{A}\times\ldots\times\mathsf{A}\to\mathsf{A}$,
which is $n$-linear. This operation is $n$-ary commutative if,

$[x_{1},\ldots,x_{n}]=[x_{\sigma(1)},\ldots,x_{\sigma(n)}],\mbox{ for all
}\sigma\in\mathbb{S}_{n}.$

There are notions of partial commutativity and partial associativity for
$n$-ary algebras. A generalization of a variety of binary algebras defined by
a family of polynomial identities to the $n$-ary case is not an easy task. So,
there are many types of generalizations of associative and Lie algebras to
$n$-ary case. For example, there are partially associative and totally
associative $n$-ary algebras, which give associative algebras in the binary
case. In the case of Lie algebras, the known generalizations are the
following: reduced Lie ternary algebras introduced by Pojidaev and $n$-Lie
algebras introduced by Filippov in [95]. Recently, the theory of Filippov
algebras has attracted much attention due to its close connection with Nambu
mechanics, proposed by Nambu as a generalization of classical Hamiltonian
mechanics. In his article, Filippov introduced the notion of a $n$-Lie algebra
and proved some of its properties. He also obtained the classification of
anticommutative $n$-ary algebras of dimension $n$ and $n+1$.

Let us give the definition of $n$-Lie (Filippov) algebras. Let $\mathsf{A}$ be
an algebra equipped with one $n$-ary operation
$[\cdot,\ldots,\cdot]:\mathsf{A}\times\ldots\times\mathsf{A}\to\mathsf{A}$,
which is $n$-linear. This algebra is $n$-Lie if,

$[x_{1},\ldots,x_{n}]=(-1)^{\sigma}[x_{\sigma(1)},\ldots,x_{\sigma(n)}],\mbox{
for all }\sigma\in\mathbb{S}_{n};$
$[[x_{1},\ldots,x_{n}],y_{2},\ldots,y_{n}]=\sum\limits_{i=1}^{n}[x_{1},\ldots,x_{i-1},[x_{i},y_{2},\ldots,y_{n}],x_{i+1},\ldots,x_{n}].$

The first identity gives an $n$-ary analog of anticommutative identity, and
the second gives the $n$-ary version of the Jacobi identity. An algebra
$\mathsf{A}$ is called $n$-ary Leibniz algebra if it satisfies only the second
identity from the identities mentioned above.

##  2 Classifications of non-associative algebras

The present part is based on the papers written together with Abror
Khudoyberdiyev, Alexandre Pozhidaev, Aloberdi Sattarov, Amir Fernández
Ouaridi, Antonio Jesús Calderón, Crislaine Kuster, Doston Jumaniyozov, Farukh
Mashurov, Iqboljon Karimjanov, Ilya Gorshkov, Isabel Hernández, Isamiddin
Rakhimov, Jobir Adashev, Luisa Camacho, Manat Mustafa, Manuel Ladra, María
Alejandra Alvarez, Mikhail Ignatyev, Mohamed Salim, Mykola Khrypchenko, Nurlan
Ismailov, Pilar Páez-Guillán, Samuel Lopes, Sharifah Kartini Said Husain,
Thiago Castilho de Mello, Vasily Voronin, Yury Popov and Yury Volkov [119,
164, 46, 7, 88, 121, 144, 146, 162, 57, 155, 173, 131, 9, 3, 4, 130, 129, 150,
151, 176, 177, 171, 170, 160, 163, 153, 137, 128, 123, 120, 122, 56, 138, 161,
110, 109, 108, 89, 47, 8, 1, 152, 48, 178, 58].

###  2.1 The algebraic classification of algebras

The algebraic classification (up to isomorphism) of algebras of dimension $n$
from a certain variety defined by a certain family of polynomial identities is
a classic problem in the theory of non-associative algebras. There are many
results related to the algebraic classification of small-dimensional algebras
in many varieties of non-associative algebras [63, 67, 70, 72, 2, 177]. So,
algebraic classifications of $2$-dimensional algebras [177, 222],
$3$-dimensional evolution algebras [41], $3$-dimensional anticommutative
algebras [189], $4$-dimensional division algebras [75, 71], have been given.

Our method for classifying nilpotent commutative algebras is based on the
calculation of central extensions of nilpotent algebras of smaller dimensions
from the same variety (first, this method has been developed by Skjelbred and
Sund for the Lie algebra case in [235]).

Our big program in the classification of complex small-dimensional nilpotent
algebras results in three big classifications of the following algebras:

  * •

$4$-dimensional nilpotent algebras [151];

  * •

$5$-dimensional commutative nilpotent algebras [129];

  * •

$6$-dimensional anticommutative nilpotent algebras [150].

On the other hand, during the realization of our classification program, we
classified small-dimensional algebras in the following varieties:

  * I.

$4$-dimensional nilpotent:

    * —

assosymmetric algebras [120];

    * —

bicommutative algebras [163];

    * —

$\mathfrak{CD}$-algebras [146];

    * —

commutative algebras [89];

    * —

left-symmetric algebras [3];

    * —

noncommutative Jordan algebras [128];

    * —

Novikov algebras [137];

    * —

right commutative algebras [4];

    * —

right alternative algebras [121];

    * —

terminal algebras [153];

    * —

weakly associative algebras [9];

  * II.

$5$-dimensional nilpotent:

    * —

antiassociative algebras [88];

    * —

anticommutative algebras [89];

    * —

commutative associative algebras [173];

    * —

commutative $\mathfrak{CD}$-algebras [130];

    * —

symmetric Leibniz algebras [9];

    * —

one-generated assosymmetric algebras [162];

    * —

one-generated bicommutative algebras [164];

    * —

one-generated Novikov algebras [57];

    * —

one-generated terminal algebras [155];

  * III.

$6$-dimensional nilpotent:

    * —

anticommutative $\mathfrak{CD}$-algebras [1];

    * —

binary Lie algebras [1];

    * —

Tortkara algebras [108];

    * —

one-generated assosymmetric algebras [162];

    * —

one-generated bicommutative algebras [164];

    * —

one-generated Novikov algebras [57].

The classification of non-nilpotent algebras is given by other methods, which
were developed in our papers. Let us now summarize the rest of the results
obtained in the algebraic classification of algebras.

  * $\bullet$

$2$-dimensional terminal, conservative and rigid algebras [47];

  * $\bullet$

$2$-dimensional algebras with left quasi-units [175];

  * $\bullet$

$2$-dimensional algebras [177];

  * $\bullet$

$3$-dimensional anticommutative algebras[123];

  * $\bullet$

$8$-dimensional dual Mock Lie algebras [58];

  * $\bullet$

central extensions of $n$-dimensional filiform associative algebras [138];

  * $\bullet$

central extensions of $n$-dimensional filiform Zinbiel algebras [56];

  * $\bullet$

non-associative central extensions of $n$-dimensional null-filiform
associative algebras [161];

  * $\bullet$

$n$-dimensional algebras with $(n-2)$-dimensional annihilator [48];

  * $\bullet$

$n$-dimensional anticommutative algebras with $(n-3)$-dimensional annihilator
[46];

  * $\bullet$

$n$-dimensional Zinbiel algebras with $(n-3)$-dimensional annihilator [7].

###  2.2 The geometric classification of algebras

Given a complex $n$-dimensional vector space $\mathbb{V}$, the set

${\rm
Hom}(\mathbb{V}\otimes\mathbb{V},\mathbb{V})\cong\mathbb{V}^{*}\otimes\mathbb{V}^{*}\otimes\mathbb{V}$

is a complex vector space of dimension $n^{3}$. This space has the structure
of the affine variety $\mathbb{C}^{n^{3}}$. Indeed, if we fix a basis
$\\{e_{1},\dots,e_{n}\\}$ of $\mathbb{V}$, then any $\mu\in{\rm
Hom}(\mathbb{V}\otimes\mathbb{V},\mathbb{V})$ is determined by $n^{3}$
structure constants $c_{ij}^{k}\in\mathbb{C}$ such that $\mu(e_{i}\otimes
e_{j})=\sum\limits_{k=1}^{n}c_{ij}^{k}e_{k}$. A subset of ${\rm
Hom}(\mathbb{V}\otimes\mathbb{V},\mathbb{V})$ is Zariski-closed if it is the
set of solutions of a system of polynomial equations in the variables
$c_{ij}^{k}$ ($1\leq i,j,k\leq n$).

Let $T$ be a set of polynomial identities. Every algebra structure on
$\mathbb{V}$ satisfying polynomial identities from $T$ forms a Zariski-closed
subset of the variety ${\rm Hom}(\mathbb{V}\otimes\mathbb{V},\mathbb{V})$. We
denote this subset by $\mathbb{L}(T)$. The general linear group ${\rm
GL}(\mathbb{V})$ acts on $\mathbb{L}(T)$ by conjugation:

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

for $x,y\in\mathbb{V}$, $\mu\in\mathbb{L}(T)\subset{\rm
Hom}(\mathbb{V}\otimes\mathbb{V},\mathbb{V})$ and $g\in{\rm GL}(\mathbb{V})$.
Thus, $\mathbb{L}(T)$ decomposes into ${\rm GL}(\mathbb{V})$-orbits that
correspond to the isomorphism classes of the algebras. We shall denote by
$O(\mu)$ the orbit of $\mu\in\mathbb{L}(T)$ under the action of ${\rm
GL}(\mathbb{V})$ and by $\overline{O(\mu)}$ its Zariski closure.

One of the main problems of the geometric classification of a variety of
algebras is a description of its irreducible components. In [102], Gabriel
described the irreducible components of the variety of $4$-dimensional unital
associative algebras, and the variety of $5$-dimensional unital associative
algebras were classified algebraically and geometrically by Mazzola [212,
213]. Later, Cibils [66] considered rigid associative algebras with 2-step
nilpotent radical. Goze and Ancochea-Bermúdez proved that the varieties of 7
and 8-dimensional nilpotent Lie algebras are reducible [13]. The irreducible
components of $2$-step nilpotent commutative associative algebras were
described in [228]. Often, the irreducible components of the variety are
determined by the rigid algebras, although this is not always the case.
Indeed, Flanigan showed in [98] that the variety of $3$-dimensional nilpotent
associative algebras has an irreducible component which does not contain any
rigid algebras — it is instead defined by the closure of a union of a one-
parameter family of algebras.

Let $\mathsf{A}_{1}$ and $\mathsf{A}_{2}$ be two $n$-dimensional algebras
satisfying the identities from $T$, and let $\mu,\lambda\in\mathbb{L}(T)$
represent $\mathsf{A}_{1}$ and $\mathsf{A}_{2}$, respectively. We say that
$\mathsf{A}_{1}$ degenerates to $\mathsf{A}_{2}$, and write
$\mathsf{A}_{1}\to\mathsf{A}_{2}$, if $\lambda\in\overline{O(\mu)}$. Note that
this implies $\overline{O(\lambda)}\subset\overline{O(\mu)}$. Hence, the
definition of degeneration does not depend on the choice of $\mu$ and
$\lambda$. If $\mathsf{A}_{1}\not\cong\mathsf{A}_{2}$, then the assertion
$\mathsf{A}_{1}\to\mathsf{A}_{2}$ is called a proper degeneration. Following
Gorbatsevich [106], we say that $\mathsf{A}$ has level $m$ if there exists a
chain of proper degenerations of length $m$ starting in $\mathsf{A}$ and there
is no such chain of length $m+1$. Also, in [107] it was introduced the notion
of _infinite level_ of an algebra $\mathsf{A}$ as the limit of the usual
levels of $\mathsf{A}\oplus\mathbb{C}^{m}$.

Let $\mathsf{A}$ be represented by $\mu\in\mathbb{L}(T)$. Then $\mathsf{A}$ is
rigid in $\mathbb{L}(T)$ if $O(\mu)$ is an open subset of $\mathbb{L}(T)$.
Recall that a subset of a variety is called irreducible if it cannot be
represented as a union of two non-trivial closed subsets. A maximal
irreducible closed subset of a variety is called an irreducible component. It
is well known that any affine variety can be uniquely represented as a finite
union of its irreducible components. Note that the algebra $\mathsf{A}$ is
rigid in $\mathbb{L}(T)$ if and only if $\overline{O(\mu)}$ is an irreducible
component of $\mathbb{L}(T)$.

In our work, we discuss the following problems:

######  Problem 2.1 (Geometric classification).

Let $\mathfrak{V}^{n}$ be a variety of $n$-dimensional algebras defined by a
family of identities $T$. Which are the irreducible components of
$\mathfrak{V}^{n}$?

######  Problem 2.2 (Degeneration classification).

Let $\mathfrak{V}^{n}$ be a variety of $n$-dimensional algebras defined by a
family of identities $T$. Is there a degeneration from $\mathsf{A}_{1}$ to
$\mathsf{A}_{2}$ for each pair of algebras from $\mathfrak{V}^{n}$?

######  Problem 2.3 (Level classification).

Let $\mathfrak{V}^{n}$ be a variety of $n$-dimensional algebras defined by a
family of identities $T$. Which algebras from $\mathfrak{V}^{n}$ have level
$m$?

In the following list, we summarize our main results related to the geometric
classification of algebras. Namely, we gave the geometric classification of
the following varieties of algebras:

  * I.

$4$-dimensional nilpotent:

    * —

assosymmetric algebras [120];

    * —

bicommutative algebras [163];

    * —

$\mathfrak{CD}$-algebras [144];

    * —

commutative algebras [89];

    * —

left-symmetric algebras [3];

    * —

Leibniz algebras [170];

    * —

noncommutative Jordan algebras [128];

    * —

Novikov algebras [137];

    * —

right commutative algebras [4];

    * —

right alternative algebras [121];

    * —

terminal algebras [153];

    * —

weakly associative algebras [9];

  * II.

$5$-dimensional nilpotent:

    * —

antiassociative algebras [88];

    * —

associative algebras [119];

    * —

commutative $\mathfrak{CD}$-algebras [131];

    * —

symmetric Leibniz algebras [9];

  * III.

$6$-dimensional nilpotent:

    * —

anticommutative $\mathfrak{CD}$-algebras [1];

    * —

binary Lie algebras [1];

    * —

Malcev algebras [171];

    * —

Tortkara algebras [109];

  * IV.

$4$-dimensional:

    * —

binary Lie algebras [171];

    * —

Leibniz algebras [122];

    * —

Zinbiel algebras [170];

  * V.

$7$-dimensional:

    * —

dual Mock Lie algebras [58].

In [152, 119] we completely solve the geometric classification problem for
nilpotent and $2$-step nilpotent, commutative nilpotent and anticommutative
nilpotent algebras of arbitrary dimension.

######  Theorem 2.4 (Theorem A, [152]).

For any $n\geq 2$, the variety of all $n$-dimensional nilpotent algebras is
irreducible and has dimension $\frac{n(n-1)(n+1)}{3}$.

Moreover, we show that the family $\mathcal{R}_{n}$ given in [152, Definition
10] is generic in the variety of $n$-dimensional nilpotent algebras and
inductively gives an algorithmic procedure to obtain any $n$-dimensional
nilpotent algebra as a degeneration from $\mathcal{R}_{n}$.

######  Theorem 2.5 (Theorem B, [152]).

For any $n\geq 2$, the variety of all $n$-dimensional commutative nilpotent
algebras is irreducible and has dimension $\frac{n(n-1)(n+4)}{6}$.

As above, we show that the family $\mathcal{S}_{n}$ given in [152, Definition
15] is generic in the variety of $n$-dimensional commutative nilpotent
algebras and inductively give an algorithmic procedure to obtain any
$n$-dimensional nilpotent commutative algebra as a degeneration from
$\mathcal{S}_{n}$.

######  Theorem 2.6 (Theorem C, [152]).

For any $n\geq 2$, the variety of all $n$-dimensional anticommutative
nilpotent algebras is irreducible and has dimension
$\frac{(n-2)(n^{2}+2n+3)}{6}$.

We show also that the family $\mathcal{T}_{n}$ given in [152, Definition 33],
is generic in the variety of $n$-dimensional anticommutative nilpotent
algebras and inductively give an algorithmic procedure to obtain any
$n$-dimensional nilpotent anticommutative algebra as a degeneration from
$\mathcal{T}_{n}$.

The notion of length for non-associative algebras has been recently introduced
in [113], generalizing the corresponding notion for associative algebras.
Using the above result, we show in [152, Corollary 39] that the length of an
arbitrary (i.e., not necessarily nilpotent) $n$-dimensional anticommutative
algebra is bounded above by the $n^{\text{th}}$ Fibonacci number, and prove
that our bound is sharp.

For $k\leq n$ consider the (algebraic) subset $\mathfrak{Nil}_{n,k}^{2}$ of
the variety $\mathfrak{Nil}_{n}^{2}$ of 2-step nilpotent $n$-dimensional
algebras defined by

$\mathfrak{Nil}_{n,k}^{2}=\\{\mathsf{A}\in\mathfrak{Nil}_{n}^{2}:\dim\mathsf{A}^{2}\leq
k,\ \dim{\rm ann}\ \mathsf{A}\geq k\\}.$

It is easy to see that
$\mathfrak{Nil}_{n}^{2}=\cup_{k=1}^{n}\mathfrak{Nil}_{n,k}^{2}$. Analogously,
for the varieties
$\mathfrak{Nil}_{n}^{2}{}^{\mathrm{c}},\mathfrak{Nil}_{n}^{2}{}^{\mathrm{ac}}$
of commutative and anticommutative 2-step nilpotent algebras we define the
subsets $\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{c}}$ and
$\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{ac}}$, respectively.

######  Theorem 2.7 (Theorem A, [119]).

The sets $\mathfrak{Nil}_{n,k}^{2}$ are irreducible and

$\mathfrak{Nil}_{n}^{2}$ | $=$ |  $\bigcup_{k}\mathfrak{Nil}_{n,k}^{2}$, | for | $1\leq k\leq\left\lfloor n+\frac{1-\sqrt{4n+1}}{2}\right\rfloor$
---|---|---|---|---
$\mathfrak{Nil}_{n}^{2}{}^{\mathrm{c}}$ | $=$ |  $\bigcup_{k}\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{c}}$, | for |  $1\leq k\leq\left\lfloor n+\frac{3-\sqrt{8n+9}}{2}\right\rfloor$,
$\mathfrak{Nil}_{n}^{2}{}^{\mathrm{ac}}$ | $=$ |  $\bigcup_{k}\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{ac}}$, | for |  $1+(n+1)\operatorname{mod}2\leq k\leq\left\lfloor n+\frac{1-\sqrt{8n+1}}{2}\right\rfloor\text{ for }n\geq 3$.

Moreover,

$\dim\mathfrak{Nil}_{n,k}^{2}$ | $=$ |  $(n-k)^{2}k+(n-k)k$,
---|---|---
$\dim\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{c}}$ | $=$ |  $\frac{(n-k)(n-k+1)}{2}k+(n-k)k$,
$\dim\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{ac}}$ | $=$ |  $\frac{(n-k)(n-k-1)}{2}k+(n-k)k$.

###  2.3 Degenerations of algebras

The present part is dedicated to the results obtained regarding Problem 2
(Degeneration classification) and Problem 3 (Level classification). Typically,
the results of degeneration classification are given by construction of the
graph of primary degenerations of the algebras from the variety. So, Grunewald
and O’Halloran constructed the graph of primary degenerations for
$5$-dimensional nilpotent Lie algebras [112]; Seeley obtained a similar graph
for $6$-dimensional nilpotent Lie algebras in [227]. The degenerations graphs
for $2$-dimensional Jordan algebras and $4$-dimensional nilpotent Jordan
algebras are presented in [11, 12]. Other graphs of degenerations were
constructed by Burde and co-authors for $4$-dimensional Lie algebras [40],
$2$-dimensional Pre-Lie algebras [30] and $3$-dimensional Novikov algebras
[31].

In the following list, we summarize our main results related to the
degeneration classification of algebras. Namely, we constructed graphs of
primary degenerations of the following varieties of algebras:

  * $\bullet$

$2$-dimensional algebras [177];

  * $\bullet$

$3$-dimensional anticommutative algebras [123];

  * $\bullet$

$3$-dimensional Jordan algebras [110];

  * $\bullet$

$3$-dimensional Jordan superalgebras [8];

  * $\bullet$

$3$-dimensional Leibniz algebras [123];

  * $\bullet$

$3$-dimensional nilpotent algebras [89];

  * $\bullet$

$4$-dimensional binary Lie algebras [171];

  * $\bullet$

$4$-dimensional nilpotent commutative algebras [89];

  * $\bullet$

$4$-dimensional nilpotent Leibniz algebras [170];

  * $\bullet$

$4$-dimensional Zinbiel algebras [170].

  * $\bullet$

$5$-dimensional nilpotent anticommutative algebras [89];

  * $\bullet$

$5$-dimensional nilpotent commutative associative algebras [160].

  * $\bullet$

$6$-dimensional nilpotent Malcev algebras [171];

  * $\bullet$

$7$-dimensional dual Mock Lie algebras [58];

  * $\bullet$

$(n+1)$-dimensional $n$-Lie (Filippov) algebras [178].

The first attempt at classification of algebras of level one is given by
Gorbatsevich [106], but he omitted some main cases. Later, his classification
was completed in a paper by Khudoyberdiyev and Omirov [185]. After this, the
classification of algebras of level two for some particular cases, such that
associative algebras, Jordan algebras, Lie algebras, Leibniz algebras and
nilpotent algebras has been given in [184, 101]. In [176] we completely solve
the level classification problem for all algebras in the case of level two,
generalizing all the cited results of Khudoyberdiyev and co-authors. Following
our ideas on the level classification of algebras, Volkov recently obtained
some results on the classifications of $n$-ary algebras of level one and on
classification of anticommutative algebras of levels $3$, $4$ and $5$ [240,
241].

##  3 Conservative algebras and superalgebras

The present part is based on the papers written together with Alexandre
Pozhidaev, Artem Lopatin, Renato Fehlberg Júnior, Yury Popov, and Yury Volkov
[157, 142, 87, 169, 175].

###  3.1 Kantor product

The idea of obtaining new objects from old ones by using derivative operations
has long been known in algebra [5]. In its most general form, the idea was
realized by Malcev [203]. Let $M_{n}$ be an associative algebra of matrices of
order $n$ over a field $\mathbb{F}$. Assume that some finite collection
$\Lambda=(a_{ij},b_{ij},c_{ij})$ of matrices in $M_{n}$ is given. Denote by
$M^{(\Lambda)}_{n}$ the algebra defined on a space of matrices in $M_{n}$ with
respect to new multiplication

$x\cdot_{\Lambda}y=\sum_{i,j}a_{ij}xb_{ij}yc_{ij}.$

It was proved that every $n$-dimensional algebra over $\mathbb{F}$ is
isomorphic to a subalgebra of $M^{(\Lambda)}_{n}$ [203]. Other interesting
ways to derive the initial multiplication are isotopes, homotopes, and
mutations [84, 6, 220]. The concept of an isotope was introduced by Albert
[5]. Let $\mathsf{A}$ and $\mathsf{A}_{0}$ be algebras with a common
underlying linear space on which right multiplication operators $R_{x}$ and
$R^{(0)}_{x}$ are defined (for $\mathsf{A}$ and $\mathsf{A}_{0}$, resp.). We
say that $\mathsf{A}_{0}$ and $\mathsf{A}$ are isotopic if there exist
invertible linear operators $\phi,\psi,\xi$ such that $R^{(0)}_{x}=\phi
R_{x\psi}\xi$. We call $\mathsf{A}_{0}$ an isotope of $\mathsf{A}$. Let
$\mathsf{A}$ be an arbitrary associative algebra, and let $p$, $q$ be two
fixed elements of $\mathsf{A}$. Then a new algebra is derived from
$\mathsf{A}$ by using the same vector space structure of $\mathsf{A}$ but
defining a new multiplication $x*y=xpy-yqx$. The resulting algebra is called
the $(p,q)$-mutation of the algebra $\mathsf{A}$.

The definition of the Kantor product of multiplications comes from the study
of a certain class of algebras. In 1972, Kantor introduced the class of
conservative algebras [133], which contains many important classes of algebras
(see [157]), for example, associative, Lie, Jordan, and Leibniz algebras. To
define what will be called the Kantor product, we need to introduce the
algebras $U(n)$ (see, for more details, [135, 157]). Consider the space $U(n)$
of all bilinear multiplications on the $n$-dimensional vector space $V_{n}$.
Now, fix a vector $u\in V_{n}$. For $A,B\in U(n)$ (two multiplications) and
$x,y\in V_{n}$, we set

$x\ast y=\llbracket A,B\rrbracket(x,y)=A(u,B(x,y))-B(A(u,x),y)-B(x,A(u,y)).$

This new multiplication is called the (left) Kantor product of the
multiplications $A$ and $B$ (it is also possible to define the right Kantor
product). The Kantor product of a multiplication “$\cdot$” by itself will be
the Kantor square of “$\cdot$”:

$x\ast y=u\cdot(x\cdot y)-(u\cdot x)\cdot y-x\cdot(u\cdot y).$

It is easy to see that the Kantor square of multiplication is a particular
case of the Malcev construction in a non-associative sense. On the other hand,

  * •

in the commutative associative case it coincides with a mutation;

  * •

in the left commutative and left-symmetric cases it coincides with an isotope.

As in [142], we will assume that the Kantor product is always the left Kantor
product. In [142], we studied the Kantor product and Kantor square of many
well-known algebras, for example, associative, (anti)-commutative, Lie,
Leibniz, Novikov, Poisson algebras, and dialgebras.

We summarize the main results of the present part in the following theorems.

######  Theorem 3.1 (Theorem 29, [142]).

Let $(\mathsf{A},\cdot)$ be a finite-dimensional associative algebra. Then
$(\mathsf{A},\cdot)$ is isomorphic to
$(\mathsf{A},\llbracket\cdot,\cdot\rrbracket)$, if and only if $A$ is a skew
field.

######  Theorem 3.2 (Theorem 10, [142]).

Let $(\mathsf{A};\cdot)$ be an alternative algebra. Then
$(\mathsf{A};\llbracket\cdot,\cdot\rrbracket)$ is a flexible algebra.
Furthermore,

  * •

$(\mathsf{A};\llbracket\cdot,\cdot\rrbracket)$ is an alternative algebra if
and only if $\mathsf{A}$ satisfies the identity

$(x,u,(x,u,y))=0$;

  * •

$(\mathsf{A};\llbracket\cdot,\cdot\rrbracket)$ is a noncommutative Jordan
algebra if and only if $\mathsf{A}$ satisfies the identity

$[L_{u}L_{x}L_{u}L_{x},R_{u}R_{x}]=[L_{xuxu},R_{ux}]$;

  * •

$(\mathsf{A};\llbracket\cdot,\cdot\rrbracket)$ is a Jordan algebra if
$(\mathsf{A},\cdot)$ is a commutative alternative algebra;

  * •

$(\textbf{C};\llbracket\cdot,\cdot\rrbracket)$ is alternative for a Cayley —
Dickson algebra C, if and only if $u=u_{0}\cdot 1$.

######  Theorem 3.3 (Theorem 19, [142]).

Let $(\mathsf{A};\cdot,\\{\cdot,\cdot\\})$ be a generalized Poisson algebra.
Then $(\mathsf{A};\llbracket\\{\cdot,\cdot\\},\cdot\rrbracket)$ is an
associative-commutative algebra, and
$(\mathsf{A};\llbracket\cdot,\\{\cdot,\cdot\\}\rrbracket)$ is a Lie algebra.

######  Theorem 3.4 (Proposition 18, [87]).

Let $(\mathsf{A};\cdot,\\{\cdot,\cdot\\})$ be a Poisson algebra. Then

$(\mathsf{A};\llbracket\cdot+\\{\cdot,\cdot\\},\cdot+\\{\cdot,\cdot\\}\rrbracket)$

is a noncommutative Jordan algebra.

######  Theorem 3.5 (Corollary 31, [87]).

Let $(\mathsf{A};\cdot,\circ)$ be a left Novikov-Poisson algebra. Then
$(\mathsf{A},\llbracket\circ,\cdot\rrbracket,\llbracket\cdot,\circ\rrbracket)$
is a left Novikov-Poisson algebra.

Also, we give constructive methods for obtaining new transposed Poisson
algebras and Poisson-Novikov algebras; and for classifying Poisson structures
and commutative post-Lie structures on a given algebra and construct some new
transposed Poisson algebras.

Let us now consider a special example of transposed Poisson algebras
constructed in [93, Theorem 25]. The transposed Poisson algebra
$(\mathcal{W},\cdot,[\cdot,\cdot])$ is spanned by the generators $\\{L_{i},\
I_{j}\\}_{i,j\in\mathbb{Z}}$. These generators satisfy

$[L_{m},L_{n}]$ | $=$ |  $(m-n)L_{m+n}$, | $[L_{m},I_{n}]$ | $=$ |  $(m-n-a~{})I_{m+n}$,
---|---|---|---|---|---
$L_{m}\cdot L_{n}$ | $=$ |  $wL_{m+n}$, | $L_{m}\cdot I_{n}$ | $=$ |  $wI_{m+n}$,

where $w$ is a fixed element from the vector space generated by $\\{L_{i},\
I_{j}\\}_{i,j\in\mathbb{Z}}$ and the multiplication given by juxtaposition
satisfies $L_{i}L_{j}=L_{i+j}$ and $L_{i}I_{j}=I_{i+j}$.

######  Theorem 3.6 (Proposition 27, [87]).

Let $\star=\llbracket[\cdot,\cdot],\cdot\rrbracket$ and
$\\{\cdot,\cdot\\}=\llbracket\ \cdot,[\cdot,\cdot]\rrbracket$ be new
multiplications defined on multiplications of the transposed Poisson algebra
$(\mathcal{W},\cdot,[\cdot,\cdot])$ defined above. Then
$(\mathcal{W},\star,\\{\cdot,\cdot\\})$ is a transposed Poisson algebra.

###  3.2 Universal conservative superalgebras

Kantor introduced the class of conservative algebras in [133]. This class
includes some well-known classes of algebras, such as associative, Jordan,
Lie, Leibniz and Zinbiel [157]. To define conservative algebras we first
introduce some notations. Let $V$ be a vector space over a field $\mathbb{F}$,
let $\varphi$ be a linear map on $V$, and let $B$ be a bilinear map on $V$ (i.
e., an algebra). Then we can consider a product of $\varphi$ and $B$, which is
the bilinear map $[\varphi,B]$ on $V$ given by

$[\varphi,B](x,y)=\varphi(B(x,y))-B(\varphi(x),y)-B(x,\varphi(y)).$

Note that this product measures how far is $\varphi$ from being a derivation
in the algebra $B$.

We say that an algebra $\mathsf{A}$ with a multiplication $\cdot$ is called a
(left) conservative algebra if there exists a (possibly different) algebra
structure $*$ (called an associated algebra) on the underlying space of
$\mathsf{A}$ such that

$[L_{a},[L_{b},\cdot]]=-[L_{a*b},\cdot].$

By replacing the left multiplications with the right multiplications, we can
define the right Kantor product and obtain a similar theory. In the theory of
conservative algebras, the conservative algebra $U(n)$ introduced in Section
3.1 has high importance. One can verify that the algebra $U(n)$ is
conservative with the associated multiplication $\ast$ given, for example, by
$A\ast B(x,y)=-B(u,A(x,y))$ (there are other associated multiplications). In
the theory of conservative algebras, the algebra $U(n)$ plays a role analogous
to the role of $\mathfrak{gl}_{n}$ in the theory of Lie algebras, that is, any
conservative algebra (modulo its maximal Jacobian ideal) embeds in $U(n)$ for
certain $n$. In 1989, Kantor defined one generalization of conservative
algebras, which he called quasi-conservative algebras (now known as “rigid”
algebras, see [61]). In 2010, Kac and Cantarini considered “rigid”
(commutative and anticommutative) superalgebras and described simple
superalgebras under some special conditions [61].

One may also use the general approach to define conservative superalgebras.
Namely, let $\Gamma:=\Gamma_{0}\oplus\Gamma_{1}$ be the Grassmann superalgebra
in generators $1,\ \xi_{i},\ i\in\mathbb{N},$

$\Gamma_{0}=\left<1,\xi_{i_{1}}\ldots\xi_{i_{2k}}:k\in\mathbb{N}\right>,\Gamma_{1}=\left<\xi_{i_{1}}\ldots\xi_{i_{2k-1}}:k\in\mathbb{N}\right>.$

Let $\mathsf{A}:=\mathsf{A}_{0}\oplus\mathsf{A}_{1}$ be a superalgebra and
$\cdot$ and $*$ be two products on $\mathsf{A}$. Consider its Grassmann
envelope
$\Gamma({\mathsf{A}}):=(\mathsf{A}_{0}\otimes\Gamma_{0})\oplus(\mathsf{A}_{1}\otimes\Gamma_{1})$
and extend the products $\cdot$ and $*$ to $\Gamma(\mathsf{A})$ as follows:

$(a\otimes f)\cdot(b\otimes g)=(-1)^{ab}ab\otimes fg,$ $(a\otimes f)*(b\otimes
g)=(-1)^{ab}a*b\otimes fg$

for all homogeneous $a,b\in{\mathsf{A}},f,g\in\Gamma\
(p(a)=p(f),\,p(b)=p(g))$. Then $(\mathsf{A},\cdot)$ is conservative with an
associated multiplication $*$ if and only if $(\Gamma(\mathsf{A}),\cdot)$ is a
conservative algebra with an associated multiplication $*$.

The following theorem provides different examples of conservative
superalgebras.

######  Theorem 3.7 (Theorem 2, [169]).

Let ${\cal V}$ be a homogeneous variety of algebras. Assume that there exist
$\alpha,\beta\in\mathbb{F}$ such that every ${\cal V}$-algebra is conservative
with the associated multiplication is given by the rule $a\,*\,b~{}=~{}\alpha
ab~{}+~{}\beta ba$. Then every ${\cal V}$-superalgebra is conservative with
the associated multiplication $a\,*\,b~{}=~{}\alpha ab+(-1)^{ab}\beta ba$.

It follows that associative, quasi-associative, Jordan, terminal, Lie,
Leibniz, and Zinbiel superalgebras are conservative (see [157]). In
particular, a superalgebra $M$ is terminal if and only if it is conservative
and the multiplication in the associated superalgebra $M^{*}$ can be given by

$M^{*}(x,y)=\frac{2}{3}xy+(-1)^{xy}\frac{1}{3}yx.$

######  Theorem 3.8 (Proposition 4, [169]).

A conservative superalgebra $\mathsf{A}$ with a unity is a noncommutative
Jordan superalgebra.

Let $V$ be a superspace. The space of the superalgebra $U(V)$ is the
superspace of all bilinear operations on $V$. Fix a nonzero homogeneous $a\in
V$. Define the multiplication ${\Delta}_{a}$ in $U(V)$ by the rule

$(A{\Delta}_{a}B)(x,y)=A(a,B(x,y))-(-1)^{\scriptscriptstyle
B(A+a)}B(A(a,x),y)-(-1)^{\scriptscriptstyle(A+a)(B+x)}B(x,A(a,y))$.

Consider the natural action of the group ${\rm GL}(V)$ of even automorphisms
of $V$ on $U(V):$

$\varphi(A)(x,y)=\varphi(A(\varphi^{-1}(x),\varphi^{-1}(y)))$

(note that we denote an automorphism and its action by the same symbol
$\varphi$). A direct computation shows that the mapping $A\mapsto\varphi(A)$
is an isomorphism between $(U(V),\Delta_{a})$ and $(U(V),\Delta_{\varphi(a)})$
Therefore, different nonzero even (respectively, odd) vectors $a$ give rise to
isomorphic even (respectively, odd) superalgebras, which we denote by
$U(V)^{0}$ and $U(V)^{1}$, respectively. Moreover, consider the opposite
superspace $V^{\Pi}$ given by
$V^{\Pi}_{\bar{0}}=V_{\bar{1}},V^{\Pi}_{\bar{1}}=V_{\bar{0}}$. Then the
parity-reversing isomorphism $V\cong V^{\Pi}$ induces an isomorphism between
$U(V^{\Pi})^{1}$ and the odd superalgebra obtained from $U(V)^{0}$ by
reversing the parity. Therefore, it suffices to consider only the
superalgebras $U(V)^{0}$. For the sake of simplicity, we denote them by
$U(V)$. If $V=V_{n,m}$ is a finite-dimensional superspace with ${\rm
dim}V_{\bar{0}}=n$ and ${\rm dim}V_{\bar{1}}=m$ then we denote $U(V)$ by
$U(n,m)$. Further, in this case, we say that $V$ is of dimension $n+m$.

######  Theorem 3.9 (Theorem 8, [169]).

Let $V$ be a superspace, and let $a\in V_{\bar{0}}$. The superalgebra
$(U(V),\Delta_{a})$ is conservative, and the associated multiplication can be
given by

$A\bigtriangledown^{1}_{a}B(x,y)=-(-1)^{AB}B(a,A(x,y))$

or

$A\bigtriangledown^{2}_{a}B(x,y)=\frac{1}{3}(A^{*}\Delta_{a}B+(-1)^{AB}\tilde{B}\Delta_{a}A)$,

where $A^{*}(x,y)=A(x,y)+(-1)^{xy}A(y,x)$ and
$\tilde{B}(x,y)=2(-1)^{xy}B(y,x)-B(x,y)$.

An even element $e\in\mathsf{A}$ is said to be a left quasiunity if the
equality

$e(xy)=(ex)y+x(ey)-xy$

holds.

An element $a$ in a superalgebra $\mathsf{A}$ is called a Jacobi element
provided that

$a(xy)=(ax)y+(-1)^{ax}x(ay).$

Let $\mathsf{A}$ be a conservative superalgebra on a space $V$ with the Jacobi
subspace $J$. Consider the space $W$, which we define as $W=V/J$ if
$\mathsf{A}$ has a left quasiunity, and $W=V/J\oplus E$ in the opposite case,
where $E$ is the one-dimensional even space with a basis element $\epsilon$.

Assume that $\mathsf{A}$ possesses a quasiunity. Define the adjoint mapping
$\operatorname{ad}:\mathsf{A}\to U(W)$ as follows:

$\operatorname{ad}(a)(\alpha,\beta)=(-1)^{\beta(a+\alpha)}((\beta*a)*\alpha+(-1)^{\alpha
a}\beta*(\alpha a)-(-1)^{\alpha a}(\beta*\alpha)*a)$.

If $\mathsf{A}$ does not have a quasiunity, we define the adjoint mapping
$\operatorname{ad}:\mathsf{A}\to U(W)$ by the equation above and the following
equations:

$\operatorname{ad}(a)(\alpha,\epsilon)=a*\alpha+(-1)^{\alpha a}\alpha
a~{}-(-1)^{\alpha a}\alpha*a,$

$\operatorname{ad}(a)(\epsilon,\beta)=(-1)^{a\beta}\beta*a,\
\operatorname{ad}(a)(\epsilon,\epsilon)=a$.

######  Theorem 3.10 (Theorem 11, [169]).

Let $\mathsf{A}$ be a conservative superalgebra on a vector space $V$ with the
Jacobi subspace $J$. Let either $W=V/J$ or $W=V/J\oplus\left<\epsilon\right>$
as above. The adjoint mapping
$\operatorname{ad}:\mathsf{A}\to(U(W),\Delta_{-\epsilon})$ is a homomorphism
whose kernel is the maximal Jacobi ideal. In particular, if $V$ is finite-
dimensional and $J$ is of codimension $n+m$, then we have a homomorphism
$\operatorname{ad}:\mathsf{A}\to U(k,m)$, where $k=n$ if $\mathsf{A}$ has a
quasiunity and $k=n+1$ otherwise.

###  3.3 Conservative algebras of $2$-dimensional algebras

Multiplication on a 2-dimensional vector space is defined by a $2\times
2\times 2$ matrix. The classification of algebras of dimension $2$ was given
in many papers (see, for example, [177]). Let us consider the space $U(2)$ of
all multiplications on the $2$-dimensional space $V_{2}$ with a basis
$v_{1},v_{2}$. The definition of the multiplication on the algebra $U(2)$ can
be found in Section 3.1. The algebra $U(2)$ is conservative. Let us consider
the multiplications $\alpha_{i,j}^{k}$ ($i,j,k=1,2$) on $V_{2}$ defined by the
formula $\alpha_{i,j}^{k}(v_{t},v_{l})=\delta_{it}\delta_{jl}v_{k}$ for all
$t,l$. It is easy to see that $\\{\alpha_{i,j}^{k}|i,j,k=1,2\\}$ is a basis of
the algebra $U(2)$. The multiplication table of $U(2)$ in this basis is given
in [157]. But in the next work [175], we found another useful basis that will
be interesting to consider. Let’s introduce the notation

$e_{1}=\alpha_{11}^{1}-\alpha_{12}^{2}-\alpha_{21}^{2}$, |  $e_{2}=\alpha_{11}^{2}$, |  $e_{3}=\alpha_{22}^{2}-\alpha_{12}^{1}-\alpha_{21}^{1}$, |  $e_{4}=\alpha_{22}^{1}$,
---|---|---|---
$e_{5}=2\alpha_{11}^{1}+\alpha_{12}^{2}+\alpha_{21}^{2}$, |  $e_{6}=2\alpha_{22}^{2}+\alpha_{12}^{1}+\alpha_{21}^{1}$, |  $e_{7}=\alpha_{12}^{1}-\alpha_{21}^{1}$, |  $e_{8}=\alpha_{12}^{2}-\alpha_{21}^{2}$.

It is easy to see that the multiplication table of $U(2)$ in the basis
$e_{1},\dots,e_{8}$ is the following.

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

The subalgebra spanned by the elements $e_{1},\ldots,e_{6}$ is the terminal
algebra $W_{2}$ of commutative 2-dimensional algebras. The subalgebra spanned
by the elements $e_{1},\ldots,e_{4}$ is the conservative (and, moreover,
terminal) algebra $S_{2}$ of all commutative 2-dimensional algebras with trace
zero multiplication [157]. In a series of our papers devoted to the study of
$U(2)$ [157, 175], we described one-sided ideals, subalgebras of codimension
$1$, derivations and automorphisms of $U(2)$, $W_{2}$ and $S_{2}$.

We summarize all the main results of the present part in the following
theorems.

######  Theorem 3.11 (Theorem 11, [175]).

${\rm Aut}\big{(}U(2)\big{)}\cong{\rm Aut}(W_{2})\cong{\rm Aut}(S_{2})$. Also,
this group is isomorphic to the matrix group $\begin{pmatrix}1&\mathbb{F}\\\
0&\mathbb{F}^{*}\end{pmatrix}$ (the one-dimensional affine group over
$\mathbb{F}$).

######  Theorem 3.12 (Theorem 11, [175]).

The set of nonzero idempotents of the algebra $U(2)$ equals the disjoint union
of the following sets (where $c,d\in\mathbb{F}$, $q\in\bar{\mathbb{F}}$):

$\mathcal{O}(e_{8}+e_{2}-e_{1}+c(3e_{8}+e_{5}-2e_{1}))$
---
$\mathcal{O}(-e_{1}+c(e_{5}-2e_{1})+de_{8})$
$\mathcal{O}(-e_{1}-2e_{8}+4e_{3}+e_{6}+3e_{7}+c(3e_{8}-e_{5}+2e_{1})+de_{4})$
$\mathcal{O}(-e_{1}-2e_{8}+c(3e_{8}-e_{5}+2e_{1})+qe_{4})$.

##  4 Non-associative algebras and superalgebras

The present part is based on the papers written together with Antonio Jesús
Calderón, Artem Lopatin, Bakhrom Omirov, Gulkhayo Solijanova, Luisa Camacho,
Pasha Zusmanovich and Yury Popov [159, 180, 59, 44].

###  4.1 Leibniz algebras

Gradings by abelian groups have played a key role in the study of Lie algebras
and superalgebras, starting with the root space decomposition of the
semisimple Lie algebras over the complex field, which is an essential
ingredient in the Killing-Cartan classification of these algebras. Gradings by
a cyclic group appear in the connection between Jordan algebras and Lie
algebras through the Tits-Kantor-Koecher construction and in the theory of
Kac-Moody Lie algebras. Gradings by the integers or the integers modulo $2$
are ubiquitous in Geometry. In 1989, Patera and Zassenhaus [219] began a
systematic study of gradings by abelian groups on Lie algebras. They raised
the problem of classifying the fine gradings, up to equivalence, on the simple
Lie algebras over the complex numbers. After that, gradings of simple
alternative and simple Malcev algebras [80], the simple Kac Jordan
superalgebra [45], countless simple Lie algebras [77, 81] and nilpotent Lie
algebras [19] were described.

In the past years, Leibniz algebras have been under active research (see, for
example, [74, 123, 170, 186, 187]). Recently, they have appeared in many
geometric and physics applications (see, for example, [35, 74, 194, 238] and
references therein). The main result on the structure of finite-dimensional
Leibniz algebras asserts that a Leibniz algebra decomposes into a semidirect
sum of a solvable radical and a semisimple Lie algebra. Therefore, the main
problem of the description of finite-dimensional Leibniz algebras consists of
the study of solvable Leibniz algebras. Similarly to the case of Lie algebras,
the study of solvable Leibniz algebras is reduced to nilpotent ones. Since the
description of all $n$-dimensional nilpotent Leibniz algebras is an unsolvable
task (even in the case of Lie algebras), we have to study nilpotent Leibniz
algebras under certain conditions (conditions on index of nilpotency, various
types of grading, characteristic sequence etc.) [186, 187]. The well-known
natural grading of nilpotent Lie and Leibniz algebras is very helpful when
investigating the properties of those algebras without restrictions on the
grading. Indeed, we can always choose a homogeneous basis and thus the grading
allows us to obtain more explicit conditions for the structural constants.
Moreover, such grading is useful for the investigation of cohomologies for the
considered algebras, because it induces the corresponding grading of the group
of cohomologies. Thus, it is crucial to know what kind of grading a nilpotent
Leibniz algebra admits.

In our work [44] we begin the study of gradings on Leibniz algebras by
classifying, up to equivalence, all abelian groups gradings of null-filiform
and one-parametric filiform Leibniz algebras. Let us define all the considered
algebras.

######  Definition 4.1.

An arbitrary complex $n$-dimensional null-filiform Leibniz algebra is
isomorphic to the algebra

$e_{i}e_{1}=e_{i+1},\quad 1\leq i\leq n-1.$

######  Definition 4.2.

A complex $n$-dimensional one-parametric filiform Leibniz algebra is
isomorphic to the algebra of the following form:

$e_{1}e_{2}=\theta e_{n},\ e_{i}e_{1}=e_{i+1},\ 2\leq i\leq{n-1}.$

We described all abelian gradings of null-filiform Leibniz algebras and one-
parametric filiform Leibniz algebras in [44, Theorems 13, 17 and 19].

A very powerful tool in the study of nilpotent algebras is the characteristic
sequence, in which a priori gives the multiplication on one element of the
basis.

######  Definition 4.3.

For a nilpotent Leibniz (Lie) algebra $\mathsf{A}$ and
$x\in\mathsf{A}\setminus\mathsf{A}^{2}$ we consider the decreasing sequence
$C(x)=(n_{1},n_{2},\ldots,n_{k})$ with respect to the lexicographical order of
the dimensions Jordan’s blocks of the operator $R_{x}$. The sequence
$C(L)=\max\limits_{x\in\mathsf{A}\setminus\mathsf{A}^{2}}C(x)$ is called the
characteristic sequence of the Leibniz algebra $\mathsf{A}$.

Recently, in the paper [10] it was considered a finite-dimensional solvable
Lie algebra $\mathfrak{r}_{c}$ whose nilpotent radical $\mathfrak{n}_{c}$ has
the simplest structure with a given characteristic sequence
$c=(n_{1},n_{2},\ldots,n_{k},1)$. Using the Hochschild – Serre factorization
theorem, the authors established that the low order cohomology groups of the
algebra $\mathfrak{r}_{c}$ with coefficients in the adjoint representation are
trivial. In our paper [59], we study a family of nilpotent Leibniz algebras
whose corresponding Lie algebra is $\mathfrak{n}_{c}$. Further, solvable
Leibniz algebras with such nilpotent radicals and $(k+1)$-dimensional
complementary subspaces to the nilpotent radicals are described. Namely, we
prove that such solvable Leibniz algebra is unique and centerless.

$\mathbf{R}:\left\\{\begin{array}[]{lll}[e_{1},e_{1}]=h,\quad[h,x_{1}]=2h,&\\\\[2.84526pt]
[e_{i},e_{1}]=-[e_{1},e_{i}]=e_{i+1},&2\leq i\leq n_{1},&\\\\[2.84526pt]
[e_{n_{1}+\ldots+n_{j}+i},e_{1}]=-[e_{1},e_{n_{1}+\ldots+n_{j}+i}]=e_{n_{1}+\ldots+n_{j}+1+i},&2\leq
i\leq n_{j+1},\\\\[2.84526pt]
[e_{1},x_{1}]=-[x_{1},e_{1}]=e_{1},&\\\\[2.84526pt]
[e_{i},x_{1}]=-[x_{1},e_{i}]=(i-2)e_{i},&3\leq i\leq n_{1}+1,\\\\[2.84526pt]
[e_{n_{1}+\ldots+n_{j}+i},x_{1}]=-[x_{1},e_{n_{1}+\ldots+n_{j}+i}]=(i-2)e_{n_{1}+\ldots+n_{j}+i}&2\leq
i\leq n_{j+1},\\\\[2.84526pt] [e_{i},x_{2}]=-[x_{2},e_{i}]=e_{i},&2\leq i\leq
n_{1}+1,\\\\[2.84526pt]
[e_{n_{1}+\ldots+n_{j}+i},x_{j+2}]=-[x_{j+2},e_{n_{1}+\ldots+n_{j}+i}]=e_{n_{1}+\ldots+n_{j}+i},&2\leq
i\leq n_{j+1}.\\\\[2.84526pt] \end{array}\right.$

where $1\leq j\leq k-1$. For this Leibniz algebra, the triviality of the first
and the second cohomology groups with coefficients in the adjoint
representation is established as well.

######  Theorem 4.4 (Theorem 21, [59]).

The first and second groups of cohomologies of the solvable Leibniz algebra
$\mathbf{R}$ in coefficient itself are trivial.

######  Theorem 4.5 (Corollary 22, [59]).

The solvable Leibniz algebra $\mathbf{R}$ is rigid.

######  Remark 4.6.

Note that the structure of the rigid algebra $\mathbf{R}$ depends on the given
decreasing sequence $(n_{1},n_{2},\ldots,n_{k})$. Set $p(n)$ to be the number
of such sequences, that is, $p(x)$ is the number of integer solutions of the
equation

$n_{1}+n_{2}+\ldots+n_{k}=n$ with $n_{1}\geq n_{2}\geq\ldots\geq n_{k}\geq 0$.

The asymptotic value of $p(n)$, given by the expression

$p(n)\approx\frac{1}{4n\sqrt{3}}e^{\pi\sqrt{2n/3}}$ (where $a(n)\approx b(n)$
means that $\lim\limits_{n\rightarrow\infty}\frac{a(n)}{b(n)}=1$),

yields the existence of at least $p(n)$ irreducible components of the variety
of Leibniz algebras of dimension $n+k+3$.

###  4.2 Anticommutative $\mathfrak{CD}$-algebras

The idea of the definition of the variety of $\mathfrak{CD}$-algebras comes
from the following property of Jordan and Lie algebras: the commutator of any
pair of multiplication operators is a derivation. This gives three identities
of degree four, which reduces to only one identity of degree four in the
commutative or anticommutative case. Commutative and anticommutative
$\mathfrak{CD}$-algebras are related to many interesting varieties of
algebras. Thus, anticommutative $\mathfrak{CD}$-algebras are a generalization
of Lie algebras, containing the intersection of Malcev and Sagle algebras as a
proper subvariety. Moreover, the following intersections of varieties
coincide: Malcev and Sagle algebras; Malcev and anticommutative
$\mathfrak{CD}$-algebras; and Sagle and anticommutative
$\mathfrak{CD}$-algebras. On the other hand, the variety of anticommutative
$\mathfrak{CD}$-algebras is a proper subvariety of the varieties of binary Lie
algebras and almost Lie algebras [180]. The variety of anticommutative
$\mathfrak{CD}$-algebras coincides with the intersection of the varieties of
binary Lie algebras and almost Lie algebras. Commutative
$\mathfrak{CD}$-algebras are a generalization of Jordan algebras, which are a
generalization of commutative associative algebras. On the other hand, the
variety of commutative $\mathfrak{CD}$-algebras is also known as the variety
of almost-Jordan algebras, which lies in the bigger variety of generalized
almost-Jordan algebras [99]. The $n$-ary version of commutative
$\mathfrak{CD}$-algebras was introduced in a recent paper by Kaygorodov,
Pozhidaev and Saraiva [172]. The variety of almost-Jordan algebras is the
variety of commutative algebras satisfying

$2((yx)x)x+yx^{3}=3(yx^{2})x.$

This present identity appeared in 1965 in a paper by Osborn during the study
of identities of degrees less than or equal to $4$ of non-associative
algebras. The identity is a linearized form of the Jordan identity. The
systematic study of almost-Jordan algebras was initiated in the following
paper by Osborn and it was continued in some papers by Petersson [221], Osborn
[217], and Sidorov (sometimes, they were called Lie triple algebras). Hentzel
and Peresi proved that every semiprime almost-Jordan ring is Jordan [116].
After that, Labra and Correa proved that a finite-dimensional almost-Jordan
right-nilalgebra is nilpotent. Assosymmetric algebras under the symmetric
product give almost-Jordan algebras [79].

An anticommutative algebra $\mathsf{A}$ is called a $\mathfrak{CD}$-algebra if
it satisfies the property that for any $a,b\in\mathsf{A}$, the commutator
$[R_{a},R_{b}]$ is a derivation of $\mathsf{A}$. This condition can be written
as a homogeneous identity of degree $4$ comprising $6$ monomials:

$((xy)a)b-((xy)b)a-((xa)b)y+((xb)a)y+((ya)b)x-((yb)a)x=0.$

We introduced the notion of anticommutative $\mathfrak{CD}$-algebras and
studied their properties in [180]. Firstly, we compare the variety of
anticommutative $\mathfrak{CD}$-algebras with other known varieties, namely
Lie, binary Lie, Malcev, and Sagle. Secondly, we consider the cohomology
theory of anticommutative $\mathfrak{CD}$-algebras and obtain some relations
with Lie algebras. Namely, we prove the following results.

######  Theorem 4.7 (Proposition 3, [180]).

For any $\mathfrak{CD}$-algebra $\mathsf{A}$, the quotient
$\mathsf{A}/Z(\mathsf{A})$ is a Lie algebra. In particular, any centerless
(and, in particular, simple) $\mathfrak{CD}$-algebra is a Lie algebra.

######  Theorem 4.8 (Theorem, [180]).

For any simple finite-dimensional Lie algebra $\mathsf{A}$ over a field of
characteristic zero, and any finite-dimensional $\mathsf{A}$-module $M$, ${\rm
H}_{\mathfrak{CD}}^{2}(\mathsf{A},M)=0$.

###  4.3 Noncommutative Jordan superalgebras

The class of noncommutative Jordan algebras is vast. As an example, it
includes alternative algebras, Jordan algebras, quasiassociative algebras,
quadratic flexible algebras, and anticommutative algebras. Schafer proved that
a simple noncommutative Jordan algebra is either a simple Jordan algebra, a
simple quasiassociative algebra, or a simple flexible algebra of degree $2$.
Oehmke proved the analog of Schafer’s classification for simple flexible
algebras with strictly associative powers in the case of characteristic
different from $2$, $3$, McCrimmon classified simple noncommutative Jordan
algebras of degree greater than $2$ in the case of characteristic different
from $2$, and Smith described such algebras of degree $2$. The case of nodal
simple algebras of positive characteristic was considered in the papers by
Kokoris. The case of simple finite-dimensional Jordan superalgebras over
algebraically closed fields of characteristic zero was studied by Kac and
Kantor. Racine and Zelmanov classified the finite-dimensional Jordan
superalgebras of characteristic different from $2$ with a semisimple even
part. The case when the even part is not semisimple was considered by Martínez
and Zelmanov in [207] and Cantarini and Kac described all linearly compact
simple Jordan superalgebras [60]. Simple noncommutative Jordan superalgebras
were described by Pozhidaev and Shestakov in [225, 226]. Representations of
simple noncommutative superalgebras were described by Popov [223]. Nowadays,
the study of properties of simple non-associative algebras and superalgebras
has aroused a strong interest. For example, Popov determined the structure of
differentiably simple Jordan algebras, and Barreiro, Elduque, and Martínez
described the derivations of the Cheng-Kac Jordan superalgebra [23]. Moreover,
Kaygorodov, Shestakov, and Zhelyabin studied generalized derivations of Jordan
algebras and superalgebras [231, 232]. Another interesting problem in the
study of Jordan algebras and superalgebras is a description of maximal
subalgebras and automorphisms [82, 83].

A superalgebra $\mathsf{A}$ is called a noncommutative Jordan superalgebra if
it satisfies the following operator identities:

$[R_{x\circ y},L_{z}]+(-1)^{|x|(|y|+|z|)}[R_{y\circ
z},L_{x}]+(-1)^{|z|(|x|+|y|)}[R_{z\circ x},L_{y}]=0,$
$[R_{x},L_{y}]=[L_{x},R_{y}].$

A binary linear operation $\\{\cdot,\cdot\\}$ is called a generic Poisson
bracket [174] on a superalgebra $(\mathsf{A},\cdot)$ if for arbitrary
homogeneous $a,b,c\in\mathsf{A}$ we have

$\\{a\cdot b,c\\}=(-1)^{|b||c|}\\{a,c\\}\cdot b+a~{}\cdot\\{b,c\\}.$

We notice that there is a one-to-one correspondence between noncommutative
Jordan superalgebras and superanticommutative Poisson brackets on adjoint
Jordan superalgebras. Let $(\mathsf{A},\bullet)$ be a Jordan superalgebra and
$[\cdot,\cdot]$ be a generic Poisson bracket on $\mathsf{A}$. Then the
operation $ab=\frac{1}{2}(a\bullet b+[a,b])$ turns
$(\mathsf{A},\bullet,[\cdot,\cdot])$ into a noncommutative Jordan
superalgebra. Conversely, if $\mathsf{A}$ is a noncommutative Jordan
superalgebra, then the supercommutator $[\cdot,\cdot]$ is a generic Poisson
bracket on a Jordan superalgebra $\mathsf{A}^{(+)}$. Moreover, the
multiplication in $\mathsf{A}$ can be recovered by the Jordan multiplication
in $\mathsf{A}^{(+)}$ and the generic Poisson bracket
$[\cdot,\cdot]:ab=\frac{1}{2}(a\bullet b+[a,b])$.

In our paper with Lopatin and Popov, we study simple noncommutative Jordan
superalgebras constructed in some papers by Pozhidaev and Shestakov [225]. We
describe all subalgebras and automorphisms of simple noncommutative Jordan
superalgebras $K_{3}(\alpha,\beta,\gamma)$ and $D_{t}(\alpha,\beta,\gamma)$
[159]. We also compute the derivations of the nontrivial simple finite-
dimensional noncommutative Jordan superalgebras [159, Theorems 9 and 10]].

##  5 Maps on associative and non-associative algebras

The present part is based on the papers written together with Artem Lopatin,
Bruno Ferreira, Feng Wei, Henrique Guzzo, Mykola Khrypchenko and Yury Popov
[158, 154, 91, 90, 165].

###  5.1 Alternative and Jordan algebras with invertible derivations

In 1983, Bergen, Herstein, and Lanski initiated a study whose purpose was to
relate the structure of a ring to the special behavior of one of its
derivations. Namely, in their article [34] they described associative rings
admitting derivations with invertible values, i.e., there is a derivation $d$
such that $d(a)$ is an invertible element or zero for every $a$ from our
algebra. They proved that such ring must be either a division ring, or the
ring of $2\times 2$ matrices over a division ring, or a factor of a polynomial
ring over a division ring of characteristic 2. They also characterized the
division rings such that the $2\times 2$ matrix ring over them has an inner
derivation with invertible values. Further, associative rings with derivations
with invertible values (and also their generalizations) were discussed in a
variety of works (see, for instance, [104]). So, in [104] semiprime
associative rings with involution, allowing a derivation with invertible
values on the set of symmetric elements, were examined. In their work, Bergen
and Carini studied associative rings, admitting a derivation with invertible
values on some non-central Lie ideal. Also, in the papers by Chang, Hongan and
Komatsu, the structure of associative rings that admit $\alpha$-derivations
with invertible values and their natural generalizations —
$(\sigma,\tau)$-derivations with invertible values, was described. Komatsu and
Nakajima described associative rings that allow generalized derivations with
invertible values. The case of associative superalgebras with derivations with
invertible values was studied in the paper by Demir, Albaş, Argaç, and Fosner.

The description of non-associative algebras admitting derivations with
invertible values began in our paper with Popov [165], where it was proved
that every alternative (non-associative) algebra admitting derivation with
invertible values is a Cayley — Dickson division algebra over their center or
a factor-algebra of a polynomial algebra $C[x]/(x^{2})$ over a Cayley —
Dickson division algebra. Our research was continued in a joint paper with
Lopatin and Popov [158], where we describe Jordan algebras admitting
derivations with invertible values. Let us summarize the main results from
[158].

######  Theorem 5.1 (Theorem 3, [158]).

Let $\mathsf{A}$ be a Jordan algebra admitting a derivation with invertible
values $d$. Then $\mathsf{A}$ is an extension of a simple Jordan algebra with
derivation with invertible values by an ideal $M$ such that $M^{2}=0,d(M)=0$
and $M$ is the largest ideal of $\mathsf{A}$.

######  Theorem 5.2 (Theorem 11, [158]).

Let $\mathsf{A}$ be a Jordan algebra of characteristic not $2$ admitting a
derivation with invertible values $d$. Then one of the following holds:

  * $(1)$

$\mathsf{A}$ is an algebra $A^{(+)}$, where $A=D$ or $D_{2}$, and $D$ is an
associative division algebra;

  * $(2)$

$\mathsf{A}$ is an algebra $H(A,*)$, where $A=D$ or $D_{2}$, and $D$ is an
associative division algebra;

  * $(3)$

$\mathsf{A}$ is an algebra of symmetric nondegenerate bilinear form $J(V,f)$;

  * $(4)$

$\mathsf{A}$ is a division algebra of Albert type;

  * $(5)$

$\mathsf{A}$ is an extension of cases $(1)$ – $(4)$ by
$M=\mathbf{P}(\mathsf{A})$ (where $\mathbf{P}(\mathsf{A})$ stands for the
prime radical of $\mathsf{A}$), where $M\subseteq\ker d$ and $M$ is the
largest ideal of $\mathsf{A}$.

######  Theorem 5.3 (Theorem 20, [158]).

Let $\mathsf{A}$ be a finite-dimensional Jordan algebra admitting a derivation
with invertible values. Then $\mathsf{A}$ is either simple or an algebra of a
symmetric bilinear form (possibly degenerated).

###  5.2 Maps on alternative algebras

The study of Lie isomorphisms of rings was originally inspired by Herstein’s
generalization of generalizing classical theorems on the Lie structure of
total matrix rings to the Lie structure of arbitrary simple rings [117]. In
this paper, Herstein formulated some open questions that gave rise to big
research in Lie and Jordan maps on various structures. The main questions from
Herstein’s program were solved for associative algebras in papers by Beidar,
Brešar, Chebotar and Martindale [27, 26]. In [205], Martindale studied Lie
isomorphisms between primitive rings $\mathsf{A}$, $\mathsf{A}^{\prime}$,
where he assumed that the characteristic of $\mathsf{A}$ is different from $2$
and $3$ and that $\mathsf{A}$ contains three nonzero orthogonal idempotents
whose sum is the identity. A few years later, he studied Lie isomorphisms
between two simple rings $\mathsf{A}$, $\mathsf{A}^{\prime}$. Alternative
algebras, as well as associative algebras, admit a Peirce decomposition. The
recently introduced axial algebras reflect an active interest in the study of
algebras admitting a Peirce-type decomposition [181]. In our works together
with Ferreira and Guzzo, we continue the study of the Herstein program for
some special types of alternative algebras admitting Peirce decomposition.

In the following theorems, we summarize the main results of the present part.

######  Theorem 5.4 (Theorem 2.1, [90]).

Let $\mathsf{A}$ be a unital prime (simple, simple associative) alternative
algebra, let $\mathsf{A}^{\prime}$ be another prime (simple, simple
associative) alternative algebra and let
$\varphi:\mathsf{A}\rightarrow\mathsf{A}^{\prime}$ be a surjective Lie
multiplicative map that preserves idempotents. Assume that $\mathsf{A}$ has a
nontrivial idempotent $e_{1}$ with associated Peirce decomposition
$\mathsf{A}=\mathsf{A}_{11}\oplus\mathsf{A}_{12}\oplus\mathsf{A}_{21}\oplus\mathsf{A}_{22}$,
such that

  * (1)

if $x_{ij}\mathsf{A}_{ji}=0$, then $x_{ij}=0$ $(i\neq j)$;

  * (2)

if $x_{11}\mathsf{A}_{12}=0$ or $\mathsf{A}_{21}x_{11}=0$, then $x_{11}=0$;

  * (3)

if $\mathsf{A}_{12}x_{22}=0$ or $x_{22}\mathsf{A}_{21}=0$, then $x_{22}=0$;

  * (4)

if $z\in Z(\mathsf{A})$ with $z\neq 0$, then $z\mathsf{A}=\mathsf{A}$.

Then, the following holds.

  * •

If $f_{i}\varphi(\mathsf{A}_{jj})f_{i}\subseteq Z(\mathsf{A}^{\prime})f_{i}$,
then $\varphi$ is of the form $\psi+\tau$, where $\psi$ is an additive
isomorphism between $\mathsf{A}$ and $\mathsf{A}^{\prime}$ and $\tau$ is a map
from $\mathsf{A}$ to $Z(\mathsf{A}^{\prime})$, which maps commutators into
zero.

  * •

If $f_{i}\varphi(\mathsf{A}_{ii})f_{i}\subseteq Z(\mathsf{A}^{\prime})f_{i}$,
then $\varphi$ is of the form $\psi+\tau$, where -$\psi$ is additive anti-
isomorphism between $\mathsf{A}$ and $\mathsf{A}^{\prime}$ and $\tau$ is a map
from $\mathsf{A}$ to $Z(\mathsf{A}^{\prime})$, which maps commutators into
zero. Observe $f_{i}=\varphi(e_{i})$ and
$f_{j}=1_{\mathsf{A}^{\prime}}-f_{i}$, $i\neq j$.

######  Definition 5.5.

Let be $\varphi$ an additive map on $\mathsf{A}$. We call $\varphi$ a
commuting map if $[\varphi(x),x]=0$ for all $x\in\mathsf{A}$.

######  Theorem 5.6 (Theorem 4, [91]).

Let $\mathsf{A}$ be a unital alternative algebra. Assume that $\mathsf{A}$ has
a nontrivial idempotent $e_{1}$ with associated Peirce decomposition
$\mathsf{A}=\mathsf{A}_{11}\oplus\mathsf{A}_{12}\oplus\mathsf{A}_{21}\oplus\mathsf{A}_{22}$,
such that $(x\mathsf{A})e_{i}={0}$ implies $x=0$ $\left(i=1,2\right)$. Let
$\varphi:\mathsf{A}\rightarrow\mathsf{A}$ be an additive map. Then the
following statements are equivalent:

  * $(\spadesuit)$

$\varphi$ is commuting;

  * $(\clubsuit)$

there exists $z\in Z(\mathsf{A})$ and an additive map
$\Xi:\mathsf{A}\rightarrow Z(\mathsf{A})$ such that $\varphi(x)=zx+\Xi(x)$ for
all $x\in\mathsf{A}$.

###  5.3 Higher derivations of incidence algebras

Let $(P,\leq)$ be a preordered set and $R$ be a commutative unital ring.
Assume that $P$ is locally finite, i.e. for any $x\leq y$ in $P$ there are
only finitely many $z\in P$ such that $x\leq z\leq y$. The incidence algebra
$I(P,R)$ of $P$ over $R$ is the set of functions

$\\{f:P\times P\to R\mid f(x,y)=0\ \text{if}\ x\not\leq y\\}$

with the natural structure of an $R$-module and multiplication given by the
convolution

$\displaystyle(fg)(x,y)$ $\displaystyle=\sum_{x\leq z\leq y}f(x,z)g(z,y)$

for all $f,g\in I(P,R)$ and $x,y\in P$. It would be helpful to point out that
the full matrix algebra $M_{n}(R)$, as well as the upper triangular matrix
algebra $T_{n}(R)$, are particular cases of incidence algebras. In addition,
in the theory of operator algebras, the incidence algebra $I(P,R)$ of a finite
poset $P$ is referred to as a bigraph algebra or a finite-dimensional
commutative subspace lattice algebra. Incidence algebras appeared in the early
work by Ward [243] as generalized algebras of arithmetic functions. Later,
they were extensively used as the fundamental tool of enumerative
combinatorics in the series of works “On the foundations of combinatorial
theory”. The study of algebraic mappings on incidence algebras was initiated
by Stanley [237]. Since then, automorphisms, involutions, and derivations (and
their generalizations) on incidence algebras have been actively investigated,
see [218, 103, 100] and the references therein.

Let $P$ be a poset and $R$ a commutative unital ring. Recall from [182] that a
finitary series is a formal sum of the form

$\alpha=\sum_{x\leq y}\alpha_{xy}e_{xy},$

where $x,y\in P$, $\alpha_{xy}\in R$ and $e_{xy}$ is a symbol, such that for
any pair $x<y$ there exists only a finite number of $x\leq u<v\leq y$ with
$\alpha_{uv}\neq 0$. The set of finitary series, denoted by $FI(P,R)$,
possesses a natural structure of an $R$-module. Moreover, it is closed under
the convolution

$\alpha\beta=\sum_{x\leq y}\left(\sum_{x\leq z\leq
y}\alpha_{xz}\beta_{zy}\right)e_{xy}.$

Thus, $FI(P,R)$ is an algebra, called the finitary incidence algebra of $P$
over $R$. The identity element of $FI(P,R)$ is the series $\delta=\sum_{x\in
P}1_{R}e_{xx}$. Here and in what follows we adopt the following convention: if
in the formal sum the indices run through a subset $X$ of the ordered pairs
$(x,y)$, $x,y\in P$, $x\leq y$, then $\alpha_{xy}$ is meant to be zero for
$(x,y)\not\in X$.

Higher derivations are closely related to derivations. It should be remarked
that the first component $d_{1}$ of each higher derivation
$D=\\{d_{n}\\}_{n=0}^{\infty}$ of an algebra $\mathsf{A}$ is itself a
derivation of $\mathsf{A}$. Conversely, let $d:\mathsf{A}\to\mathsf{A}$ be an
ordinary derivation of an algebra $\mathsf{A}$ over a field of characteristic
zero. Then, $D=\\{\frac{1}{n!}d^{n}\\}_{n=0}^{\infty}$ is a higher derivation
of $\mathsf{A}$. Heerema, Mirzavaziri and Saymeh independently proved that
each higher derivation of an algebra $\mathsf{A}$ over a field of
characteristic zero is a combination of compositions of derivations, and hence
one can characterize all higher derivations on $\mathsf{A}$ in terms of the
derivations on $\mathsf{A}$. Ribenboim systemically studied higher derivations
of arbitrary rings and those of arbitrary modules, where some familiar
properties of derivations are generalized to the case of higher derivations.
Ferrero and Haetinger found the conditions under which Jordan higher
derivations (or Jordan triple higher derivations) of a $2$-torsion-free
(semi-)prime ring are higher derivations, and the same authors studied higher
derivations on (semi-)prime rings satisfying linear relations. Wei and Xiao
[244] described higher derivations of triangular algebras and related
mappings, such as inner higher derivations, Jordan higher derivations, Jordan
triple higher derivations, and their generalizations.

######  Definition 5.7.

A sequence $d=\\{{d}_{n}\\}_{n=0}^{\infty}$ of additive maps $R\to R$ is a
_higher derivation_ of $R$ _(of infinite order)_ if it satisfies the
conditions

1\. $d_{0}={\rm id}_{R}$; 2. $d_{n}(rs)=\sum_{i+j=n}d_{i}(r)d_{j}(s)$

for all $n\in\mathbb{N}=\\{1,2,\dots\\}$ and all $r,s\in R$. If definition 5.7
holds for all $1\leq n\leq N$, then the sequence $\\{d_{n}\\}_{n=0}^{N}$ is
called a _higher derivation of order $N$_. Evidently,
$\\{{d}_{n}\\}_{n=0}^{\infty}$ is a higher derivation if and only if
$\\{d_{n}\\}_{n=0}^{N}$ is a higher derivation of order $N$ for all
$N\in\mathbb{N}$. In particular, $d_{1}$ is always a usual derivation of $R$.

Denote by $\operatorname{\mathrm{HDer}}{R}$ the set of higher derivations of
$R$ and consider the following operation on $\operatorname{\mathrm{HDer}}{R}$

$(d^{\prime}*d^{\prime\prime})_{n}=\sum_{i+j=n}d^{\prime}_{i}\circ
d^{\prime\prime}_{j}.$

In particular,

$(d^{\prime}*d^{\prime\prime})_{1}=d^{\prime}_{1}+d^{\prime\prime}_{1}.$

It was proved in [115] that $\operatorname{\mathrm{HDer}}{R}$ forms a group
with respect to $*$, whose identity is the sequence
$\\{{\epsilon}_{n}\\}_{n=0}^{\infty}$ with $\epsilon_{0}=id_{R}$ and
$\epsilon_{n}=0$ for $n\in\mathbb{N}$.

Given $r\in R$ and $k\in\mathbb{N}$, define

$\displaystyle[r,k]_{0}$ $\displaystyle=id_{R},$ $\displaystyle[r,k]_{n}(x)$
$\displaystyle=\begin{cases}0,&k\nmid n,\\\
r^{l}x-r^{l-1}xr,&n=kl,\end{cases}$

for all $n\in\mathbb{N}$ and $x\in R$. It was proved in [216] that
$\\{{[r,k]}_{n}\\}_{n=0}^{\infty}\in\operatorname{\mathrm{HDer}}{R}$, so that
for any sequence $\\{r_{n}\\}_{n=1}^{\infty}\subseteq R$ one may define
$\\{{(\Delta_{r})}_{n}\\}_{n=0}^{\infty}$ by means of
$(\Delta_{r})_{0}=id_{R}$ and
$(\Delta_{r})_{n}=([r_{1},1]*\dots*[r_{n},n])_{n}$, where $n\in\mathbb{N}$.
Higher derivations of the form $\Delta_{r}$ are called _inner_. By [216,
Corollary 3.3] the set of inner higher derivations forms a normal subgroup in
$\operatorname{\mathrm{HDer}}{R}$, which will be denoted by
$\operatorname{\mathrm{IHDer}}{R}$. In particular,

$(\Delta_{r})_{1}(x)=[r_{1},1]_{1}(x)=r_{1}x-xr_{1}$

is the usual inner derivation of $R$ associated with $r_{1}\in R$, which we
denote by $\mathrm{ad}_{r_{1}}$.

######  Definition 5.8.

A sequence $\sigma=\\{{\sigma}_{n}\\}_{n=0}^{\infty}$ of maps on
$I=\\{(x,y)\in P\times P\mid x\leq y\\}$ with values in $R$ is called a
_higher transitive map_ if

  * •

$\sigma_{0}(x,y)=1_{R}$ for all $x\leq y$;

  * •

$\sigma_{n}(x,y)=\sum_{i+j=n}\sigma_{i}(x,z)\sigma_{j}(z,y)$ for all $x\leq
z\leq y$.

######  Theorem 5.9 (Lemma 2.6, [154]).

Given a higher transitive map $\sigma$, denote by $\tilde{\sigma}$ the
following sequence of maps $FI(P,R)\to FI(P,R)$:

$\displaystyle\tilde{\sigma}_{n}(\alpha)=\sum_{x\leq
y}\sigma_{n}(x,y)\alpha_{xy}e_{xy},$

where $n\in\mathbb{N}\cup\\{0\\}$ and $\alpha\in FI(P,R)$. Then,
$\tilde{\sigma}\in\operatorname{\mathrm{HDer}}{FI(P,R)}$.

######  Theorem 5.10 (Theorem 2.8, [154]).

Every $R$-linear higher derivation of $FI(P,R)$ is of the form
$\Delta_{\rho}*\tilde{\sigma}$ for some
$\rho=\\{\rho_{n}\\}_{n=1}^{\infty}\subseteq FI(P,R)$ and some higher
transitive map $\sigma$.

##  6 Generalized derivations of non-associative algebras

The present part is based on the papers written together with Bruno Ferreira,
Inomjon Yuldashev, Karimbergan Kudaybergenov, Patrícia Beites, and Yury Popov
[156, 139, 140, 28, 167, 165, 166, 92].

###  6.1 Local derivations of $n$-ary algebras

The study of local derivations started in 1990 with Kadison’s article [132]. A
similar notion, which characterizes non-linear generalizations of derivations,
was introduced by Šemrl as $2$-local derivations. In his paper [242], it was
proved that a $2$-local derivation of the algebra $B(H)$ of all bounded linear
operators on the infinite-dimensional separable Hilbert space $H$ is a
derivation. After these works, numerous new results related to the description
of local and $2$-local derivations of associative algebras appeared (see, for
example, [183]). The study of local and $2$-local derivations of non-
associative algebras was initiated in some papers by Ayupov and Kudaybergenov
(for the case of Lie algebras, see [16, 17]). In particular, they proved that
there are no pure local and $2$-local derivations on semisimple finite-
dimensional Lie algebras. Also, [18] provides examples of $2$-local
derivations on nilpotent Lie algebras that are not derivations. After the
cited works, the study of local and $2$-local derivations was continued for
Leibniz algebras and Jordan algebras [15]. Local automorphisms and $2$-local
automorphisms were also studied in many cases; for example, they were studied
on Lie algebras [16, 68].

The description of local and $2$-local derivations of $n$-ary algebras started
in our paper with Ferreira and Kudaybergenov [92]. Namely, we gave the first
example of a complex simple finite-dimensional (ternary) algebra with non-
trivial local derivations. The idea of introducing a generalization of
Filippov algebras comes from binary Malcev algebras and it was carried out in
a paper by Pozhidaev [224]. He defined $n$-ary Malcev algebras, generalizing
Malcev algebras and $n$-ary Filippov algebras. Let us summarize the
construction of the most important example of $n$-ary Malcev (non-Filippov).
We denote by $\mathsf{A}$ a composition $8$-dimensional algebra with an
involution $\bar{}:a~{}\mapsto\bar{a}$ and unity $1$. The symmetric bilinear
form $\left\langle x,y\right\rangle=\frac{1}{2}(x\bar{y}+y\bar{x})$ defined on
$\mathsf{A}$ is assumed to be nonsingular. If $\mathsf{A}$ is equipped with a
ternary multiplication $[\cdot,\cdot,\cdot]$ by the rule

$[x,y,z]=(x\bar{y})z-\left\langle y,z\right\rangle x+\left\langle
x,z\right\rangle y-\left\langle x,y\right\rangle z$,

then $\mathsf{A}$ becomes a ternary Malcev algebra [224], which will be
denoted by ${\rm M}_{8}$.

Let us give two main definitions necessary for this part.

######  Definition 6.1.

Let $\mathsf{A}$ be an $n$-ary algebra. A linear map
$\nabla:\mathsf{A}\rightarrow\mathsf{A}$ is called a local derivation if for
any element $x\in\mathsf{A}$ there exists a derivation
${\mathfrak{D}}_{x}:\mathsf{A}\rightarrow\mathsf{A}$ such that
$\nabla(x)={\mathfrak{D}}_{x}(x)$.

######  Definition 6.2.

A (not necessary linear) map $\Delta:\mathsf{A}\rightarrow\mathsf{A}$ is
called a $2$-local derivation, if for any two elements $x,y\in\mathsf{A}$
there exists an derivation
${\mathfrak{D}}_{x,y}:\mathsf{A}\rightarrow\mathsf{A}$ such that
$\Delta(x)={\mathfrak{D}}_{x,y}(x)$, $\Delta(y)={\mathfrak{D}}_{x,y}(y)$.

In the following theorems, we summarize all the main results of the present
part.

######  Theorem 6.3 (Theorem 3, [156]).

Every local derivation of a complex finite-dimensional semisimple Leibniz
algebra is a derivation.

######  Theorem 6.4 (Theorems 4 and 5, [92]).

Every local (and $2$-local) derivation of a complex finite-dimensional simple
$n$-Lie algebra is a derivation.

######  Theorem 6.5 (Theorem 8, [92]).

A linear mapping $\nabla$ on $M_{8}$ is a local derivation if and only if its
matrix is antisymmetric. In particular, the dimension of the space
$LocDerM_{8}$ of all local derivations of ${\rm M}_{8}$ is equal to $28$.

###  6.2 Leibniz-derivations of non-associative algebras

The theory of Lie algebras having a nonsingular derivation has a rich history
and is still an active research area. Such Lie algebras appear in many
different situations, such as in the studies of pro-$p$ groups of finite
coclass by Shalev and Zelmanov [229, 230] and in the problems concerning the
existence of left-invariant affine structures on Lie groups (see Burde’s
survey [38] for details).

In 1955, Jacobson [124] proved that a finite-dimensional Lie algebra over a
field of characteristic zero admitting a nonsingular (invertible) derivation
is nilpotent. The problem of whether the inverse of this statement is correct
remained open until the work [76], where an example of a nilpotent Lie algebra
in which all derivations are nilpotent (and hence, singular) was constructed.
For Lie algebras in prime characteristic, the situation is more complicated.
In that case, there exist non-nilpotent Lie algebras, even simple ones, which
admit nonsingular derivations [32]. The main examples of nonsingular
derivations are periodic derivations. Kostrikin and Kuznetsov [193] noted that
a Lie algebra admitting a nondegenerate derivation admits a periodic
derivation, that is, a derivation $d$ such that $d^{N}={\rm id}$ for some $N$,
and proved that a Lie algebra admitting a derivation of period $N$ is abelian
provided that $N\not\equiv 0\text{ (mod 6)}$. Burde and Moens proved that a
finite-dimensional complex Lie algebra $\mathsf{A}$ admits a periodic
derivation if and only if $\mathsf{A}$ admits a nonsingular derivation whose
inverse is again a derivation if and only if $\mathsf{A}$ is hexagonally
graded [39]. In the case of positive characteristic $p$, Shalev proved that if
a Lie algebra $\mathsf{A}$ admits a nonsingular derivation of order
$n=p^{s}m$, where $(m,p)=1$ and $m<p^{2}-1$, then $\mathsf{A}$ is nilpotent.
The study of periodic derivations was continued by Mattarei [209, 210, 211].

The study of derivations of Lie algebras led to the appearance of the notion
of their natural generalization — a pre-derivation of a Lie algebra, which is
a derivation of a Lie triple system induced by that algebra. In [21] it was
proved that Jacobson’s result remains true in terms of pre-derivations.
Several examples of nilpotent Lie algebras whose pre-derivations are nilpotent
were presented in [21].

A generalization of derivations and pre-derivations of Lie algebras is defined
in the paper [214] as a Leibniz-derivation of order $k$. Moens proved that a
finite-dimensional Lie algebra over a field of characteristic zero is
nilpotent if and only if it admits an invertible Leibniz-derivation. After
that, Fialowski, Khudoyberdiyev and Omirov [94] showed that with the
definition of Leibniz-derivations from [214], the analogous result for non-Lie
Leibniz algebras is not true. Namely, they gave an example of non-nilpotent
Leibniz algebra, which admits an invertible Leibniz-derivation. To extend the
results of the paper [214] for Leibniz algebras they introduced the definition
of Leibniz-derivation of a Leibniz algebra which is coherent with the
definition of Leibniz-derivation of a Lie algebra and proved that a finite-
dimensional Leibniz algebra is nilpotent if and only if it admits an
invertible Leibniz-derivation. In the paper [165], the authors showed that the
same result holds for alternative algebras (particularly, for associative
algebras). Also, in this article an example of a nilpotent alternative (non-
associative) algebra over a field of positive characteristic possessing only
singular derivations was provided.

It is well known that the radicals of finite-dimensional algebras belonging to
the classical varieties (such as varieties of Jordan algebras, Lie algebras,
alternative algebras, and many others) are invariant under their derivations
[236]. Therefore, it is natural to state another interesting problem: is the
radical of an algebra invariant under its Leibniz-derivations? Moens proved
that the solvable radical of a Lie algebra is invariant under all its Leibniz-
derivations, Fialowski, Khudoyberdiev and Omirov showed the invariance of
solvable and nilpotent radicals of a Leibniz algebra, and the authors of [165]
proved analogous results for alternative algebras. Another interesting task is
to describe the Leibniz-derivations of algebras belonging to certain “nice”
classes, such as semisimple and perfect algebras. Moens described all Leibniz-
derivations of semisimple Lie algebras [214] and Zhou described all pre-
derivations of perfect centerless Lie algebras of characteristic $\neq 2$.

######  Definition 6.6.

Let $\mathsf{A}$ be an algebra, $n$ be a natural number $\geq 2$, and $f$ be
an arrangement of brackets on a product of length $n$. A linear mapping $d$ on
$\mathsf{A}$ is called an $f$-Leibniz derivation of $\mathsf{A}$, if for any
$a_{1},\dots a_{n}\in\mathsf{A}$ we have

$d([a_{1},\ldots,a_{n}]_{f})=\sum\limits_{i=1}^{n}[a_{1},a_{2},\ldots,d(a_{i}),\ldots
a_{n}]_{f}.$

Particularly, if $f=l(n)$ ($f=r(n)$) is the left (right) arrangement of
brackets of length $n$, that is,

$[x_{1},\ldots,x_{n}]_{l(n)}=((\ldots(x_{1}x_{2})\ldots)x_{n-1})x_{n},\ \
[x_{1},\ldots,x_{n}]_{r(n)}=(x_{1}(x_{2}\ldots(x_{n-1}x_{n})\ldots)),$

then an $l(n)$-Leibniz derivation ($r(n)$-Leibniz derivation) of $\mathsf{A}$
will be called a  _left (right) Leibniz-derivation of_ $\mathsf{A}$ of order
$n$. If $d$ is an $f$-derivation for any arrangement $f$ of length $n$, then
$d$ will be called a  _Leibniz-derivation of $\mathsf{A}$ of order $n$._ One
can see that in our terms a Leibniz-derivation by Moens is a right Leibniz-
derivation of a Lie algebra $\mathsf{A}$, and a Leibniz-derivation by
Fialowski-Khudoyberdiev-Omirov is a left Leibniz-derivation of a Leibniz
algebra $\mathsf{A}$. It is easy to see that for

  * I.

(anti)commutative algebras, the notion of a left Leibniz-derivation coincides
with the notion of a right Leibniz derivation;

  * II.

left Leibniz algebras, every right Leibniz-derivation is a left Leibniz-
derivation;

  * III.

left Zinbiel algebras, every left Leibniz-derivation is a right Leibniz-
derivation;

  * IV.

associative and Lie algebras, the notion of a Leibniz-derivation coincides
with the notion of an $f$-Leibniz derivation for any $f$.

We summarize the main results of the present part in the following theorems.

######  Theorem 6.7 (Theorem 18, [166]).

Let $\mathsf{A}$ be a Malcev algebra over a field of characteristic zero. Then
${\rm Rad}(\mathsf{A})$ is invariant under all the left Leibniz-derivations of
$\mathsf{A}$.

######  Theorem 6.8 (Theorem 18, [166]).

Let $\mathsf{A}$ be a semisimple Malcev algebra over a field of characteristic
0. Then $\mathfrak{Der}(\mathsf{A})={\rm LDer}_{l}(\mathsf{A})$.

######  Theorem 6.9 (Theorem 28, [166]).

A Malcev algebra over a field of characteristic zero is nilpotent if and only
if it admits an invertible left Leibniz-derivation.

######  Theorem 6.10 (Theorem 30, [166]).

A Jordan algebra over a field of characteristic zero is nilpotent if and only
if it admits an invertible left Leibniz–derivation.

######  Theorem 6.11 (Theorem 32, [166]).

A $(-1,1)$-algebra over a field of characteristic zero is nilpotent if and
only if it admits an invertible left Leibniz–derivation.

######  Theorem 6.12 (Theorem 33, [166]).

A right alternative algebra over a field of characteristic zero admitting an
invertible Leibniz-derivation is right nilpotent.

###  6.3 Generalized derivations of non-associative algebras

In 1998, the notion of $\delta$-derivations appeared in the paper by Filippov
[96]. He studied $\delta$-derivations of prime Lie algebras [97]. After that,
$\delta$-derivations of structurable algebras, $n$-ary algebras, and Jordan
and Lie superalgebras were studied (see [250] and references therein). The
notion of $\frac{1}{2}$-derivation plays an important role in the description
of transposed Poisson structures on a certain Lie algebra [93]. The notion of
generalized derivations is a generalization of $\delta$-derivations and
$(\alpha,\beta,\gamma)$-derivations. The most important and systematic
research on the generalized derivations algebras of a Lie algebra and their
subalgebras was in a paper by Leger and Luks [197]. In their article, they
studied the properties of generalized derivation algebras and their
subalgebras, for example, the quasiderivation algebras. They determined the
structure of algebras of quasiderivations and generalized derivations and
proved that the quasiderivation algebra of a Lie algebra can be embedded into
the derivation algebra of a larger Lie algebra. Their results were generalized
by many authors. For example, Zhang and Zhang [249] generalized the above
results to the case of Lie superalgebras; Chen, Ma, Ni and Zhou considered the
generalized derivations of color Lie algebras, Hom-Lie superalgebras and Lie
triple systems [64]. Generalized derivations of simple algebras and
superalgebras were investigated in [231, 232, 126]. Pérez-Izquierdo and
Jiménez-Gestal used the generalized derivations to study non-associative
algebras [127]. Derivations and generalized derivations of $n$-ary algebras
were considered in many papers. For example, Williams proved that, unlike the
case of binary algebras, for any $n\geq 3$ there exists a non-nilpotent
$n$-Lie algebra with invertible derivation [245].

The main purpose of the results contained in the present part is to generalize
the results of Leger and Luks [197] to the case of color $n$-ary algebras and
${\rm Hom}$-algebras. Particularly, we proved some properties of generalized
derivations of color $n$-ary algebras and ${\rm Hom}$-algebras, that the
quasiderivation algebra of a color $n$-ary $\Omega$-algebra (resp.,
$\Omega$-${\rm Hom}$-algebra) can be embedded into the derivation algebra of a
large color $n$-ary $\Omega$-algebra (resp., $\Omega$-${\rm Hom}$-algebra)
[167, 28], and obtained the classification of all $n$-ary algebras with the
property ${\rm End=QDer}$. Let us give the main definitions of this part.

######  Definition 6.13.

A linear mapping $\mathfrak{D}\in{\rm End}(\mathsf{A})$ is called a
generalized derivation of an $n$-ary algebra $\mathsf{A}$ if there exists
linear mappings
$\mathfrak{D}^{\prime},\mathfrak{D}^{\prime\prime},\ldots,\mathfrak{D}^{(n-1)},\mathfrak{D}^{(n)}\in{\rm
End}(\mathsf{A})$ such that

$\sum[x_{1},\ldots,\mathfrak{D}^{(i-1)}(x_{i}),\ldots,x_{n}]=\mathfrak{D}^{(n)}([x_{1},\ldots,x_{n}])$.

An $(n+1)$-tuple
$(\mathfrak{D},\mathfrak{D}^{\prime},\ldots,\mathfrak{D}^{(i-1)},\ldots,\mathfrak{D}^{(n-1)},\mathfrak{D}^{(n)})$
is called an $(n+1)$-ary derivation. An $(n+1)$-ary derivation is trivial if
it is a sum of $(d,\ldots,d)$ and $(\phi_{1},\ldots,\phi_{n},\sum\phi_{i})$,
where $d$ is a derivation and $\phi_{i}$ are elements from the centroid.

######  Definition 6.14.

A linear mapping $\mathfrak{D}\in{\rm End}(\mathsf{A})$ is said to be a
quasiderivation if there exists a $\mathfrak{D}^{\prime}\in{\rm
End}(\mathsf{A})$ such that

$\sum[x_{1},\ldots,\mathfrak{D}(x_{i}),\ldots,x_{n}]=\mathfrak{D}^{{}^{\prime}}([x_{1},\ldots,x_{n}])$.

######  Definition 6.15.

A pair of linear mappings $(d,f)$ satisfying the condition of the previous
definition is called a pair of quasiderivations of $\mathsf{A}$. The image of
the projection of ${\rm QDer}(\mathsf{A})$ onto the first coordinate will be
denoted as ${\rm QDer}_{KS}(\mathsf{A})$ and will be called a space of
quasiderivations in the sense of Kaygorodov and Shestakov [139, 231], and the
image of the projection of ${\rm QDer}(\mathsf{A})$ onto the second coordinate
will be denoted as ${\rm QDer}_{LL}(\mathsf{A})$ and will be called a space of
quasiderivations in the sense of Leger and Luks [197].

In the following theorems, we summarize the main results of the present part.

######  Theorem 6.16 (Theorem 1, [140]).

There are no nontrivial ternary derivations of the simple Malcev algebra ${\rm
M}_{7}$.

######  Theorem 6.17 (Theorem 5, [140]).

There are no nontrivial $4$-ary derivations of the simple ternary Malcev
algebra ${\rm M}_{8}$.

######  Theorem 6.18 (Theorem 7, [139]).

Let $\mathsf{A}$ be a semisimple finite-dimensional $n$-ary Filippov algebra
over an algebraically closed field of characteristic zero. Then, the algebra
of generalized derivations of $\mathsf{A}$ is isomorphic to
$\oplus\mathfrak{sl}_{n+1}$.

To understand the main result of this part, let us give some additional
information about $n$-ary algebras (see [95]).

$\bullet$ Up to isomorphism, there is only one $n$-ary anticommutative algebra
$A_{n}$ in dimension $n$. The product of the basis elements
$e_{1},\dots,e_{n}$ of $A_{n}$ is given in the following way:

$[e_{1},\dots,e_{n}]=e_{1}.$

$\bullet$ Up to isomorphism, there is only one perfect (i.e.,
$\mathsf{A}^{2}=\mathsf{A}$) $n$-Lie algebra $D_{n+1}$ of dimension $n+1$. The
product of the basis elements $e_{1},\dots,e_{n+1}$ of $D_{n+1}$ is given in
the following way:

$[e_{1},\dots,\hat{e}_{i},\dots,e_{n+1}]=(-1)^{n+i+1}e_{i}.$

$\bullet$ Let $\mathsf{A}$ be an $n$-ary $(n+1)$-dimensional algebra with the
basis $e_{1},\dots,e_{n+1}$. Let

$e^{i}=(-1)^{n+i+1}[e_{1},\dots,\hat{e}_{i},\dots,e_{n}],i=1,\dots,n+1.$ (1)

Then, the multiplication in $\mathsf{A}$ is defined by the matrix
$B=(\beta_{ij})$ which is given by

$e^{i}=\beta_{1i}e_{1}+\ldots+\beta_{n+1i}e_{n+1},$

or in terms of matrices as $(e^{1},\ldots,e^{n+1})=(e_{1},\ldots,e_{n+1})B.$
The rank of $B$ is equal to the dimension of
$\mathsf{A}^{2}=[\mathsf{A},\ldots,\mathsf{A}]$. An $(n+1)$-dimensional
anticommutative algebra with multiplication defined by (1) will be denoted by
$\mathsf{A}_{B}$. It is easy to see that $D_{n+1}$ is an algebra
$\mathsf{A}_{I}$, where $I$ is an identity matrix of order $n+1$.

Later in the discussion we will also need the description of $1$-dimensional
$n$-ary algebras. One can easily see that the multiplication in such algebras
is completely defined by an element $\alpha\in\mathbb{F}$, where $\mathbb{F}$
is the base field, for it is enough to determine $[v,v,\ldots,v]=\alpha v$ for
any nonzero $v\in\mathsf{A}$ and extend the multiplication linearly. We denote
such algebras by $\mathsf{A}_{\alpha}$. It is also easy to see that
$\mathsf{A}_{\alpha}\cong\mathsf{A}_{\beta}$ for $\alpha,\beta\neq 0$ if and
only if the polynomial $x^{n-1}-\frac{\alpha}{\beta}$ has a root in
$\mathbb{F}$.

######  Theorem 6.19 (Theorem 5.6, [167]).

Let $\mathsf{A}$ be an $n$-ary algebra such that ${\rm
QDer}_{LL}(\mathsf{A})=End(\mathsf{A})$. Then

$\bullet$ If $\mathsf{A}$ is commutative, then either $\mathsf{A}$ has zero
product or $\mathsf{A}\cong\mathsf{A}_{\alpha}$ for some $\alpha$ in base
field $\mathbb{F}$. Moreover, if the base field $\mathbb{F}$ of $\mathsf{A}$
is algebraically closed, then one can define a binary multiplication $\cdot$
on $\mathsf{A}$ such that $(\mathsf{A},\cdot)\cong\mathbb{F}$ and
$[x_{1},\ldots,x_{n}]=x_{1}\cdot\ldots\cdot x_{n}$, where
$[\cdot,\ldots,\cdot]$ is multiplication in $\mathsf{A}$ and
$x_{1},\dots,x_{n}\in\mathsf{A}$.

$\bullet$ If $\mathsf{A}$ is anticommutative, then either $\mathsf{A}$ has
zero product or $\mathsf{A}$ is isomorphic to either $A_{n}$ or
$\mathsf{A}_{B}$ for a nondegenerate $(n+1)\times(n+1)$-matrix $B$. Moreover,
in the last case if $\mathsf{A}$ is a Filippov algebra and the base field is
algebraically closed, then $\mathsf{A}\cong D_{n+1}$.

######  Theorem 6.20 (Theorem 5.9, [167]).

Let $\mathsf{A}$ be an $n$-ary algebra such that ${\rm
QDer}_{KS}(\mathsf{A})={\rm End}(\mathsf{A})$. Then

$\bullet$ If $\mathsf{A}$ is commutative, then
$\mathsf{A}\cong\mathsf{A}_{\alpha}$ for some $\alpha\in\mathbb{F}$. Moreover,
if the base field $\mathbb{F}$ of $\mathsf{A}$ is algebraically closed, then
one can define a binary multiplication $\cdot$ in $\mathsf{A}$ such that
$(\mathsf{A},\cdot)\cong\mathbb{F}$ and
$[x_{1},\ldots,x_{n}]=x_{1}\cdot\ldots\cdot x_{n}$, where
$[\cdot,\ldots,\cdot]$ is multiplication in $\mathsf{A}$ and
$x_{1},\dots,x_{n}\in\mathsf{A}$.

$\bullet$ If $\mathsf{A}$ is anticommutative, then
$\mathsf{A}\cong\mathsf{A}_{B}$ for a nondegenerate $(n+1)\times(n+1)$-matrix
$B$ or $\mathsf{A}$ has zero multiplication. Moreover, if $\mathsf{A}$ is a
Filippov algebra and the base field is algebraically closed, then
$\mathsf{A}\cong D_{n+1}$.

##  7 Poisson type algebras and superalgebras

The present part is based on the papers written together with Bruno Ferreira,
Ivan Shestakov, Mykola Khrypchenko, Ualbai Umirbaev and Viktor Lopatkin [93,
145, 174, 141].

###  7.1 Poisson structures on finitary incidence algebras

The systematic study of noncommutative Poisson algebra structures began in the
paper by Kubo [195]. He obtained a description of all the Poisson structures
on the full and upper triangular matrix algebras, which was later generalized
to prime associative algebras in [86]. Namely, it was proved in [86] that any
Poisson bracket on a prime noncommutative associative algebra is the
commutator bracket multiplied by an element from the extended centroid of the
algebra. On the other hand, in his next paper, Kubo studied noncommutative
Poisson algebra structures on affine Kac-Moody algebras. The investigation of
Poisson structures on associative algebras continued in some papers by Yao, Ye
and Zhang [246]; Mroczyńska, Jaworska-Pastuszak, and Pogorzały [125, 215],
where Poisson structures on finite-dimensional path algebras and on canonical
algebras were studied. In our joint work with Khrypchenko we described Poisson
structures on finitary incidence algebras [145].

######  Definition 7.1.

Let $(\mathsf{A},\cdot)$ be an associative algebra and $\\{\cdot,\cdot\\}$ an
additional multiplication on the same vector space. We call
$\\{\cdot,\cdot\\}$ a Poisson structure on $(\mathsf{A},\cdot)$ if
$(\mathsf{A},\cdot,\\{\cdot,\cdot\\})$ is a Poisson algebra.

The main tool for the description of Poisson structures on associative algebra
is the study of biderivations.

######  Definition 7.2.

Let $\mathsf{A}$ be an associative algebra and $M$ an $\mathsf{A}$-bimodule. A
bilinear map $B:\mathsf{A}\times\mathsf{A}\to M$ is called an antisymmetric
biderivation of $\mathsf{A}$ with values in $M$, if it is anticommutative and
a derivation $\mathsf{A}\to M$ with respect to each of its two variables, i.e.

$B(xy,z)=B(x,z)y+xB(y,z),\ B(x,x)=0.$

For any $\lambda\in C(\mathsf{A})$, the map $B(x,y)=\lambda[x,y],$ where
$[x,y]$ is the commutator $xy-yx$, is an antisymmetric biderivation of
$\mathsf{A}$. Such biderivations will be called inner.

For any $\lambda\in C(\mathsf{A})$, the inner biderivation
$\\{a,b\\}=\lambda[a,b]$ is clearly a Poisson structure on $\mathsf{A}$.
Following [246], we call such Poisson structures standard. There is a
generalization of this notion introduced in the same paper [246]. Namely, a
Poisson structure on $\mathsf{A}$ is said to be piecewise standard if
$\mathsf{A}$ decomposes into a direct sum $\bigoplus_{i=1}^{m}\mathsf{A}_{i}$
of indecomposable Lie ideals in such a way that $\\{a,b\\}=\lambda_{i}[a,b]$
for all $a\in\mathsf{A}_{i}$ and $b\in\mathsf{A}$, where $\lambda_{i}\in
C(\mathsf{A})$.

It is well-known that the upper triangular matrix algebra $T_{n}(R)$ is the
incidence algebra $I(C_{n},R)$ of a chain $C_{n}$ of cardinality $n$. There is
a description of (not necessarily antisymmetric) biderivations of $T_{n}(R)$
given by Benkovič in [33, Corollary 4.13] (case $n\geq 3$) and [33,
Proposition 4.16] (case $n=2$). A straightforward calculation, based on this
description, shows that all the antisymmetric biderivations of $T_{n}(R)$ are
inner. Consequently, we have the following (see also [195]).

######  Example 1.

All the Poisson structures on $T_{n}(R)$ are standard.

This is not the case for a general incidence algebra, as the next easy example
shows.

######  Example 2.

Let $R$ be a commutative ring and consider $P=\\{1,2,3,4\\}$ with the
following Hasse diagram (a crown).

$1$$3$$4$$2$

######  Remark 7.3 (Remark 1.3, [145]).

If $R$ is a field, then the Poisson structure $B$ from 2 is not even piecewise
standard.

Let us recall that the definition and some main properties of incidence
algebras were considered in Section 5.3. For any pair $x\leq y$ we shall
identify $e_{xy}$ with $1_{R}e_{xy}\in FI(P,R)$ and denote by $E$ the set
$\\{e_{xy}\mid x\leq y\\}\subseteq FI(P,R)$. Denote by $\tilde{I}(P,R)$ the
subalgebra of $FI(P,R)$ generated by $E$. Clearly,
$\tilde{I}(P,R)=\textup{span}\\{E\\}$ as an $R$-module. We will first deal
with biderivations defined on $\tilde{I}(P,R)$.

######  Theorem 7.4 (Proposition 3.1, [145]).

For all $x\leq y$ and $u\leq v$ we have

$B(e_{xy},e_{uv})=\lambda(e_{xy},e_{uv})[e_{xy},e_{uv}]$

for some map $\lambda:E\times E\to R$.

######  Theorem 7.5 (Proposition 3.5, [145]).

Let $\lambda:E\times E\to R$ be a symmetric map. Then the bilinear map $B$
given by $B(e_{xy},e_{uv})=\lambda(e_{xy},e_{uv})[e_{xy},e_{uv}]$ is an
antisymmetric biderivation of $\tilde{I}(P,R)$ with values in $FI(P,R)$ if and
only if

$[e_{xy},e_{x^{\prime}y^{\prime}}]\neq 0\ \&\
[e_{uv},e_{u^{\prime}v^{\prime}}]\neq 0\ \Rightarrow\
\lambda(e_{xy},e_{x^{\prime}y^{\prime}})=\lambda(e_{uv},e_{u^{\prime}v^{\prime}})$

holds for any chain $C\subseteq P$.

######  Definition 7.6.

A map $\sigma:(P\times P)_{<}\to R$ is said to be constant on chains if for
any chain $C\subseteq P$ and for all $x<y$, $u<v$ from $C$ one has
$\sigma(x,y)=\sigma(u,v)$.

######  Theorem 7.7 (Theorem 3.7, [145]).

There is a one-to-one correspondence between the antisymmetric biderivations
$B$ of $\tilde{I}(P,R)$ with values in $FI(P,R)$ and the maps $\sigma:(P\times
P)_{<}\to R$, which are constant on chains. More precisely,

$\displaystyle B(f,g)(x,y)=\begin{cases}0,&x=y,\\\
\sigma(x,y)[f,g](x,y),&x<y,\end{cases}$ (2)

where $f,g\in\tilde{I}(P,R)$.

######  Theorem 7.8 (Theorem 4.3, [145]).

The Poisson structures $B$ on $FI(P,R)$ are in a one-to-one correspondence
with the maps $\sigma:(P\times P)_{<}\to R$ which are constant on chains. The
correspondence is given by eq. 2, in which $f,g\in FI(P,R)$.

###  7.2 Generalized and generic Poisson algebras and superalgebras

A generalization of Poisson algebras known as generic Poisson algebras was
studied in the papers by Kolesnikov, Makar-Limanov, Shestakov [190] and
Kaygorodov, Shestakov, Umirbaev [174].

######  Definition 7.9.

The triple $(\mathsf{A},\cdot,\\{\cdot,\cdot\\})$ is a generic Poisson algebra
if $(\mathsf{A},\cdot)$ is a commutative associative algebra,
$(\mathsf{A},\\{\cdot,\cdot\\})$ is an anticommutative algebra and

$\\{a,bc\\}=\\{a,b\\}c+b\\{a,c\\}.$

In our paper written, together with Shestakov and Umirbaev [174], we establish
that many results known true for Poisson algebras also hold for generic
Poisson algebras. For example, we study the general properties of generic
Poisson modules and of universal multiplicative enveloping algebras (Section
2, [174]); we determine the structure of free generic Poisson algebras and of
free generic Poisson fields (Section 3, [174]); we prove an analog of
Makar–Limanov–Shestakov’s Theorem, namely that two Poisson dependent elements
in a free generic Poisson field are polynomial dependent (Section 4, [174]);
we apply the obtained results to the study of automorphisms of the free
generic Poisson algebra $GP\\{x,y\\}$ with two generators and prove that the
groups of automorphisms of the free generic Poisson algebra with two
generators, of the free Poisson algebra with two generators, of the free
associative algebra with two generators and of the free commutative
associative algebra with two generators are isomorphic (Section 5, [174]).

Other generalizations of Poisson superalgebras include generalized Poisson
superalgebras and superalgebras of Jordan brackets. Every Poisson superalgebra
is a superalgebra of Jordan brackets [134, 136]. The identities defining
unital superalgebras of Jordan brackets were described in [188]. Superalgebras
of Jordan brackets are important in the classification of finite-dimensional
simple Jordan superalgebras [207]. Kac and Cantarini studied linearly compact
simple superalgebras of Jordan brackets [60], Kaygorodov and Zhelyabin studied
$\delta$-superderivations of simple superalgebras of Jordan brackets [179],
Zelmanov, Shestakov, and Martínez studied the relation between Jordan brackets
and Poisson brackets [206], and special superalgebras of Jordan brackets were
considered in [179]. Kantor’s construction gives interesting relations between
Novikov–Poisson algebras and Jordan superalgebras [248].

A superalgebra $\mathsf{A}$ is called a superalgebra of Jordan brackets
whenever its Kantor double is a Jordan superalgebra [134, 136]. As [188]
implies, a superalgebra $\mathsf{A}$ whose associative-supercommutative
multiplication $\cdot$ and superanticommutative multiplication
$\\{\cdot,\cdot\\}$ satisfy

$\\{a,bc\\}=\\{a,b\\}c+(-1)^{|a||b|}b\\{a,c\\}-D(a)bc,$
$\\{a,\\{b,c\\}\\}=\\{\\{a,b\\},c\\}+(-1)^{|a||b|}\\{b,\\{a,c\\}\\}+D(a)\\{b,c\\}+(-1)^{|a||bc|}D(b)\\{c,a\\}+(-1)^{|c||ab|}D(c)\\{a,b\\},$

with $D(a)=\\{a,1\\}$, is a superalgebra of Jordan brackets. Observe that $D$
is an even derivation of the superalgebra $(\mathsf{A},\cdot)$. If $D=0$, then
$(\mathsf{A},\cdot,\\{\cdot,\cdot\\})$ is a Poisson superalgebra [134]. As
[60] implies, on the same vector space we can consider the new multiplication
$\\{a,b\\}_{D}=\\{a,b\\}+\frac{1}{2}(aD(b)-D(a)b)$ and the new linear mapping
$D_{D}=\frac{1}{2}D$. It is a new superalgebra with the identities

$\\{a,bc\\}_{D}=\\{a,b\\}_{D}c+(-1)^{|a||b|}b\\{a,c\\}_{D}-D_{D}(a)bc,$
$\\{a,\\{b,c\\}_{D}\\}_{D}=\\{\\{a,b\\}_{D},c\\}_{D}+(-1)^{|a||b|}\\{b,\\{a,c\\}_{D}\\}_{D}.$

Every superalgebra satisfying the last two identities is called a generalized
Poisson superalgebra (also known as a superalgebra of contact brackets).

In our paper, we construct bases for the $n$-generated free unital
superalgebra of Jordan brakets and the $n$-generated free unital generalized
Poisson superalgebra in [141, Theorems 2 and 5].

The celebrated Amitsur–Levitsky theorem states that the algebra $M_{k}(R)$ of
$k\times k$ matrices over a commutative ring $R$ satisfies the identity
$s_{2k}=0$. Furthermore, every associative PI algebra satisfies
$(s_{k})^{l}=0$ by Amitsur’s theorem. Farkas defined customary polynomials

$\displaystyle
g=\sum_{\sigma\in{\mathbb{S}}_{m}}c_{\sigma}\\{x_{\sigma(1)},x_{\sigma(2)}\\}\ldots\\{x_{\sigma(2i-1)},x_{\sigma(2i)}\\}$

and proved that every Poisson PI algebra satisfies some customary identity
[85]. Farkas’ theorem was established for generic Poisson algebras in [190].

We have found an analog of customary identities for generalized Poisson
algebras and algebras of Jordan brackets [141]:

$\displaystyle
g_{*}=\sum\limits_{i=0}^{[m/2]}\sum_{\sigma\in{\mathbb{S}}_{m}}c_{\sigma,i}\langle
x_{\sigma(1)},x_{\sigma(2)}\rangle\ldots\langle
x_{\sigma(2i-1)},x_{\sigma(2i)}\rangle D(x_{\sigma(2i+1)})\ldots
D(x_{\sigma(m)}),$ (3)

where

$\langle x,y\rangle:=\\{x,y\\}-(D(x)y-xD(y)).$

######  Theorem 7.10 (Theorem 13, [141]).

If a unital generalized Poisson algebra $\mathsf{A}$ satisfies a polynomial
identity $g_{0}$, then $\mathsf{A}$ satisfies a polynomial identity $g_{*}$ of
type (3).

Consequently, we obtain the following statement.

######  Theorem 7.11 (Corollary 14, [141]).

Every PI generalized Poisson algebra satisfies an identity of the type

$\displaystyle
f_{*}=\sum\limits_{i=0}^{[m/2]}\sum_{\sigma\in\mathbb{S}_{m}}c_{\sigma,i}\prod\limits_{k=1}^{i}[x_{\sigma(2k-1)};x_{\sigma(2k)};z_{2k-1};z_{2k}]\cdot\prod\limits_{k=1}^{m-2i}\\{x_{\sigma(2i+k)};z_{2i+2k-1};z_{2i+2k}\\}\cdot\prod\limits_{k=1}^{2i}z_{2m-2i+k},$
(4)

where

$[u_{1};u_{2};w_{1};w_{2}]=\\{u_{1},u_{2}\\}w_{1}w_{2}+\\{u_{1},w_{1}w_{2}\\}u_{2}+u_{1}\\{w_{1}w_{2},u_{2}\\}+\sum_{\sigma_{1},\sigma_{2}\in\mathbb{S}_{2}}\\{u_{\sigma_{1}(1)},w_{\sigma_{2}(1)}\\}u_{\sigma_{1}(2)}w_{\sigma_{2}(2)},$
$\\{t_{1};t_{2};t_{3}\\}=\\{t_{2}t_{3},t_{1}\\}-\\{t_{2},t_{1}\\}t_{3}-\\{t_{3},t_{1}\\}t_{2}.$

######  Theorem 7.12 (Theorem 15, [141]).

If a unital algebra of Jordan brackets $\mathsf{A}$ satisfies a polynomial
identity $g_{0}$, then $\mathsf{A}$ satisfies polynomial identities of types
(3) and (4).

###  7.3 Transposed Poisson algebra structures

Recently, a dual notion of the Poisson algebra (transposed Poisson algebra)
has been introduced in the paper by Bai, Bai, Guo, and Wu [20]. Roughly
speaking, this new type of algebra is defined by exchanging the roles of the
two binary operations in the Leibniz rule defining the Poisson algebra. The
authors showed that the transposed Poisson algebras defined in this way not
only share common properties with Poisson algebras, including the closure
under taking tensor products and the Koszul self-duality as an operad but also
admit a rich class of identities. More significantly, a transposed Poisson
algebra naturally arises from a Novikov-Poisson algebra by taking the
commutator Lie algebra of the Novikov algebra. Consequently, we find a
parallelism between the classical construction of a Poisson algebra from a
commutative associative algebra with a pair of commuting derivations and the
construction of a transposed Poisson algebra when there is some non-zero
derivation. Furthermore, the transposed Poisson algebras also capture the
algebraic structure when the commutator is taken in pre-Lie Poisson algebras
and two other algebras of Poisson type.

######  Definition 7.13.

Let $\mathsf{A}$ be a vector space equipped with two nonzero bilinear
operations $\cdot$ and $[\cdot,\cdot]$. The triple
$(\mathsf{A},\cdot,[\cdot,\cdot])$ is called a transposed Poisson algebra if
$(\mathsf{A},\cdot)$ is a commutative associative algebra and
$(\mathsf{A},[\cdot,\cdot])$ is a Lie algebra that satisfies the following
compatibility condition

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

The study of $\delta$-derivations of Lie algebras was initiated by Filippov in
1998 [96, 97]. Let us define $\frac{1}{2}$-derivations which will be very
useful for our work.

######  Definition 7.14.

Let $(\mathsf{A},[\cdot,\cdot])$ be an algebra with multiplication
$[\cdot,\cdot]$ and $\varphi$ be a linear map. Then $\varphi$ is a
$\frac{1}{2}$-derivation if it satisfies
$\varphi[x,y]=\frac{1}{2}\left([\varphi(x),y]+[x,\varphi(y)]\right)$.

Summarizing definitions 7.13 and 7.14, we have the following key statement.

######  Theorem 7.15 (Lemma 7, [93]).

Let $(\mathsf{A},\cdot,[\cdot,\cdot])$ be a transposed Poisson algebra and $z$
an arbitrary element from $\mathsf{A}$. Then the right multiplication $R_{z}$
in the commutative associative algebra $(\mathsf{A},\cdot)$ gives a
$\frac{1}{2}$-derivation of the Lie algebra $(\mathsf{A},[\cdot,\cdot])$.

In our joint work with Ferreira and Lopatkin [93], we found a way to describe
all transposed Poisson algebra structures with a certain Lie algebra. Our main
tools are the description of the space of $\frac{1}{2}$-derivations of Lie
algebras and the connection between the space of $\frac{1}{2}$-derivations of
the Lie part of a transposed Poisson algebra and the space of right
multiplications of the associative part of this transposed Poisson algebra.
Namely, every right associative multiplication is a Lie
$\frac{1}{2}$-derivation. Using the known description of $\delta$-derivations
of semisimple finite-dimensional Lie algebras, we have found that there are no
transposed Poisson algebras with a semisimple finite-dimensional Lie part. In
the case of simple infinite-dimensional algebras, we have a different
situation: it has been proved that the Witt algebra admits many nontrivial
structures of transposed Poisson algebras. Later we studied structures of
transposed Poisson algebras defined on one of the most interesting
generalizations of the Witt algebra. Namely, we considered
$\frac{1}{2}$-derivations of the algebra ${\mathcal{W}}(a,b)$ and proved that
the algebra ${\mathcal{W}}(a,b)$ does not admit structures of transposed
Poisson algebras if and only if $b\neq-1$. All the transposed Poisson algebra
structures defined on ${\mathcal{W}}(a,-1)$ have been described. In the
following section, we considered another example of an algebra related to the
Witt algebra. We proved that there are no transposed Poisson algebras defined
on the Virasoro algebra. As for some corollaries, we proved that there are no
transposed Poisson algebra structures defined on the $N=1$ and $N=2$
superconformal algebras. The rest of the paper is dedicated to a
classification of $\frac{1}{2}$-derivations and constructions of transposed
Poisson algebras defined on the thin Lie algebra and the solvable Lie algebra
with the abelian nilpotent radical of codimension $1$. Other interesting
examples of transposed Poisson algebras constructed in a series of papers
together with Khrypchenko [147, 149, 148, 143].

##  8 $n$-Ary algebras

The present part is based on the papers written together with Alexandre
Pozhidaev, Antonio Jesús Calderón, Elisabete Barreiro, José María Sánchez,
Paulo Saraiva, and Yury Popov [24, 22, 50, 168, 172].

###  8.1 Ternary Jordan algebras

Based on the relation between the notions of Lie triple systems and Jordan
algebras, we introduce the $n$-ary Jordan algebras [172], an $n$-ary
generalization of Jordan algebras obtained via the generalization of the
following property
$\left[R_{x},R_{y}\right]\in{\mathfrak{D}er}\left(\mathsf{A}\right)$, where
$\mathsf{A}$ is an $n$-ary algebra. As we can see, the present generalization
of algebras is an $n$-ary generalization of almost Jordan and commutative
$\mathfrak{CD}$-algebras.

In [172], we give the first example of ternary Jordan algebras defined in this
way and study some properties of this algebra. Namely, let $\mathbb{V}$ be an
$n$-dimensional vector space over a field $\mathbb{F}$, and denote the
bilinear form by $\left(\cdot,\cdot\right)$. Consider the following ternary
multiplication defined on $\mathbb{V}$:

$\llbracket
x,y,z\rrbracket=\left(y,z\right)x+\left(x,z\right)y+\left(x,y\right)z.$

Denote the obtained ternary algebra by $\mathbb{A}$. The following three
theorems give us a characterization of $\mathbb{A}$.

######  Theorem 8.1 (Theorem 5, [172]).

$\mathbb{A}$ is a ternary Jordan algebra.

######  Theorem 8.2 (Theorem 6, [172]).

The ternary Jordan algebra $\mathbb{A}$ is simple, except if $\dim\
\mathbb{V}=2$ and $char\left(\mathbb{F}\right)=2$.

######  Theorem 8.3 (Theorem 11, [172]).

$\mathfrak{Der}\left(\mathbb{A}\right)={\rm
Inder}\left(\mathbb{A}\right)=\mathfrak{so}(n)$.

######  Remark 8.4 (Remark 12, [172]).

In $1955$ Jacobson proved that if a finite-dimensional Lie algebra over a
field of characteristic zero has an invertible derivation, then it is a
nilpotent algebra [124]. The same result was proved for Jordan algebras [166],
but as we can see from Theorem 8.3, the Theorem of Jacobson is not true for
ternary Jordan algebras. We can take the ternary Jordan algebra $\mathbb{A}$
(as in Theorem 8.3) with dimension $4$ and consider the map defined by the
matrix $\sum_{1\leq i<j\leq 4}(e_{ij}-e_{ji})$. As follows, there is a simple
ternary Jordan algebra with an invertible derivation.

Later in [172], we study a ternary example of these algebras. Finally, based
on the construction of a family of ternary algebras defined using the Cayley —
Dickson algebras, we present an example of a ternary $\mathfrak{CD}$-algebra
($n$-ary $\mathfrak{CD}$-algebras are the $n$-ary version of (not necessarily
commutative) $\mathfrak{CD}$-algebras) in [172, Section 5].

Let $\mathcal{U}_{2}$ be the generalized quaternions and $\mathcal{U}_{3}$ be
the generalized octonions. Define on $\mathcal{U}_{t}$, for $t=2,3$, the
ternary multiplication:

$\llbracket x,y,z\rrbracket=\left(x\overline{y}\right)z,$

and take
$\mathcal{D}_{t}=\left(\mathcal{U}_{t},\llbracket\cdot,\cdot,\cdot\rrbracket\right)$.

This ternary multiplication is not totally commutative, so these algebras are
not ternary Jordan algebras.

######  Theorem 8.5 (Theorem 15, [172]).

$\mathcal{D}_{2}$ is a simple ternary $\mathfrak{CD}$-algebra.

######  Theorem 8.6 (Lemma 15, [172]).

$\mathcal{D}_{3}$ is not a ternary $\mathfrak{CD}$-algebra.

###  8.2 Split regular ${\rm Hom}$-Leibniz color $3$-algebras

The study of ${\rm Hom}$-structures began in the paper by Hartwig, Larsson,
and Silvestrov [114]. The notion of ${\rm Hom}$-Lie triple systems was
introduced in [202]. In the paper [247] Yau gave a general method for
constructing ${\rm Hom}$-type algebras starting from usual algebras and a
twisting self-map.

In our joint work with Popov [168], we study the structure of split regular
${\rm Hom}$-Leibniz $3$-algebras of arbitrary dimension and over an arbitrary
base field ${\mathbb{F}}$. Split structures first appeared in the classical
theory of (finite-dimensional) Lie algebras, but have been extended to more
general settings like, for example, Leibniz algebras [53], Poisson algebras,
Leibniz superalgebras, regular ${\rm Hom}$-Lie algebras, regular ${\rm
Hom}$-Lie superalgebras, regular ${\rm Hom}$-Lie color algebras, regular ${\rm
Hom}$-Poisson algebras [14], regular ${\rm Hom}$-Leibniz algebras, regular
${\rm BiHom}$-Lie algebras [54], and regular ${\rm BiHom}$-Lie superalgebras,
among many others. As for the study of split ternary structures, see [49] for
Lie triple systems, twisted inner derivation triple systems, Lie $3$-algebras
[49], Leibniz $3$-algebras [55], and for Leibniz triple systems.

######  Definition 8.7.

Let ${\mathbb{F}}$ be a field and $\mathbb{G}$ be an abelian group. A map
$\epsilon:\mathbb{G}\times\mathbb{G}\rightarrow{\mathbb{F}}^{\times}$ is
called a bicharacter on $\mathbb{G}$ if the following relations hold for all
$f,g,h\in\mathbb{G}:$

(1) $\epsilon(f,g+h)=\epsilon(f,g)\epsilon(f,h)$;

(2) $\epsilon(g+h,f)=\epsilon(g,f)\epsilon(h,f)$;

(3) $\epsilon(g,h)\epsilon(h,g)=1$.

######  Definition 8.8.

A $\mathbb{G}$-graded color $n$-ary algebra $T$ is a vector space
$T=\bigoplus_{g\in\mathbb{G}}T_{g}$ with an $n$-linear map
$[\cdot,\ldots,\cdot]:T\times\ldots\times T\rightarrow T$ satisfying

$[T_{\theta_{1}},\ldots,T_{\theta_{n}}]\subseteq
T_{\theta_{1}+\ldots+\theta_{n}},\ \theta_{i}\in\mathbb{G}.$

######  Definition 8.9.

A regular ${\rm Hom}$-Leibniz color $3$-algebra
$(T,[\cdot,\cdot,\cdot],\epsilon,\phi)$ is a $\mathbb{G}$-graded vector space
$T$ with a bicharacter $\epsilon$, an even trilinear map $[\cdot,\cdot,\cdot]$
and an even automorphism $\phi$ satisfying

$[\phi(x_{1}),\phi(x_{2}),[y_{1},y_{2},y_{3}]]=[[x_{1},x_{2},y_{1}],\phi(y_{2}),\phi(y_{3})]+$
$\epsilon(x_{1}+x_{2},y_{1})[\phi(y_{1}),[x_{1},x_{2},y_{2}],\phi(y_{3})]+\epsilon(x_{1}+x_{2},y_{1}+y_{2})[\phi(y_{1}),\phi(y_{2}),[x_{1},x_{2},y_{3}]].$

Additionally, if the identities

$[x_{1},x_{2},x_{3}]=-\epsilon(x_{1},x_{2})[x_{2},x_{1},x_{3}],\
[x_{1},x_{2},x_{3}]=-\epsilon(x_{2},x_{3})[x_{1},x_{3},x_{2}]$

hold in $T$, then $T$ is called a regular ${\rm Hom}$-Lie color $3$-algebra,
and if the identities

$\displaystyle[x_{1},x_{2},x_{3}]=-\epsilon(x_{1},x_{2})[x_{2},x_{1},x_{3}],$
$\displaystyle\epsilon(x_{3},x_{1})[x_{1},x_{2},x_{3}]+\epsilon(x_{1},x_{2})[x_{2},x_{3},x_{1}]+\epsilon(x_{2},x_{3})[x_{3},x_{1},x_{2}]=0$

hold in $T$, then $T$ is called a regular ${\rm Hom}$-Lie triple color system.

######  Definition 8.10.

Let $T=(T,[\cdot,\cdot,\cdot],\epsilon,\phi)$ be a ${\rm Hom}$-Leibniz color
$3$-algebra. Consider the space $\mathfrak{L}={\rm
span}\\{\mathrm{ad}(x,y):x,y\in T\\}$, where $\mathrm{ad}(x,y)(z):=[x,y,z]$.
The  standard embedding of a regular ${\rm Hom}$-Leibniz color $3$-algebra
$(T,[\cdot,\cdot,\cdot],\epsilon,\phi)$ is a color $2$-graded algebra with an
automorphism $(\mathsf{A},\epsilon,\Phi)$, where
${\mathsf{A}}^{0}:={\mathfrak{L}}$, ${\mathsf{A}}^{1}:=T$, the product is
given by

$(\mathrm{ad}(x,y),z)\cdot(\mathrm{ad}(u,v),w)=$
$(\mathrm{ad}([x,y,u]^{\phi^{-1}},v)+\epsilon(x+y,u)\mathrm{ad}(u,[x,y,v]^{\phi^{-1}})+\mathrm{ad}(z,w)\,\
[x,y,w]^{\phi^{-1}}-\epsilon(z,u+v)[u,v,z]^{\phi^{-1}}),$

the automorphism $\Phi$ by

$\Phi:\begin{cases}x&\mapsto\phi(x),\\\
\mathrm{ad}(x,y)&\mapsto\phi\mathrm{ad}(x,y)\phi^{-1}=\mathrm{ad}(\phi(x),\phi(y)),\end{cases}\text{
for }x,y\in T,$

and the $\mathbb{G}$-grading is induced by the $\mathbb{G}$-gradings of $T$
and $\mathfrak{L}$.

######  Definition 8.11.

Let $T=(T,[\cdot,\cdot,\cdot],\epsilon,\phi)$ be a ${\rm Hom}$-Leibniz color
$3$-algebra (or a Leibniz color $3$-algebra with an automorphism) and let
${\mathsf{A}}=({\mathfrak{L}}\oplus T,\cdot,\epsilon,\Phi)$ be its standard
embedding. Let $H$ be a maximal abelian subalgebra (shortly MASA) of
$\mathfrak{L}_{0}$, the zeroth $\mathbb{G}$-component of $\mathfrak{L}$. The
root space of $T$ with respect to $H$ associated to a linear functional
$\alpha\in H^{*}$ is the subspace

$T_{\alpha}:=\\{v\in{T}:h\cdot v=\alpha(h)v\,\mbox{ for any }\,h\in H\\}.$

The elements $\alpha\in H^{*}$ such that $T_{\alpha}\neq 0$ are called roots
of $T$ (with respect to $H$), and we write $\Lambda^{T}:=\\{\alpha\in
H^{*}\backslash\\{0\\}:T_{\alpha}\neq 0\\}$. Analogously, by
$\Lambda^{{\mathfrak{L}}}$ we denote the set of all non-zero $\alpha\in H^{*}$
such that ${\mathfrak{L}}_{\alpha}\neq 0$, where

${\mathfrak{L}}_{\alpha}:=\\{e\in{\mathfrak{L}}:[h,e]=\alpha(h)e\,\mbox{ for
any }\,h\in H\\}$

are the root subspaces of $\mathfrak{L}$ with respect to $T$.

######  Definition 8.12.

Let $T$ be a ${\rm Hom}$-Leibniz color $3$-algebra or a Leibniz color