# Weak Representability of Actions
of Non-Associative Algebras

J. Brox  ,  X. García-Martínez  ,  M. Mancini  ,  T. Van der Linden  and  C.
Vienne<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Departamento de Álgebra, Análisis Matemático,
Geometría y Topología, Universidad de Valladolid, Palacio de Santa Cruz,
E–47002 Valladolid, Spain  CITMAga & Universidade de Vigo, Departamento de
Matemáticas, Esc. Sup. de Enx. Informática, Campus de Ourense, E–32004
Ourense, Spain  Dipartimento di Matematica e Informatica, Università degli
Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy  Institut de
Recherche en Mathématique et Physique, Université catholique de Louvain,
chemin du cyclotron 2 bte L7.01.02, B–1348 Louvain-la-Neuve, Belgium

###### Abstract.

We study the categorical-algebraic condition that _internal actions are weakly
representable_ (WRA) in the context of varieties of (non-associative) algebras
over a field.

Our first aim is to give a complete characterization of action accessible,
operadic quadratic varieties of non-associative algebras which satisfy an
identity of degree two and to study the representability of actions for them.
Here we prove that the varieties of two-step nilpotent (anti-)commutative
algebras and that of commutative associative algebras are weakly action
representable, and we explain that the condition (WRA) is closely connected to
the existence of a so-called _amalgam_.

Our second aim is to work towards the construction, still within the context
of algebras over a field, of a weakly representing object $\mathscr{E}(X)$ for
the actions on (or split extensions of) an object $X$. We actually obtain a
_partial_ algebra $\mathscr{E}(X)$, which we call _external weak actor_ of
$X$, together with a monomorphism of functors
${\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}(U(-),\mathscr{E}(X))}$,
which we study in detail in the case of quadratic varieties. Furthermore, the
relations between the construction of the _universal strict general actor_
$\operatorname{USGA}(X)$ and that of $\mathscr{E}(X)$ are described in detail.
We end with some open questions.

###### Key words and phrases:

Action representable category, amalgamation property, split extension, non-
associative algebra, partial algebra, quadratic operad

###### 2020 Mathematics Subject Classification:

08A35; 08C05; 16B50; 16W25; 17A32; 17A36; 18C05; 18E13

The first author is supported by a postdoctoral fellowship “Convocatoria 2021”
funded by Universidad de Valladolid, and partially supported by grant
PID2022-137283NB-C22 funded by MCIN/AEI/10.13039/501100011033 and ERDF “A way
of making Europe”. The second author is supported by Ministerio de Economía y
Competitividad (Spain) with grant number PID2021-127075NA-I00. The third
author is supported by the University of Palermo and by the “National Group
for Algebraic and Geometric Structures, and their Applications” (GNSAGA –
INdAM). The fourth author is a Senior Research Associate of the Fonds de la
Recherche Scientifique–FNRS. The fifth author is supported by the Fonds Thelam
of the Fondation Roi Baudouin.

## Introduction

In the article [3], F. Borceux, G. Janelidze and G. M. Kelly introduce the
concept of an _internal object action_ , with the aim of extending the
correspondence between actions and split extensions from the context of groups
and Lie algebras to arbitrary semi-abelian categories [19]. In certain of
those categories, internal actions are exceptionally well behaved, in the
sense that the actions on each object $X$ are _representable_ : this means
that there exists an object $[X]$, called the _actor_ of $X$, such that the
functor $\operatorname{Act}(-,X)\cong\operatorname{SplExt}(-,X)$, which sends
an object $B$ to the set of actions/split extensions of $B$ on/by $X$, is
naturally isomorphic to the functor $\operatorname{Hom}(-,[X])$. The context
of action representable semi-abelian categories is further studied in [4],
where it is for instance explained that the category of commutative
associative algebras over a field is not action representable. Later it was
shown that the only action representable variety of non-associative algebras
over an infinite field $\mathbb{F}$ of characteristic different from $2$ is
the variety of Lie algebras [14]. The relative strength of the notion
naturally led to the definition of closely related weaker notions.

The first of these was the concept of an _action accessible_ category due to
D. Bourn and G. Janelidze [6]: it is weak enough to include all _Orzech
categories of interest_ [28], as proved by A. Montoli in [27].

Alternatively, the properties of the representing object $[X]$ may be
weakened; this is the aim in [8], where it is shown that each Orzech category
of interest admits a so-called _universal strict general actor_ (USGA for
short).

Our present article focuses on a concept which was more recently introduced,
by G. Janelidze in [17]: _weak representability of actions_ (WRA). Instead of
asking that for each object $X$ in a semi-abelian category $\mathscr{C}$ we
have an object $[X]$ and a natural isomorphism
$\operatorname{SplExt}(-,X)\cong\operatorname{Hom}_{\mathscr{C}}(-,[X])$, we
require the existence of an object $T$ and a monomorphism of functors

$\tau\colon\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}_{\mathscr{C}}(-,T).$

Such an object $T$ is then called a _weak actor_ of $X$, and when each $X$
admits a weak actor, $\mathscr{C}$ is said to be _weakly action
representable_. For instance, if in an Orzech category of interest, each
$\operatorname{USGA}(X)$ is an object of the category, then this category is
weakly action representable [10]. This is the case of the category
$\mathbf{Assoc}$ of associative algebras [17] or the category $\mathbf{Leib}$
of Leibniz algebras [10] over a field.

J. R. A. Gray observed in [15] that an Orzech category of interest need not be
weakly action representable. One of our aims in the present article is to
study the condition (WRA) in the context of varieties of (non-associative)
algebras over a field. (We recall basic definitions and results concerning
this setting in Section 1.) It is known that such a variety is action
accessible if and only if it is algebraically coherent [14], and it is also
known [17] that action accessibility is implied by (WRA). In Section 2 we give
a complete classification of the action accessible, operadic quadratic
varieties of non-associative algebras with an identity of degree $2$ (so
commutative or anti-commutative algebras) and we study the representability of
actions of each of them. Moreover, we prove that the variety of commutative
associative algebras, the variety of two-step nilpotent commutative algebras
and that of two-step nilpotent anti-commutative algebras are weakly action
representable categories. For the variety of commutative associative algebras,
we show that the existence of a weak representation is closely connected to
the _amalgamation property_ (AP) [20] which already appeared in [4] in
relation to action representability. In Section 3 the study of (WRA) and its
relations with the condition (AP) is extended to a general variety of algebras
over a field.

Our second aim is to work towards the construction, still within the context
of algebras over a field, of a weakly representing object $\mathscr{E}(X)$ for
the actions on/split extensions of an object $X$ of a variety of non-
associative algebras $\mathscr{V}$. We believe that in certain settings, this
object may be easier to work with than the more abstract weak actor. In
Definition 3.3 we actually obtain a _partial_ algebra $\mathscr{E}(X)$, which
we call _external weak actor_ of $X$, together with a monomorphism of functors
${\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}(U(-),\mathscr{E}(X))}$,
where $U$ is the forgetful functor from $\mathscr{V}$ to the category of
partial algebras, which we study in detail in the case of quadratic varieties
of algebras (Section 4).

We end the article with some open questions (Section 5).

##  1\. Preliminaries

The present work takes place in _semi-abelian categories_ which were
introduced in [19] in order to capture categorical-algebraic properties of
non-abelian algebraic structures. A category is _semi-abelian_ if it is
pointed, admits binary coproducts, is protomodular and Barr-exact. Well-known
examples are the category $\mathbf{Grp}$ of groups, the category
$\mathbf{Rng}$ of not necessarily unitary rings, any variety $\mathscr{V}$ of
non-associative algebras over a field $\mathbb{F}$, as well as all abelian
categories. Throughout the remainder of the paper, when we consider a category
$\mathscr{C}$, we assume it to be semi-abelian; when we consider a variety
$\mathscr{V}$, we assume that $\mathscr{V}$ is a variety of non-associative
algebra over a field $\mathbb{F}$. We fix the field $\mathbb{F}$, so that we
may drop it from our notation.

### Internal actions and their representability

A central notion which appears in the semi-abelian context is that of _split
extensions_. Let $X$, $B$ be objects of a semi-abelian category $\mathscr{C}$;
a _split extension_ of $B$ by $X$ is a diagram

${0}$${X}$${A}$${B}$${0}$$\scriptstyle{k}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$
(1.1)

in $\mathscr{C}$ such that $\alpha\circ\beta=1_{B}$ and $(X,k)$ is a kernel of
$\alpha$. We observe that since protomodularity implies that the pair
$(k,\beta)$ is jointly strongly epic, the morphism $\alpha$ is indeed the
cokernel of $k$ and diagram (1.1) represents an extension of $B$ by $X$ in the
usual sense. Morphisms of split extensions are morphisms of extensions that
commute with the sections. Let us observe that, again by protomodularity, a
morphism of split extensions fixing $X$ and $B$ is necessarily an isomorphism.
For an object $X$ of $\mathscr{C}$, we define the functor

$\operatorname{SplExt}_{\mathscr{C}}(-,X)\colon\mathscr{C}^{\operatorname{op}}\rightarrow\mathbf{Set}$

which assigns to any object $B$ of $\mathscr{C}$, the set
$\operatorname{SplExt}_{\mathscr{C}}(B,X)$ of isomorphism classes of split
extensions of $B$ by $X$ in $\mathscr{C}$, and to any arrow $f\colon
B^{\prime}\to B$ the _change of base_ function
$f^{*}\colon\operatorname{SplExt}_{\mathscr{C}}(B,X)\to\operatorname{SplExt}_{\mathscr{C}}(B^{\prime},X)$
given by pulling back along $f$. When there is no ambiguity on the category
$\mathscr{C}$, we will use the notation $\operatorname{SplExt}(-,X)$.

A feature of semi-abelian categories is that one can define a notion of
_internal action_. Internal actions correspond to split extensions via a
semidirect product construction; it turns out that, as a result, for our
purposes we need no explicit description of what is an internal action. We
refer the interested reader to [4], where the equivalence between the two
concepts is described in detail. For us here, it suffices to note that if we
fix an object $X$, internal actions on $X$ in $\mathscr{C}$ give rise to a
functor

$\operatorname{Act}(-,X)=\operatorname{Act}_{\mathscr{C}}(-,X)\colon\mathscr{C}^{\operatorname{op}}\rightarrow\mathbf{Set}$

and a natural isomorphism of functors
$\operatorname{Act}(-,X)\cong\operatorname{SplExt}(-,X)$. This justifies the
terminology in the definition that follows.

######  Definition 1.1 ([4]).

A semi-abelian category $\mathscr{C}$ is said to be action representable if
for every object $X$ in it the functor $\operatorname{Act}(-,X)$ is
representable. In other words, there exists an object $[X]$ in $\mathscr{C}$,
called the actor of $X$, and a natural isomorphism

$\displaystyle\operatorname{SplExt}(-,X)\cong\operatorname{Hom}_{\mathscr{C}}(-,[X]).$

Basic examples of semi-abelian categories which satisfy action
representability are the category $\mathbf{Grp}$ of groups with the actor of
$X$ being the group of automorphisms $\operatorname{Aut}(X)$, the category
$\mathbf{Lie}$ of Lie algebras with the actor of $X$ being the Lie algebra of
derivations $\operatorname{Der}(X)$, and any abelian category with the actor
of $X$ being the zero object. For the categories $\mathbf{Assoc}$ of
associative algebras and $\mathbf{CAssoc}$ of commutative associative
algebras, representability of actions was studied in [4], where the authors
proved that they are not action representable.

It is explained in [3] that action representability is equivalent to the
condition that for every object $X$ in $\mathscr{C}$ the category
$\mathbf{SplExt}(X)$ of split extensions in $\mathscr{C}$ with kernel $X$ has
a terminal object

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[X]\ltimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[X]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We can weaken this condition assuming instead that for any $X$, every object
in $\mathbf{SplExt}(X)$ is accessible (i.e. it has a morphism into a
subterminal or so-called _faithful_ object, see [6]). In this way, we
encompass a wider class of examples that did not satisfy representability of
actions such as the category $\mathbf{Pois}$ of (non-commutative) Poisson
algebras, the category $\mathbf{Assoc}$ of associative algebras or the
category $\mathbf{CAssoc}$ of commutative associative algebras. This notion
called action accessibility was introduced by D. Bourn and G. Janelidze [6] in
order to calculate centralisers of normal subobjects or of equivalence
relations. It was then shown by A. Montoli that any Orzech category of
interest is an action accessible category [27]. This explains why all of the
varieties of non-associative algebras mentioned above are action accessible.

Since by definition the existence of a terminal object in $\mathbf{SplExt}(X)$
is stronger than every object being accessible, it is immediate that

$\textit{action representability}\Rightarrow\textit{action accessibility.}$

Recently, in [17], G. Janelidze introduced an intermediate notion: _weak
representability of actions_.

######  Definition 1.2.

A semi-abelian category $\mathscr{C}$ is said to be weakly action
representable (WRA) if for every object $X$ in it, there exists an object $T$
of $\mathscr{C}$ and a monomorphism of functors

$\tau\colon\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}_{\mathscr{C}}(-,T).$

We call such an object $T$ a weak actor of $X$, and a morphism $\varphi\colon
B\rightarrow T$ in the image of $\tau_{B}$ an acting morphism.

It is clear from the definitions that every action representable category is
weakly action representable. Also in [17], it is proven that the category
$\mathbf{Assoc}$ is weakly action representable with a weak actor of $X$ given
by the associative algebra

$\operatorname{Bim}(X)=\\{(f*-,-*f)\in\operatorname{End}(X)\times\operatorname{End}(X)^{\operatorname{op}}\mid
f*(xy)=(f*x)y,\\\ (xy)*f=x(y*f),\;x(f*y)=(x*f)y,\;\forall x,y\in X\\}$

of bimultipliers of $X$ (see [24]). The case of the category $\mathbf{Leib}$
of Leibniz algebras was studied in [10]. There the authors showed that a weak
actor of a Leibniz algebra $X$ is the Leibniz algebra

$\operatorname{Bider}(X)=\\{(d,D)\in\operatorname{End}(X)^{2}\mid
d(xy)=d(x)y+xd(y),\\\ D(xy)=D(x)y-D(y)x,\;xd(y)=xD(y),\;\forall x,y\in X\\}$

of biderivations of $X$ (see [23] and [25]), where the bilinear operation is
defined by

$[(d,D),(d^{\prime},D^{\prime})]=(d\circ d^{\prime}-d^{\prime}\circ d,D\circ
d^{\prime}-d^{\prime}\circ D).$

In the same paper, the representability of actions in the categories
$\mathbf{Pois}$ and $\mathbf{CPois}$ of (commutative) Poisson algebras was
studied.

Another important observation made by G. Janelidze is that every weakly action
representable category is action accessible. We thus have that

$\textit{action representability}\Rightarrow\textit{weak action
representability}\Rightarrow\textit{action accessibility.}$

J. R. A. Gray proved in [15] that the varieties of $n$-solvable groups where
$n>3$ are action accessible but not weakly action representable. This
partially answers a question asked by G. Janelidze in [17], whether reasonably
mild conditions may be found on a semi-abelian category under which the second
implication may be reversed: already it makes clear that not all action
accessible semi-abelian varieties are weakly action representable. Our aim
here is to study what happens for a different class of categories, namely
varieties of not necessarily associative algebras over a field.

### Varieties of non-associative algebras

We now recall the algebraic setting we are working in: _varieties of non-
associative algebras_ over a field $\mathbb{F}$. We think of those as
collections of algebras satisfying a chosen set of polynomial equations. The
interested reader can find a more detailed presentation of the subject in
[32].

By a (non-associative) algebra $A$ we mean a vector space $A$ equipped with a
bilinear operation $A\times A\rightarrow A\colon(x,y)\mapsto xy$ which we call
the multiplication. The existence of a unit element is not assumed, nor are
any other conditions on the multiplication besides its bilinearity. Let
$\mathbf{Alg}$ denote the category of non-associative algebras, where
morphisms are linear maps which preserve the multiplication.

We consider the _free algebra_ functor $\mathbf{Set}\rightarrow\mathbf{Alg}$
which sends a set $S$ to the _free algebra_ generated by elements of $S$. This
functor has the forgetful functor as a right adjoint. Moreover, it factorises
through the _free magma functor_
$\operatorname{FM}\colon\mathbf{Set}\rightarrow\mathbf{Mag}$, which sends a
set $S$ to the magma $\operatorname{FM}(S)$ of non-associative words in $S$,
and the _magma algebra functor_
$\mathbb{F}[-]\colon\mathbf{Mag}\rightarrow\mathbf{Alg}$.

Let $S$ be a set. An element $\varphi$ of $\mathbb{F}[\operatorname{FM}(S)]$
is called a non-associative polynomial on $S$. We say that such a polynomial
is a monomial when it is a scalar multiple of an element in
$\operatorname{FM}(S)$. For example, if $S=\\{x,y,z,t\\}$, then $(xy)t+(zy)x$,
$xx+yz$ and $(xt)(yz)$ are polynomials in $S$ and only the last one is a
monomial. For a monomial $\varphi$ on a set $\\{x_{1},\dots,x_{n}\\}$, we
define its type as the $n$-tuple $(k_{1},\dots,k_{n})\in\mathbb{N}^{n}$ where
$k_{i}$ is the number of times $x_{i}$ appears in $\varphi$, and its degree as
the natural number $k_{1}+\dots+k_{n}$. A polynomial is said to be multilinear
if all monomials composing it have the same type of the form $(1,\dots,1)$.
Among the examples above, only the last one is multilinear.

######  Definition 1.3.

An identity of an algebra $A$ is a non-associative polynomial
$\varphi=\varphi(x_{1},\dots,x_{n})$ such that $\varphi(a_{1},\dots,a_{n})=0$
for all $a_{1}$, …, $a_{n}\in A$. We say that the algebra $A$ _satisfies_ the
identity $\varphi$.

Let $I$ be a subset of $\mathbb{F}[\operatorname{FM}(S)]$ with $S$ being a set
of variables. The variety of algebras determined by $I$ is the class of all
algebras which satisfy all the identities in $I$. We say that a variety
satisfies the identities in $I$ if all algebras in this variety satisfy the
given identities. In particular, if the variety is determined by a set of
multilinear polynomials, then we say that the variety is operadic. If there
exists a set of identities of degree $2$ or $3$ that generate all the
identities of $\mathscr{V}$, we say that the variety is quadratic. Recall—see
for instance [11] where this is explained in detail—that an operadic,
quadratic variety of algebras over a field can be viewed as a variety
determined by a quadratic operad.

Any variety of non-associative algebras can, of course, be seen as a category
where the morphisms are the same as in $\mathbf{Alg}$. In particular, any such
variety is a semi-abelian category.

######  Remark 1.4.

Whenever the characteristic of the field $\mathbb{F}$ is zero, any variety of
non-associative algebras over $\mathbb{F}$ is operadic. This is due to the
well-known multilinearisation process, see [29, Corollary 3.7]. The reason
behind the name “operadic” is explained in [30, Section 2].

######  Examples 1.5.

  1. (1)

We write $\mathbf{AbAlg}$ for the variety of _abelian_ algebras determined by
the identity $xy=0$. Seen as a category, this variety is isomorphic to the
category $\mathbf{Vec}$ of vector spaces over $\mathbb{F}$. It is the only
non-trivial variety of non-associative algebras which is an abelian category;
this explains the terminology.

  2. (2)

We write $\mathbf{Assoc}$ for the variety of _associative_ algebras determined
by the identity of _associativity_ which is $x(yz)-(xy)z=0$, or equivalently
$x(yz)=(xy)z$.

  3. (3)

We write $\mathbf{AAssoc}$ for the variety of _anti-associative_ algebras,
determined by the _anti-associative_ identity $x(yz)=-(xy)z$.

  4. (4)

We write $\mathbf{Com}$ for the variety of _commutative_ algebras determined
by the identity of _commutativity_ which is $xy-yx=0$, or equivalently
$xy=yx$.

  5. (5)

We write $\mathbf{ACom}$ for the variety of _anti-commutative_ algebras
determined by _anti-commutativity_ which is $xy+yx=0$, or equivalently
$xy=-yx$.

  6. (6)

We write $\mathbf{CAssoc}$ for the variety of commutative associative
algebras.

  7. (7)

We write $\mathbf{ACAAssoc}$ for the variety of anti-commutative anti-
associative algebras.

  8. (8)

We write $\mathbf{Lie}$ for the variety of _Lie algebras_ determined by _anti-
commutativity_ and the _Jacobi identity_ , which respectively are $xy+yx=0$
and $x(yz)+y(zx)+z(xy)=0$.

  9. (9)

One can see that all the previous examples are operadic varieties. Let us
provide a non-operadic example: the variety $\mathbf{Bool}$ of _Boolean rings_
, which may be seen as associative $\mathbb{Z}_{2}$-algebras satisfying
$xx=x$. This variety is action representable.

  10. (10)

We write $\mathbf{JJord}$ for the variety of _Jacobi–Jordan algebras_ which is
determined by commutativity and the Jacobi identity. Jacobi–Jordan algebras,
also known as _mock-Lie_ algebras, are the commutative counterpart of Lie
algebras. The name of Jordan in the definition is justified by the fact that
every Jacobi–Jordan algebra is a Jordan algebra (see [7]).

  11. (11)

We write $\mathbf{Leib}$ for the variety of _(right) Leibniz algebras_
determined by the _(right) Leibniz identity_ which is $(xy)z-(xz)y-x(yz)=0$.

  12. (12)

We write $\mathbf{Alt}$ for the variety of _alternative algebras_ , which is
determined by the identities $(yx)x-yx^{2}=0$ and $x(xy)-x^{2}y=0$ . Every
associative algebra is obviously alternative and an example of an alternative
algebra which is not associative is given by the _octonions_ $\mathbb{O}$,
that is the eight-dimensional algebra with basis
$\\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8}\\}$ and multiplication
table

$e_{i}e_{j}=\begin{cases}e_{j},&\text{if }i=1\\\ e_{i},&\text{if }j=1\\\
-\delta_{ij}e_{1}+\varepsilon_{ijk}e_{k},&\text{otherwise},\end{cases}$

where $\delta_{ij}$ is the _Kronecker delta_ and $\varepsilon_{ijk}$ a
_completely antisymmetric tensor_ with value 1 when
$ijk=123,145,176,246,257,347,365$. Notice that $e_{1}$ is the unit of the
algebra $\mathbb{O}$.

When $\operatorname{char}(\mathbb{F})\neq 2$, the multilinearisation process
shows that $\mathbf{Alt}$ is equivalent to the variety defined by

$(xy)z+(xz)y-x(yz)-x(zy)=0$

and

$(xy)z+(yx)z-x(yz)-y(xz)=0.$

  13. (13)

Taking any variety $\mathscr{V}$, one can look at a subvariety of it by adding
further identities to be satisfied. For example, let $\mathscr{V}$ be a
variety determined by a set of identities $I$ and let $k$ be any positive
natural number, then we write $\mathbf{Nil}_{k}(\mathscr{V})$ for the variety
of _$k$ -step nilpotent algebras in $\mathscr{V}$_ determined by the
identities in $I$ and the identities of the form $x_{1}\cdots x_{k+1}=0$ with
all possible choices of parentheses.

We now want to explain how we may describe actions in a variety of non-
associative algebras. As we already mentioned before, in a semi-abelian
category, actions are split extensions.

######  Definition 1.6.

Let

${0}$${X}$${A}$${B}$${0}$$\scriptstyle{i}$$\scriptstyle{\pi}$$\scriptstyle{s}$
(1.2)

be a split extension in the variety $\mathscr{V}$. The pair of bilinear maps

$l\colon B\times X\rightarrow X,\qquad r\colon X\times B\rightarrow X$

defined by

$b\ast x=s(b)i(x),\quad x\ast b=i(x)s(b),\quad\forall b\in B,\;\forall x\in X$

where $b\ast-=l(b,-)$ and $-\ast b=r(-,b)$, is called the _derived action_ of
$B$ on $X$ associated with (1.2).

Given a pair of bilinear maps

$l\colon B\times X\rightarrow X,\qquad r\colon X\times B\rightarrow X$

with $B$, $X$ objects of $\mathscr{V}$, we may define a multiplication on the
direct sum of vector spaces $B\oplus X$ by

$(b,x)\cdot(b^{\prime},x^{\prime})=(bb^{\prime},xx^{\prime}+b\ast
x^{\prime}+x\ast b^{\prime})$ (1.3)

with $b\ast x^{\prime}\coloneqq l(b,x^{\prime})$ and $x\ast
b^{\prime}\coloneqq r(x,b^{\prime})$. This construction allows us to build the
split extension in $\mathbf{Alg}$

${0}$${X}$${B\oplus
X}$${B}$${0}$$\scriptstyle{i_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{i_{1}}$
(1.4)

with $i_{2}(x)=(0,x)$, $i_{1}(b)=(b,0)$ and $\pi_{1}(b,x)=b$. This is a split
extension in $\mathscr{V}$ if and only if $(B\oplus X,\cdot)$ is an object of
$\mathscr{V}$, i.e. it satisfies the identities which determine $\mathscr{V}$.
In other words, we have the following result analogous to [28, Theorem 2.4]
and [14, Lemma 1.8]:

######  Lemma 1.7.

In a variety of non-associative algebras $\mathscr{V}$, given a pair of
bilinear maps

$l\colon B\times X\rightarrow X,\qquad r\colon X\times B\rightarrow X,$

we define the multiplication on $B\oplus X$ as above in (1.3). Then, the pair
$(l,r)$ is a derived action of $B$ on $X$ if and only if $(B\oplus X,\cdot)$
is in $\mathscr{V}$. In this case, we call $B\oplus X$ the semi-direct product
of $B$ and $X$ (with respect to the derived action) and we denote it by
$B\ltimes X$.

######  Remark 1.8.

Notice that, for any split extension (1.2) and the corresponding derived
action $(l,r)$, there is an isomorphism of split extensions

${0}$${X}$${B\ltimes
X}$${B}$${0}$${0}$${X}$${A}$${B}$${0}$$\scriptstyle{i_{2}}$$\scriptstyle{1_{X}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\theta}$$\scriptstyle{i_{1}}$$\scriptstyle{1_{B}}$$\scriptstyle{i}$$\scriptstyle{\pi}$$\scriptstyle{s}$

where $\theta\colon B\ltimes X\rightarrow A\colon(b,x)\mapsto s(b)+i(x)$.
Thus, when we write $b\ast x$ (resp. $x\ast b$), one can think of it as the
multiplication $(b,0)\cdot(0,x)$ (resp. $(0,x)\cdot(b,0$)) in $B\ltimes X$.

### Categorical consequences

Let $\mathscr{V}$ be an operadic variety of non-associative algebras. We
recall two results which will be useful for understanding the rest of the
paper.

######  Theorem 1.9 ([12, 13]).

The following conditions are equivalent:

  1. (i)

$\mathscr{V}$ is algebraically coherent [9];

  2. (ii)

$\mathscr{V}$ is an Orzech category of interest;

  3. (iii)

$\mathscr{V}$ is action accessible;

  4. (iv)

there exist $\lambda_{1}$, …, $\lambda_{8}$, $\mu_{1}$, …, $\mu_{8}$ in
$\mathbb{F}$ such that

$\displaystyle x(yz)=\lambda_{1}(xy)z$
$\displaystyle+\lambda_{2}(yx)z+\lambda_{3}z(xy)+\lambda_{4}z(yx)$
$\displaystyle+\lambda_{5}(xz)y+\lambda_{6}(zx)y+\lambda_{7}y(xz)+\lambda_{8}y(zx)$
and $\displaystyle(yz)x=\mu_{1}(xy)z$
$\displaystyle+\mu_{2}(yx)z+\mu_{3}z(xy)+\mu_{4}z(yx)$
$\displaystyle+\mu_{5}(xz)y+\mu_{6}(zx)y+\mu_{7}y(xz)+\mu_{8}y(zx)$

are identities in $\mathscr{V}$. ∎

We call the two previous identities together the $\lambda/\mu$-rules. Since
(WRA) implies action accessibility in general, the existence of the
$\lambda/\mu$-rules is a necessary condition for the variety $\mathscr{V}$ to
be weakly action representable.

######  Theorem 1.10 ([14]).

The following conditions are equivalent:

  1. (i)

$\mathscr{V}$ is action representable;

  2. (ii)

$\mathscr{V}$ is either the variety $\mathbf{Lie}$ or the variety
$\mathbf{AbAlg}$. ∎

Theorem 1.10 helps motivating our interest in the condition (WRA). In fact, in
our context, there is only one non-trivial example of a variety which is
action representable. This suggests to study a generalisation of the notion of
representability of actions. On the other hand, action accessibility may not
be enough to study some kind of (weak) actor. The next result, which is
closely related to [17, Proposition 4.5], explains one way of understanding
weak action representability for any variety of non-associative algebras over
a field.

######  Proposition 1.11.

A variety of non-associative algebras $\mathscr{V}$ is weakly action
representable if and only if for any object $X$ in it, there exists an object
$T$ of $\mathscr{V}$ such that for every derived action of an object $B$ of
$\mathscr{V}$ on $X$

$l\colon B\times X\rightarrow X,\qquad r\colon X\times B\rightarrow X,$

there exists a unique morphism
$\varphi\in\operatorname{Hom}_{\mathscr{V}}(B,T)$ and a derived action
$(l^{\prime},r^{\prime})$ of $\varphi(B)$ on $X$ such that

$l^{\prime}(\varphi(b),x)=l(b,x),\quad r^{\prime}(x,\varphi(b))=r(x,b),$

for every $b\in B$ and for every $x\in X$.

###### Proof.

($\Rightarrow$) If $\mathscr{V}$ is weakly action representable, then for any
object $X$ in it there exists a weak representation $(T,\tau)$. Let $B$ be an
object of $\mathscr{V}$ which acts on $X$ and let $\varphi\colon B\rightarrow
T$ be the corresponding acting morphism. Consider the split extension diagram

${0}$${X}$${B\ltimes X}$${B}$${0}$${0}$${X}$${\varphi(B)\ltimes
X}$${\varphi(B)}$${0}$$\scriptstyle{i}$$\scriptstyle{1_{X}}$$\scriptstyle{\pi}$$\scriptstyle{\exists!f}$$\scriptstyle{s}$$\scriptstyle{\widetilde{\varphi}}$$\scriptstyle{i^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{s^{\prime}}$

where $\widetilde{\varphi}$ is the corestriction of $\varphi$ to its image,
$i^{\prime}(x)=(0,x)$, $s^{\prime}(\varphi(b))=(\varphi(c),0)$, where
$(c,0)=s(b)$, and $f(b,x)=(\varphi(b),x)$. Then the action of $\varphi(B)$ on
$X$ is defined by the pair of bilinear maps

$l^{\prime}\colon\varphi(B)\times X\rightarrow X,\qquad r^{\prime}\colon
X\times\varphi(B)\rightarrow X$

where

$l^{\prime}(\varphi(b),x)=s^{\prime}(\varphi(b))i^{\prime}(x)=s(b)i(x)=l(b,x)$

and

$r^{\prime}(\varphi(b),x)=i(x)s^{\prime}(\varphi(b))=i(x)s(b)=r(b,x),$

for every $b\in B$ and for every $x\in X$ (we use [17, Proposition 4.5] to see
that $l^{\prime}$ and $r^{\prime}$ are well defined).

($\Leftarrow$) Conversely, given an object $X$ of $\mathscr{V}$, a weak
representation of $\operatorname{SplExt}(-,X)$ is given by $(T,\tau)$, where
the component

$\tau_{B}\colon\operatorname{SplExt}(B,X)\rightarrowtail\operatorname{Hom}_{\mathscr{V}}(B,T)$

sends every action of $B$ on $X$ to the corresponding morphism $\varphi$.
Moreover, $\tau_{B}$ is an injection since the morphism $\varphi$ is uniquely
determined by the action of $B$ on $X$. Thus $\tau$ is a monomorphism of
functors. ∎

### Partial Algebras

We end this chapter with a notion we shall use throughout the text.

######  Definition 1.12.

Let $X$ be an $\mathbb{F}$-vector space. A _bilinear partial operation_ on $X$
is a map

$\cdot\colon\Omega\rightarrow X,$

where $\Omega$ is a vector subspace of $X\times X$, which is bilinear on
$\Omega$, i.e.

$(\alpha_{1}x_{1}+\alpha_{2}x_{2})\cdot y=\alpha_{1}x_{1}\cdot
y+\alpha_{2}x_{2}\cdot y$

for any $\alpha_{1},\alpha_{2}\in\mathbb{F}$ and $x_{1},x_{2},y\in X$ such
that $(x_{1},y),(x_{2},y)\in\Omega$ and

$x\cdot(\beta_{1}y_{1}+\beta_{2}y_{2})=\beta_{1}x\cdot y_{1}+\beta_{2}x\cdot
y_{2}$

for any $\beta_{1},\beta_{2}\in\mathbb{F}$ and $x,y_{1},y_{2}\in X$ such that
$(x,y_{1}),(x,y_{2})\in\Omega$.

######  Definition 1.13.

A _partial algebra_ over $\mathbb{F}$ is an $\mathbb{F}$-vector space $X$
endowed with a bilinear partial operation

$\cdot\colon\Omega\rightarrow X.$

We denote it by $(X,\cdot,\Omega)$. When $\Omega=X\times X$ we say that the
algebra is _total_.

Let $(X,\cdot,\Omega)$ and $(X^{\prime},\ast,\Omega^{\prime})$ be partial
algebras over $\mathbb{F}$. A homomorphism of partial algebras is an
$\mathbb{F}$-linear map $f\colon X\rightarrow X^{\prime}$ such that $f(x\cdot
y)=f(x)\ast f(y)$ whenever $(x,y)\in\Omega$, which tacitly implies that
$(f(x),f(y))\in\Omega^{\prime}$ (i.e. both $x\cdot y$ and $f(x)\ast f(y)$ are
defined). We denote by $\mathbf{PAlg}$ the category whose objects are partial
algebras and whose morphisms are partial algebra homomorphisms.

######  Definition 1.14.

We say that a partial algebra $(X,\cdot,\Omega)$ _satisfies an identity_ when
that identity holds wherever the bilinear partial operation is well defined.

For instance, a partial algebra $(X,\cdot,\Omega)$ is associative if

$x\cdot(y\cdot z)=(x\cdot y)\cdot z$

for every $x$, $y$, $z\in X$ such that $(x,y)$, $(y,z)$, $(x,yz)$,
$(xy,z)\in\Omega$.

######  Remark 1.15.

We observe that any variety of non-associative algebras $\mathscr{V}$ has an
obvious forgetful functor $U\colon\mathscr{V}\rightarrow\mathbf{PAlg}$.

##  2\. Commutative and anti-commutative algebras

In this section we aim to study the (weak) representability of actions of some
varieties of non-associative algebras which satisfy the commutative law or the
anti-commutative law. As explained in Section 1, we may assume our variety
satisfies the $\lambda/\mu$-rules, or equivalently is action accessible.

When $\mathscr{V}$ is either a variety of commutative or anti-commutative
algebras, i.e. $xy=\varepsilon yx$ is an identity of $\mathscr{V}$, with
$\varepsilon=\pm 1$, the $\lambda/\mu$-rules reduce to

$x(yz)=\alpha(xy)z+\beta(xz)y,$

for some $\alpha$, $\beta\in\mathbb{F}$. The following proposition is a
representation theory exercise:

######  Proposition 2.1.

Let $\mathscr{V}$ be a non-abelian, action accessible, operadic variety of
non-associative algebras.

  1. (1)

If $\mathscr{V}$ is a variety of commutative algebras, then $\mathscr{V}$ is a
either a subvariety of $\mathbf{CAssoc}$ or a subvariety of $\mathbf{JJord}$.

  2. (2)

If $\mathscr{V}$ is a variety of anti-commutative algebras, then $\mathscr{V}$
is either a subvariety of $\mathbf{Lie}$ or a subvariety of
$\mathbf{ACAAssoc}$. ∎

######  Remark 2.2.

We observe that $\mathbf{Nil}_{2}(\mathbf{Com})$ is a subvariety of both
$\mathbf{CAssoc}$ and $\mathbf{JJord}$: in fact, from $x(yz)=(xy)z=0$ we may
deduce that associativity holds and the Jacobi identity is satisfied:

$x(yz)+y(zx)+z(xy)=0+0+0=0.$

If $\operatorname{char}(\mathbb{F})\neq 3$, then
$\mathbf{Nil}_{2}(\mathbf{Com})$ is precisely the intersection of the
varieties $\mathbf{CAssoc}$ and $\mathbf{JJord}$. Indeed, let $\mathscr{V}$ be
a subvariety of both $\mathbf{CAssoc}$ and $\mathbf{JJord}$. Since
commutativity, associativity and the Jacobi identity hold in $\mathscr{V}$, we
have

$(xy)z=x(yz)=-y(zx)-z(xy)=-x(yz)-(xy)z=-2(xy)z$

and thus $3(xy)z=3x(yz)=0$.

An example of an algebra which lies in the intersection of $\mathbf{CAssoc}$
and $\mathbf{JJord}$ but which is not two-step nilpotent is the two-
dimensional $\mathbb{F}_{3}$-algebra with basis $\\{e_{1},e_{2}\\}$ and
bilinear multiplication determined by

$e_{1}^{2}=e_{1}e_{2}=e_{2}e_{1}=e_{2}^{2}=e_{2}.$

Likewise, $\mathbf{Nil}_{2}(\mathbf{ACom})$ is a subvariety of both
$\mathbf{Lie}$ and $\mathbf{ACAAssoc}$: from $x(yz)=(xy)z=0$ we may deduce
anti-associativity and the Jacobi identity. If
$\operatorname{char}(\mathbb{F})\neq 3$, then
$\mathbf{Nil}_{2}(\mathbf{ACom})$ coincides with the intersection of the
varieties $\mathbf{Lie}$ and $\mathbf{ACAAssoc}$. Indeed, let $\mathscr{V}$ be
a subvariety of both $\mathbf{Lie}$ and $\mathbf{ACAAssoc}$. Since anti-
commutativity, anti-associativity and the Jacobi identity hold in
$\mathscr{V}$, we have

$(xy)z=-x(yz)=-(xy)z-y(xz)=-(xy)z+(yx)z=-2(xy)z$

and thus $3(xy)z=-3x(yz)=0$.

When $\operatorname{char}(\mathbb{F})=3$, it is possible to construct an
algebra that lies in the intersection of $\mathbf{Lie}$ and
$\mathbf{ACAAssoc}$ but which is not two-step nilpotent. Let $X$ be the
algebra of dimension $7$ over $\mathbb{F}_{3}$ with basis

$\\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7}\\}$

and bilinear multiplication determined by

$e_{1}e_{2}=-e_{2}e_{1}=e_{4},\;e_{1}e_{3}=-e_{3}e_{1}=-e_{6},\;e_{2}e_{3}=-e_{3}e_{2}=e_{5}$

and

$e_{1}e_{5}=-e_{5}e_{1}=e_{2}e_{6}=-e_{6}e_{2}=e_{3},e_{4}=-e_{4}e_{3}=e_{7}.$

Then $X$ is a Lie algebra such that

$x(xy)=0$

for any $x$, $y\in X$ and, using the multi-linearisation process, one can
check this identity is equivalent to anti-associativity if the characteristic
of the field is different from $2$. This $X$ is not two-step nilpotent, since

$e_{1}(e_{2}e_{3})=e_{1}e_{5}=e_{7}.$

######  Corollary 2.3.

Let $\mathscr{V}$ be an action accessible, operadic, quadratic variety of non-
associative algebras and suppose that $\mathscr{V}$ is not the variety
$\mathbf{AbAlg}$ of abelian algebras.

  1. (1)

If $\mathscr{V}$ is commutative, then it has to be one of the following
varieties: $\mathbf{JJord}$, $\mathbf{CAssoc}$, their intersection, or
$\mathbf{Nil}_{2}(\mathbf{Com})$.

  2. (2)

If $\mathscr{V}$ is anti-commutative, then it has to be one of the following
varieties: $\mathbf{Lie}$, $\mathbf{ACAAssoc}$, their intersection, or
$\mathbf{Nil}_{2}(\mathbf{ACom})$.

∎

We already know that $\mathbf{Lie}$ is action representable and that the actor
of a Lie algebra $X$ is the Lie algebra $\operatorname{Der}(X)$ of derivations
of $X$. Therefore, we shall study the representability of actions of the
varieties $\mathbf{CAssoc}$, $\mathbf{JJord}$,
$\mathbf{Nil}_{2}(\mathbf{Com})$, $\mathbf{ACAAssoc}$ and
$\mathbf{Nil}_{2}(\mathbf{ACom})$.

### Commutative associative algebras

The representability of actions of the variety of commutative associative
algebras over a field was studied in [4], where F. Borceux, G. Janelidze and
G. M. Kelly proved that it is not action representable. We want to extend this
result proving that the variety $\mathbf{CAssoc}$ is weakly action
representable. In Section 3 this is further extended to general algebras over
a field. We start by recalling the following result, where
$U\colon\mathbf{CAssoc}\to\mathbf{Assoc}$ denotes the forgetful functor.

######  Lemma 2.4 ([4], proof of Theorem 2.6).

Let $X$ be a commutative associative algebra. There exists a natural
isomorphism of functors from
$\mathbf{CAssoc}^{\operatorname{op}}\to\mathbf{Set}$ which we denote

$\rho\colon\operatorname{SplExt}(-,X)\cong\operatorname{Hom}_{\mathbf{Assoc}}(U(-),\operatorname{M}(X)),$

where
$\operatorname{SplExt}(-,X)=\operatorname{SplExt}_{\mathbf{CAssoc}}(-,X)$ and

$\operatorname{M}(X)=\\{f\in\operatorname{End}(X)\mid f(xy)=f(x)y,\quad\forall
x,y\in X\\}$

is the associative algebra of _multipliers_ of $X$, endowed with the product
induced by the usual composition of functions (see [8, 24]). ∎

We recall that $\operatorname{M}(X)$ in general does not need to be a
commutative algebra. For instance, let $X=\mathbb{F}^{2}$ be the abelian two-
dimensional algebra, then $\operatorname{M}(X)=\operatorname{End}(X)$ which is
not commutative. However there are special cases where $\operatorname{M}(X)$
is an object of $\mathbf{CAssoc}$, such as when the _annihilator_ of $X$
(which coincides with the categorical notion of center)

$\operatorname{Ann}(X)=\\{x\in X\mid xy=0,\;\forall y\in X\\}$

is trivial or when $X^{2}=X$, where $X^{2}$ denotes the subalgebra of $X$
generated by the products $xy$ where $x$, $y\in X$. We refer the reader to [8]
for further details.

######  Theorem 2.5 ([4], Theorem 2.6).

Let $X$ be a commutative associative algebra. The following statements are
equivalent:

  1. (i)

$\operatorname{M}(X)$ is a commutative associative algebra;

  2. (ii)

the functor $\operatorname{SplExt}(-,X)$ is representable. ∎

Since we have examples where $\operatorname{M}(X)$ is not commutative, we
conclude that $\mathbf{CAssoc}$ is not action representable. We now want to
prove that it is a weakly action representable category. We analyse what this
means and then prove that the category does indeed fulfil these requirements.

For any commutative associative algebra $T$, the fully faithful embedding $U$
of the category $\mathbf{CAssoc}$ into $\mathbf{Assoc}$ induces a natural
isomorphism

$i\colon\operatorname{Hom}_{\mathbf{CAssoc}}(-,T)\cong\operatorname{Hom}_{\mathbf{Assoc}}(U(-),U(T))\colon\mathbf{CAssoc}^{\operatorname{op}}\to\mathbf{Set}\text{.}$

######  Lemma 2.6.

If the functor $\operatorname{SplExt}(-,X)$ admits a weak representation
$(T,\tau)$, then there exists an injective function
$j\colon\operatorname{M}(X)\to T$ such that for each commutative associative
algebra $B$, the square

$\textstyle{\operatorname{SplExt}(B,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{B}}$$\scriptstyle{\tau_{B}}$$\textstyle{\operatorname{Hom}_{\mathbf{CAssoc}}(B,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{i_{B}}$$\textstyle{\operatorname{Hom}_{\mathbf{Assoc}}(U(B),\operatorname{M}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j\circ(-)}$$\textstyle{\operatorname{Hom}_{\mathbf{Assoc}}(U(B),U(T))}$

commutes.

###### Proof.

The free associative $\mathbb{F}$-algebra on a single generator is the algebra
$\mathbb{F}[x]$ of non-constant polynomials in a single variable $x$, which
since it is commutative is also the free algebra on a single generator in
$\mathbf{CAssoc}$. We find $j$ as the injective function

$V(\operatorname{M}(X))\cong\operatorname{Hom}_{\mathbf{Assoc}}(\mathbb{F}[x],\operatorname{M}(X))\to\operatorname{Hom}_{\mathbf{Assoc}}(\mathbb{F}[x],U(T))\cong
V(U(T))$

where $V(A)$ denotes the underlying set of an algebra $A$ and the function in
the middle is the $\mathbb{F}[x]$-component of the monomorphism of functors

$i\circ\tau\circ\rho^{-1}\colon\operatorname{Hom}_{\mathbf{Assoc}}(U(-),\operatorname{M}(X))\rightarrowtail\operatorname{Hom}_{\mathbf{Assoc}}(U(-),U(T)).$

Now each $b\in B$ induces a morphism $b\colon\mathbb{F}[x]\to B$, and the
collection of morphisms ${(b\colon\mathbb{F}[x]\to B)_{b\in B}}$ is jointly
epic. Hence its image

$(\operatorname{Hom}_{\mathbf{Assoc}}(U(B),U(T))\to\operatorname{Hom}_{\mathbf{Assoc}}(U(\mathbb{F}[x]),U(T)))_{b\in
B}$

through the contravariant functor
$\operatorname{Hom}_{\mathbf{Assoc}}(U(-),U(T))$ is a jointly monic collection
of arrows. It thus suffices that for each $b\in B$, the outer rectangle in the
diagram

$\textstyle{\operatorname{SplExt}(B,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{B}}$$\scriptstyle{\tau_{B}}$$\textstyle{\operatorname{Hom}_{\mathbf{CAssoc}}(B,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{i_{B}}$$\textstyle{\operatorname{Hom}_{\mathbf{Assoc}}(U(B),\operatorname{M}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)\circ
U(b)}$$\scriptstyle{j\circ(-)}$$\textstyle{\operatorname{Hom}_{\mathbf{Assoc}}(U(B),U(T))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)\circ
U(b)}$$\textstyle{\operatorname{Hom}_{\mathbf{Assoc}}(U(\mathbb{F}[x]),\operatorname{M}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{i_{\mathbb{F}[x]}\circ\tau_{\mathbb{F}[x]}\circ\rho^{-1}_{\mathbb{F}[x]}}$$\textstyle{\operatorname{Hom}_{\mathbf{Assoc}}(U(\mathbb{F}[x]),U(T))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{V(\operatorname{M}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{V(U(T))}$

commutes in $\mathbf{Set}$. This is an immediate consequence of the naturality
of the transformations involved. ∎

######  Remark 2.7.

The above proof can be modified to show that the function $j$ is in fact a
vector space monomorphism. If it were moreover an algebra monomorphism, then
this would yield a proof that all $\operatorname{M}(X)$ are commutative, which
is false by the above-mentioned example. Thus we would be able to conclude
that $\mathbf{CAssoc}$ is not weakly action representable. Theorem 2.11 below
proves that this is wrong. Hence $j$ cannot preserve the algebra
multiplication in general.

Each action $\xi$ of a commutative associative algebra $B$ on $X$ gives rise
to a morphism $\rho_{B}(\xi)\colon U(B)\to\operatorname{M}(X)$ in
$\mathbf{Assoc}$. If the actions in $\mathbf{CAssoc}$ are weakly
representable, then $\xi$ is also determined by a morphism of commutative
associative algebras $\tau_{B}(\xi)\colon B\to T$. The above lemma tells us
that $j\circ\rho_{B}(\xi)=\tau_{B}(\xi)$. Note that here we drop the $i_{B}$
for the sake of clarity.

Each $\rho_{B}(\xi)\colon U(B)\to\operatorname{M}(X)$ is the composite of a
surjective commutative associative algebra map $\rho^{\prime}_{B}(\xi)\colon
U(B)\to M_{\xi}$ and the canonical inclusion of a subalgebra $M_{\xi}$ of
$\operatorname{M}(X)$. We find a diagram of subalgebras of
$\operatorname{M}(X)$ indexed over the commutative associative algebra actions
on $X$. Note that since trivial actions exist, the image in $\mathbf{Assoc}$
of the diagram $(M_{\xi})_{\xi}$ actually consists of all commutative
subalgebras of $\operatorname{M}(X)$, with the canonical inclusions between
them. We may re-index and view $(M_{\xi})_{\xi}$ as a diagram in
$\mathbf{Assoc}$ over the thin category of commutative subalgebras of
$\operatorname{M}(X)$.

By the above, the $M_{\xi}$ further include into $T$ via $j$. For each $\xi$,
an image factorisation of $\tau_{B}(\xi)\colon B\to T$ is given by the
surjective algebra map $\rho^{\prime}_{B}(\xi)\colon U(B)\to M_{\xi}$ followed
by the inclusion of $M_{\xi}$ into $\operatorname{M}(X)$ composed with $j$. We
denote this function $\mu_{M_{\xi}}\colon M_{\xi}\to T$ and note that it only
depends on the object $M_{\xi}$. (That is to say, if $\xi$ and $\psi$ are two
$B$-actions such that $M_{\xi}=M_{\psi}$, then the induced inclusions into $T$
coincide as well.) _A priori_ this $\mu_{M_{\xi}}$ is only an injective _map_
, but since $\tau_{B}(\xi)$ and $\rho^{\prime}_{B}(\xi)$ are morphisms of
algebras and $\rho^{\prime}_{B}(\xi)$ is a surjection, that injection is a
monomorphism of commutative associative algebras. Furthermore, the
$\mu_{M_{\xi}}$ form a cocone on the diagram of all commutative subalgebras of
$\operatorname{M}(X)$ with vertex $T$.

Recall that for a diagram in a category, an _amalgam_ is a monic cocone, i.e.
a cocone which is a monomorphic natural transformation. This means that each
component of that cocone is a monomorphism, which implies that all the
morphisms of the given diagram were monomorphisms to begin with. Note that in
a category with colimits, an amalgam for a diagram exists if and only if its
colimit cocone is such an amalgam. A category is said to have the
_amalgamation property_ (AP) when each span of monomorphisms admits an
amalgam; equivalently, for each pushout square

$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{s}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{T}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{S}}$$\textstyle{S+_{I}T}$

if $s$ and $t$ are monomorphisms then so are $\iota_{S}$ and $\iota_{T}$. It
is known that neither the category of associative algebras, nor the category
of commutative associative algebras satisfies the condition (AP)—see [20] for
an overview of examples and references to the rich literature on the subject.

This is as follows related to the problem at hand. The associative algebra
$\operatorname{M}(X)$ is an amalgam in $\mathbf{Assoc}$ of the diagram
consisting of the commutative subalgebras $M_{\xi}$ of $\operatorname{M}(X)$.
So if the functor $\operatorname{SplExt}(-,X)$ admits a weak representation
$(T,\tau)$, then the natural transformation $\tau$ factors through the diagram
$(M_{\xi})_{\xi}$ as explained above, and we see that $(T,\tau)$ restricts to
an amalgam of that diagram in the category $\mathbf{CAssoc}$.

Thus we find a necessary condition for weak representability of actions in the
category $\mathbf{CAssoc}$: we need that for each commutative associative
algebra $X$, the diagram $(M_{\xi})_{\xi}$ of commutative subalgebras of the
associative algebra $\operatorname{M}(X)$ not only admits the amalgam
$\operatorname{M}(X)$ in the category $\mathbf{Assoc}$; it should also admit
an amalgam $T$ in $\mathbf{CAssoc}$. Actually, the converse also holds:

######  Proposition 2.8.

For a commutative associative algebra $X$, a weak representation $(T,\tau)$ of
$\operatorname{SplExt}(-,X)$ exists if and only if an amalgam in
$\mathbf{CAssoc}$ exists for the diagram of commutative subalgebras of
$\operatorname{M}(X)$.

###### Proof.

We already explained that any weak representation of
$\operatorname{SplExt}(-,X)$ restricts to such an amalgam. So let us assume
that a commutative amalgam for the diagram of commutative subalgebras of
$\operatorname{M}(X)$ exists. For each commutative associative algebra action
$\xi$ of an object $B$ on $X$, we let $\tau_{B}(\xi)\colon B\to T$ be the
composite of $\rho^{\prime}_{B}(\xi)\colon U(B)\to M_{\xi}$ with the inclusion
$\mu_{M_{\xi}}\colon M_{\xi}\to T$ of $M_{\xi}$ into the amalgam $T$.

The thus defined $\tau$ is a natural transformation by the naturality of both
$\rho^{\prime}$ and the cocone components in the amalgam. Note that if two
maps, say $\rho_{B}(\xi)$ and $\rho_{C}(\psi)$, to $\operatorname{M}(X)$ have
the same image subalgebra $M_{\xi}=M_{\psi}$ of $\operatorname{M}(X)$, then by
naturality of $\rho$ and the fact that the inclusion of $M_{\xi}$ into
$\operatorname{M}(X)$ is a monomorphism, for any equivariant map $f\colon B\to
C$ we have that the square on the left

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\rho^{\prime}_{B}(\xi)}$$\textstyle{M_{\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{M_{\xi}}}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\prime}_{C}(\psi)}$$\textstyle{M_{\psi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{M_{\psi}}}$$\textstyle{T}$

commutes. The commutativity of the entire diagram proves naturality of $\tau$.

We still have to prove that the components of $\tau$ are monomorphisms: two
different actions $\xi$ and $\psi$ of $B$ on $X$ give rise to two different
maps $\tau_{B}(\xi),\tau_{B}(\psi)\colon B\to T$. Suppose, on the contrary,
that $\tau_{B}(\xi)=\tau_{B}(\psi)$. Then by uniqueness of image
factorisations, the images of $\mu_{M_{\xi}}\colon M_{\xi}\to T$ and
$\mu_{M_{\psi}}\colon M_{\psi}\to T$ are isomorphic subobjects of $T$. Now the
image in $\mathbf{Assoc}$ of the diagram $(M_{\xi})_{\xi}$ is a thin category,
so that $M_{\xi}=M_{\psi}$. Hence
$\mu_{M_{\xi}}\circ\rho^{\prime}_{B}(\xi)=\tau_{B}(\xi)=\tau_{B}(\psi)=\mu_{M_{\psi}}\circ\rho^{\prime}_{B}(\psi)=\mu_{M_{\xi}}\circ\rho^{\prime}_{B}(\psi)$,
which implies $\rho^{\prime}_{B}(\xi)=\rho^{\prime}_{B}(\psi)$. But then the
actions $\xi$ and $\psi$ are equal, since $\rho$ is a natural isomorphism by
Lemma 2.4. ∎

Thus we see that the problem of weak representability of actions of
$\mathbf{CAssoc}$ amounts to proving that an amalgam in $\mathbf{CAssoc}$
exists for the diagram of commutative subalgebras of $\operatorname{M}(X)$ for
any object $X$. We are actually going to prove something a bit stronger:
namely, that an amalgam in $\mathbf{CAssoc}$ exists for any diagram of
commutative associative algebras for which an amalgam exists in
$\mathbf{Assoc}$. The essence of the proof is contained in the following
special case.

######  Theorem 2.9.

If $S\leftarrow I\to T$ is a span of commutative associative algebras for
which an amalgam exists in $\mathbf{Assoc}$, then it has an amalgam in
$\mathbf{CAssoc}$.

The proof depends on the following lemma.

######  Lemma 2.10.

Let $f\colon X\to Y$ and $g\colon Y\to Z$ be morphisms in a semi-abelian
category. The composite $g\circ f\colon X\to Z$ is a monomorphism if and only
if $f$ is a monomorphism and $\operatorname{Im}(f)\cap\operatorname{Ker}(g)$
is trivial.

###### Proof.

Note that if $g\circ f$ is a monomorphism, then so is $f$. So we may assume
that $f$ is a monomorphism in either case. The composite $g\circ f$ is a
monomorphism precisely when $\operatorname{Ker}(g\circ f)$ is trivial. Now
this kernel is a pullback of $\operatorname{Ker}(g)$ along $f$. Since $f$ is a
monomorphism, that pullback is
$\operatorname{Im}(f)\cap\operatorname{Ker}(g)$. So $\operatorname{Ker}(g\circ
f)$ is zero if and only if $\operatorname{Im}(f)\cap\operatorname{Ker}(g)$ is
zero. ∎

###### Proof of Theorem 2.9.

Let $S\leftarrow I\to T$ be such a span. Recall that an amalgam in either
category exists if and only if the $S$ and $T$ components of the induced
pushout cocone in either category are monic. We focus on the case
$\iota_{S}\colon S\to S+_{I}T=S+^{\mathbf{Assoc}}_{I}T$ which we assume to be
monic. The question is whether its composite with the reflection unit
$\eta_{S+_{I}T}\colon S+^{\mathbf{Assoc}}_{I}T\to S+^{\mathbf{CAssoc}}_{I}T$
is still monic. We are going to prove that the answer is yes, indeed it is.

Consider the following morphism of short exact sequences in $\mathbf{Assoc}$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{S+T}|_{J}}$$\textstyle{S+T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{S+T}}$$\textstyle{S+_{I}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{S+_{I}T}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S+^{\mathbf{CAssoc}}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S+^{\mathbf{CAssoc}}_{I}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

As a vector space, the coproduct $S+T$ of $S$ and $T$ in $\mathbf{Assoc}$ is
$S\oplus T\oplus U$ where $U=(S\otimes T)\oplus(T\otimes S)\oplus(S\otimes
T\otimes S)\oplus\cdots$ contains all the tensors. The coproduct
$S+^{\mathbf{CAssoc}}T$ of $S$ and $T$ in $\mathbf{CAssoc}$ is $S\oplus
T\oplus(S\otimes T)$, so that $\eta_{S+T}$ admits a canonical splitting
$\sigma\colon S+^{\mathbf{CAssoc}}T\to S+T$ in $\mathbf{Vec}$ which commutes
with the inclusions of $S$, $T$ and $S\otimes T$.

We note that $K$ is the ideal of $S+^{\mathbf{CAssoc}}T=S\oplus
T\oplus(S\otimes T)$ generated by the elements of the form
$\underline{i}-\overline{i}$, where $\underline{i}$ is $i\in I$ viewed as an
element of $S$, while $\overline{i}$ is $i$ viewed as an element of $T$. The
algebra $J$ is generated by the same elements, but now as an ideal of
$S+T=S\oplus T\oplus U$. It follows that $\sigma(K)\subseteq J$.

Thanks to Lemma 2.10, inside the vector space $S+T$, the intersection $S\cap
J$ is trivial, by the assumption that the composite ${\iota_{S}\colon S\to
S+T\to S+_{I}T}$ is monic. But then the smaller space $S\cap\sigma(K)$ is
trivial as well, so that the composite

$S\to S+^{\mathbf{CAssoc}}T\to S+^{\mathbf{CAssoc}}_{I}T$

is a monomorphism by Lemma 2.10. ∎

For arbitrary diagrams of monomorphisms of commutative associative algebras,
the proof stays essentially the same. This allows us to conclude:

######  Theorem 2.11.

The category $\mathbf{CAssoc}$ of commutative associative algebras is weakly
action representable. ∎

######  Remark 2.12.

For a given diagram of commutative associative algebras as above, the amalgam
$T$ in $\mathbf{CAssoc}$ is also an amalgam in $\mathbf{Assoc}$.

######  Remark 2.13.

Note that by its construction as a colimit, the weak representation $(T,\tau)$
is automatically an _initial weak representation_ (see [17, Section 5]). As
explained in [17, Corollary 5.3, Corollary 5.4], the existence of the initial
weak representation also follows from the existence of a weak representation
and the fact that $\mathbf{CAssoc}$, as a semi-abelian variety of _universal_
algebras, is a total category [31].

### Jacobi–Jordan algebras

We now want to study the representability of actions of the variety
$\mathbf{JJord}$ of Jacobi–Jordan algebras. As already mentioned in Section 1,
every split extension of $B$ by $X$ in $\mathbf{Lie}$ is represented by a
homomorphism ${B\rightarrow\operatorname{Der}(X)}$. For Jacobi–Jordan
algebras, the role the derivations have in $\mathbf{Lie}$ is played by the so-
called _anti-derivations_.

######  Definition 2.14.

Let $X$ be a Jacobi–Jordan algebra. An _anti-derivation_ is a linear map
$d\colon X\rightarrow X$ such that

$d(xy)=-d(x)y-d(y)x,\quad\forall x,y\in X.$

The (left) multiplications $L_{x}$ for $x\in X$ are particular anti-
derivations, called _inner anti-derivations_. We denote by
$\operatorname{ADer}(X)$ the space of anti-derivations of $X$ and by
$\operatorname{Inn}(X)$ the subspace of the inner anti-derivations. Anti-
derivations play a significant role in the study of cohomology of
Jacobi–Jordan algebras: see [1] for further details.

We now want to make explicit what are the derived actions in the category
$\mathbf{JJord}$ and how they are related with the anti-derivations. The
following is an easy application of Lemma 1.7.

######  Proposition 2.15.

Let $X$ and $B$ be two Jacobi–Jordan algebras. Given a pair of bilinear maps

$l\colon B\times X\rightarrow X,\qquad r\colon X\times B\rightarrow X$

defined by

$b\ast x=l(b,x),\qquad x\ast b=r(x,b),$

we construct $(B\oplus X,\cdot)$ as in (1.3). Then $(B\oplus X,\cdot)$ is a
Jacobi–Jordan algebra if and only if

  1. (1)

$b\ast x=x\ast b$;

  2. (2)

$b\ast(xx^{\prime})=-(b\ast x)x^{\prime}-(b\ast x)\ast x^{\prime}$;

  3. (3)

$(bb^{\prime})\ast x=-b\ast(b^{\prime}\ast x)-b^{\prime}\ast(b\ast x)$;

for all $b$, $b^{\prime}\in B$ and $x$, $x^{\prime}\in X$. ∎

In an equivalent way, a derived action of $B$ on $X$ in the variety
$\mathbf{JJord}$ is given by a linear map

$B\rightarrow\operatorname{ADer}(X)\colon b\mapsto b\ast-$

which satisfies

$(bb^{\prime})\ast x=-b\ast(b^{\prime}\ast x)-b^{\prime}\ast(b\ast
x),\quad\forall b,b^{\prime}\in B,\;\forall x\in X.$ (2.1)

######  Remark 2.16.

The vector space $\operatorname{ADer}(X)$ endowed with the _anti-commutator_

$\langle-,-\rangle\colon\operatorname{ADer}(X)\times\operatorname{ADer}(X)\rightarrow\operatorname{End}(X),\quad\langle
f,f^{\prime}\rangle=-f\circ f^{\prime}-f^{\prime}\circ f$

is not in general an algebra, since the anti-commutator of two anti-
derivations is not in general an anti-derivation: in [2, Remark 2.2], the
authors proved that $\langle f,f^{\prime}\rangle\in\operatorname{ADer}(X)$ if
and only if

$\langle
f,f^{\prime}\rangle(xy)=-f(x)f^{\prime}(y)-f^{\prime}(x)f(y),\quad\forall
x,y\in X.$

Moreover, it can happen that the anti-commutator $\langle-,-\rangle$ is a well
defined bilinear operation on the space $\operatorname{ADer}(X)$ but it does
not define a Jacobi–Jordan algebra structure: for instance, if $X=\mathbb{F}$
is the abelian one-dimensional algebra, then
$\operatorname{ADer}(X)=\operatorname{End}(X)\cong\mathbb{F}$ (every linear
endomorphism of $X$ is of the form $\varphi_{\alpha}\colon x\mapsto\alpha x$,
for some $\alpha\in\mathbb{F}$) and the Jacobi identity is not satisfied.
Nevertheless, there are some subspaces of $\operatorname{ADer}(X)$ that are
Jacobi–Jordan algebras. For instance, the subspace $\operatorname{Inn}(X)$ of
all inner anti-derivations of $X$. Indeed, the linear map

$X\rightarrow\operatorname{ADer}(X)\colon x\mapsto L_{x},$

restricts to a Jacobi–Jordan algebra homomorphism
$X\rightarrow\operatorname{Inn}(X)$. This is true in general for the image of
any linear map $B\rightarrow\operatorname{ADer}(X)$ satisfying equation (2.1).

Thus we need to use an algebraic structure which includes the space of anti-
derivations endowed with the anti-commutator and which allows us to describe
categorically the representability of actions of the variety $\mathbf{JJord}$.
One possible solution is given by _partial algebras_.

Indeed, the vector space $\operatorname{ADer}(X)$ endowed with the anti-
commutator $\langle-,-\rangle$ is a commutative partial algebra. In this case
$\Omega$ is the preimage

$\langle-,-\rangle^{-1}(\operatorname{ADer}(X))$

of the inclusion $\operatorname{ADer}(X)\hookrightarrow\operatorname{End}(X)$.

######  Theorem 2.17.

Let $X$ be a Jacobi–Jordan algebra and let
$U\colon\mathbf{JJord}^{\operatorname{op}}\rightarrow\mathbf{PAlg}$ denote the
forgetful functor.

  1. (1)

There exists a natural isomorphism of functors from
$\mathbf{JJord}^{\operatorname{op}}\rightarrow\mathbf{Set}$

$\rho\colon\operatorname{SplExt}(-,X)\cong\operatorname{Hom}_{\mathbf{PAlg}}(U(-),\operatorname{ADer}(X)),$

where
$\operatorname{SplExt}(-,X)=\operatorname{SplExt}_{\mathbf{JJord}}(-,X)$;

  2. (2)

if $\operatorname{ADer}(X)$ is a Jacobi–Jordan algebra, then the functor
$\operatorname{SplExt}(-,X)$ is representable and $\operatorname{ADer}(X)$ is
the actor of $X$;

###### Proof.

(1) For a Jacobi–Jordan algebra $B$, we define the component

$\rho_{B}\colon\operatorname{SplExt}(B,X)\rightarrow\operatorname{Hom}_{\mathbf{PAlg}}(U(B),\operatorname{ADer}(X))$

as the functor which sends any split extension

${0}$${X}$${A}$${B}$${0}$$\scriptstyle{i}$$\scriptstyle{\pi}$$\scriptstyle{s}$

to the morphism $B\rightarrow\operatorname{ADer}(X)\colon b\mapsto b\ast-$.
The transformation $\rho$ is natural. Indeed, for any Jacobi–Jordan algebra
homomorphism $f\colon B^{\prime}\rightarrow B$, it is easy to check that the
diagram in $\mathbf{Set}$

${\operatorname{SplExt}(B,X)}$${\operatorname{Hom}(U(B),\operatorname{ADer}(X))}$${\operatorname{SplExt}(B^{\prime},X)}$${\operatorname{Hom}(U(B^{\prime}),\operatorname{ADer}(X))}$$\scriptstyle{\rho_{B}}$$\scriptstyle{\operatorname{SplExt}(f,X)}$$\scriptstyle{\operatorname{Hom}(U(f),\operatorname{ADer}(X))}$$\scriptstyle{\rho_{B^{\prime}}}$

where $\operatorname{Hom}(U(-),-)=\operatorname{Hom}_{\mathbf{PAlg}}(U(-),-)$,
is commutative. Moreover, for any Jacobi–Jordan algebra $B$, the morphism
$\rho_{B}$ is an injection, as each element of $\operatorname{SplExt}(B,X)$ is
uniquely determined by the corresponding action of $B$ on $X$. Thus $\rho$ is
a monomorphism of functors. Finally $\rho$ is a natural isomorphism since,
given any Jacobi–Jordan algebra $B$ and any homomorphism of partial algebras
$\varphi\colon B\rightarrow\operatorname{ADer}(X)$, the bilinear maps
$l_{\varphi}\colon B\times X\rightarrow X\colon(b,x)\mapsto\varphi(b)(x)$,
$r_{\varphi}=l_{\varphi}$ define a (unique) derived action of $B$ on $X$ such
that $\rho_{B}(l_{\varphi},r_{\varphi})=\varphi$.

(2) If $\operatorname{ADer}(X)$ is a Jacobi–Jordan algebra, then by (1) we
have a natural isomorphism

$\operatorname{SplExt}(-,X)\cong\operatorname{Hom}_{\mathbf{JJord}}(-,\operatorname{ADer}(X)),$

hence $\operatorname{ADer}(X)$ is the actor of $X$. ∎

### Two-step nilpotent commutative algebras

We now analyse the case where $\mathscr{V}$ is a subvariety of both
$\mathbf{CAssoc}$ and $\mathbf{JJord}$, i.e. $\mathscr{V}$ is the variety
$\mathbf{Nil}_{2}(\mathbf{Com})$ of two-step nilpotent commutative algebras.
We recall this means that $xyz=0$ is an identity of $\mathscr{V}$. An example
of such an algebra is the _Kronecker algebra_ $\mathfrak{k}_{1}$ (see [21]),
which is the three-dimensional algebra with basis $\\{e_{1},e_{2},e_{3}\\}$
and multiplication determined by $e_{1}e_{2}=e_{2}e_{1}=e_{3}$.

We shall show that $\mathbf{Nil}_{2}(\mathbf{Com})$ is an example of a weakly
action representable, operadic, quadratic variety of commutative algebras.

######  Proposition 2.18.

Let $X$ and $B$ be two algebras in $\mathbf{Nil}_{2}(\mathbf{Com})$. Given a
pair of bilinear maps

$l\colon B\times X\rightarrow X,\qquad r\colon X\times B\rightarrow X,$

we construct $(B\oplus X,\cdot)$ as in (1.3). Then $(B\oplus X,\cdot)$ is in
$\mathbf{Nil}_{2}(\mathbf{Com})$ if and only if

  1. (1)

$b\ast x=x\ast b$;

  2. (2)

$b\ast(xx^{\prime})=(b\ast x)x^{\prime}=0$;

  3. (3)

$(bb^{\prime})\ast x=b\ast(b^{\prime}\ast x)=0$;

for any $b$, $b^{\prime}\in B$ and $x$, $x^{\prime}\in X$. ∎

The second equation of Proposition 2.18 states that, for every $b\in B$, the
linear map $b\ast-$ belongs to the vector space

$[X]_{2}=\\{f\in\operatorname{End}(X)\mid f(xy)=f(x)y=0,\;\forall x\in X\\}.$

Moreover, seeing $[X]_{2}$ as an abelian algebra (i.e. $\langle
f,g\rangle=0_{\operatorname{End}(X)}$, for every $f,g\in[X]_{2}$), from the
third equation we deduce that the linear map

$B\rightarrow[X]_{2}\colon b\mapsto b\ast-$

is an algebra homomorphism.

On the other hand, given a morphism of algebras

$\varphi\colon B\rightarrow[X]_{2},\quad\varphi(b)=b\ast-$

satisfying

$b\ast(b^{\prime}\ast x)=0,\quad\forall b,b^{\prime}\in B,\;\forall x\in X,$

we can consider the split extension

${0}$${X}$${(B\oplus
X,\ast_{\varphi})}$${B}$${0}$$\scriptstyle{i}$$\scriptstyle{\pi}$$\scriptstyle{s}$

where the two-step nilpotent commutative algebra structure of $B\oplus X$ is
given by

$(b,x)\ast_{\varphi}(b^{\prime},x^{\prime})=(bb^{\prime},xx^{\prime}+b\ast
x^{\prime}+b^{\prime}\ast x),\quad\forall(b,x),(b^{\prime},x^{\prime})\in
B\oplus X.$

We can now claim the following result.

######  Theorem 2.19.

  1. (1)

Let $B$ and $X$ be two-step nilpotent commutative algebras. The isomorphism
classes of split extensions of $B$ by $X$ are in bijection with the algebra
homomorphisms

$B\rightarrow[X]_{2}\colon b\mapsto b\ast-$

satisfying

$b\ast(b^{\prime}\ast x)=0,\quad\forall b,b^{\prime}\in B,\;\forall x\in X.$
(2.2)

  2. (2)

The variety $\mathbf{Nil}_{2}(\mathbf{Com})$ is weakly action representable. A
weak representation of the functor
$\operatorname{SplExt}(-,X)=\operatorname{SplExt}_{\mathbf{Nil}_{2}(\mathbf{Com})}(-,X)$
is given by

$\tau\colon\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}_{\mathbf{Nil}_{2}(\mathbf{Com})}(-,[X]_{2}),$

where $\tau_{B}$ is the injection which sends any split extension of $B$ by
$X$ to the corresponding homomorphism $B\rightarrow[X]_{2}$, defined by
$b\mapsto b\ast-$ as above.

  3. (3)

A homomorphism $B\rightarrow[X]_{2}$ is an acting morphism if and only if it
satisfies Equation 2.2.

###### Proof.

(1) It follows from the analysis above.

(2) We observe that $\tau$ is a natural transformation. Indeed, for every
morphism $f\colon B^{\prime}\rightarrow B$ in
$\mathbf{Nil}_{2}(\mathbf{Com})$, we can check that the diagram in
$\mathbf{Set}$

${\operatorname{SplExt}(B,X)}$${\operatorname{Hom}(B,[X]_{2})}$${\operatorname{SplExt}(B^{\prime},X)}$${\operatorname{Hom}(B^{\prime},[X]_{2})}$$\scriptstyle{\tau_{B}}$$\scriptstyle{\operatorname{SplExt}(f,X)}$$\scriptstyle{\operatorname{Hom}(f,[X]_{2})}$$\scriptstyle{\tau_{B^{\prime}}}$

is commutative. Moreover $\tau_{B}$ is an injection since every isomorphism
class of split extensions of $B$ by $X$ is uniquely determined by the
corresponding derived action. Thus $\tau$ is a monomorphism of functors and
$\mathbf{Nil}_{2}(\mathbf{Com})$ is a weakly action representable category.

(3) Finally, $\varphi\colon B\rightarrow[X]_{2}$ is an acting morphism if and
only if it defines a split extension of $B$ by $X$ in
$\mathbf{Nil}_{2}(\mathbf{Com})$, i.e. it satisfies equation (2.2). ∎

Let us observe that not every morphism $B\rightarrow[X]_{2}$ defines a split
extension of $B$ by $X$. For instance, if $B=\mathbb{F}\\{b,b^{\prime}\\}$ and
$X=\mathbb{F}\\{x\\}\cong\mathbb{F}$ are abelian algebras, then
$[X]_{2}=\operatorname{End}(X)$ and the homomorphism $\varphi\colon
B\rightarrow[X]_{2}$, defined by

$\varphi(b)=\varphi(b^{\prime})=1_{X}$

is not an acting morphism. Indeed,

$\varphi(b)(\varphi(b^{\prime})(x))=1_{X}(1_{X}(x))=x\neq 0.$

### Anti-commutative anti-associative algebras

For the variety $\mathbf{ACAAssoc}$ of anti-commutative anti-associative
algebras, a similar description of split extensions and derived actions can be
made as for the variety $\mathbf{JJord}$. The role of the anti-derivations is
played here by the endomorphisms in the associative partial algebra

$[X]\coloneqq\\{f\in\operatorname{End}(X)\;|\;f(xy)=-f(x)y,\;\forall x\in
X\\},$

whose bilinear partial operation is given by

$\langle f,g\rangle=-f\circ g.$

It is easy to see that $\langle-,-\rangle$ does not define, in general, a
total algebra structure on $[X]$, nor need it be anti-commutative or anti-
associative. An example is given by the abelian two-dimensional algebra
$X=\mathbb{F}^{2}$, where $[X]=\operatorname{End}(X)$.

We may check that a derived action of $B$ by $X$ in the variety
$\mathbf{ACAAssoc}$ is the same thing as a partial algebra homomorphism

$B\rightarrow[X]\colon b\mapsto b\ast-$

which satisfies

$(bb^{\prime})\ast-=-b\ast(b^{\prime}\ast-),\quad\forall b,b^{\prime}\in B.$

Moreover, we obtain the following result whose proof is similar to the one of
Theorem 2.17.

######  Theorem 2.20.

Let $X$ be a an object of $\mathbf{ACAAssoc}$ and let
$U\colon\mathbf{ACAAssoc}\rightarrow\mathbf{PAlg}$ denote the forgetful
functor.

  1. (1)

There exists a natural isomorphism

$\operatorname{SplExt}(-,X)\cong\operatorname{Hom}_{\mathbf{PAlg}}(U(-),[X]),$

where
$\operatorname{SplExt}(-,X)=\operatorname{SplExt}_{\mathbf{ACAAssoc}}(-,X)$;

  2. (2)

if $[X]$ is an anti-commutative anti-associative algebra, then the functor
$\operatorname{SplExt}(-,X)$ is representable and $[X]$ is the actor of $X$; ∎

### Two-step nilpotent anti-commutative algebras

We conclude this section by studying the representability of actions of the
variety $\mathbf{Nil}_{2}(\mathbf{ACom})$. An important example of a two-step
nilpotent anti-commutative algebra is the $(2n+1)$-dimensional _Heisenberg
algebra_ $\mathfrak{h}_{2n+1}$, that is the algebra with basis

$\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},h\\}$

and non-trivial products $e_{i}f_{j}=-f_{j}e_{i}=\delta_{ij}h$, for all $i$,
$j=1,\ldots,n$, where $\delta_{ij}$ is the _Kronecker delta_.

An analysis, similar to the case of two-step nilpotent commutative algebras
can be made, so we simply state the following theorem:

######  Theorem 2.21.

  1. (1)

Let $B$ and $X$ be two-step nilpotent anti-commutative algebras. The
isomorphism classes of split extensions of $B$ by $X$ are in bijection with
the algebra homomorphisms

$B\rightarrow[X]_{2}\colon b\mapsto b\ast-$

where $[X]_{2}$ is defined as in the commutative case, which satisfy the
condition

$b\ast(b^{\prime}\ast x)=0,\quad\forall b,b^{\prime}\in B,\;\forall x\in X.$
(2.3)

  2. (2)

The variety $\mathbf{Nil}_{2}(\mathbf{ACom})$ is weakly action representable.
A weak representation of
$\operatorname{SplExt}(-,X)=\operatorname{SplExt}_{\mathbf{Nil}_{2}(\mathbf{ACom})}$
is given by

$\tau\colon\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}_{\mathbf{Nil}_{2}(\mathbf{ACom})}(-,[X]_{2}),$

where $\tau_{B}$ is the injection which associates with any split extension of
$B$ by $X$, the corresponding homomorphism $B\rightarrow[X]_{2}\colon b\mapsto
b\ast-$ as in (1).

  3. (3)

A homomorphism $B\rightarrow[X]_{2}$ is an acting morphism if and only if it
satisfies Equation 2.3. ∎

Again, if $B=\mathbb{F}\\{b,b^{\prime}\\}$ is the abelian two-dimensional
algebra and $X=\mathbb{F}$ is the abelian one-dimensional algebra, the linear
map $\varphi\colon B\rightarrow[X]_{2}=\operatorname{End}(X)$, defined by
$\varphi(b)=\varphi(b^{\prime})=1_{X}$ is an example of a morphism in
$\mathbf{Nil}_{2}(\mathbf{ACom})$ which is not an acting morphism.

##  3\. Representability of actions of non-associative algebras

We want to extend the results obtained in the previous section by studying the
(weak) representability of actions of a general variety of non-associative
algebras over a field $\mathbb{F}$. Again, we assume that $\mathscr{V}$ is an
action accessible, operadic variety of non-associative algebras over
$\mathbb{F}$. Thus $\mathscr{V}$ satisfies a set of multilinear identities

$\Phi_{k,i}(x_{1},\ldots,x_{k})=0,\quad i=1,\ldots,n,$

where $k$ is the degree of the polynomial $\Phi_{k,i}$. We fix $\lambda_{1}$,
…, $\lambda_{8}$, $\mu_{1}$, …, $\mu_{8}\in\mathbb{F}$ which determine a
choice of $\lambda/\mu$ rules, i.e.

$\displaystyle x(yz)=\lambda_{1}(xy)z$
$\displaystyle+\lambda_{2}(yx)z+\lambda_{3}z(xy)+\lambda_{4}z(yx)$
$\displaystyle+\lambda_{5}(xz)y+\lambda_{6}(zx)y+\lambda_{7}y(xz)+\lambda_{8}y(zx)$
and $\displaystyle(yz)x=\mu_{1}(xy)z$
$\displaystyle+\mu_{2}(yx)z+\mu_{3}z(xy)+\mu_{4}z(yx)$
$\displaystyle+\mu_{5}(xz)y+\mu_{6}(zx)y+\mu_{7}y(xz)+\mu_{8}y(zx)$

which are identities in $\mathscr{V}$. Note that these are not unique, but
fixed for our purposes.

For any object $X$ of $\mathscr{V}$, we want to define a vector space
$\mathscr{E}(X)$ such that

$\operatorname{Inn}(X)\leq\mathscr{E}(X)\leq\operatorname{End}(X)^{2},$

where $\operatorname{Inn}(X)=\\{(L_{x},R_{x})\mid x\in X\\}$ is the vector
space of left and right multiplications of $X$, and we want to endow it with a
bilinear partial operation

$\langle-,-\rangle\colon\Omega\subseteq X\times X\rightarrow X,$

such that we can associate in a natural way a homomorphism of partial algebras
$B\rightarrow\mathscr{E}(X)$, with every split extension of $B$ by $X$ in
$\mathscr{V}$. To do this, we describe derived actions in $\mathscr{V}$ in a
similar fashion as in the previous section.

######  Proposition 3.1.

Let $X$ and $B$ be two algebras in $\mathscr{V}$. Given a pair of bilinear
maps

$l\colon B\times X\rightarrow X,\qquad r\colon X\times B\rightarrow X,$

we construct $(B\oplus X,\cdot)$ as in (1.3). Then $(B\oplus X,\cdot)$ is an
object of $\mathscr{V}$ if and only if

$\Phi_{k,i}(\alpha_{1},\ldots,\alpha_{k})=0,\quad\forall i=1,\ldots,n,$

where at least one of the $\alpha_{1}$, …, $\alpha_{k}$ is an element of of
the form $(0,x)$, with $x\in X$, and the others are of the form $(b,0)$, with
$b\in B$. The resulting algebra is the _semi-direct product_ of $B$ and $X$,
denoted by $B\ltimes X$. ∎

Using the same notation of Remark 1.8, we obtain the following:

######  Corollary 3.2.

When every identity of $\mathscr{V}$ can be deduced from the $\lambda/\mu$
rules, $(B\oplus X,\cdot)$ is an object of $\mathscr{V}$ if and only if

  1. (1)

$b\ast(xx^{\prime})=\lambda_{1}(b\ast
x)x^{\prime}+\cdots+\lambda_{8}x(x^{\prime}\ast b)$;

  2. (2)

$(xx^{\prime})\ast b=\mu_{1}(b\ast x)x^{\prime}+\cdots+\mu_{8}x(x^{\prime}\ast
b)$;

  3. (3)

$x(x^{\prime}\ast b)=\lambda_{1}(xx^{\prime})\ast
b+\cdots+\lambda_{8}x^{\prime}(b\ast x)$;

  4. (4)

$(x^{\prime}\ast b)x=\mu_{1}(xx^{\prime})\ast b+\cdots+\mu_{8}x^{\prime}(b\ast
x)$;

  5. (5)

$x(b\ast x^{\prime})=\lambda_{1}(x\ast
b)x^{\prime}+\cdots+\lambda_{8}b\ast(x^{\prime}x)$;

  6. (6)

$(b\ast x^{\prime})x=\mu_{1}(x\ast
b)x^{\prime}+\cdots+\mu_{8}b\ast(x^{\prime}x)$;

  7. (7)

$x\ast(bb^{\prime})=\lambda_{1}(x\ast b)\ast
b^{\prime}+\cdots+\lambda_{8}b\ast(b^{\prime}\ast x)$;

  8. (8)

$(bb^{\prime})\ast x=\mu_{1}(x\ast b)\ast
b^{\prime}+\cdots+\mu_{8}b\ast(b^{\prime}\ast x)$;

  9. (9)

$b\ast(b^{\prime}\ast x)=\lambda_{1}(bb^{\prime})\ast
x+\cdots+\lambda_{8}b^{\prime}\ast(x\ast b)$;

  10. (10)

$(b^{\prime}\ast x)\ast b=\mu_{1}(bb^{\prime})\ast
x+\cdots+\mu_{8}b^{\prime}\ast(x\ast b)$;

  11. (11)

$b\ast(x\ast b^{\prime})=\lambda_{1}(b\ast x)\ast
b^{\prime}+\cdots+\lambda_{8}x\ast(b^{\prime}b)$;

  12. (12)

$(x\ast b)\ast b^{\prime}=\mu_{1}(b\ast x)\ast
b^{\prime}+\cdots+\mu_{8}x\ast(b^{\prime}b)$,

for all $b$, $b^{\prime}\in B$ and $x$, $x^{\prime}\in X$. ∎

######  Definition 3.3.

For every object $X$ of $\mathscr{V}$, we define $\mathscr{E}(X)$ as the
subspace of all pairs $(f\ast-,-\ast f)\in\operatorname{End}(X)^{2}$
satisfying

$\Phi_{k,i}(\alpha_{1},\ldots,\alpha_{k})=0,\quad\forall i=1,\ldots,n,$

for each choice of $\alpha_{j}=f$ and $\alpha_{t}\in X$, where $t\neq
j\in\\{1,\ldots,k\\}$ and $fx\coloneq f\ast x$, $xf\coloneq x\ast f$. We endow
it with the bilinear map
$\langle-,-\rangle\colon\mathscr{E}(X)\times\mathscr{E}(X)\rightarrow\operatorname{End}(X)^{2}$

$\langle(f\ast-,-\ast f),(g\ast-,-\ast g)\rangle=(h\ast-,-\ast h),$

where

$\displaystyle x\ast h=\lambda_{1}(x\ast f)\ast g$
$\displaystyle+\lambda_{2}(f\ast x)\ast g+\lambda_{3}g\ast(x\ast
f)+\lambda_{4}g\ast(f\ast x)$ $\displaystyle+\lambda_{5}(x\ast g)\ast
f+\lambda_{6}(g\ast x)\ast f+\lambda_{7}f\ast(x\ast g)+\lambda_{8}f\ast(g\ast
x)$ and $\displaystyle h\ast x=\mu_{1}(x\ast f)\ast g$
$\displaystyle+\mu_{2}(f\ast x)\ast g+\mu_{3}g\ast(x\ast f)+\mu_{4}g\ast(f\ast
x)$ $\displaystyle+\mu_{5}(x\ast g)\ast f+\mu_{6}(g\ast x)\ast
f+\mu_{7}f\ast(x\ast g)+\mu_{8}f\ast(g\ast x).$

When every identity of $\mathscr{V}$ is a consequence of the $\lambda/\mu$
rules, $\mathscr{E}(X)$ becomes the subspace of all pairs $(f\ast-,-\ast
f)\in\operatorname{End}(X)^{2}$ satisfying

  1. (1)

$f\ast(xx^{\prime})=\lambda_{1}(f\ast
x)x^{\prime}+\cdots+\lambda_{8}x(x^{\prime}\ast f)$;

  2. (2)

$(xx^{\prime})\ast f=\mu_{1}(f\ast x)x^{\prime}+\cdots+\mu_{8}x(x^{\prime}\ast
f)$;

  3. (3)

$x(x^{\prime}\ast f)=\lambda_{1}(xx^{\prime})\ast
f+\cdots+\lambda_{8}x^{\prime}(f\ast x)$;

  4. (4)

$(x^{\prime}\ast f)x=\mu_{1}(xx^{\prime})\ast f+\cdots+\mu_{8}x^{\prime}(f\ast
x)$;

  5. (5)

$x(f\ast x^{\prime})=\lambda_{1}(x\ast
f)x^{\prime}+\cdots+\lambda_{8}f\ast(x^{\prime}x)$;

  6. (6)

$(f\ast x^{\prime})x=\mu_{1}(x\ast
f)x^{\prime}+\cdots+\mu_{8}f\ast(x^{\prime}x)$,

for every $x$, $x^{\prime}\in X$.

Note that the choice of $\lambda/\mu$ rules does not affect to the definition
of the underlying vector space of $\mathscr{E}(X)$, but it does play an
important role in the bilinear map $\langle-,-\rangle$. In general, the vector
space $\mathscr{E}(X)$ endowed with the bilinear map $\langle-,-\rangle$ is
not an object of $\mathscr{V}$. It may happen that $\langle-,-\rangle$ does
not even define a bilinear operation on $\mathscr{E}(X)$, i.e. there exist
$(f\ast-,-\ast f)$,$(g\ast-,-\ast g)\in\mathscr{E}(X)$ such that

$\langle(f\ast-,-\ast f),(g\ast-,-\ast g)\rangle\not\in\mathscr{E}(X)$

or that $(\mathscr{E}(X),\langle-,-\rangle)$ is a non-associative algebra
which does not satisfy some identity of $\mathscr{V}$.

######  Example 3.4.

We may check that, if $\mathscr{V}=\mathbf{Assoc}$, then
$\mathscr{E}(X)\cong\operatorname{Bim}(X)$ as vector spaces. Moreover, with
the standard choice of $\lambda/\mu$ rules $\lambda_{1}=\mu_{8}=1$ and the
rest equal to zero, it is also an isomorphism of associative algebras.

######  Example 3.5.

Let $\mathscr{V}=\mathbf{Leib}$, it is easy to see that
$\mathscr{E}(X)\cong\operatorname{Bider}(X)$ as vector spaces. Choosing the
$\lambda/\mu$ rules as

$\displaystyle x(yz)$ $\displaystyle=(xy)z-(xz)y,$ $\displaystyle(yz)x$
$\displaystyle=(yx)z-y(xz),$

we get the standard multiplication defined in $\operatorname{Bider}(X)$ as in
[23], that defines a weak actor in $\mathbf{Leib}$. On the other hand,
choosing the $\lambda/\mu$ rules as

$\displaystyle x(yz)$ $\displaystyle=(xy)z-(xz)y,$ $\displaystyle(yz)x$
$\displaystyle=(yx)z+y(zx),$

we get the non-associative algebra structure defined in [8, Definition 5.2],
which, in general, is not a Leibniz algebra.

######  Example 3.6.

If $\mathscr{V}=\mathbf{Nil}_{k}(\mathbf{Assoc})$, with $k\geq 3$, then

$\mathscr{E}(X)=\\{(f\ast-,-\ast f)\in\operatorname{Bim}(X)\mid
f\ast(x_{1}\cdots x_{k})=(x_{1}\cdots x_{k})\ast f=0\\}.$

With the same choice of $\lambda/\mu$ rules as in Example 3.4, the bilinear
operation $\langle-,-\rangle$ becomes

$\langle(f\ast-,-\ast f),(g\ast-,-\ast g)\rangle=(f\ast(g\ast-),(-\ast f)\ast
g)$

which makes $\mathscr{E}(X)$ an associative algebra, but not a $k$-step
nilpotent algebra. For instance, let $X$ be the abelian one-dimensional
algebra, then

$\mathscr{E}(X)=\operatorname{End}(X)\times\operatorname{End}(X)^{\operatorname{op}}\cong\mathbb{F}^{2}$

which is not nilpotent. Indeed, every linear endomorphism of $X$ is of the
form $\varphi_{\alpha}\colon x\mapsto\alpha x$, for some $\alpha\in\mathbb{F}$
and

$\langle(\varphi_{\alpha},\varphi_{\beta}),(\varphi_{\alpha^{\prime}},\varphi_{\beta^{\prime}})\rangle=(\varphi_{\alpha}\circ\varphi_{\alpha^{\prime}},\varphi_{\beta^{\prime}}\circ\varphi_{\beta})=(\varphi_{\alpha\alpha^{\prime}},\varphi_{\beta\beta^{\prime}}).$

######  Example 3.7.

If $\mathscr{V}=\mathbf{Alt}$ is the variety of alternative algebras over a
field $\mathbb{F}$ with $\operatorname{char}(\mathbb{F})\neq 2$, then
$\mathscr{E}(X)$ consists of the pairs $(f\ast-,-\ast
f)\in\operatorname{End}(X)^{2}$ satisfying

$f\ast(xy)=(x\ast f)y+(f\ast x)y-x(f\ast y),$ $(xy)\ast f=x(f\ast y)+x(y\ast
f)-(x\ast f)y,$ $x(y\ast f)=(yx)\ast f+(xy)\ast f-y(x\ast f)$

and

$(f\ast x)y=f\ast(yx)+f\ast(xy)-(f\ast y)x$

for any $x,y\in X$, and the bilinear map

$\langle(f\ast-,-\ast f),(g\ast-,-\ast g)\rangle=(h\ast-,-\ast h)$

is given by

$h\ast x=-(f\ast x)\ast g+f\ast(g\ast x)+f\ast(x\ast g)$

and

$x\ast h=(x\ast f)\ast g+(f\ast x)\ast g-f\ast(x\ast g).$

One can check that $\langle-,-\rangle$ does not define an algebra structure.
Nevertheless, it is possible to find examples where $\mathscr{E}(X)$ is an
alternative algebra.

For instance, if $X$ is a _unitary_ , or _unital_ , alternative algebra (i.e.
there exists an element $e\in X$ such that $xe=ex=x$, for any $x\in X$), such
as the algebra of octonions $\mathbb{O}$, then the elements of
$\mathscr{E}(X)$ satisfy the following set of equations

$\displaystyle f\ast x=$ $\displaystyle(x\ast f)e+(f\ast x)e-x(f\ast e),$
$\displaystyle x\ast f=$ $\displaystyle e(f\ast x)+e(x\ast f)-(e\ast f)x,$
$\displaystyle x\ast f=$ $\displaystyle x\ast f+x\ast f-x(e\ast f),$
$\displaystyle f\ast x=$ $\displaystyle f\ast x+f\ast x-(f\ast e)x,$

for any $x\in X$. Thus, if $\alpha\coloneqq f\ast e$ and $\beta\coloneqq e\ast
f$, one has

$f\ast x=\alpha x=\beta x,\qquad x\ast f=x\alpha=x\beta$

and, for $x=e$, one obtains $\alpha=\beta$. In other words, an element of
$\mathscr{E}(X)$ is uniquely determined by an element $\alpha=f\ast e=e\ast f$
of $X$, i.e.

$\mathscr{E}(X)\cong\\{(\alpha,\alpha)\mid\alpha\in X\\}\cong X$

is an object of $\mathbf{Alt}$.

######  Remark 3.8.

The same result can be obtained for unitary algebras in the variety
$\mathbf{Assoc}$. In fact, let $X$ be a unitary associative algebra and let
$(f\ast-,-\ast f)\in\operatorname{Bim}(X)$. Thus

$f\ast x=f\ast(ex)=\alpha x,$ $x\ast f=(xe)\ast f=x\beta$

and

$x\alpha=(x\ast f)e=x\beta,$

where $\alpha\coloneqq f\ast e$ and $\beta\coloneqq e\ast f$. For $x=e$, we
obtain $\alpha=\beta$ and

$\operatorname{Bim}(X)\cong\\{(\alpha,\alpha)\mid\alpha\in X\\}\cong X.$

Since unitary algebras are perfect and have trivial center, from [8] we have a
natural isomorphism

$\operatorname{SplExt}(-,X)\cong\operatorname{Hom}_{\mathbf{Assoc}}(-,X)$

for any unitary associative algebra $X$, i.e. $X$ is its own actor.

The construction of $\mathscr{E}(X)$ gives rise to an alternative
characterisation of the split extensions in $\mathscr{V}$. In fact, a split
extension of $B$ by $X$ in $\mathscr{V}$ is the same as a linear map

$B\rightarrow\mathscr{E}(X)\colon b\mapsto(b\ast-,-\ast b),$

such that $((bb^{\prime})\ast-,-\ast(bb^{\prime}))=\langle(b\ast-,-\ast
b),(b^{\prime}\ast-,-\ast b^{\prime})\rangle$ and

$\Phi_{k,i}(\alpha_{1},\ldots,\alpha_{k})=0,\quad i=1,\ldots,n,$

where $\alpha_{1}$, …, $\alpha_{k}$ are as in Proposition 3.1.

We remark also that the bilinear map

$\langle-,-\rangle\colon\mathscr{E}(X)\times\mathscr{E}(X)\rightarrow\operatorname{End}(X)^{2}$

defines a partial operation
$\langle-,-\rangle\colon\Omega\rightarrow\mathscr{E}(X)$, where $\Omega$ is
the preimage

$\langle-,-\rangle^{-1}(\mathscr{E}(X))$

of the inclusion $\mathscr{E}(X)\hookrightarrow\operatorname{End}(X)^{2}$.

Now we are ready to announce and prove our main result about the weak
representability of actions of non-associative algebras.

######  Theorem 3.9.

Let $\mathscr{V}$ be an action accessible, operadic variety of non-associative
algebras over a field $\mathbb{F}$ and let
$U\colon\mathscr{V}\rightarrow\mathbf{PAlg}$ denote the forgetful functor.

  1. (1)

Let $X$ be an object of $\mathscr{V}$. There exists a monomorphism of functors

$\tau\colon\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}_{\mathbf{PAlg}}(U(-),\mathscr{E}(X)),$

where $\operatorname{SplExt}(-,X)=\operatorname{SplExt}_{\mathscr{V}}(-,X)$
and, for every object $B$ of $\mathscr{V}$, $\tau_{B}$ is the injection which
sends an element of $\operatorname{SplExt}(B,X)$ to the corresponding partial
algebra homomorphism

$B\rightarrow\mathscr{E}(X)\colon b\mapsto(b\ast-,-\ast b).$

  2. (2)

Let $B$, $X$ be objects of $\mathscr{V}$. The homomorphism of partial algebras

$B\rightarrow\mathscr{E}(X)\colon b\mapsto(b\ast-,-\ast b)$

belongs to $\operatorname{Im}(\tau_{B})$ if and only if
$\Phi_{k,i}(\alpha_{1},\ldots,\alpha_{k})=0$, as in Proposition 3.1.

  3. (3)

If $(\mathscr{E}(X),\langle-,-\rangle)$ is an object of $\mathscr{V}$, then
$(\mathscr{E}(X),\tau)$ becomes a weak representation of
$\operatorname{SplExt}(-,X)$.

  4. (4)

If the diagram in $\mathscr{V}$ given by those subalgebras of $\mathscr{E}(X)$
which occur as a codomain of a morphism
$(B\rightarrow\mathscr{E}(X))\in\operatorname{Im}(\tau_{B})$ admits an amalgam
in $\mathscr{V}$, then the colimit of that diagram determines an (initial)
weak representation of $\operatorname{SplExt}(-,X)$.

  5. (5)

If $\mathscr{V}$ is a variety of commutative or anti-commutative algebras,
then $\mathscr{E}(X)$ is isomorphic to the partial algebra

$\\{f\in\operatorname{End}(X)\mid\Phi_{k,i}(f,x_{2},\ldots,x_{k})=0,\;\forall
x_{2},\ldots,x_{k}\in X\\}$

endowed with the bilinear partial operation $\langle f,g\rangle=\alpha(f\circ
g)+\beta(g\circ f)$, where $\alpha$, $\beta\in\mathbb{F}$ are given by the
$\lambda/\mu$ rules.

Because of these results, we can give the following definitions.

######  Definition 3.10.

Let $X$ be an object of an action accessible, operadic variety of non-
associative algebras $\mathscr{V}$ with a choice of $\lambda/\mu$ rules. The
partial algebra $\mathscr{E}(X)$ is called _external weak actor_ of $X$. The
pair $(\mathscr{E}(X),\tau)$ is called _external weak representation_ of the
functor $\operatorname{SplExt}(-,X)$. When $\tau$ is a natural isomorphism, we
say that $\mathscr{E}(X)$ is an _external actor_ of $X$.

###### Proof.

(1) The collection $\\{\tau_{B}\\}_{B}$ gives rise to a natural transformation
since, for every algebra homomorphism $f\colon B^{\prime}\rightarrow B$, the
diagram in $\mathbf{Set}$

${\operatorname{SplExt}(B,X)}$${\operatorname{Hom}(U(B),\mathscr{E}(X))}$${\operatorname{SplExt}(B^{\prime},X)}$${\operatorname{Hom}(U(B^{\prime}),\mathscr{E}(X))}$$\scriptstyle{\tau_{B}}$$\scriptstyle{\operatorname{SplExt}(f,X)}$$\scriptstyle{\operatorname{Hom}(U(f),\mathscr{E}(X))}$$\scriptstyle{\tau_{B^{\prime}}}$

where $\operatorname{Hom}(U(-),-)=\operatorname{Hom}_{\mathbf{PAlg}}(U(-),-)$,
is commutative. Moreover, for every object $B$ of $\mathscr{V}$, the map
$\tau_{B}$ is an injection, since every element of
$\operatorname{SplExt}(B,X)$ is uniquely determined by the corresponding
derived action of $B$ on $X$, i.e. by the pair of bilinear maps

$l\colon B\times X\rightarrow X,\qquad r\colon X\times B\rightarrow X$

defined as in Definition 1.6. Thus $\tau$ is a monomorphism of functors.

(2) Let $B$, $X$ be objects of $\mathscr{V}$. A homomorphism of partial
algebras $B\rightarrow\mathscr{E}(X)$ belongs to $\operatorname{Im}(\tau_{B})$
if and only if it defines a split extension of $B$ by $X$ in $\mathscr{V}$.
This is equivalent to saying that

$\Phi_{k,i}(\alpha_{1},\ldots,\alpha_{k})=0,\quad\forall i=1,\ldots,n,$

where $\alpha_{1},\ldots,\alpha_{k}$ are as in Proposition 3.1

(3) If $(\mathscr{E}(X),\langle-,-\rangle)$ is an object of $\mathscr{V}$,
then we have a monomorphism of functors

$\tau\colon\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}_{\mathscr{V}}(-,\mathscr{E}(X)),$

and $(\mathscr{E}(X),\tau)$ is a weak representation of
$\operatorname{SplExt}(-,X)$.

(4) We may copy the “if” part of the proof of Proposition 2.8, replacing the
subalgebras of $\operatorname{M}(X)$ in $\mathbf{CAssoc}$ with those
subalgebras of $\mathscr{E}(X)$ in $\mathscr{V}$ which occur as codomain of a
morphism $(B\rightarrow\mathscr{E}(X))\in\operatorname{Im}(\tau_{B})$. As in
Remark 2.13, by its construction as a colimit, the weak representation thus
obtained is automatically an initial weak representation [17].

(5) If $\mathscr{V}$ is a variety of commutative (resp. anti-commutative)
algebras, then for every object $X$ of $\mathscr{V}$, $\mathscr{E}(X)$
consists of pairs of the form $(f\ast-,-\ast f)$ with $x\ast f=f\ast x$ (resp.
$x\ast f=-f\ast x$), for every $x\in X$. Thus, an explicit isomorphism

$\\{f\in\operatorname{End}(X)\mid\Phi_{k,i}(f,x_{2},\ldots,x_{k})=0\\}\rightarrow\mathscr{E}(X)$

is given by $f\mapsto(f,\pm f)$. ∎

######  Example 3.11.

We may check that, with the _obvious_ choices of the $\lambda/\mu$ rules,

  1. (1)

if $\mathscr{V}=\mathbf{AbAlg}$, then $\mathscr{E}(X)=0$ is the actor of $X$;

  2. (2)

if $\mathscr{V}=\mathbf{CAssoc}$, then
$\mathscr{E}(X)\cong\operatorname{M}(X)$ is an external actor of $X$ (see
Lemma 2.4);

  3. (3)

if $\mathscr{V}=\mathbf{JJord}$, then as observed in Theorem 2.17, the
external actor $\mathscr{E}(X)$ is isomorphic to the partial algebra
$\operatorname{ADer}(X)$ of anti-derivations of $X$;

  4. (4)

if $\mathscr{V}=\mathbf{Lie}$, then $\mathscr{E}(X)\cong\operatorname{Der}(X)$
is the actor of $X$;

  5. (5)

if $\mathscr{V}=\mathbf{ACAAssoc}$, then as observed in Theorem 2.20, the
external actor $\mathscr{E}(X)$ is isomorphic to the partial algebra $[X]$;

  6. (6)

if $\mathscr{V}=\mathbf{Nil}_{2}(\mathbf{Com})$ or
$\mathscr{V}=\mathbf{Nil}_{2}(\mathbf{ACom})$, then
$\mathscr{E}(X)\cong[X]_{2}$ is a weak actor of $X$;

  7. (7)

if $\mathscr{V}=\mathbf{Nil}_{2}(\mathbf{Alg})$, then $\mathscr{E}(X)$ is an
abelian algebra and and it is a weak actor of $X$;

  8. (8)

if $\mathscr{V}=\mathbf{Alt}$ over a field $\mathbb{F}$ with
$\operatorname{char}(\mathbb{F})\neq 2$ and $X$ is a unitary alternative
algebra, then $\mathscr{E}(X)\cong X$ is an alternative algebra and we have a
natural isomorphism

$\operatorname{SplExt}(-,X)\cong\operatorname{Hom}_{\mathbf{Alt}}(-,X)$

i.e. $X$ is the actor of itself. In particular, the algebra of octonions
$\mathbb{O}$ has representable actions.

######  Remark 3.12.

The construction of the vector space $\mathscr{E}(X)$ can be done also in a
variety of non-associative algebras $\mathscr{V}$ which is not action
accessible. However, there is no canonical way to endow $\mathscr{E}(X)$ with
a bilinear map $\langle-,-\rangle$ as in Definition 3.3 so we only have a
monomorphism of functors

$\tau\colon\operatorname{SplExt}(-,X)\rightarrow\operatorname{Hom}_{\mathbf{Vec}}(U(-),\mathscr{E}(X)),$

where $U\colon\mathscr{V}\rightarrow\mathbf{Vec}$ denotes the forgetful
functor.

######  Remark 3.13.

As described in [10, Section 3], for every Orzech category of interest
$\mathscr{C}$ and for every object $X$ of $\mathscr{C}$, it is possible to
define a monomorphism of functors

$\mu\colon\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}_{\mathscr{C}^{\prime}}(V(-),\operatorname{USGA}(X)),$

where $\mathscr{C}^{\prime}$ is a category which contains $\mathscr{C}$ as a
full subcategory,$\operatorname{USGA}(X)$ is an object of
$\mathscr{C}^{\prime}$ called the _universal strict general actor_ of $X$ [8]
and $V\colon\mathscr{C}\rightarrow\mathscr{C}^{\prime}$ denotes the forgetful
functor. We further recall that $\operatorname{USGA}(X)$ is unique up to
isomorphism, once the presentation of the Orzech category of interest
$\mathscr{C}$ is fixed.

For a variety of non-associative algebras $\mathscr{V}$, a presentation is
given by a choice of constants $\lambda_{1}$, …, $\lambda_{8}$, $\mu_{1}$, …,
$\mu_{8}\in\mathbb{F}$ which determine the $\lambda/\mu$ rules. In this case,
it turns out that $\mathscr{V}^{\prime}=\mathbf{Alg}$. Thus we have
monomorphism of functors

$\mu\colon\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}_{\mathbf{Alg}}(V(-),\operatorname{USGA}(X))$

and, by Theorem 3.9, another monomorphism of functors

$\tau\colon\operatorname{SplExt}(-,X)\rightarrowtail\operatorname{Hom}_{\mathbf{PAlg}}(U(-),\mathscr{E}(X)).$

As explained at the beginning of [8, Section 4], $\operatorname{USGA}(X)$ is
the algebraic closure of the external weak actor $\mathscr{E}(X)$ with respect
to the bilinear partial operation $\langle-,-\rangle$. When
$\langle-,-\rangle$ is well defined on $\mathscr{E}(X)\times\mathscr{E}(X)$,
then $\operatorname{USGA}(X)=\mathscr{E}(X)$ and $\mu=\tau$.

However, it is often more convenient to work with the external weak actor
$\mathscr{E}(X)$, since it is easier to construct than the universal strict
general actor $\operatorname{USGA}(X)$. In fact, in the next section we shall
present the construction of $\mathscr{E}(X)$ in different varieties of non-
associative algebras.

##  4\. The quadratic case

In this section we introduce a systematic approach to finding the explicit
structure of $\mathscr{E}(X)$ in the setting of operadic, quadratic varieties
of algebras. Here we shall denote an element $(f\ast-,-\ast f)$ of
$\mathscr{E}(X)$ by the symbol $f$; this means that $fx\coloneq f\ast x$ and
$xf\coloneq x\ast f$.

Let $\mathscr{V}$ be a action accessible, operadic, quadratic variety of non-
associative algebras with no identities of degree $2$. Let us consider the
free non-associative algebra generated by the symbols $f$, $x$ and $y$, and
let us focus on its multilinear component of degree $3$. There are $12$
possible monomials which we order as follows:

$\displaystyle f(xy)$ $\displaystyle>f(yx)>(xy)f>(yx)f>(fy)x>(fx)y$
$\displaystyle>(yf)x>(xf)y>x(fy)>y(fx)>x(yf)>y(xf).$

Permuting the variables determines an action of the symmetric group
$\mathbb{S}_{3}$ on this space. For a given variety of algebras $\mathscr{V}$,
we can write the orbit under the $\mathbb{S}_{3}$-action of its defining
equations in matrix form, where each row corresponds to an equation and each
column corresponds to a monomial, ordered as above. Let us denote this matrix
by $M_{3}$, and its reduced row echelon form by $RM_{3}$. Action accessibility
implies the following:

######  Lemma 4.1.

The rank of $M_{3}$ is at least $4$. Moreover, the $4\times 4$ minor located
on the top left of $RM_{3}$ is the identity matrix. ∎

The vector space $\mathscr{E}(X)$ will be the subspace of
$\operatorname{End}(X)^{2}$ formed by the pairs that satisfy the identities
coming from $RM_{3}$.

Our task now is to endow this vector space with a partial multiplication,
induced by action accessibility, and to provide strategies to check

  1. (1)

when this multiplication is total;

  2. (2)

when it induces a $\mathscr{V}$-algebra structure on $\mathscr{E}(X)$.

Let us rename the tags on the columns of $M_{3}$ by the following rule:
$f\mapsto x$, $x\mapsto f$ and $y\mapsto g$. Then, the third and first columns
of $RM_{3}$ will give us equations of the form

$\displaystyle(fg)x=\lambda_{1}(fg)x$
$\displaystyle+\lambda_{2}(fx)g+\lambda_{3}(gf)x+\lambda_{4}(xf)g$
$\displaystyle+\lambda_{5}x(fg)+\lambda_{6}y(fx)+\lambda_{7}x(gf)+\lambda_{8}g(xf)$
and $\displaystyle x(fg)=\mu_{1}(fg)x$
$\displaystyle+\mu_{2}(fx)g+\mu_{3}(gf)x+\mu_{4}(xf)g$
$\displaystyle+\mu_{5}x(fg)+\mu_{6}g(fx)+\mu_{7}x(gf)+\mu_{8}g(xf).$

At a first glance, these rules seem to yield a way of multiplying two elements
$f$ and $g$ belonging to $\mathscr{E}(X)$. However, this choice might not be
unique. If the rank of $M_{3}$ is strictly larger than $4$, the lower rows
will have zeroes in the first four positions, so adding any linear combination
of them will produce a new bracket in $\mathscr{E}(X)$. Let us exemplify this
with a concrete variety:

######  Example 4.2.

The most common presentation of the variety of right Leibniz algebras is given
by the identity $(xy)z-(xz)y-x(yz)=0$. Then, $M_{3}$ will be the matrix

$\blockarray{cccccccccccc}f(xy)&f(yx)(xy)f(yx)f(fy)x(fx)y(yf)x(xf)yx(fy)y(fx)x(yf)y(xf)\\\
\block{(cccccccccccc)}-1000-11000000\\\ 0-1001-1000000\\\ 00-100001-1000\\\
0010000-100-10\\\ 000-100100-100\\\ 000100-10000-1\\\ $

while its reduced row echelon form is

$\blockarray{cccccccccccc}f(xy)&f(yx)(xy)f(yx)f(fy)x(fx)y(yf)x(xf)yx(fy)y(fx)x(yf)y(xf)\\\
\block{(cccccccccccc)}10001-1000000\\\ 0100-11000000\\\ 0010000-100-10\\\
000100-10000-1\\\ 000000001010\\\ 000000000101\\\ $

Removing the rows in odd position—which we are entitled to, thanks to the
obvious symmetry—we obtain that $\mathscr{E}(X)$ is formed by the elements of
$\operatorname{End}(X)^{2}$ satisfying the following identities:

$\displaystyle f(xy)$ $\displaystyle=(fx)y-(fy)x$ (4.1) $\displaystyle(xy)f$
$\displaystyle=(xf)y+x(yf)$ $\displaystyle x(fy)$ $\displaystyle=x(yf)$

These are exactly the identities satisfied by biderivations. With the change
of tag in the columns described before, we obtain

$\blockarray{cccccccccccc}x(fg)&x(gf)(fg)x(gf)x(xg)f(xf)g(gx)f(fx)gf(xg)g(xf)f(gx)g(fx)\\\
\block{(cccccccccccc)}10001-1000000\\\ 0100-11000000\\\ 0010000-100-10\\\
000100-10000-1\\\ 000000001010\\\ 000000000101\\\ $

Therefore, the multiplication

$\displaystyle(fg)x=(fx)g+f(gx)+\alpha_{1}\big{(}f(xg)+f(gx)\big{)}+\alpha_{2}\big{(}g(xf)+g(fx)\big{)}$
(4.2) $\displaystyle
x(fg)=(xf)g-(xg)f+\beta_{1}\big{(}f(xg)+f(gx)\big{)}+\beta_{2}\big{(}g(xf)+g(fx)\big{)}$

induces a partial algebra structure on $\mathscr{E}(X)$, for any choice of
$\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in\mathbb{F}$.

Now that we have a partial algebra structure induced on a general
$\mathscr{E}(X)$, the next step is to verify when it is total. To do so, we
have to focus on a partial subset of the set of consequences of degree $4$.
Let us consider the $120$-dimensional space formed by the multilinear
monomials of degree $4$ in the free non-associative algebra generated by the
symbols $f,g,x,y$. To gather all the consequences of the identities in degree
$3$, we have three different ways of operating. Let us take any identity from
$RM_{3}$. The first way is to multiply it from the right or from the left by
$g$. The second way, is to substitute $x$ by $(gx)$ or $(xg)$. Finally, we can
substitute $y$ by $(gy)$ or by $(yg)$. Doing all these substitutions together
with the permutations of $f$ and $g$, we obtain all the consequences. Note
that in none of these identities the terms $(fg)$ or $(gf)$ will appear.

Now we need to check if the defining bracket satisfies the identities of
$\mathscr{E}(X)$. To do so, we take again the identities from $RM_{3}$,
substitute $f$ by $(fg)$ and expand it by the already defined product. The
bracket will be closed if and only if these new obtained equations are linear
combination of the previously obtained consequences.

To conclude, we shall check when the bracket satisfies the identities of the
variety. This can be done just by directly substituting elements of
$\mathscr{E}(X)$ in the defining equations of the variety. After applying them
to a generic element $x$, once on the left and once on the right, it is a
matter of substituting the bracket on $\mathscr{E}(X)$ when necessary.

######  Example 4.3.

Continuing with the Leibniz algebras example 4.2, applying the procedure
described before to the first equation in (4.1) yields:

$\displaystyle g(f(xy))$ $\displaystyle=g((fx)y)-g((fy)x),$
$\displaystyle(f(xy))g$ $\displaystyle=((fx)y)g-((fy)x)g,$ $\displaystyle
f((gx)y)$ $\displaystyle=(f(gx))y-(fy)(gx),$ $\displaystyle f((xg)y)$
$\displaystyle=(f(xg))y-(fy)(xg),$ $\displaystyle f(x(gy))$
$\displaystyle=(fx)(gy)-(f(gy))x,$ $\displaystyle f(x(yg))$
$\displaystyle=(fx)(yg)-(f(yg))x.$

It is a straightforward computation to check that the rank of the matrix
formed by all the consequences is $72$. Then, we have to compare it with the
multiplication defined in Equation 4.2. For instance, taking again the first
equation in Equation 4.1 we have to expand the identity

$(fg)(xy)=((fg)x)y-((fg)y)x,$

which gives us

$\displaystyle
f(g(xy))+(f(xy))g+\alpha_{1}\big{(}f(g(xy))+f((xy)g)\big{)}+\alpha_{2}\big{(}g(f(xy))+g((xy)f)\big{)}$
$\displaystyle=(f(gx))y+((fx)g)y+\alpha_{1}\big{(}(f(gx))y+(f(xg))y\big{)}+\alpha_{2}\big{(}(g(fx))y+(g(xf))y\big{)}$
$\displaystyle-(f(gy))x+((fy)g)x+\alpha_{1}\big{(}(f(gy))x+(f(yg))x\big{)}+\alpha_{2}\big{(}(g(fy))x+(g(yf))x\big{)}.$

After a linear algebra computation it can be checked that no matter which
$\alpha_{1}$ and $\alpha_{2}$ we choose, it belongs to the subspace generated
by the consequences. In fact, this will be true for all the identities (4.1),
so any $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$, $\beta_{2}\in\mathbb{F}$ will
produce a total multiplication on $\mathscr{E}(X)$.

To check whether the induced bracket endows $\mathscr{E}(X)$ with a Leibniz
algebra structure, we just need to check when the following identities hold

$\displaystyle(f(gh))x$ $\displaystyle=((fg)h)x-((fh)g)x$ $\displaystyle
x(f(gh))$ $\displaystyle=x((fg)h)-x((fh)g).$

A quick computation tells us that this is only true when
$(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})=(1,0,0,0)$, so that we recover
exactly the multiplication defined in [8, Definition 5.1].

We consider some further examples.

######  Example 4.4.

In the case of associative algebras any choice of bracket will induce a total
algebra structure, but only the already known example of bimultipliers will be
an associative algebra.

######  Example 4.5.

The variety of _symmetric Leibniz algebras_ is formed by the intersection
between the varieties of right and left Leibniz algebras, i.e. the variety
determined by $(xy)z-(xz)y-x(yz)=0$ (right Leibniz identity) and
$z(xy)-(xz)y-x(zy)=0$ (left Leibniz identity). The space generated by its
bilinear identities of degree $3$ has dimension $10$, which means that there
are $12$ parameters to define a product in $\mathscr{E}(X)$. With the help of
a computer algebra system such as Macaulay2 [16] we check that any choice will
give us a total algebra structure, and the set of variables that induces a
symmetric Leibniz algebra structure on $\mathscr{E}(X)$ forms an affine
variety of dimension $2$.

######  Example 4.6.

Following the algorithm proposed before, it can be checked easily that the
variety of two-step nilpotent (non-commutative) algebras is weakly action
representable. In fact, a weak actor may be given by the expected structure

$\mathscr{E}(X)=\\{f\in\operatorname{End}(X)^{2}\mid
f(xy)=(xy)f=0=(fx)y=x(yf)\\}$

with product $fg=0=gf$. Nevertheless, this is not the only product that can be
induced. Since the space generated by its bilinear identities of degree $3$
has maximum dimension $12$, there are $16$ parameters that can be taken into
account to define a product in $\mathscr{E}(X)$. All of them induce a total
multiplication on it, and the set of parameters which induce a two-step
nilpotent algebra on $\mathscr{E}(X)$ forms an affine variety of dimension
$3$. Note that these algebras were studied and classified in [21] and [22].

######  Example 4.7.

Although commutative Poisson algebras are usually defined as a variety with
two operations, in [26] it was shown that with the depolarisation technique
they can be seen as a quadratic variety (with one operation), so they fit in
the scope of this section. The algorithmic approach presented before shows
that it is possible to induce several total algebra structures on
$\mathscr{E}(X)$, more precisely it gives rise to a $3$-parametric family.

######  Example 4.8.

The varieties of Novikov algebras or anti-associative algebras do not allow a
total algebra structure on their respective $\mathscr{E}(X)$ induced by action
accessibility, but it is still an open problem whether or not these varieties
are weakly action representable.

##  5\. Open Questions and further directions

### Converse of the implication “weakly action representable category
$\Rightarrow$ action accessible category”

We studied the representability of actions of a general operadic variety of
non-associative algebras over a field but we were not able to find an example
of an action accessible variety which is not weakly action representable. Does
the converse of the implication

$\textit{weakly action representable category}\Rightarrow\textit{action
accessible category}$

hold in this context?

### Subvarieties

We do not know how the condition (WRA) behaves under taking subvarieties
(especially in the non-quadratic case, when the degree of the identities may
be higher than $3$). For instance, we know that the variety $\mathbf{Assoc}$
is weakly action representable, but we do not know whether the subvariety
$\mathbf{Nil}_{k}(\mathbf{Assoc})$ for $k\geq 3$ satisfies the same condition.
We recall that in this case, $\mathscr{E}(X)$ is an associative algebra, but
it is not $k$-step nilpotent in general (see Example 3.6).

### Initial weak representation

As already mentioned in Remark 2.13, a variety of non-associative algebras
$\mathscr{V}$ is weakly action representable if and only if it is _initially
weakly action representable_ , which means that for every object $X$, the
functor $\operatorname{SplExt}(-,X)$ admits an initial weak representation. We
do not know whether or not the weak representations that occur in this article
when the external weak actor $\mathscr{E}(X)$ is an object of $\mathscr{V}$
(such as for Leibniz algebras or associative algebras) are initial, or how we
would check this in practice.

### Representability of actions of _unitary_ algebras

In the recent article [18], G. Janelidze introduced the notion of _ideally
exact_ category, with the aim of generalising semi-abelian categories in a way
which includes relevant examples of _non-pointed_ categories, such as the
categories Ring and CRing of (commutative) rings with unit.

A category $\mathscr{C}$ is _ideally exact_ when it is Barr-exact and Bourn-
protomodular with finite coproducts, such that the unique morphism $0\to 1$ in
$\mathscr{C}$ is a regular epimorphism. Thus, semi-abelian categories are
precisely the pointed ideally exact categories.

G. Janelidze also extended the notions of action representability and weak
action representability to ideally exact categories, showing Ring and CRing
are action representable, with the actor of a (commutative) unitary ring $X$
being isomorphic to $X$ itself. We do not recall the construction here, since
it is essentially the same as the one for alternative algebras with unit given
in Example 3.7.

We recall that a variety of non-associative algebras $\mathscr{V}$ is said to
be _unitary closed_ if for any object $X$ in it, the algebra $\tilde{X}$
spanned by $X$ and the element $1$, equipped with the multiplication $x\cdot
1=1\cdot x=x$ for any $x\in X$, is still an object of $\mathscr{V}$. For
instance, $\mathbf{Assoc}$ and $\mathbf{Alt}$ are unitary closed, and the
category $\mathbf{Leib}$, or any variety of anti-commutative algebras over a
field of characteristic different from $2$, such as $\mathbf{Lie}$, are
examples of varieties which are not. Thus, the condition of being unitary
closed depends on the set of identities which determine the variety
$\mathscr{V}$.

When a variety of algebras $\mathscr{V}$ is unitary closed, one can consider
the subcategory $\mathscr{V}_{1}$ of unitary algebras of $\mathscr{V}$ with
the arrows being the algebra morphisms of $\mathscr{V}$ that preserve the
unit. Of course, $\mathscr{V}_{1}$ is an ideally-exact category and it is not
pointed.

Examples 3.7 and 3.8 suggest that for a unitary closed variety $\mathscr{V}$,
one may use the construction of the external weak actor $\mathscr{E}(X)$ to
study the representability of actions of the subcategory $\mathscr{V}_{1}$.
For instance, it follows easily that $\mathbf{Alt}_{1}$ and
$\mathbf{Assoc}_{1}$ are action representable, with the actor of an object $X$
in both cases being the object $X$ itself.

## Acknowledgements

First we would like to thank the anonymous referee, whose careful reading and
invaluable comments helped us correct a crucial mistake in the first version
of the text.

We are grateful to Abdenacer Makhlouf for recommending us to study the
representability of actions of Jacobi–Jordan algebras, to Giuseppe Metere for
suggesting us the name _external weak actor_ , and to Gabor P. Nagy for
helping us with finding an explicit example of a $2$-Engel Lie algebra in
characteristic $3$ which is not two-step nilpotent.

We would like to express our sincere gratitude to the Institut de Recherche en
Mathématique et Physique (IRMP) for the warm reception we received during our
visits to Louvain-la-Neuve. We would also like to extend our heartfelt
appreciation to the Universities of Santiago de Compostela and Vigo for the
generous support and welcoming atmosphere provided during our time there.

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