# Crossed products of $4$-algebras. Applications

G. Militaru  Faculty of Mathematics and Computer Science, University of
Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania and Simion
Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764,
014700 Bucharest, Romania<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract.

A $4$-algebra is a commutative algebra $A$ over a field $k$ such that
$(a^{2})^{2}=0$, for all $a\in A$. We have proved recently [22] that
$4$-algebras play a prominent role in the classification of finite dimensional
Bernstein algebras. Let $A$ be a $4$-algebra, $E$ a vector space and $\pi:E\to
A$ a surjective linear map with $V={\rm Ker}(\pi)$. All $4$-algebra structures
on $E$ such that $\pi:E\to A$ is an algebra map are described and classified
by a global cohomological object ${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)$. Any
such $4$-algebra is isomorphic to a crossed product $V\\#A$ and
${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)$ is a coproduct, over all $4$-algebras
structures $\cdot_{V}$ on $V$, of all non-abelian cohomologies
${\mathbb{H}}^{2}_{\rm nab}\,\bigl{(}A,\,(V,\,\cdot_{V})\bigl{)}$, which are
the classifying objects for all extensions of $A$ by $V$. Several applications
and examples are provided: in particular,
${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,k)$ and
${\mathbb{G}}{\mathbb{H}}^{2}\,(k,\,V)$ are explicitly computed and the Galois
group ${\rm Gal}\,(V\\#A/V)$ of the extension $V\hookrightarrow V\\#A$ is
described.

###### Key words and phrases:

Non-associative algebra, extension problem, non-abelian cohomology.

###### 2010 Mathematics Subject Classification:

16T10, 16T05, 16S40

###### 2020 Mathematics Subject Classification:

17A60, 17A30, 17D92

This work was supported by a grant of the Ministry of Research, Innovation and
Digitization, CNCS/CCCDI–UEFISCDI, project number PN-III-P4-ID-PCE-2020-0458,
within PNCDI III

## Introduction

Bernstein algebras were introduced independently by Lyubich [19] and Holgate
[17] as an algebraic tool to answer the Bernstein problem [5] which consists
in classifying all possible situations of a population that attains genetic
equilibrium after one generation ([23, Section 4], [25, Chapter 9]). A
Bernstein algebra is a commutative algebra $B$ over a field $k$ of
characteristic $\neq 2$ such that there exists a non-zero morphism of algebras
$\omega:B\to k$ such that $(x^{2})^{2}=\omega(x)^{2}\,x^{2}$, for all $x\in
B$. For algebrists, the Bernstein problem can now be rephrased as follows:
_for a given positive integer $n$, describle and classify all simplicial
stochastic Bernstein ${\mathbb{R}}$-algebras of dimension $n$_. For more
details we refer to the work of Gutierrez Fernandez [13] which completely
solved this problem. Now, leaving aside the simplicial stochastic condition
and replacing ${\mathbb{R}}$ by an arbitrary field $k$, the problem of
classifying finite dimensional Bernstein algebras is still open: it was solved
only up to dimension $4$ and there are partial answers in dimensions $5$ or
$6$ (see [8, 9, 14, 17, 20, 25]).

If $(B,\omega)$ is a Bernstein algebra, then its barideal $A:={\rm
Ker}(\omega)$ is itself a commutative algebra satisfying the compatibilities
$(a^{2})^{2}=0$, for all $a\in A$: we called this class of algebras _$4$
-algebras_ [22] and we have proved that they play the key role in the
structure and classification of Bernstein algebras. It is worth pointing out
that $4$-algebras are a special case of _admissible cubic algebras_ as
introduced by Elduque and Okubo [11]: these are commutative algebras
satisfying the identity $(a^{2})^{2}=N(a)\,a$, where $N:A\to k$ is a cubic
form. In [22, Theorem 2.6] we prove that any Bernstein algebra $B$ is
isomorphic to a semidirect product $A\ltimes_{(\cdot,\,\Omega)}\,k$, where
$(A,\cdot)$ is a $4$-algebra and $\Omega$ is a _Bernstein operator_ on $A$,
i.e. $\Omega=\Omega^{2}\in{\rm End}_{k}(A)$ is an idempotent endomorphism of
$A$ such that for any $x\in A$:

$x^{2}\cdot\Omega(x)=0,\qquad\Omega(x)^{2}+\Omega(x^{2})=x^{2}.$

The classification of two such Bernstein algebras
$A\ltimes_{(\cdot,\,\Omega)}\,k$ and
$A^{\prime}\ltimes_{(\cdot^{\prime},\,\Omega^{\prime})}\,k$ is also proven in
[22, Theorem 2.7 and Theorem 2.11]. Based on this, the first step we have to
take in order to classify all Bernstein algebras is to classify all
$4$-algebras of a given finite dimension $n$. For this reason the paper is
devoted to the study of this class of algebras. An efficient tool for
classifying finite objects and a source for developing cohomology theories is
the _extension problem_ introduced by Hölder [16] at the level of groups and
intesively studied in the last 100 years for many categories of algebras such
as associative algebras [12], Lie algebras [7], Hopf algebras [4], Poisson
algebras [18], Lie-Rinehart algebras [6], etc. For $4$-algebras the extension
problem consists of the following question: _Let $A$ and $V$ be two given
$4$-algebras. Describe and classify all extensions of $A$ by $V$, i.e. all
triples $(E,i,\pi)$ consisting of a $4$-algebra $E$ and two morphisms of
algebras that fit into an exact sequence of the form:_
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$.

Two extensions $(E,\,i,\,\pi)$ and $(E^{\prime},\,i^{\prime},\,\pi^{\prime})$
of $A$ by $V$ are called _equivalent_ (or _cohomologous_) if there exists a
morphism $\varphi:E\to E^{\prime}$ of algebras that stabilizes $V$ and co-
stabilizes $A$, i.e. $\varphi\circ i=i^{\prime}$ and
$\pi^{\prime}\circ\varphi=\pi$. Any such a morphism is an isomorphism and we
denote by ${\rm Ext}\,(A,\,V)$ the set of equivalence classes of all
$4$-algebras that are extensions of $A$ by $V$; an answer to the extension
problem means to calculate explicitly ${\rm Ext}\,(A,\,V)$ for two given
$4$-algebras $A$ and $V$. The Schreirer [24] approach to the extension problem
for groups works, mutatis-mutandis, also for $4$-algebras as for others
varieties of algebras: the classifying object ${\rm Ext}\,(A,\,V)$ is
parameterized by the non-abelian cohomology ${\mathbb{H}}^{2}_{\rm
nab}\,\bigl{(}A,\,V\bigl{)}$ (Corollary 2.12). More general than the extension
problem is what we have called [21] _global extension problem_ (GE-problem)
and was studied for Leibniz algebras, associative algebras, Poisson algebras
or Jacobi-Jordan algebras [1, 2, 3]. The GE-problem, formulated for
$4$-algebras, is the following question:

Global Extension Problem. _Let $A$ be a $4$-algebra, $E$ a vector space and
$\pi:E\to A$ a linear epimorphism of vector spaces. Describe and classify the
set of all $4$-algebra structures that can be defined on $E$ such that
$\pi:E\to A$ becomes a morphism of algebras._

The difference between the GE-problem and the classical extension problem is
explained in detail in [1]. Proposition 2.4 proves that any such a $4$-algebra
structure $\cdot_{E}$ on $E$ is isomorphic to a _crossed product_
$V\\#A=V\\#_{(\triangleright,\,f,\,\cdot_{V})}A$, which is a $4$-algebra
associated to $A$ and $V:={\rm Ker}(\pi)$ connected by a _weak action_
$\triangleright:A\times V\to V$, a _symmetric non-abelian $2$-cocycle_
$f:A\times A\to V$ and a $4$-algebra structure $\cdot_{V}$ on $V$ satisfying
the axioms of Proposition 2.2. The crossed product $V\\#A$ is a $4$-algebra
containg $V$ as an ideal and the Galois group ${\rm Gal}\,(V\\#A/V)$ of the
extension $V\hookrightarrow V\\#A$ is described in Corollary 2.9 as a subgroup
of the semidirect product of groups ${\rm Hom}_{k}(A,\,V)\ltimes{\rm
GL}_{k}(A)$. The main result of the paper is Theorem 2.11 that gives the
theoretical answer to the GE-problem: the classifying object for the GE-
problem is parameterized by a global non-abelian cohomological object denoted
by ${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)$. Corollary 2.12 proves that
${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)$ is the coproduct of all non-abelian
cohomologies ${\mathbb{H}}^{2}\,\,(A,\,(V,\cdot_{V}))$, the latter being the
classifying object for the classical extension problem at the level of
$4$-algebras. Several examples and applications are given in Section 3: in
particular, ${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,k)$ and
${\mathbb{G}}{\mathbb{H}}^{2}\,(k,\,V)$ are computed and the structure of
metabelian $4$-algebras is given in Corollary 3.6.

##  1\. Preliminaries

Throughout this paper all vector spaces, linear or bilinear maps are over a
field $k$ of characteristic $\neq 2$. For a family of sets $(A_{i})_{i\in I}$
we shall denote by $\amalg_{i\in I}\,A_{i}$ their coproduct, i.e.
$\amalg_{i\in I}\,A_{i}$ is the disjoint union of all sets $A_{i}$. If $A$ and
$V$ are two vector spaces, ${\rm Hom}_{k}(A,\,V)$ denotes the vector space of
all linear maps $A\to V$ and ${\rm Sym}(A\times A;\,V)$ the set of all
symmetric bilinear maps $f:A\times A\to V$; ${\rm End}_{k}\,(A)$ is the usual
associative and unital endomorphisms algebra of $A$ and ${\rm GL}_{k}(A)$ is
the automorphisms group of $A$. We denote by ${\rm Hom}_{k}(A,\,V)\ltimes{\rm
GL}_{k}(A):={\rm Hom}_{k}(A,\,V)\times{\rm GL}_{k}(A)$ the semidirect product
of groups having the multiplication defined for any $(r,\,\alpha)$ and
$(r^{\prime},\,\alpha^{\prime})\in{\rm Hom}_{k}(A,\,V)\times{\rm GL}_{k}(A)$
by:

$(r,\,\alpha)\bullet(r^{\prime},\,\alpha^{\prime}):=(r^{\prime}+r\circ\alpha^{\prime},\,\alpha\circ\alpha^{\prime}).$
(1)

A _$4$ -algebra_ [22] is a vector space $A$ together with a bilinear map
$\cdot:A\times A\to A$, called multiplication, such that for any $a$, $b\in A$
we have:

$a\cdot b=b\cdot a,\qquad(a^{2})^{2}=0.$ (2)

The concepts of subalgebras, ideals, morphisms of algebras, etc. for
$4$-algebras are defined in the obvious way. The class of $4$-algebras were
studied before in [15] where it was proved that any $4$-algebra of dimension
$\leq 7$ is solvable and it was conjectured that any finite dimensional
$4$-algebra is solvable. Any vector space $V$ is a $4$-algebra with the
trivial multiplication $x\cdot y:=0$, for all $x$, $y\in V$: we call this
algebra _abelian_ and it will be denoted by $V_{0}$. If $(A,\omega)$ is a
Bernstein algebra, then the barideal ${\rm Ker}(\omega)$ is a $4$-algebra.
Conversely, if $A$ is a $4$-algebra and $\Omega=\Omega^{2}\in{\rm End}_{k}(A)$
is a Bernstein operator on $A$, then $A\times k$ has a canonical structure of
Bernstein algebra with the barideal $A$ (for details see [22, Proposition
2.3]). Linearizing several times the second compatibility of (2) we obtain
that in a $4$-algebra over a field of characteristic $\neq 2,\,3$ the
following relations hold [15]:

$\displaystyle a^{2}\cdot(a\cdot b)=0,\qquad a^{2}\cdot b^{2}+2\,(a\cdot
b)^{2}=0,\qquad a^{2}\cdot(b\cdot c)+2\,(a\cdot b)\cdot(a\cdot c)=0,$
$\displaystyle(a\cdot b)\cdot(c\cdot d)+(a\cdot c)\cdot(b\cdot d)+(a\cdot
d)\cdot(b\cdot c)=0$

for all $a$, $b$, $c$, $d\in A$. For a $4$-algebra $A$ we denote by
$A^{\prime}:=A\cdot A$ its derived algebra, i.e. $A^{\prime}$ is the
$k$-subspace of $A$ generated by all $a\cdot b$, for any $a$, $b\in A$.
Similarly to the groups or Lie algebras theory, a $4$-algebra $A$ is called
_metabelian_ if $A^{\prime}$ is an abelian subalgebra of $A$, i.e. $(a\cdot
b)\cdot(c\cdot d)=0$, for all $a$, $b$, $c$, $d\in A$. Throughout this paper
we use the following convention: the multiplication of a $4$-algebra $A$ will
be written on the elements of a $k$-basis $\\{e_{i}\,|\,i\in I\\}$ of $A$ and
undefined multiplications are all zero.

######  Examples 1.1.

(1) Any $1$-dimensional $4$-algebra is isomorphic to the abelian one:
$e_{1}\cdot e_{1}:=0$. We can easily prove that, up to an isomorphism, there
are exactly three $2$-dimensional $4$-algebras, namely the abelian one and the
algebras with the basis $\\{e_{1},\,e_{2}\\}$ and the multiplication given by:

$\displaystyle A_{1}:$ $\displaystyle e_{1}^{2}=e_{2},\quad
e_{1}e_{2}=e_{2}^{2}=0,$ $\displaystyle A_{2}:$ $\displaystyle
e_{1}e_{2}=e_{2},\quad e_{1}^{2}=e_{2}^{2}=0.$

(2) Let $n$ be a positive integer and $\mathfrak{h}(2n+1)$ the
$(2n+1)$-dimensional algebra having
$\\{e_{1},\cdots,e_{n},f_{1},\cdots,f_{n},z\\}$ as a basis and multiplication
defined by $e_{i}\cdot f_{i}=f_{i}\cdot e_{i}:=z$, for all $i=1,\cdots,n$ and
the other products of basis elements are zero. Then $\mathfrak{h}(2n+1)$ is a
$4$-algebra called the commutative Heisenberg $4$-algebra.

(3) Let $m$ and $n$ be two positive integers and two bilinear maps

$\triangleright:k^{m}\times k^{n}\to k^{n},\qquad f:k^{m}\times k^{m}\to
k^{n}$

such that $f$ is symmetric. Let $\\{e_{i}\,|\,i=1,\cdots,n\\}$ be a basis of
$k^{n}$ and $\\{f_{j}\,|\,j=1,\cdots,m\\}$ a basis of $k^{m}$. Let $A$ be the
$(n+m)$-dimensional algebra with the basis
$\\{e_{i},\,f_{j}\,|\,i=1,\cdots,n,\,j=1,\cdots,m\\}$ and the multiplication
defined by:

$e_{i}\cdot f_{j}=f_{j}\cdot e_{i}:=f_{j}\triangleright e_{i},\qquad
f_{j}\cdot f_{l}:=f(f_{j},\,f_{l})$

for all $i=1,\cdots,n$, $j$, $l=1,\cdots m$, and the other products of basis
elements are zero. Then $A$ is a metabelian $4$-algebra and will be denoted by
${\rm Met}_{n}^{m}\,(\triangleright,\,f)$: Corollary 3.6 will prove that any
finite dimensional metabelian $4$-algebra is isomorphic to such an algebra
${\rm Met}_{n}^{m}\,(\triangleright,\,f)$.

Similarly, with other classes of non-associative algebras we define the
concept of modules/representations over $4$-algebras as follows:

######  Definition 1.2.

An _$A$ -module_ over a $4$-algebra $A$ is a vector space $V$ with a bilinear
map $\triangleright:A\times V\to V$, called action of $A$ on $V$, such that
for any $a\in A$ and $x\in V$ we have:

$\displaystyle a^{2}\triangleright(a\triangleright x)=0.$ (3)

A _representation_ of a $4$-algebra $A$ on a vector space $V$ is a linear map
$\varphi:A\to{\rm End}_{k}\,(V)$ such that for any $a\in A$ we have
$\varphi(a^{2})\circ\varphi(a)=0$, in the endomorphism algebra of $V$.

######  Remark 1.3.

The axiom (3) of defining modules over a $4$-algebra was influenced by the
view point of Eilenberg [10] of defining modules over a given object
${\mathcal{O}}$ in a $k$-linear category ${\mathcal{C}}$: that is, a vector
space $V$ with a bilinear map $\triangleright:{\mathcal{O}}\times V\to V$ such
that the trivial extension $V\times{\mathcal{O}}$, with the multiplication
given by $(x,a)\cdot(y,b):=(a\triangleright y+b\triangleright x,\,a\cdot b)$,
has to be an object _inside_ the $k$-linear category ${\mathcal{C}}$. For
details see Example 2.3 below.

Of course, representations and modules over a $4$-algebras are equivalent
concepts and $A$ is a module over itself with $a\triangleright b:=a\cdot b$,
for all $a$, $b\in A$.

##  2\. Crossed products and the global extension problem for $4$-algebras

Let $A$ be a $4$-algebra, $E$ a vector space, $\pi:E\to A$ a linear
epimorphism of vector spaces with $V:={\rm Ker}(\pi)$ and denote by $i:V\to E$
the inclusion map. We recall that a linear map $\varphi:E\to E$ _stabilizes_
$V$ (resp. _co-stabilizes_ $A$) if $\varphi\circ i=i$ (resp.
$\pi\circ\varphi=\pi$). Two $4$-algebra structures $\cdot_{E}$ and
$\cdot_{E}^{\prime}$ on $E$ such that $\pi:E\to A$ is a morphism of
$4$-algebras are called _cohomologous_ and we denote this by
$(E,\cdot_{E})\approx(E,\cdot_{E}^{\prime})$, if there exists an algebra map
$\varphi:(E,\cdot_{E})\to(E,\cdot_{E}^{\prime})$ which stabilizes $V$ and co-
stabilizes $A$. Any such morphism is an isomorphism and therefore $\approx$ is
an equivalence relation on the set of all $4$-algebra structures on $E$ such
that $\pi:E\to A$ is an algebra map. The set of all equivalence classes via
the equivalence relation $\approx$ will be denoted by ${\rm Gext}\,(A,\,E)$
and it is the classifying object for the GE-problem for $4$-algebras. In this
section we will prove that ${\rm Gext}\,(A,\,E)$ is parameterized by a global
non-abelian cohomological object ${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)$ which
will be explicitly constructed. First, we need to introduce the crossed
product of $4$-algebras:

######  Definition 2.1.

Let $A=(A,\cdot)$ be a $4$-algebra and $V$ a vector space. A _crossed data_ of
$A$ by $V$ is a system
${\mathcal{C}}\,(A,\,V)=(\triangleright,\,f,\,\cdot_{V})$ consisting of three
bilinear maps

$\triangleright\,\,:A\times V\to V,\quad f\,\,:A\times A\to
V,\quad\cdot_{V}\,:V\times V\to V.$

For a crossed data ${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ we
denote by $V\\#A=V\,\\#_{(\triangleright,\,f,\,\cdot_{V})}\,A$ the vector
space $V\times A$ with the multiplication $\circ$ defined for any $a$, $b\in
A$ and $x$, $y\in V$ by:

$(x,\,a)\circ(y,\,b):=(x\cdot_{V}y+a\triangleright y+b\triangleright
x+f(a,\,b),\,\,a\cdot b).$ (4)

$V\\#A$ is called the _crossed product_ associated to ${\mathcal{C}}\,(A,V)$
if it is a $4$-algebra with the multiplication given by (4). In this case
${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ is called a _crossed
system_ of $A$ by $V$. The set of all crossed systems of $A$ by $V$ will be
denoted by ${\mathcal{C}}{\mathcal{S}}\,(A,V)$. Our first result provides the
necessary and sufficient conditions for $V\\#A$ to be a crossed product:

######  Proposition 2.2.

Let $A=(A,\cdot)$ be a $4$-algebra, $V$ a vector space and
${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ a crossed data of $A$
by $V$. Then $V\\#A$ is a crossed product if and only if the following
compatibility conditions hold:

  1. (CS1)

$(V,\,\cdot_{V})$ is a $4$-algebra and the bilinear map $f:A\times A\to V$ is
symmetric;

  2. (CS2)

$f(a^{2},a^{2})+f(a,a)\cdot_{V}f(a,a)+2\,a^{2}\triangleright f(a,a)=0$;

  3. (CS3)

$a^{2}\triangleright
x^{2}+x^{2}\cdot_{V}f(a,a)+2\,x^{2}\cdot_{V}\,(a\triangleright
x)+2\,(a\triangleright x)^{2}+2\,(a\triangleright
x)\cdot_{V}\,f(a,a)+2\,a^{2}\triangleright(a\triangleright x)=0$,

for all $a\in A$ and $x\in V$.

Borrowing the terminology from crossed products of Hopf algebras [4], the
symmetric bilinear map $f:A\times A\to V$ satisfying (CS2) is called a _non-
abelian $2$-cocycle_ while the axiom (CS3) we called the _twisted module
condition_ for $\triangleright\,:A\times V\to V$.

###### Proof.

First of all we can easily prove that the multiplication $\circ$ defined by
(4) is commutative if and only if $f:A\times A\to V$ is symmetric and
$\cdot_{V}:V\times V\to V$ is commutative algebra structure on $V$. Assume
that $\circ$ is commutative. Then for any $x\in V$ and $a\in A$ we have that
$(x,a)^{2}=(x^{2}+2\,a\triangleright x+f(a,a),\,a^{2})$. Based on this and
taking into account that $A$ is a $4$-algebra we obtain that
$\bigl{(}(x,a)^{2}\bigl{)}^{2}=(0,0)$ if and only if

$\displaystyle(x^{2})^{2}+4\,(a\triangleright
x)^{2}+f(a,a)^{2}+4\,x^{2}\cdot_{V}\,(a\triangleright
x)+2\,x^{2}\cdot_{V}f(a,a)+$ $\displaystyle+4\,(a\triangleright
x)\cdot_{V}\,f(a,a)+2\,a^{2}\triangleright
x^{2}+4\,a^{2}\triangleright(a\triangleright x)+2\,a^{2}\triangleright
f(a,a)+f(a^{2},a^{2})=0.$

This equation holds for $a:=0$ if and only if $(x^{2})^{2}=0$, for all $x\in
V$ (i.e. $(V,\cdot_{V})$ is a $4$-algebra) and it holds for $x:=0$ if and only
if (CS2) holds. Assuming these conditions, we obtain that the above equation
holds if and only if (CS3) holds and this finishes the proof. ∎

From now on a crossed system of a $4$-algebra $A$ by a vector space $V$ will
be seen as a system of bilinear maps
${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ satisfying axioms
(CS1)-(CS3) of Proposition 2.2.

######  Examples 2.3.

(1) Let ${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ be a crossed
data of a $4$-algebra $A$ by $V$ such that $f$ is the trivial map, i.e.
$f(a,b):=0$, for all $a$, $b\in A$. Then, applying Proposition 2.2 we obtain
that $(\triangleright,\,f:=0,\,\cdot_{V})$ is a crossed system if and only if
$(V,\cdot)$ is a $4$-algebra and for all $a\in A$, $x\in V$ we have:

$\displaystyle a^{2}\triangleright x^{2}+2\,x^{2}\cdot_{V}\,(a\triangleright
x)+2\,(a\triangleright x)^{2}+2\,a^{2}\triangleright(a\triangleright x)=0.$
(5)

The associated crossed product will be called a _semidirect product_ of
$4$-algebras $A$ and $V$ and will be denoted by $V\ltimes
A:=V\ltimes_{(\triangleright,\,\cdot_{V})}A$. The terminology is motivated
below in Corollary 2.7: exactly as in the case of groups or Lie algebras, this
construction describes split epimorphisms in the category of $4$-algebras.

Assume, in addition, that the multiplication $\cdot_{V}$ is also the trivial
map. Then, the compatibility condition (5) becomes
$a^{2}\triangleright(a\triangleright x)=0$, for all $a\in A$ and $x\in V$,
i.e. $(V,\triangleright)$ is a $A$-module as we introduced in Definition 1.2.
In this case, the semidirect product $V\ltimes A$ is called the _trivial
extension_ of the $4$-algebra $A$ by the $A$-module $V$.

(2) The following special case of crossed products play an important role in
the classification of finite dimensional $4$-algebras. Assume that $A$ is the
abelian $4$-algebra. Then a crossed data
${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ of $A$ by $V$ is a
crossed system if and only if $(V,\cdot)$ is a $4$-algebra, $f:A\times A\to V$
is a symmetric bilinear map such that:

$\displaystyle f(a,a)^{2}=0,\quad
x^{2}\cdot_{V}f(a,a)+2\,x^{2}\cdot_{V}\,(a\triangleright
x)+2\,(a\triangleright x)^{2}+2\,(a\triangleright x)\cdot_{V}\,f(a,a)=0$ (6)

for all $a\in A$ and $x\in V$. In this case, the associated crossed product
$V\\#A$ will be called the _twisted product_ of $V$ and $A$. If
$\\{e_{i}\,|\,i\in I\\}$ is a basis of $V$ and $\\{f_{j}\,|\,j\in J\\}$ is a
basis of $A$, then the twisted product $V\\#A$ is the $4$-algebra having
$\\{e_{i},\,f_{j}\,|\,i\in I,\,j\in J\\}$ as a basis and the multiplication
given by:

$e_{i}\circ e_{j}:=e_{j}\cdot_{V}e_{j},\quad e_{i}\circ f_{l}=f_{l}\circ
e_{i}:=f_{l}\triangleright e_{i},\quad f_{l}\circ f_{m}:=f(f_{l},\,f_{m})$ (7)

for all $i$, $j\in I$ and $l$, $m\in J$. In Corollary 2.5 we shall prove that
any $4$-algebra is isomorphic to a such a twisted product.

(3) Let ${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ be a crossed
data of a $4$-algebra $A$ by $V$ such that $\cdot_{V}$ is the trivial map. By
applying Proposition 2.2 we obtain that $(\triangleright,\,f,\,\cdot_{V}:=0)$
is a crossed system if and only if $(V,\triangleright)$ is an $A$-module and
$f:A\times A\to V$ is _symmetric abelian $2$-cocycle_, i.e.

$f(a^{2},a^{2})+2\,a^{2}\triangleright f(a,a)=0$

for all $a\in A$. This case will appear in the study of the classical
extension problem for $4$-algebras, namely those with an abelian kernel (see
Corollary 2.13 below).

(4) Let ${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ be a crossed
data of a $4$-algebra $A$ by $V$ such that $\triangleright$ is the trivial
action, i.e. $a\triangleright x:=0$, for all $a\in A$ and $x\in V$. Then
$(\triangleright:=0,\,f,\,\cdot_{V})$ is a crossed system if and only if
$(V,\cdot)$ is a $4$-algebra, $f:A\times A\to V$ is symmetric and

$f(a^{2},a^{2})+f(a,a)^{2}=0,\qquad x^{2}\cdot_{V}f(a,a)=0$

for all $a\in A$ and $x\in V$.

Let now ${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ be a crossed
system of a $4$-algebra $A$ by $V$. Then, the canonical projection
$\pi_{A}:V\\#A\to A$, $\pi_{A}(x,a):=a$ is a surjective algebra map and ${\rm
Ker}(\pi_{A})=V\times\\{0\\}\cong V$ is an ideal in the $4$-algebra $V\\#A$.
Thus, we obtain that the $4$-algebra $V\\#A$ is an extension of the
$4$-algebra $A$ by the $4$-algebra $(V,\cdot_{V})$ via

(10)

where $i_{V}(x)=(x,\,0)$, for all $x\in V$. Conversely, we have:

######  Proposition 2.4.

Let $A$ be a $4$-algebra, $E$ a vector space and $\pi:E\to A$ an epimorphism
of vector spaces with $V={\rm Ker}(\pi)$. Then any $4$-algebra structure
$\cdot_{E}$ which can be defined on $E$ such that $\pi:(E,\cdot_{E})\to A$ is
a morphism of $4$-algebras is isomorphic to a crossed product $V\\#A$.
Furthermore, the isomorphism of $4$-algebras $(E,\cdot_{E})\cong V\\#A$ can be
chosen such that it stabilizes $V$ and co-stabilizes $A$.

In particular, any $4$-algebra structure on $E$ such that $\pi:E\to A$ is an
algebra map is cohomologous to an extension of the form (10).

###### Proof.

Let $\cdot_{E}$ be a $4$-algebra structure of $E$ such that
$\pi:(E,\cdot_{E})\to A$ is an algebra map and let $s:A\to E$ be a $k$-linear
section of $\pi$, i.e. $\pi\circ s={\rm Id}_{A}$. Then $\varphi:V\times A\to
E$, $\varphi(x,a):=x+s(a)$ is an isomorphism of vector spaces with the inverse
$\varphi^{-1}(y)=\bigl{(}y-s(\pi(y)),\,\pi(y)\bigl{)}$, for all $y\in E$.
Using the section $s$ we define the following bilinear maps:

$\displaystyle\triangleright$ $\displaystyle=$
$\displaystyle\triangleright_{s}\,\,:A\times V\to V,\,\,\,\,a\triangleright
x:=s(a)\cdot_{E}x$ (11) $\displaystyle f$ $\displaystyle=$ $\displaystyle
f_{s}\,\,\,:A\times A\to V,\,\,\,\,f(a,b):=s(a)\cdot_{E}s(b)-s(a\cdot b)$ (12)
$\displaystyle\cdot_{V}$ $\displaystyle\,\,\,\,\,\,\,\,\,:V\times V\to
V,\,\,\,\,x\cdot_{V}y:=x\cdot_{E}y$ (13)

for all $a$, $b\in A$ and $x$, $y\in V$. These are well-defined maps since
$\pi$ is an algebra map and $s$ a section of $\pi$. Using this crossed data
$(\triangleright,\,f,\,\cdot_{V})$ connecting $A$ and $V$ we can prove that
the unique $4$-algebra structure $\circ$ that can be defined on the direct
product of vector spaces $V\times A$ such that $\varphi:V\times
E\to(E,\cdot_{E})$ is an isomorphism of $4$-algebras is given by:

$(x,\,a)\circ(y,\,b):=(x\cdot_{V}y+a\triangleright y+b\triangleright
x+f(a,\,b),\,a\cdot b)$

for all $a$, $b\in A$, $x$, $y\in V$. Indeed, let $\circ$ be such a
$4$-algebra structure on $V\times A$. Then we have:

$\displaystyle(x,a)\circ(y,b)$ $\displaystyle=$
$\displaystyle\varphi^{-1}\bigl{(}\varphi(x,a)\cdot_{E}\varphi(y,b)\bigl{)}=\varphi^{-1}\bigl{(}(x+s(a))\cdot_{E}(y+s(b))\bigl{)}$
$\displaystyle=$
$\displaystyle\varphi^{-1}(x\cdot_{E}y+s(a)\cdot_{E}y+x\cdot_{E}s(b)+s(a)\cdot_{E}s(b))$
$\displaystyle=$
$\displaystyle\bigl{(}x\cdot_{E}y+s(a)\cdot_{E}y+s(b)\cdot_{E}x+s(a)\cdot_{E}s(b)-s(a\cdot
b),\,a\cdot b\bigl{)}$ $\displaystyle=$
$\displaystyle\bigl{(}x\cdot_{V}y+a\triangleright y+b\triangleright
x+f(a,\,b),\,a\cdot b)$

as desired. Thus, $\varphi:V\\#A\to(E,\cdot_{E})$ is an isomorphism of
$4$-algebras which stabilizes $V$ and co-stabilizes $A$ since the diagram is
obviously commutative. ∎

As a first consequence we obtain that any $4$-algebra is isomorphic to a
twisted product as constructed in (2) of Example 2.3.

######  Corollary 2.5.

Any $4$-algebra $E=(E,\cdot_{E})$ is isomorphic to a twisted product $V\\#A$
associated to a $4$-algebra $V$ with ${\rm dim}_{k}(V)={\rm dim}_{k}(E^{2})$
and an abelian algebra $A$.

###### Proof.

Indeed, let $(E,\cdot_{E})$ be a $4$-algebra and $V:=E^{\prime}$ its derived
algebra. Then $A:=E/E^{\prime}$ is an abelian algebra and the canonical
projection $\pi:E\to A$ is a surjective $4$-algebra map. Now, we apply
Proposition 2.4. ∎

Corollary 2.5 is a tool that can be used for the classification of
$4$-algebras of a given finite dimension. Let us explain briefly how it works:
let $E$ be an $(m+n)$-dimensional $4$-algebra. An invariant of such algebras
is the dimension $m$ of the derived algebra. Thus, we have to fix a positive
integer $m$. Based on Corollary 2.5 we obtain that $E$ is isomorphic to a
twisted product $V\\#A$, where $V$ is an $m$-dimensional $4$-algebra and $A$
is a vector space (viewed with the abelian $4$-algebra structure) of dimension
$n$ (explicitely, the multiplication on $V\\#A$ is given by (7) in Example
2.3).

######  Example 2.6.

Let $V=(V,\cdot_{V})$ be an $m$-dimensional $4$-algebra having
$\\{e_{1},\cdots,e_{m}\\}$ as a basis. Then any $(m+1)$-dimensional
$4$-algebra having the derived algebra $V$ is isomorphic to the $4$-algebra
having $\\{e_{i},\,f_{0}\,|\,i=1,\cdots,m\\}$ as a basis and the
multiplication given for any $i$, $j=1,\cdots,m$ by:

$e_{i}\circ e_{j}:=e_{j}\cdot_{V}e_{j},\quad e_{i}\circ
f_{0}:=\xi(e_{i}),\quad f_{0}\circ f_{0}:=F$ (15)

for some pair $(F,\xi)\in V\times{\rm End}_{k}(V)$ satisfying the following
compatibility conditions

$F^{2}=0,\qquad
x\cdot_{V}F+2\,x^{2}\cdot_{V}\xi(x)+2\,\xi(x)^{2}+2\,\xi(x)\cdot_{V}F=0$

for all $x\in V$.

The semidirect product of $4$-algebras characterizes split epimorphisms in the
category on $4$-algebras, as we mentioned in Example 2.3:

######  Corollary 2.7.

An algebra map $\pi:B\to A$ between two $4$-algebras $A$ and $B$ is a split
epimorphism in the category of $4$-algebras if and only if there exists an
isomorphism of $4$-algebras $B\cong V\ltimes A$, where $V={\rm Ker}(\pi)$ and
$V\ltimes A$ is a semidirect product of $4$-algebras as constructed in Example
2.3.

###### Proof.

For a semidirect product $V\ltimes A$, the canonical projection
$p_{A}:V\ltimes A\to A$, $p_{A}(x,\,a)=a$ has a section that is an algebra map
defined by $s_{A}(a)=(0,\,a)$, for all $a\in A$. Conversely, let $s:A\to B$ be
an algebra map with $\pi\circ s={\rm Id}_{A}$. Then, the bilinear map $f_{s}$
given by (12) from Proposition 2.4 is the trivial map and hence the
corresponding crossed product $V\\#A$ reduces to a semidirect product
$V\ltimes A$. ∎

Now we shall describe the morphisms that stabilize $V$ between two arbitrary
crossed products $V\\#A$ and $V\\#^{\prime}A$, for two reasons: to compute the
Galois group ${\rm Gal}(V\\#A/A)$ of the extension $V\hookrightarrow V\\#A$
and then to answer the classification part of the GE problem for $4$-algebras.

######  Lemma 2.8.

Let ${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ and
${\mathcal{C}}^{\prime}\,(A,V)=(\triangleright^{\prime},\,f^{\prime},\,\cdot_{V}^{\prime})$
be two crossed systems of a $4$-algebra $A$ by $V$, with $V\\#A$, respectively
$V\\#^{\prime}A$, the corresponding crossed products. If
$\cdot_{V}=\cdot_{V}^{\prime}$, then there exists a bijection between the set
of all $4$-algebra morphisms $\psi:V\\#A\to V\\#^{\prime}A$ which stabilize
$V$ and the set of all pairs $(r,\alpha)\in{\rm Hom}_{k}(A,\,V)\times{\rm
End}_{\rm alg}(A)$, where $r:A\to V$ is a linear map, $\alpha:A\to A$ is an
algebra morphism satisfying the following compatibilities for all $a$, $b\in
A$ and $x\in V$:

  1. (M1)

$a\triangleright x=\alpha(a)\triangleright^{\prime}x+r(a)\cdot_{V}^{\prime}x$;

  2. (M2)

$f(a,\,b)=f^{\prime}(\alpha(a),\,\alpha(b))+\alpha(a)\triangleright^{\prime}r(b)+\alpha(b)\triangleright^{\prime}r(a)+r(a)\cdot_{V}^{\prime}r(b)-r(a\cdot
b)$

Under the above bijection the $4$-algebra morphism
$\psi=\psi_{(r,\alpha)}:V\\#A\to V\\#^{\prime}A$ corresponding to the pair
$(r,\alpha)$ is given by $\psi(x,a)=(x+r(a),\,\alpha(a))$, for all $a\in A$,
$x\in V$. Furthermore, $\psi=\psi_{(r,\alpha)}$ is an isomorphism of
$4$-algebras if and only if $\alpha$ is bijective and $\psi=\psi_{(r,\alpha)}$
co-stabilize $A$ if and only if $\alpha={\rm Id}_{A}$.

###### Proof.

First of all we observe that for any linear map $\psi:V\,\\#\,A\to
V\,\\#\,^{\prime}A$ that makes the following diagram commutative

there exists a unique pair of linear maps $(r,\alpha)\in{\rm
Hom}_{k}(A,\,V)\times{\rm End}_{k}(A)$ such that
$\psi(x,a)=(x+r(a),\,\alpha(a))$, for all $a\in A$, and $x\in V$. Let
$\psi=\psi_{(r,\alpha)}$ be such a linear map, i.e.
$\psi(x,a)=(x+r(a),\,\alpha(a))$, for some linear maps $r:A\to V$,
$\alpha:A\to A$. We will prove that $\psi$ is a morphism of $4$-algebras if
and only if $\cdot_{V}=\cdot_{V}^{\prime}$, $\alpha:A\to A$ is an algebra map
and the compatibility conditions (M1)-(M2) hold. To this end, it is enough to
prove that the compatibility

$\psi\bigl{(}(x,a)\circ(y,b)\bigl{)}=\psi(x,a)\circ^{\prime}\psi(y,b)$ (16)

holds on all generators of $V\,\\#\,A$. We leave out the detailed computations
and only indicate the main steps of this verification. First, it is easy to
see that (16) holds for the pair $(x,0)$, $(y,0)$ if and only if
$\cdot_{V}=\cdot_{V}^{\prime}$. Secondly, we can prove that (16) holds for the
pair $(a,0)$, $(0,b)$ if and only if $\alpha:A\to A$ is an algebra maps and
(M2) hold. Finally, (16) holds for the pair $(x,0)$, $(0,a)$ if and only if
(M1) holds. The last two statements are elementary: we just note that if
$\alpha:A\to A$ is bijective, then $\psi_{(r,v)}$ is an isomorphism of
$4$-algebras with the inverse given for any $a\in A$ and $x\in V$ by:

$\psi_{(r,\alpha)}^{-1}(x,\,a)=\bigl{(}x-r(\alpha^{-1}(a)),\,\alpha^{-1}(a)\bigl{)}.$

This finishes the proof. ∎

It can be easily observed from the proof of Lemma 2.8 that if
$\cdot_{V}\neq\cdot_{V}^{\prime}$, then there are no $4$-algebra morphisms
$\psi:V\\#A\to V\\#^{\prime}A$ which stabilize $V$. As a first application of
Lemma 2.8 we can describe the Galois group ${\rm Gal}\,(V\\#A/V)$ of the
extension $V\hookrightarrow V\\#A$, consisting of all automorphisms of the
crossed product $4$-algebra $V\\#A$ acting as identity on $V$. Let
${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ be a crossed system of
a $4$-algebra $A$ by $V$. Let
${\mathcal{G}}\,_{A}^{V}(\triangleright,\,f,\,\cdot_{V})$ be the set of all
pairs $(r,\alpha)\in{\rm Hom}_{k}(A,\,V)\times{\rm Aut}_{\rm Alg}(A)$
consisting of a linear map $r:A\to V$ and an automorphism $\alpha:A\to A$ of
the $4$-algebra $(A,\cdot)$ such that:

$\displaystyle a\triangleright x$ $\displaystyle=$
$\displaystyle\alpha(a)\triangleright x+r(a)\cdot_{V}x$ $\displaystyle
f(a,\,b)$ $\displaystyle=$ $\displaystyle
f(\alpha(a),\,\alpha(b))+\alpha(a)\triangleright r(b)+\alpha(b)\triangleright
r(a)+r(a)\cdot_{V}r(b)-r(a\cdot b)$

for all $x\in V$, $a$, $b\in A$. Then, we can easily prove that
${\mathcal{G}}\,_{A}^{V}(\triangleright,\,f,\,\cdot_{V})$ has a group
structure via the multiplication given by:

$(r,\,\alpha)\bullet(r^{\prime},\,\alpha^{\prime}):=(r^{\prime}+r\circ\alpha^{\prime},\,\alpha\circ\alpha^{\prime})$
(17)

for all $(r,\,\alpha)$,
$(r^{\prime},\,\alpha^{\prime})\in{\mathcal{G}}\,_{A}^{V}(\triangleright,\,f,\,\cdot_{V})$
and, moreover, ${\mathcal{G}}\,_{A}^{V}(\triangleright,\,f,\,\cdot_{V})$ is a
subgroup in the semidirect product of groups ${\rm Hom}_{k}(A,\,V)\ltimes{\rm
GL}_{k}(A)$ as defined by (1). Applying Lemma 2.8 for
$(\triangleright^{\prime},\,f^{\prime},\,\cdot_{V}^{\prime}):=(\triangleright,\,f,\,\cdot_{V})$
we obtain:

######  Corollary 2.9.

Let ${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ be a crossed
system of a $4$-algebra $A$ by $V$. Then the map defined by:

$\vartheta:({\mathcal{G}}\,_{A}^{V}(\triangleright,\,f,\,\cdot_{V}),\,\bullet)\to{\rm
Gal}\,(V\\#A/V),\qquad\vartheta(r,\alpha)(x,\,a):=\bigl{(}x+r(a),\,\alpha(a)\bigl{)}$

for all $(r,\alpha)\in{\mathcal{G}}\,_{A}^{V}(\triangleright,\,f,\,\cdot_{V})$
and $(x,a)\in V\times A$ is an isomorphism of groups.

Arising from Lemma 2.8 we introduce the following concept needed for the
classification part of the GE-problem :

######  Definition 2.10.

Let $A$ be a $4$-algebra and $V$ a vector space. Two crossed systems
${\mathcal{C}}\,(A,V)=(\triangleright,\,f,\,\cdot_{V})$ and
${\mathcal{C}}^{\prime}\,(A,V)=(\triangleright^{\prime},\,f^{\prime},\,\cdot_{V}^{\prime})$
of $A$ by $V$ are called _cohomologous_ , and we denote this by
${\mathcal{C}}\,(A,V)\approx{\mathcal{C}}^{\prime}\,(A,V)$, if and only if
$\cdot_{V}=\cdot_{V}^{\prime}$ and there exists a linear map $r:A\to V$ such
that for any $a$, $b\in A$, $x\in V$ we have:

$\displaystyle a\triangleright x$ $\displaystyle=$ $\displaystyle
a\triangleright^{\prime}x+r(a)\cdot_{V}^{\prime}x$ (18) $\displaystyle
f(a,\,b)$ $\displaystyle=$ $\displaystyle
f^{\prime}(a,\,b)+a\triangleright^{\prime}r(b)+b\triangleright^{\prime}r(a)+r(a)\cdot_{V}^{\prime}r(b)-r(a\cdot
b).$ (19)

As a conclusion of the results obtained so far we obtain the theoretical
answer to the GE-problem for $4$-algebras:

######  Theorem 2.11.

Let $A$ be a $4$-algebra, $E$ a vector space and $\pi:E\to A$ a linear
epimorphism of vector spaces with $V={\rm Ker}(\pi)$. Then $\approx$ defined
in Definition 2.10 is an equivalence relation on the set
${\mathcal{C}}{\mathcal{S}}(A,\,V)$ of all crossed systems of $A$ by $V$. If
we denote by
${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V):={\mathcal{C}}\mathcal{S}(A,V)/\approx$,
then the map

${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)\to{\rm
Gext}\,(A,\,E),\qquad\overline{(\triangleright,\,f,\,\cdot_{V})}\,\longmapsto\,V\\#_{(\triangleright,\,f,\,\cdot_{V})}\,A$
(20)

is bijective, where $\overline{(\triangleright,\,f,\,\cdot_{V})}$ denotes the
equivalence class of $(\triangleright,\,f,\,\cdot_{V})$ via $\approx$.

###### Proof.

Using the last statement of Lemma 2.8 we obtain that
${\mathcal{C}}\,(A,V)\approx{\mathcal{C}}^{\prime}\,(A,V)$ if and only if
there exists an isomorphism of $4$-algebras $V\\#A\cong V\\#^{\prime}A$ that
stabilize $V$ and co-stabilize $A$. Thus, $\approx$ is an equivalence relation
on the set ${\mathcal{C}}{\mathcal{S}}(A,\,V)$. The rest of the proof follows
from Proposition 2.2 and Proposition 2.4. ∎

The classifying object ${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)$ constructed in
Theorem 2.11 will be called the _global non-abelian cohomology_ of $A$ by $V$
and its explicit computation for a given $4$-algebra $A$ and a vector space
$V$ is a very difficult problem. The first step in its calculation is inspired
by the way the equivalence relation $\approx$ is expressed in Definition 2.10:
two different $4$-algebra structures $\cdot_{V}$ and $\cdot_{V}^{{}^{\prime}}$
on $V$ give rise to two different equivalence classes with respect to the
relation $\approx$ on ${\mathcal{C}}{\mathcal{S}}(A,V)$. Hence we can fix
$\cdot_{V}$ a $4$-algebra structure on $V$ and denote by
${\mathcal{C}}{\mathcal{S}}_{\cdot_{V}}(A,\,V)$ the set of pairs
$\bigl{(}\triangleright,\,f\bigl{)}$ such that
$\bigl{(}\triangleright,\,f,\,\cdot_{V}\bigl{)}\in{\mathcal{C}}{\mathcal{S}}(A,V)$.
Two such pairs $\bigl{(}\triangleright,\,f\bigl{)}$ and
$\bigl{(}\triangleright^{\prime},\,f^{\prime}\bigl{)}\in{\mathcal{C}}{\mathcal{S}}_{\cdot_{V}}(A,\,V)$
will be called _$\cdot_{V}$ -cohomologous_ and will be denoted by
$\bigl{(}\triangleright,\,f\bigl{)}\,\approx_{\cdot_{V}}\bigl{(}\triangleright^{\prime},\,f^{\prime}\bigl{)}$
if there exists a linear map $r:A\to V$ such that for any $a$, $b\in A$, $x\in
V$ we have:

$\displaystyle a\triangleright x$ $\displaystyle=$ $\displaystyle
a\triangleright^{\prime}x+r(a)\cdot_{V}x$ (21) $\displaystyle f(a,\,b)$
$\displaystyle=$ $\displaystyle
f^{\prime}(a,\,b)+a\triangleright^{\prime}r(b)+b\triangleright^{\prime}r(a)+r(a)\cdot_{V}r(b)-r(a\cdot
b).$ (22)

Then $\approx_{\cdot_{V}}$ is an equivalence relation on
${\mathcal{C}}{\mathcal{S}}_{\cdot_{V}}(A,\,V)$ and we denote by
${\mathbb{H}}^{2}_{\rm nab}\,\bigl{(}A,\,(V,\,\cdot_{V})\bigl{)}$ the quotient
set ${\mathcal{C}}{\mathcal{S}}_{\cdot_{V}}(A,\,V)/\approx_{\cdot_{V}}$ and
call it the _non-abelian cohomology_ of the $4$-algebras $(A,\cdot)$ and
$(V,\cdot_{V})$. The object ${\mathbb{H}}^{2}_{\rm
nab}\,\bigl{(}A,\,(V,\,\cdot_{V})\bigl{)}$ classifies all extensions of the
given $4$-algebra $A$ by the given $4$-algebra $(V,\,\cdot_{V})$ and gives the
theoretical answer to the classical extension problem for $4$-algebras. We
record all these observations in the following result:

######  Corollary 2.12.

Let $A$ be a $4$-algebra and $V$ a vector space. Then

${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)=\,\amalg_{\cdot_{V}}\,{\mathbb{H}}^{2}_{\rm
nab}\,\bigl{(}A,\,(V,\,\cdot_{V})\bigl{)}$ (23)

where the coproduct on the right hand side is in the category of sets over all
possible $4$-algebra structures $\cdot_{V}$ on the vector space $V$.
Furthermore, for a given $4$-algebra structure on $V$ the map

${\mathbb{H}}^{2}_{\rm nab}\,\bigl{(}A,\,(V,\,\cdot_{V})\bigl{)}\to{\rm
Ext}\,(A,\,(V,\,\cdot_{V})),\qquad\overline{\overline{(\triangleright,\,f)}}\,\longmapsto\,V\\#_{(\triangleright,\,f,\,\cdot_{V})}\,A$
(24)

is bijective, where ${\rm Ext}\,(A,\,(V,\,\cdot_{V}))$ is the set of
equivalence classes of all $4$-algebras that are extensions of the $A$ by
$(V,\cdot_{V})$ and $\overline{\overline{(\triangleright,\,f)}}$ denotes the
equivalence class of $(\triangleright,\,f)$ via $\approx_{\cdot_{V}}$.

We continue our investigation of the object
${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)$ observing that among all components of
the coproduct in (23) the simplest one is that corresponding to the trivial
$4$-algebra structure on $V$, i.e. $x\cdot_{V}y:=0$, for all $x$, $y\in V$
which we denoted by $V_{0}$. Let ${\mathcal{C}}{\mathcal{S}}_{0}(A,\,V_{0})$
be the set of all pairs $\bigl{(}\triangleright,\,f\bigl{)}$ such that
$\bigl{(}\triangleright,\,f,\,\cdot_{V}:=0\bigl{)}\in{\mathcal{C}}{\mathcal{S}}(A,V)$.
As shown in (3) of Example 2.3, a pair
$\bigl{(}\triangleright,\,f\bigl{)}\in{\mathcal{C}}{\mathcal{S}}_{0}(A,\,V_{0})$
if and only if $(V,\triangleright)$ is an $A$-module and $f:A\times A\to V$ is
symmetric abelian $2$-cocycles, i.e.

$a^{2}\triangleright(a\triangleright x)=0,\qquad
f(a^{2},a^{2})+2\,a^{2}\triangleright f(a,a)=0$ (25)

for all $a\in A$ and $x\in V$. Applying now Definition 2.10 for the trivial
multiplication $\cdot_{V}:=0$ we obtain that two pairs
$\bigl{(}\triangleright,\,f\bigl{)}$ and
$\bigl{(}\triangleright^{\prime},\,f^{\prime}\bigl{)}\in{\mathcal{C}}{\mathcal{S}}_{0}(A,\,V_{0})$
are _$0$ -cohomologous_
$\bigl{(}\triangleright,\,f\bigl{)}\approx_{0}\bigl{(}\triangleright^{\prime},\,f^{\prime}\bigl{)}$
if and only if $\triangleright=\triangleright^{\prime}$ and there exists a
linear map $r:A\to V$ such that for all $a$, $b\in A$ we have:

$f(a,\,b)=f^{\prime}(a,\,b)+a\triangleright r(b)+b\triangleright r(a)-r(a\cdot
b).$ (26)

The equality $\triangleright=\triangleright^{\prime}$ shows that two different
$A$-module structures on $V$ give different equivalence classes in the
classifying object ${\mathbb{H}}_{\rm nab}^{2}\,\bigl{(}A,\,V_{0}\bigl{)}$.
Thus, we can apply the same strategy as before for computing
${\mathbb{H}}_{\rm nab}^{2}\,\bigl{(}A,\,V_{0}\bigl{)}$: we fix
$(V,\,\triangleright)$ a $A$-module structure on $V$ and consider the set
${\rm Z}^{2}_{\triangleright}\,(A,\,V_{0})$ of all symmetric abelian
$2$-cocycle, i.e. symmetric bilinear maps $f:A\times A\to V$ such that

$\displaystyle f(a^{2},a^{2})+2\,a^{2}\triangleright f(a,a)=0$

for all $a$, $b\in A$. Two symmetric abelian $2$-cocycles $f$ and $f^{\prime}$
are $0$-cohomologous $f\approx_{0}f^{\prime}$ if and only if there exists a
linear map $r:A\to V$ such that (26) holds. Then $\approx_{0}$ is an
equivalence relation on ${\rm Z}^{2}_{\triangleright}\,(A,\,V_{0})$ and the
quotient set ${\rm Z}^{2}_{\triangleright}\,(A,\,V_{0})/\approx_{0}$, which we
will denote by ${\rm H}^{2}_{\triangleright}\,(A,\,V_{0})$, plays the role of
the second cohomological group from the theory of groups or Lie algebras. All
in all, we have obtained the following results which classifies all extensions
of a $4$-algebra $A$ by an abelian algebra $V_{0}$.

######  Corollary 2.13.

Let $A$ be a $4$-algebra and $V$ a vector space viewed with the trivial
$4$-algebra structure $V_{0}$. Then:

${\mathbb{H}}_{\rm
nab}^{2}\,\bigl{(}A,\,V_{0}\bigl{)}\,=\,\amalg_{\triangleright}\,{\rm
H}^{2}_{\triangleright}\,(A,\,V_{0})$ (27)

where the coproduct on the right hand side is in the category of sets over all
possible $A$-module structures $\triangleright$ on the vector space $V$.

##  3\. Applications and examples

The computation of the classifying object
${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,V)$ as constructed in Theorem 2.11 and
Corollary 2.12 is a very challenging problem. In the following we shall
compute ${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,k)$ and
${\mathbb{G}}{\mathbb{H}}^{2}\,(k,\,V)$. First we need the following:

######  Proposition 3.1.

Let $A$ be a $4$-algebra. Then there exists a bijection between the set
${\mathcal{C}}{\mathcal{S}}\,(A,\,k)$ of all crossed systems of $A$ by $k$ and
the set ${\mathcal{C}}{\mathcal{F}}\,(A)$ consisting of all pairs
$(\lambda,\,f)$, where $\lambda:A\to k$ is linear map, $f:A\times A\to k$ a
symmetric bilinear form on $A$ satisfying the following compatibilities
conditions for any $a\in A$:

$f(a^{2},\,a^{2})+2\,\lambda(a^{2})f(a,\,a)=0,\qquad\lambda(a)\lambda(a^{2})=0.$
(28)

The correspondence is given such that the crossed crossed product $k\\#A$
associated to the pair $(\lambda,\,f)\in{\mathcal{C}}{\mathcal{F}}\,(A)$ is
the $4$-algebra denoted by $A_{(\lambda,f)}:=k\times A$ with the
multiplication given for any $a$, $b\in A$ and $x$, $y\in k$ by:

$(x,a)\circ(y,b)=\bigl{(}\lambda(a)y+\lambda(b)x+f(a,b),\,a\cdot b\bigl{)}.$
(29)

Furthermore, a $4$-algebra $B$ has a surjective algebra map $B\to A\to 0$
whose kernel is $1$-dimensional if and only if $B$ is isomorphic to
$A_{(\lambda,\,f)}$, for some $(\lambda,f)\in{\mathcal{C}}{\mathcal{F}}\,(A)$.

###### Proof.

Since $k$ has dimension $1$, any $4$-algebra structure $\cdot_{k}$ on $k$ is
the abelian one, $x\cdot_{k}y=0$, for all $x$, $y\in k$. Hence, there exists a
bijection between the set of all crossed datums
$\bigl{(}\triangleright,\,f,\,\cdot_{k}\bigl{)}$ of $A$ by $k$ and the set of
pairs $(\lambda,\,f)$ consisting of a linear map $\lambda:A\to k$ and a
bilinear map $f:A\times A\to k$. The bijection is given such that
$(\triangleright,\,f)$ corresponding to $(\lambda,\,f)$ is defined by
$a\triangleright x:=\lambda(a)x$ for all $a\in A$ and $x\in k$. Now the axiom
(CS1) of Proposition 2.2 is equivalennt to $f$ being symmetric, while the
axioms (CS2) and (CS3) are equivalent to (28). The algebra $A_{(\lambda,\,f)}$
is just the crossed product $k\\#A$ associated to this context and the last
statement follows from Proposition 2.4. ∎

Let $(\lambda,f)\in{\mathcal{C}}{\mathcal{F}}\,(A)$ be a pair as in
Proposition 3.1. We shall explicitly describe the multiplication of the
$4$-algebra $A_{(\lambda,\,f)}$. We will see the elements of $A$ as elements
in $k\times A$ through the identification $a=(0,\,a)$ and denote by
$g:=(1,\,0_{A})\in k\times A$. Let $\\{e_{i}\,|\,i\in I\\}$ be a basis of $A$
as a vector space over $k$. Then, the $4$-algebra $A_{(\lambda,f)}$ is the
vector space having $\\{g,\,e_{i}\,|\,i\in I\\}$ as a basis and the
multiplication $\circ$ is given for any $i$, $j\in I$ by:

$g^{2}=0,\quad e_{i}\circ g=g\circ e_{i}=\lambda(e_{i})\,g,\quad e_{i}\circ
e_{j}=e_{i}\cdot e_{j}+f(e_{i},\,e_{j})\,g.$ (30)

In order to compute ${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,k)$, we observe first
that the equivalence relation given by (18) and (19) from Definition 2.10,
written for the set ${\mathcal{C}}{\mathcal{F}}\,(A)$, via the bijection
${\mathcal{C}}{\mathcal{S}}\,(A,\,k)\cong{\mathcal{C}}{\mathcal{F}}\,(A)$
proven in Proposition 3.1 takes the following form:
$(\lambda,f)\approx(\lambda^{\prime},f^{\prime})$ if and only if
$\lambda=\lambda^{\prime}$ and there exist a linear map $r:A\to k$ such that

$f(a,\,b)=f^{\prime}(a,\,b)+\lambda^{\prime}(a)r(b)+\lambda^{\prime}(b)r(a)-r(a\cdot
b)$ (31)

for all $a$, $b\in A$. Thus, we obtain that
${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,k)\cong{\mathcal{C}}{\mathcal{F}}\,(A)/\approx$.
We continue our investigation since the equality $\lambda=\lambda^{\prime}$ in
the above equivalence relation shows that two different $\lambda$ and
$\lambda^{\prime}$ give different equivalence classes in the classifying
object ${\mathcal{C}}{\mathcal{F}}\,(A)/\approx$. Hence, we can fix a linear
map $\lambda:A\to k$ such that $\lambda(a)\lambda(a^{2})=0$, for all $a\in A$.
We denote by ${\rm Z}^{2}_{\lambda}\,(A,\,k)$ the set of all _$\lambda$
-cocycles_: that is, the set of all symmetric bilinear maps $f:A\times A\to k$
satisfying the first compatibility condition of (28). Two $\lambda$-cocycles
$f$, $f^{\prime}:A\times A\to k$ are equivalent $f\approx^{\lambda}f^{\prime}$
if and only if there exists a linear map $r:A\to k$ such that

$f(a,\,b)=f^{\prime}(a,\,b)+\lambda(a)r(b)+\lambda(b)r(a)-r(a\cdot b)$

for all $a$, $b\in A$. We denote ${\rm H}^{2}_{\lambda}\,(A,\,k):={\rm
Z}^{2}_{\lambda}\,(A,\,k)/\approx^{\lambda}$ and we record all the above
results in the following decomposition of
${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,k)$, which is a special case of Corollary
2.13 applied for $V:=k$:

######  Corollary 3.2.

Let $A$ be a $4$-algebra. Then,

${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,k)\,\cong\,{\mathcal{C}}{\mathcal{F}}\,(A)/\approx\,\,\cong\,\amalg_{\lambda}\,\,{\rm
H}^{2}_{\lambda}\,(A,\,k)$ (32)

where the coproduct on the right hand side is in the category of sets over all
possible linear maps $\lambda:A\to k$ satisfying $\lambda(a)\lambda(a^{2})=0$,
for all $a\in A$.

######  Example 3.3.

Let $n$ be a positive integer and $A$ the $4$-algebra having
$\\{e_{1},\cdots,e_{n+1}\\}$ as a basis and the multiplication given by
$e_{1}\cdot e_{2}:=e_{n+1}$ and the other products of basis elements are zero.
Then we can prove that

${\mathcal{C}}{\mathcal{F}}\,(A)\cong\,k^{n}\times{\rm Sym}_{n+1}^{0}\,(k)$

where we denoted by ${\rm Sym}_{n+1}^{0}\,(k)$ the vector space of all
$(n+1)\times(n+1)$-symmetric matrices $(f_{ij})$ such that $f_{n+1,n+1}:=0$.
The bijection is given such that
$(\lambda,\,f)\in{\mathcal{C}}{\mathcal{F}}\,(A)$ associated to a pair
$\bigl{(}\lambda_{i}\bigl{)}\times\,(f_{ij})\in k^{n}\times{\rm
Sym}_{n+1}^{0}\,(k)$ is given by:

$\lambda(e_{t}):=\lambda_{t},\quad\lambda(e_{n+1}):=0,\qquad
f(e_{i},\,e_{j}):=f_{ij}$

for all $t=1,\cdots,n$ and $i$, $j=1,\cdots,n+1$. In particular, we obtain
that

${\mathbb{G}}{\mathbb{H}}^{2}\,(A,\,k)\,\cong\,\amalg_{\lambda}\,\,{\rm
Sym}_{n+1}^{0}\,(k)/\approx_{\lambda}$

where the coproduct on the right hand side is taken over all
$\lambda=(\lambda_{1},\cdots,\lambda_{n})\in k^{n}$ and ${\rm
Sym}_{n+1}^{0}\,(k)/\approx_{\lambda}$ is the quotient set of ${\rm
Sym}_{n+1}^{0}\,(k)$ via the following equivalence relation:
$(f_{ij})\approx_{\lambda}(f_{ij}^{\prime})$ if and only if there exists a
linear map $r:A\to k$ such that

$f_{ij}=f_{ij}^{\prime}+\lambda_{i}r(e_{j})+\lambda_{j}r(e_{i})-r(e_{i}\cdot
e_{j})$

for all $i$, $j=1,\cdots,n+1$ (with $\lambda_{n+1}:=0$).

Now we describe the opposite case, namely
${\mathbb{G}}{\mathbb{H}}^{2}\,(k,\,V)$.

######  Proposition 3.4.

Let $V$ be a vector space. Then there exists a bijection between the set
${\mathcal{C}}{\mathcal{S}}\,(k,\,V)$ of all crossed systems of $k$ by $V$ and
the set ${\mathcal{C}}{\mathcal{T}}\,(V)$ consisting of all triples
$(\theta,\,F,\,\cdot_{V})$, where $\theta:V\to V$ is linear map, $F\in V$ and
$\cdot_{V}:V\times V$ is a $4$-algebra structure on $V$ satisfying the
following compatibility conditions for any $x\in V$:

$F^{2}=0,\quad\theta(x)\cdot_{V}\,x^{2}=\theta(x)\cdot_{V}\,F=0,\quad
2\,\theta(x)^{2}+x^{2}\cdot_{V}\,F=0.$ (33)

The correspondence is given such that the crossed product $V\\#k$ associated
to the triple $(\theta,\,F,\,\cdot_{V})\in{\mathcal{C}}{\mathcal{T}}\,(V)$ is
the $4$-algebra denoted by $V_{(\theta,\,F,\,\cdot_{V})}:=V\times k$ with the
multiplication given for any $a$, $b\in k$ and $x$, $y\in V$ by:

$(x,a)\circ(y,b)=\bigl{(}x\cdot_{V}y+a\theta(y)+b\theta(x)+abF,\,0\bigl{)}.$
(34)

###### Proof.

We leave it to the reader since it is similar to the one of Proposition 3.1,
taking into account that the only $4$-algebra structure on $A:=k$ is the
abelian one. ∎

We fix now $\cdot_{V}$ a $4$-algebra structure on the vector space $V$ and
denote by ${\mathcal{T}}_{\cdot_{V}}\,(V)$ the set of all pairs
$(\theta,\,F)\in{\rm End}_{k}(V)\times V$ satisfying the compatibility
conditions (33). Two pairs $(\theta,\,F)$ and
$(\theta^{\prime},\,F^{\prime})\in{\mathcal{T}}_{\cdot_{V}}\,(V)$ are
$\cdot_{V}$-cohomologous
$(\theta,\,F)\approx_{\cdot_{V}}(\theta^{\prime},\,F^{\prime})$ if and only if
there exists an element $r\in V$ such that

$\theta(x)=\theta^{\prime}(x)+r\cdot_{V}x,\qquad
F=F^{\prime}+2\,\theta^{\prime}(r)+r^{2}$

for all $x\in V$. Applying Corollary 2.12 we obtain:

######  Corollary 3.5.

Let $V$ be a vector space. Then

${\mathbb{G}}{\mathbb{H}}^{2}\,(k,\,V)=\,\amalg_{\cdot_{V}}\,{\mathcal{T}}_{\cdot_{V}}\,(V)/\approx_{\cdot_{V}}$
(35)

where the coproduct on the right hand side is in the category of sets over all
possible $4$-algebra structures $\cdot_{V}$ on $V$.

In the last part we shall apply our results to metabelian $4$-algebras. We
recall that a $4$-algebra $E$ is called metabelian if the derived algebra
$E^{\prime}$ is an abelian subalgebra of $E$, i.e. $(a\cdot b)\cdot(c\cdot
d)=0$, for all $a$, $b$, $c$, $d\in E$. Let $I$ be an ideal of a $4$-algebra
$E$: then the quotient algebra $E/I$ is an abelian algebra if and only if
$E^{\prime}\subseteq I$. Thus, $E$ is a metabelian $4$-algebra if and only if
it fits into an exact sequence of $4$-algebras

(38)

where $B$ and $A$ are abelian $4$-algebras. Indeed, if $E$ is metabelian we
can take $B:=E^{\prime}$ and $A:=E/E^{\prime}$ with the obvious morphisms.
Conversely, assume that a $4$-algebra $E$ is an extension of an abelian
algebra $A$ by an abelian algebra $B$. Since $E/i(B)=E/{\rm Ker}(\pi)\cong A$
is an abelian algebra, we obtain that $E^{\prime}\subseteq i(B)\cong B$.
Hence, $E^{\prime}$ is abelian as a subalgebra in an abelian algebra, i.e. $E$
is metabelian. Using this observation, Proposition 2.2 and Proposition 2.4 we
obtain the structure of metabelian $4$-algebras:

######  Corollary 3.6.

A $4$-algebra $E$ is metabelian if and only if there exists an isomorphism of
$4$-algebras $E\cong V\\#_{(\triangleright,\,f)}\,A$, where $A$ and $V$ are
two vector spaces and $\triangleright:A\times V\to V$, $f:A\times A\to V$ are
two bilinear maps such that $f$ is symmetric. The crossed product
$V\\#_{(\triangleright,\,f)}\,A$ is the vector space $V\times A$ with the
multiplication given for any $a$, $b\in A$ and $x$, $y\in V$ by:

$(x,\,a)\circ(y,\,b):=(a\triangleright y+b\triangleright x+f(a,\,b),\,0).$
(39)

Let now $A$ and $V$ be two fixed vector spaces viewed with the abelian
$4$-algebra structure $A_{0}$ and $V_{0}$: in the next step we shall classify
all metabelian $4$-algebras that are extensions of $A$ by $V$, that is we
classify all crossed products $V\\#_{(\triangleright,\,f)}\,A$ up to an
isomorphism that stabilizes $V$ and co-stabilizes $A$. For this purpose, we
have to compute the classifying object ${\mathbb{H}}^{2}_{\rm
nab}\,\bigl{(}A,\,(V,\,\cdot_{V})\bigl{)}$ from Corollary 2.12 and Corollary
2.13 in the case that both $\cdot_{V}$ and $\cdot_{A}$ are trivial
multiplication. Let $\triangleright:A\times V\to V$ be a fixed bilinear map
and ${\rm Sym}(A\times A;\,V)$ the set of all symmetric bilinear maps
$f:A\times A\to V$. Two elements $f$ and $f^{\prime}\in{\rm Sym}\,(A\times
A;\,V)$ are called _$\triangleright$ -cohomologous_
$f\approx_{\triangleright}\,f^{\prime}$ if and only if there exists a linear
map $r:A\to V$ such that:

$f(a,\,b)=f^{\prime}(a,\,b)+a\triangleright r(b)+b\triangleright r(a)$ (40)

for all $a$, $b\in A$ (i.e. (26) holds for the trivial multiplication on $A$).
We denote by ${\rm H}^{2}_{\triangleright}(A_{0},\,V_{0}):={\rm Sym}\,(A\times
A;\,V)/\approx_{\triangleright}$. We have obtained the following:

######  Corollary 3.7.

Let $A$ and $V$ be two vector spaces viewed with the abelian algebra structure
$A_{0}$ and $V_{0}$. Then there exists a bijection:

${\mathbb{H}}^{2}_{\rm
nab}\,\bigl{(}A_{0},\,V_{0}\bigl{)}\,\cong\,\amalg_{\triangleright}\,{\rm
H}^{2}_{\triangleright}\,(A_{0},\,V_{0})$ (41)

where the coproduct on the right hand side is taken over all bilinear maps
$\triangleright:A\times V\to V$.

######  Examples 3.8.

1\. Let $V$ be a vector space with a basis $\\{e_{i}\,|\,i\in I\\}$ viewed
with the abelian $4$-algebra structure $V_{0}$. Then

${\mathbb{H}}^{2}_{\rm
nab}\,\bigl{(}k_{0},\,V_{0}\bigl{)}\,\cong\,\amalg_{g\in{\rm
End}_{k}(V)}\,\,V/{\rm Im}(g).$

In particular, any $4$-algebra $E$ containing $V$ as an abelian ideal of
codimension $1$ is isomorphic to the $4$-algebra with the basis
$\\{e,\,e_{i}\,|\,i\in I\\}$ and the multiplication given for any $i\in I$ by:

$e\circ e_{i}:=g(e_{i}),\qquad e^{2}:=f_{0}$

for some $g\in{\rm End}_{k}(V)$ and $f_{0}\in V$. Indeed, since $A:=k$ any
bilinear map $\triangleright:k\times V\to V$ has the form
$\alpha\triangleright x=\alpha g(x)$, for a linear map $g\in{\rm End}_{k}(V)$
and the correspondence is bijective. Moreover, the set of all symmetric
bilinear maps $f:k\times k\to V$ is in bijection with the set of all elements
of $V$ (the bijection maps $f$ to $f_{0}:=f(1,1))$. The conclusion follows
from Corollary 3.7 once we observe that the equivalent relation (40) written
for the set of all elements $f_{0}\in V$ comes down to
$f_{0}\approx_{\triangleright}\,f_{0}^{\prime}$ if and only if
$f_{0}-f_{0}^{\prime}\in{\rm Im}(g)$. For the last part we use (39) of
Corollary 3.6 since any such $4$-algebra is metabelian.

2\. The other way around, let $A$ be a vector space with a basis
$\\{f_{j}\,|\,j\in J\\}$ viewed with the abelian $4$-algebra structure
$A_{0}$. Then,

${\mathbb{H}}^{2}_{\rm
nab}\,\bigl{(}A_{0},\,k_{0}\bigl{)}\,\cong\,\amalg_{\lambda\in A^{*}}\,\,{\rm
Sym}(A\times A;\,k)/\approx_{\lambda}$

where, for any linear map $\lambda\in A^{*}={\rm Hom}_{k}(A,\,k)$,
$\approx_{\lambda}$ is the following equivalent relation of the set of all
symmetric bilinear forms on $A$: $f\approx_{\lambda}\,f^{\prime}$ if and only
if there exists a linear map $r\in A^{*}$ such that for any $a$, $b\in A$ :

$f(a,\,b)=f^{\prime}(a,\,b)+r(a)\lambda(b)+r(b)\lambda(a).$

Furthermore, any $4$-algebra $E$ having the derived subalgebra of dimension
$1$ is isomorphic to the $4$-algebra with the basis $\\{f,\,f_{j}\,|\,j\in
J\\}$ and the multiplication given for any $j$, $l\in J$ by:

$f\circ f_{j}:=\lambda(f_{j})\,f,\qquad f_{j}\circ f_{l}:=f(f_{j},\,f_{l})\,f$

for some $\lambda\in A^{*}$ and $f\in{\rm Sym}(A\times A;\,k)$, where $A$ is
the abelian $4$-algebra $A:=E/E^{\prime}$.

Acknowledgment: The author warmly thanks the referee for his/her suggestions
which improved the first version of the paper.

##  4\. Competing interests declaration

The author declares none.

## References

  * [1] A.L. Agore, G. Militaru, Hochschild products and global non-abelian cohomology for algebras. Applications, J. Pure Appl. Algebra 221 (2017) 366–392.
  * [2] A.L. Agore, G. Militaru, The global extension problem, crossed products and co-flag non-commutative Poisson algebras, J. Algebra 426 (2015) 1–31.
  * [3] A.L. Agore, G. Militaru, On a type of commutative algebras, Linear Algebra Appl. 485 (2015) 222–249.
  * [4] N. Andruskiewitsch, J. Devoto, Extensions of Hopf algebras, Algebra i Analiz 7 (1995) 22–61.
  * [5] S.N. Bernstein, Mathematical problems in modern biology, Science Ukraine 1 (1922) 14–19 (in Russian).
  * [6] J.L. Castiglioni, X. García-Martínez, M. Ladra, Universal central extensions of Lie-Rinehart algebras, J. Algebra App. 17 (2018), No. 07, 1850134.
  * [7] C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63(1948) 85–124.
  * [8] T. Cortés, Classification of $4$-dimensional Bernstein algebras, Comm. in Algebra, 19 (1991) 1429–1443.
  * [9] T. Cortés, F. Montaner, Low dimensional Bernstein-Jordan algebras, J. London Math. Soc. 51 (1995) 53–61.
  * [10] S. Eilenberg, Extensions of general algebras, Ann. Soc. Math. Pol. 21 (1948) 125–134.
  * [11] A. Elduque, S. Okubo, On algebras satisfying $x^{2}x^{2}=N(x)x$, Math. Z. 235 (2000) 275–314.
  * [12] C.J. Jr. Everett, An extension theory for rings, Amer. J. Math. 64 (1942), 363–370.
  * [13] J.C. Gutierrez Fernandez, Solution of the Bernstein problem in the non-regular case, J. Algebra 223 (2000) 109–132.
  * [14] J.C. Gutierrez Fernandez, The Bernstein Problem in Dimension $6$, J. Algebra 185 (1996) 420–439.
  * [15] H. Jr. Guzzo, A. Benh, Solvability of a commutative algebra which atisfies $(x^{2})^{2}=0$, Comm. in Algebra 42 (2014) 417–422.
  * [16] O. Hölder, Bildung zusammengesetzter Gruppen, Math. Ann. 46 (1895) 321–422.
  * [17] P. Holgate, Genetic algebras satisfying Bernstein’s stationarity principle, J. London Math. Soc. 9 (1975) 613–623.
  * [18] J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990) 57–113.
  * [19] Yu.I. Lyubich, Two-level Bernstein populations, Math. USSR Sb. 24 (1974) 593–615.
  * [20] Yu.I. Lyubich, A classification of some types of Bernstein algebras, Selecta Mathematica Sovietica 6 (1987) l–14.
  * [21] G. Militaru, The global extension problem, co-flag and metabelian Leibniz algebras, Linear Multilinear Algebra 63 (2015) 601–621.
  * [22] G. Militaru, On the structure and classification of Bernstein algebras, arXiv:2203.13627.
  * [23] M.L. Reed, Algebraic structure of genetic inheritance, Bull. Amer. Math. Soc. 34 (1997) 107–130.
  * [24] O. Schreier, Uber die Erweiterung von Gruppen, I, Monatshefte für Mathematik und Physik 34 (1926) 165–180.
  * [25] A. WöRz-Busekros, Algebras in genetics, Lecture Notes in Biomathematics, Vol. 36, Springer-Verlag, 1980.