# Non-associative categories of octonionic bimodules

Qinghai Huo<EMAIL_ADDRESS>Department of Mathematics, University of
Science and Technology of China, Hefei 230026, China  and  Guangbin Ren
<EMAIL_ADDRESS>Department of Mathematics, University of Science and
Technology of China, Hefei 230026, China

###### Abstract.

Category is put to work in the non-associative realm in the article. We focus
on a typical example of non-associative category. Its objects are octonionic
bimodules, morphisms are octonionic para-linear maps, and compositions are
non-associative in general. The octonionic para-linear map is the main object
of octonionic Hilbert theory because of the octonionic Riesz representation
theorem. An octonionic para-linear map $f$ is in general not octonionic linear
since it subjects to the rule

$\text{Re}\,\big{(}f(px)-pf(x)\big{)}=0.$

The composition should be modified as

$f\circledcirc
g(x):=f(g(x))-\sum_{j=1}^{7}e_{j}\text{Re}\,\Big{(}f\big{(}g(e_{i}x)\big{)}-f\big{(}e_{i}g(x)\big{)}\Big{)}$

so that it preserves the octonionic para-linearity.

In this non-associative category, we introduce the Hom and Tensor functors
which constitute an adjoint pair. We establish the Yoneda lemma in terms of
the new notion of weak functor. To define the exactness in a non-associative
category, we introduce the notion of the enveloping category via a universal
property. This allows us to establish the exactness of the Hom functor and
Tensor functor.

###### Key words and phrases:

Octonions; category; regular composition; para-linear; adjoint functor theorem

###### 2010 Mathematics Subject Classification:

Primary: 17A35; 18A05; 46S10

This work was supported by the NNSF of China (12171448).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Octonions
    2. 2.2 $\mathbb{O}$-modules
  3. 3 $\mathbb{O}$-para-linearity
    1. 3.1 Real part operator on $\mathbb{O}$-bimodules
    2. 3.2 Second associator
    3. 3.3 $\mathbb{O}$-para-linear map
    4. 3.4 Second associator of para-linear maps
    5. 3.5 $\mathbb{O}$-module structure of $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$
    6. 3.6 Real part of $\mathbb{O}$-para-linear maps
    7. 3.7 Regular composition of $\mathbb{O}$-para-linear maps
    8. 3.8 Right para-linear maps
    9. 3.9 Semi-reflexivity
  4. 4 Non-associative categories of para-linear maps
    1. 4.1 Non-associative categories and weak functors
    2. 4.2 Conjugate functor
    3. 4.3 Hom functors
    4. 4.4 Tensor functors
    5. 4.5 Adjoint functor theorem
    6. 4.6 Enveloping categories
    7. 4.7 Natural associated functor
    8. 4.8 Exactness of functors
  5. 5 Concluding remarks

##  1\. Introduction

Category theory, initiated by Eilenberg and MacLane in 1945 [16], provides
deep insights and similarities between seemingly different areas of
mathematics [1, 16, 20, 25]. However, category theory is exclusively
restricted to the associative realm since compositions among morphisms are
always assumed to be associative.

The purpose of this article is to bring the category into the non-associative
realm by providing a typical example of the non-associative category whose
compositions are non-associative. This category comes from octonionic
functional analysis. In octonionic functional analysis, the central objects
are octonionic para-linear functionals (see [13]). This notion can be
generalized to a more general setting, which leads to a specific non-
associative category. The objects of this category are octonionic bimodules,
the morphisms are octonionic para-linear maps, and the compositions are
regular compositions that are non-associative.

We first recall the notion of para-linearity by focusing our attention on
octonionic Hilbert spaces.

######  Definition 1.1 ([13]).

Suppose that $H$ is a left $\mathbb{O}$-module and also a real Hilbert space
with a real-valued inner product $\langle\cdot,\cdot\rangle_{\mathbb{R}}$. We
call $H$ an  $\mathbb{O}$-Hilbert space if there exists an
$\mathbb{R}$-bilinear map

$\left\langle\cdot,\cdot\right\rangle:H\times H\rightarrow\mathbb{O}$

such that

$Re\left\langle\cdot,\cdot\right\rangle=\langle\cdot,\cdot\rangle_{\mathbb{R}}$

and satisfying axioms

  1. (${\mathrm{i}}$)

($\mathbb{O}$-para-linearity) $\left\langle\cdot,u\right\rangle$ is left
$\mathbb{O}$-para-linear for all $u\in H$.

  2. (${\mathrm{ii}}$)

($\mathbb{O}$-hermiticity) $\left\langle
u,v\right\rangle=\overline{\left\langle v,u\right\rangle}$ for all $u,v\in H$.

  3. (${\mathrm{iii}}$)

(Positivity) $\left\langle u,u\right\rangle\in\mathbb{R}^{+}$ and
$\left\langle u,u\right\rangle=0$ if and only if $u=0$.

The definition of octonionic Hilbert spaces can trace back to 1964 by
Goldstine and Horwitz [8]. In fact, they put an extra axiom which states

$\left\langle pu,u\right\rangle=p\left\langle u,u\right\rangle$

for all $p\in\mathbb{O}$ and $u\in H$. This axiom turns out to be
redundant[13].

Definition 1.1 provides a unified viewpoint for the theory of Hilbert spaces
over normed division algebras. In contrast to the full development of
quaternionic Hilbert spaces [11, 19, 18, 23, 24, 5, 6, 4, 7], the research on
octonionic Hilbert spaces is very rare.

The map $\left\langle\cdot,u\right\rangle$ induced by the $\mathbb{O}$-inner
product is a real linear map but not an $\mathbb{O}$-linear map. Even though
this, it satisfies

$\mbox{Re}\left\langle px,u\right\rangle=\mbox{Re}(p\left\langle
x,u\right\rangle)$

for any $p\in\mathbb{O}$. This motivates us to introduce a notion of
$\mathbb{O}$-para-linear function as a real linear map $f:H\to\mathbb{O}$ for
which

(1.1) $\text{Re}\,f(px)=\text{Re}\,(pf(x))$

for any $p\in\mathbb{O}$ and $x\in H$. The importance of octonionic para-
linear maps has been shown in the octonionic Riesz representation theorem
which provides a bijection between an $\mathbb{O}$-Hilbert space $H$ and its
dual space consisting of $\mathbb{O}$-para-linear functions [13]. We will
further study $\mathbb{O}$-para-linear operators between octonionic Hilbert
spaces. To this end, we need to generalize the notion of para-linear functions
to para-linear maps between $\mathbb{O}$-modules.

Let $M$ and $M^{\prime}$ be two $\mathbb{O}$-bimodules. We want to define the
notion of left $\mathbb{O}$-para-linear maps from $M$ to $M^{\prime}$ in terms
of the definition equality (1.1). But in contrast to the case where
$M^{\prime}=\mathbb{O}$, the real part operator in (1.1) is meaningless for a
general $\mathbb{O}$-module $M^{\prime}$. Therefore, we need to introduce a
real part structure in the target module $M^{\prime}$, which is given by a
projection map

(1.2) $\mbox{Re}:M^{\prime}\to\mathscr{A}(M^{\prime}).$

Here $\mathscr{A}(M^{\prime})$ is the set of associative elements of
$M^{\prime}$. Notice that in the specific case $M^{\prime}=\mathbb{O}$ we have

(1.3) $\text{Re}\,x=\frac{5}{12}x-\frac{1}{12}\sum_{i=1}^{7}e_{i}xe_{i}.$

It turns out that the real part operator on an $\mathbb{O}$-bimodule
$M^{\prime}$ also satisfies (1.3). Similar results also hold in quaternionic
case [17].

The set $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ of all left
$\mathbb{O}$-para-linear maps can be endowed with a canonical
$\mathbb{O}$-bimodule structure. Its $\mathbb{O}$-scalar multiplications are
defined by

$\displaystyle(r\odot f)(x)$ $\displaystyle:=$ $\displaystyle
f(xr)+A_{r}(x,f),$ $\displaystyle(f\odot r)(x)$ $\displaystyle:=$
$\displaystyle f(xr)+A_{r}(x,f),$

where

$A_{r}(x,f):=f(rx)-rf(x).$

We find that the real part of the bimodule
$\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ is

$\text{Re}\,\big{(}\text{Hom}_{\mathcal{LO}}(M,M^{\prime})\big{)}=\text{Hom}_{\mathbb{O}}(M,M^{\prime}).$

This indicates that it is too small to study the $\mathbb{O}$-linear maps in
octonionic functional analysis.

Note that the para-linear maps are not preserved under ordinary composition.
We then introduce a new notion of regular composition which preserves
octonionic para-linear morphisms. More precisely, let
$M,M^{\prime},M^{\prime\prime}$ be $\mathbb{O}$-bimodules and let
$f\in\text{Hom}_{\mathcal{LO}}(M^{\prime},M^{\prime\prime})$ and
$g\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$. We define

$(f\circledcirc
g)(x):=f(g(x))-\sum_{j=1}^{7}e_{j}\text{Re}\,\Big{(}f\big{(}g(e_{i}x)\big{)}-f\big{(}e_{i}g(x)\big{)}\Big{)}.$

Then we have

$f\circledcirc g\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime\prime}).$

The above definition originates from the specific case of the multiplication
of octonionic matrices. Notice that

(1.4)
$\displaystyle\text{Hom}_{\mathcal{LO}}(\mathbb{O},\mathbb{O})=\\{R_{p}:p\in\mathbb{O}\\},$

where $R_{p}$ is a right multiplication operator on $\mathbb{O}$ defined by

$\displaystyle R_{p}x=xp.$

It turns out that

(1.5) $\displaystyle R_{p}\circledcirc R_{q}=R_{pq}$

so that $\text{Hom}_{\mathcal{LO}}(\mathbb{O},\mathbb{O})$ is isomorphic to
$\mathbb{O}$ as an algebra. The multiplications of general matrices have the
similar properties. Notice that the regular composition is generally non-
associative. This gives rise to our specific non-associative category. We
shall denote by $\mathcal{LO}$ the typical non-associative category, whose
objects are octonionic bimodules, whose morphisms are left $\mathbb{O}$-para-
linear maps, and whose compositions are left regular compositions. The
category $\mathcal{RO}$ of right para-linear maps can be defined similarly.

Now we move on to study functors between these two non-associative categories.
We shall introduce several functors, including the conjugate, Hom, and tensor
functors.

Firstly, we define the conjugate functor. For any $\mathbb{O}$-bimodule, we
have an induced bimodule, denoted by $M^{C}$, whose bimodule structure is
defined as:

$p\cdot x=x\overline{p},\quad x\cdot p=\overline{p}x.$

This leads to a functor, called the conjugate functor,

$C:\mathcal{LO}\to\mathcal{RO},$

which is an isomorphism of categories. The effect of the conjugate functor is
that it can translate results from $\mathcal{LO}$ to $\mathcal{RO}$ and vise
versa.

Next, we define Hom functors between categories $\mathcal{LO}$ and
$\mathcal{RO}$. They will play a central role in the study of dual operators
in octonionic functional analysis.

######  Theorem 1.2.

For any $\mathbb{O}$-bimodule $M$, we have a covariant Hom functor

$\text{Hom}_{\mathcal{LO}}(M,-):\mathcal{LO}\to\mathcal{LO}$

and a contravariant Hom functor

$\text{Hom}_{\mathcal{LO}}(-,M):\mathcal{LO}\to\mathcal{RO}.$

Moreover, both functors are $\mathbb{O}$-linear.

Finally, we consider the tensor functors. We define the octonionic tensor
product of two $\mathbb{O}$-bimodules $M$, $M^{\prime}$ as

$M\otimes_{\mathbb{O}}M^{\prime}:=(\text{Re}\,M\otimes_{\mathbb{R}}\text{Re}\,M^{\prime})\otimes\mathbb{O}.$

We then have a natural tensor functor:

$\displaystyle M\otimes^{ll}_{\mathbb{O}}-:\qquad\mathcal{LO}\qquad$
$\displaystyle\xlongrightarrow[]{\hskip 28.45274pt}\qquad\mathcal{LO}$
$\displaystyle X\qquad$ $\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\qquad M\otimes_{\mathbb{O}}X$ $\displaystyle
X\stackrel{{\scriptstyle f}}{{\longrightarrow}}X^{\prime}\qquad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\qquad
l\text{-}\mathrm{ext}\,f_{M}.$

Here the map

$f_{M}:\text{Re}\,M\otimes_{\mathbb{R}}\text{Re}\,X\to\text{Re}\,M\otimes_{\mathbb{R}}\text{Re}\,X^{\prime}$

is defined by

$f_{M}(m\otimes x):=m\otimes f(x)$

for all $m\in\text{Re}\,M$ and $x\in\text{Re}\,X$. The map
$l\text{-}\mathrm{ext}\,f_{M}$ is a canonical para-linear extension map of
$f_{M}$. Similarly, we can also define functors

$M\otimes^{lr}_{\mathbb{O}}-,\qquad\quad M\otimes^{rr}_{\mathbb{O}}-,\qquad
M\otimes^{rl}_{\mathbb{O}}-.$

It turns out that the two functors

$(M\otimes^{ll}_{\mathbb{O}}-,\ \text{Hom}_{\mathcal{LO}}(M,-))$

constitute an adjoint pair.

The above preparation allows us to study further exactness for non-associative
categories. It is hard to consider exactness in the non-associativity setting.
To overcome the difficulty, we associate the non-associative category with an
associative category. This associative category is constructed due to a
universal property and is called the enveloping category. The exactness in a
non-associative category can thus be defined as the exactness of its
enveloping category.

Studying the exactness, we need to introduce the notion of natural associated
transformations. Through a natural associated isomorphism, a functor between
non-associative categories naturally corresponds to an associated functor
between their enveloping categories.

For example, we can provide the natural associated functors of the three
functors $\text{Hom}_{\mathcal{LO}}(-,M)$, $\text{Hom}_{\mathcal{LO}}(M,-)$,
and $M\otimes^{ll}_{\mathbb{O}}-$. It turns out that the natural associated
functors of $\text{Hom}_{\mathcal{LO}}(-,M)$ is the usual Hom functor

$\text{Hom}_{\mathbb{R}}(-,\text{Re}\,M):\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}\stackrel{{\scriptstyle}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}},$

where $\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}$ stands for the associative
category, whose objects are $\mathbb{O}$-bimodules, morphisms are
$\mathbb{R}$-linear maps, and compositions are usual compositions. The natural
associated functor of $\text{Hom}_{\mathcal{LO}}(M,-)$ is the usual Hom
functor

$\text{Hom}_{\mathbb{R}}(\text{Re}\,M,-):\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}\stackrel{{\scriptstyle}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}},$

and the natural associated functor of $M\otimes^{ll}_{\mathbb{O}}-$ is the
usual tensor functor

$\text{Re}\,M\otimes_{\mathbb{R}}-:\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}\stackrel{{\scriptstyle}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}.$

These yield the exactness of Hom functor and tensor functor in $\mathcal{LO}$.

In the classical theory of categories, the Yoneda lemma is arguably the most
important result. We can extend this result to non-associative categories. It
gives rise to a new notion called weak functors. It turns out that the usual
Hom functor from a non-associative category to the Sets category is merely a
weak functor other than a functor.

The organization of this article is as follows. In Section 2, we give a brief
overview of the algebra of octonions, and review some definitions and basic
properties on the theory of octonionic modules; more details can be found in
[12, 21, 22]. In Section 3, we introduce the notion of $\mathbb{O}$-para-
linear maps and give a detailed discussion on this concept. In Section 4, we
adapt the method of category to study $\mathbb{O}$-para-linear maps and
initiate the study of non-associative categories.

##  2\. Preliminaries

In this section, we review some basic properties of octonions and octonionic
modules.

###  2.1. Octonions

The algebra $\mathbb{O}$ of octonions is a non-associative, non-commutative,
normed division algebra over $\mathbb{R}$. As a real vector space, it has a
basis [2]

$e_{0}=1,\quad e_{1},\quad\dots,\quad e_{7},$

subject to the multiplication rules

$e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij},\quad i,j=1,\dots,7.$

An octonion can be written as

$x=x_{0}+\sum_{i=1}^{7}x_{i}e_{i},\quad x_{i}\in\mathbb{R}.$

Its conjugate is defined as

$\overline{x}=x_{0}-\sum_{i=1}^{7}x_{i}e_{i}.$

The norm of $x$ can be expressed as

$|x|=\sqrt{x\overline{x}}\in\mathbb{R};$

and the real part of $x$ is given by

$\text{Re}\,{x}=x_{0}=\frac{1}{2}(x+\overline{x}).$

Following the $\epsilon$-notation from [3], the multiplication rules can be
expressed as

(2.1) $\displaystyle e_{i}e_{j}=\epsilon_{ijk}e_{k}-\delta_{ij}.$

The symbol $\epsilon$ is skew-symmetric in indices. In fact, if we write
$e^{ijk}$ for the wedge product $e^{i}\wedge e^{j}\wedge e^{k}$ in
$\Lambda^{3}\left(\left(\mathbb{R}^{7}\right)^{*}\right)$, then we obtain a
$3$-form:

$\displaystyle\phi$ $\displaystyle=\frac{1}{6}\varepsilon_{ijk}e^{ijk}.$

More precisely, we have

$\displaystyle\phi$
$\displaystyle=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}.$

###  2.2. $\mathbb{O}$-modules

There are abundant results on octonionic bimodules, or more generally,
alternative modules. Already in 1952, Schafer [21] gave the birepresentations
of alternative algebras. Subsequently, Jacobson [14] determined the
irreducible representations for finite dimensional semi-simple alternative
algebras. A more general study for alternative bimodules is given in [22].
Recently, only for the octonionic case, we [12] introduce the new notions of
associative elements and conjugate associative elements, which give rise to a
new description of the structure of $\mathbb{O}$-modules.

We now recall some basic notations and results on $\mathbb{O}$-modules. A real
vector space $M$ is called a left $\mathbb{O}$-module if it is endowed with an
$\mathbb{O}$-scalar multiplication such that the left associator is left
alternative. That is, for all $p,q\in\mathbb{O},\ m\in M$,

(2.2) $\displaystyle[p,q,m]=-[q,p,m],$

where the left associator is defined by

$[p,q,m]=(pq)m-p(qm).$

It is easy to check that (2.2) is equivalent to

$r(rm)=r^{2}m\text{\ \ \ for all \ \ }r\in\mathbb{O},\,\ m\in M.$

We recall a useful identity which holds in any left $\mathbb{O}$-module $M$.

(2.3) $[p,q,r]m+p[q,r,m]=[pq,r,m]-[p,qr,m]+[p,q,rm]$

for all $p,q,r\in\mathbb{O}$ and $m\in M$. We shall generalize this identity
in many different settings.

The nucleus of a left $\mathbb{O}$-module $M$ is defined by

$\mathscr{A}(M):=\\{m\in M\mid[p,q,m]=0,\text{ for all }p,q\in\mathbb{O}\\}.$

Any element in the nucleus $\mathscr{A}(M)$ is called an associative element.
We denote by $\mathscr{A}^{-}(M)$ the set of all conjugate associative
elements

$\mathscr{A}^{-}(M):=\\{m\in M\mid(pq)m=q(pm)\text{ for all
}p,q\in\mathbb{O}\\}.$

The irreducible left $\mathbb{O}$-modules are already known to be isomorphic
to the regular or conjugate regular modules. We refer to Chapter 11 of the
monograph [26] for detailed results. We [12] describe the structure of a left
$\mathbb{O}$-module in terms of associative elements and conjugate associative
elements as follows

######  Theorem 2.1 ([12]).

Let $M$ be a left $\mathbb{O}$-module. Then

$M=\mathbb{O}\mathscr{A}(M)\oplus{\mathbb{O}}\mathscr{A}^{-}(M).$

We now recall the definition of $\mathbb{O}$-bimodules. Since $\mathbb{O}$ is
a special alternative algebra, the notion of an $\mathbb{O}$-bimodule is just
an alternative bimodule over $\mathbb{O}$ [21, 14]. Here we state the
definition as follows.

######  Definition 2.2.

A left $\mathbb{O}$-module $M$ is called an $\mathbb{O}$-bimodule if there is
a right $\mathbb{R}$-bilinear multiplications such that for any
$p,q\in\mathbb{O}$ and $m\in M$,

$[p,q,m]=[m,p,q]=[q,m,p].$

Here we denote

$[q,m,p]:=(qm)p-q(mp)$

and denote the similar notation for $[m,p,q]$.

We denote by $\text{Reg}\,{\mathbb{O}}$, or just $\mathbb{O}$ if there is no
confusion, the regular bimodule with the multiplication given by the product
in $\mathbb{O}$. Clearly, the $\mathbb{O}$-bimodule $\text{Reg}\,{\mathbb{O}}$
is irreducible. Moreover, it is the only irreducible $\mathbb{O}$-bimodule;
see the results by Schafer [21] and Jacobson [14].

##  3\. $\mathbb{O}$-para-linearity

Recently, we [13] introduce a notion of $\mathbb{O}$-para-linear functional.
In fact, we can generalize this notion to a more general case of
$\mathbb{O}$-para-linear maps between $\mathbb{O}$-modules. The
$\mathbb{O}$-para-linearity plays a central role in the study of octonionic
functional analysis. We shall give a complete discussion about this concept in
this section.

###  3.1. Real part operator on $\mathbb{O}$-bimodules

To generalize the notion of para-linear functions to para-linear maps, we need
to generalize the notion of the real part structure of $\mathbb{O}$ to general
$\mathbb{O}$-bimodules.

Let $M$ be an $\mathbb{O}$-bimodule. Since the only irreducible
$\mathbb{O}$-bimodule is the regular bimodule $\text{Reg}\,{\mathbb{O}}$, it
follows from Theorem 2.1 that

(3.1) $\displaystyle M=\mathbb{O}\mathscr{A}(M).$

One can easily check that

$e_{i}\mathscr{A}(M)\cap e_{j}\mathscr{A}(M)=\\{0\\}$

for each $i\neq j\in\\{0,\dots,7\\}$. Indeed, supposing on the contrary, we
can find

$e_{i}x=e_{j}y$

for some non-zero elements $x,y\in\mathscr{A}(M)$. Then
$x=(\overline{e_{i}}e_{j})y$. It follows that

$[p,q,x]=[p,q,\overline{e_{i}}e_{j}]y=0$

for arbitrary $p,q\in\mathbb{O}$. This forces that $y=0$, a contradiction.

Hence we have a direct sum decomposition of $M$

(3.2) $\displaystyle M=\bigoplus_{i=0}^{7}e_{i}\mathscr{A}(M).$

We thus define the real part operator as the projective operator

$\text{Re}\,:M\to\mathscr{A}(M).$

The right multiplication in an $\mathbb{O}$-bimodule is uniquely determined by
its left multiplication, which is completely different from the quaternionic
case. We denote the commutative center of $M$ by:

$\mathscr{Z}(M):=\\{x\in M\mid px=xp,\text{ for all }p\in\mathbb{O}\\}.$

Then we have

(3.3) $\displaystyle\text{Re}\,M=\mathscr{A}(M)=\mathscr{Z}(M).$

We collect some properties of the real part operator as follows.

######  Theorem 3.1.

Let $M$ be an $\mathbb{O}$-bimodule. We have a concrete expression of the real
part operator in terms of scalar multiplications

(3.4) $\text{Re}\,x=\frac{5}{12}x-\frac{1}{12}\sum_{i=1}^{7}e_{i}xe_{i}.$

Moreover, for all $x\in M$, we have

  1. (${\mathrm{i}}$)

$\text{Re}\,^{2}x=\text{Re}\,x$.

  2. (${\mathrm{ii}}$)

For all $p,q\in\mathbb{O}$,

(3.5) $\displaystyle\text{Re}\,(p\text{Re}\,(x))$ $\displaystyle=$
$\displaystyle(\text{Re}\,p)\text{Re}\,(x),$ (3.6)
$\displaystyle\text{Re}\,[p,q,x]$ $\displaystyle=$
$\displaystyle\text{Re}\,[p,x]=0.$

  3. (${\mathrm{iii}}$)

$x=\sum_{i=0}^{7}e_{i}\text{Re}\,(\overline{e_{i}}x)$.

###### Proof.

We prove (3.4). One can check directly that (3.4) holds for the case
$M=\mathbb{O}$. The general case then follows. Indeed, for any $x\in M$, write
$x=\sum_{j=0}^{7}e_{j}x_{j}$ with $x_{j}\in\mathscr{A}(M)$. Then according to
(3.3), we have

$\displaystyle\frac{5}{12}x-\frac{1}{12}\sum_{i=1}^{7}e_{i}xe_{i}$
$\displaystyle=\frac{5}{12}\sum_{j=0}^{7}e_{j}x_{j}-\frac{1}{12}\sum_{i=1}^{7}e_{i}\left(\sum_{j=0}^{7}e_{j}x_{j}\right)e_{i}$
$\displaystyle=\sum_{j=0}^{7}\left(\frac{5}{12}e_{j}-\frac{1}{12}\sum_{i=1}^{7}e_{i}e_{j}e_{i}\right)x_{j}$
$\displaystyle=\sum_{j=0}^{7}\text{Re}\,(e_{j})x_{j}$ $\displaystyle=x_{0}.$

This proves (3.4).

The proof of assertion $({\mathrm{i}})$ is trivial. We prove (3.5). Without
loss of generality we can assume $x\in\mathscr{A}(M)$. It follows from
assertion $({\mathrm{i}})$ that

$\displaystyle\text{Re}\,(px)$
$\displaystyle=\frac{5}{12}px-\frac{1}{12}\sum_{i=1}^{7}e_{i}(px)e_{i}$
$\displaystyle=\frac{5}{12}px-\frac{1}{12}\sum_{i=1}^{7}(e_{i}pe_{i})x$
$\displaystyle=(\text{Re}\,p)x.$

Then (3.6) follows from (3.5) directly.

Let $x=\sum_{i=0}^{7}e_{i}x_{i}$. Then it follows from (3.5) that

$\text{Re}\,(\overline{e_{i}}x)=\text{Re}\,\sum_{i=0}^{7}(e_{j}\overline{e_{i}})x_{j}=x_{i}.$

This proves assertion $({\mathrm{iii}})$. ∎

######  Remark 3.2.

Recall that in a quaternionic bimodule, the real part operator is defined by

$\text{Re}\,x=\frac{1}{4}\sum_{e\in B}\overline{e}xe\quad(x\in X).$

Where $B:=\\{1,i,j,k\\}$ is a basis of the quaternions $\mathbb{H}$ ([17]).
And the polarization identity holds:

(3.7) $\displaystyle x=\sum_{e\in B}e\text{Re}\,(\overline{e}x).$

(3.4) is a similar result in octonionic case.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules. A map $f:M\to M^{\prime}$
is called $\mathbb{O}$-linear if

$f(px)=pf(x),\quad f(xp)=f(x)p$

for all $x\in M,\ p\in\mathbb{O}$. Note that the right multiplication in an
$\mathbb{O}$-bimodule is uniquely determined by its left multiplication. We
conclude that

(3.8)
$\begin{split}\text{Hom}_{\mathbb{O}}(M,M^{\prime})&=\\{f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})\mid
f(px)=pf(x)\text{ for all }x\in M,\ p\in\mathbb{O}\\}\\\
&=\\{f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})\mid f(px)=pf(x)\text{ for all
}x\in M,\ p\in\mathbb{O}\\}.\end{split}$

It follows from (3.4) that any $\mathbb{O}$-linear map commutes with the real
part operator.

######  Lemma 3.3.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules and
$f\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$. Then

$f(\text{Re}\,x)=\text{Re}\,f(x)$

for all $x\in M$.

###  3.2. Second associator

To study the para-linear map, we need to introduce the notion of the second
associators related to a linear map between $\mathbb{O}$-modules and establish
some identities associated with this notion.

Let $M$, $M^{\prime}$ be two left $\mathbb{O}$-modules. For any
$f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$, $p\in\mathbb{O}$ and $x\in M$, we
denote

$A_{p}(x,f):=f(px)-pf(x)$

and call it the second left associator of $f$. Similarly we can introduce the
second right associator between right $\mathbb{O}$-modules via

$B_{p}(f,x):=f(x)p-f(xp).$

By definition, we can generalize identity (2.3) as follows.

######  Lemma 3.4.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-modules and
$f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$. Assume that
$\alpha\in\mathbb{R}$, $p,q\in\mathbb{O}$ and $x\in M$. The following
statements hold.

  1. (${\mathrm{i}}$)

If $M$, $M^{\prime}$ are two left $\mathbb{O}$-modules, then

(3.9) $A_{pq}(x,f)=f([p,q,x])-[p,q,f(x)]+pA_{q}(x,f)+A_{p}(qx,f).$

  2. (${\mathrm{ii}}$)

If $M$, $M^{\prime}$ are two right $\mathbb{O}$-modules, then

(3.10) $B_{pq}(f,x)=f([x,p,q])-[f(x),p,q]+B_{p}(f,x)q+B_{q}(f,xp).$

  3. (${\mathrm{iii}}$)

If $M$, $M^{\prime}$ are two $\mathbb{O}$-modules, then

$A_{\alpha}(x,f)=B_{\alpha}(f,x)=0.$

###### Proof.

A direct calculation shows that

$\displaystyle A_{pq}(x,f)$ $\displaystyle=f((pq)x)-(pq)f(x)$
$\displaystyle=f(p(qx)+[p,q,x])-(pq)f(x)$
$\displaystyle=pf(qx)+A_{p}(qx,f)+f([p,q,x])-(pq)f(x)$
$\displaystyle=p(qf(x))+pA_{q}(x,f)+A_{p}(qx,f)+f([p,q,x])-(pq)f(x)$
$\displaystyle=f([p,q,x])-[p,q,f(x)]+pA_{q}(x,f)+A_{p}(qx,f).$

This proves assertion $({\mathrm{i}})$. Assertion $({\mathrm{ii}})$ can be
proved similarly. Assertion $({\mathrm{iii}})$ is trivial. ∎

######  Remark 3.5.

Let $f$ be a map between two left (right) $\mathbb{O}$-modules. We denote the
evaluation of $f$ on an element $x$ by

(3.11) $\displaystyle\left<x,f\right>:=f(x)\quad(\text{resp.
}\left<f,x\right>:=f(x)).$

This notation is reminiscent of the inner product on a Hilbert space.

If we write

$A_{p}(x,f)=\left<px,f\right>-p\left<x,f\right>=:[p,x,f],$

as we have denoted in [13], then identity (3.9) can be rephrased as

$p[q,x,f]+\left<[p,q,x],f\right>=[pq,x,f]-[p,qx,f]+[p,q,\left<x,f\right>],$

which is an analogue of identity (2.3).

Similarly, we denote

$B_{p}(f,x)=\left<f,x\right>p-\left<f,xp\right>=:[f,x,p].$

Then identity (3.10) can be interpreted as an identity similar to identity
(2.3).

Although such notations provide a neat mnemonic, we shall still keep the
notations $A_{p}(x,f)$ and $B_{p}(f,x)$ throughout in order to avoid any
confusions. The notation given in (3.11) will be used many times in the
sequel.

The following identities related to second associators are crucial in
applications.

######  Proposition 3.6.

Let $M,\ M^{\prime}$ be two left $\mathbb{O}$-modules. Then for all
$p\in\mathbb{O},\ x\in M$, and $f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$,
the following identities hold:

  1. (${\mathrm{i}}$)

$A_{\overline{p}}(x,f)=-A_{p}(x,f)$;

  2. (${\mathrm{ii}}$)

$A_{p^{2}}(x,f)=pA_{p}(x,f)+A_{p}(px,f)$;

  3. (${\mathrm{iii}}$)

$A_{p}(\overline{p}x,f)=pA_{p}(x,f),\qquad
A_{p}({p}x,f)=\overline{p}A_{p}(x,f)$;

  4. (${\mathrm{iv}}$)

$A_{p}(\overline{p}^{k}x,f)=p^{k}A_{p}(x,f),k\in\mathbb{N}$.

  5. (${\mathrm{v}}$)

$A_{p^{k+1}}(x,f)=pA_{p^{k}}(x,f)+\overline{p}^{k}A_{p}(x,f),k\in\mathbb{N}$.

Similar results hold for the second right associators.

###### Proof.

Assertion $({\mathrm{i}})$ follows from assertion $({\mathrm{iii}})$ of Lemma
3.4. Assertion $({\mathrm{ii}})$ follows from identity (3.9) with $q$ replaced
by $p$. Assertion $({\mathrm{iii}})$ follows also from identity (3.9) with $q$
replaced by $\overline{p}$.

Assertion $({\mathrm{iv}})$ can be proved by induction on $k$. Assertion
$({\mathrm{v}})$ follows by induction on $k$ and using assertion
$({\mathrm{iv}})$. ∎

To show the $\mathbb{O}$-module structures of the set of para-linear maps, we
shall define several kinds of octonionic scalar multiplications on
$\text{Hom}_{\mathbb{R}}(M,M^{\prime})$ according to the types of
$\mathbb{O}$-modules $M$ and $M^{\prime}$. The notation (3.11) will be used in
the following proposition for memory convenience.

######  Proposition 3.7.

Let $M,M^{\prime}$ be two $\mathbb{O}$-modules,
$f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$ and $r,p,q\in\mathbb{O}$.

  1. (${\mathrm{i}}$)

If $M$ is left module and $M^{\prime}$ is bimodule, we define

(3.12) $\displaystyle\left<x,f\odot r\right>:=\left<x,f\right>r-A_{r}(x,f).$

Then we have

(3.13) $\displaystyle A_{p}(x,f\odot r)$
$\displaystyle=[p,f(x),r]+A_{p}(x,f)r-A_{r}(px,f)+pA_{r}(x,f),$ (3.14)
$\displaystyle[f,p,q](x)$ $\displaystyle=[f(x),p,q]-A_{p}(x,f)q-A_{q}(x,f\odot
p)+A_{pq}(x,f).$

  2. (${\mathrm{ii}}$)

If $M$ is bimodule and $M^{\prime}$ is left module, we define

(3.15) $\displaystyle\left<x,r\odot f\right>:=\left<xr,f\right>+A_{r}(x,f).$

Then we have

(3.16) $\displaystyle A_{p}(x,r\odot f)$
$\displaystyle=f([p,x,r])+A_{p}(xr,f)+A_{r}(px,f)-pA_{r}(x,f),$ (3.17)
$\displaystyle[p,q,f](x)$
$\displaystyle=A_{pq}(x,f)-A_{q}(xp,f)-A_{p}(x,q\odot f)-f([x,p,q]).$

  3. (${\mathrm{iii}}$)

If $M$ is right module and $M^{\prime}$ is bimodule, we define

(3.18) $\displaystyle\left<r\odot f,x\right>:=r\left<f,x\right>+B_{r}(f,x).$

Then we have

(3.19) $\displaystyle B_{p}(r\odot f,x)$
$\displaystyle=[r,f(x),p]+B_{r}(f,xp)+rB_{p}(f,x)-B_{r}(f,xp),$ (3.20)
$\displaystyle[p,q,f](x)$ $\displaystyle=B_{pq}(f,x)-pB_{q}(f,x)-B_{p}(q\odot
f,x)+[p,q,f(x)].$

  4. (${\mathrm{iv}}$)

If $M$ is bimodule and $M^{\prime}$ is right module, we define

(3.21) $\displaystyle\left<f\odot r,x\right>:=\left<f,rx\right>-B_{r}(f,x).$

Then we have

(3.22) $\displaystyle B_{p}(f\odot r,x)$
$\displaystyle=f[r,x,p]-B_{r}(f,x)p+B_{p}(f,rx)+B_{r}(f,xp),$ (3.23)
$\displaystyle[f,p,q](x)$ $\displaystyle=B_{pq}(f,x)-B_{q}(f\odot
p,x)-B_{p}(f,qx)+f([p,q,x]).$

  5. (${\mathrm{v}}$)

If $M$ and $M^{\prime}$ are two bimodules, then with the left or right
multiplication defined by (3.12) and (3.15) we have

(3.24) $\displaystyle[q,f,p](x)$ $\displaystyle=A_{q}(x,f)p-A_{p}(x,q\odot
f)+A_{p}(xq,f)-A_{q}(x,f\odot p).$

Similarly, with the left or right multiplication defined by (3.18) and (3.21),
we have

(3.25) $\displaystyle[q,f,p](x)$ $\displaystyle=qB_{p}(f,x)-B_{p}(q\odot
f,x)+B_{q}(f,px)-B_{q}(f\odot p,x).$

###### Proof.

The results follows by straightforward calculations. We prove assertion
$({\mathrm{i}})$ for example. By definition, we have

$\displaystyle A_{p}(x,f\odot r)$ $\displaystyle=\left<px,f\odot
r\right>-p\left<x,f\odot r\right>$
$\displaystyle=\left<px,f\right>r-A_{r}(px,f)-p\left(f(x)r-A_{r}(x,f)\right)$
$\displaystyle=\left(pf(x)+A_{p}(x,f)\right)r-A_{r}(px,f)-p\left(f(x)r-A_{r}(x,f)\right)$
$\displaystyle=[p,f(x),r]+A_{p}(x,f)r-A_{r}(px,f)+pA_{r}(x,f).$

The rest proof runs in the same manner and is omitted. ∎

######  Remark 3.8.

Although we have defined the octonionic scalar multiplication on
$\text{Hom}_{\mathbb{R}}(M,M^{\prime})$, the set
$\text{Hom}_{\mathbb{R}}(M,M^{\prime})$ is not an $\mathbb{O}$-module in
general. Indeed, by definition (3.12), it may hold

$f\odot(r^{2})\neq(f\odot r)\odot r$

for some $r\in\mathbb{O}$ and $f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$.

The identities in Proposition 3.7 are all akin to identity (2.3). Those
identities about associators together with the real part structure of modules
play a crucial role in the sequel.

###  3.3. $\mathbb{O}$-para-linear map

In this subsection, we shall generalize the notion of para-linear functions
developed in our previous work [13] to arbitrary para-linear maps between
$\mathbb{O}$-modules.

Convention: The target module is always assumed to be an $\mathbb{O}$-bimodule
from now on, except where otherwise specified.

######  Definition 3.9.

Let $M$ be a left $\mathbb{O}$-module and $M^{\prime}$ an
$\mathbb{O}$-bimodule. A map $f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$ is
called (left) $\mathbb{O}$-para-linear if

(3.26) $\displaystyle\text{Re}\,A_{p}(x,f)=0$

for all $p\in\mathbb{O}$ and $x\in M$.

Similarly, we can also define the notion of right $\mathbb{O}$-para-linear map
with the left associator $A_{p}(x,f)$ replaced by the right associator
$B_{p}(f,x)$.

We denote by

$\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$

the set of (left) $\mathbb{O}$-para-linear maps and denote

$\text{End}_{\mathcal{LO}}(M):=\text{Hom}_{\mathcal{LO}}(M,M).$

In order to avoid any confusion, we use systematically the subscripts
$\mathcal{LO}$ and $\mathcal{RO}$ to indicate that the module under
consideration is a left or a right module. But we may omit the subscripts when
the module under consideration is understood well.

For $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, we shall only use the
notation $\left<x,f\right>$ to represent the value $f(x)$. Similarly, only the
notation $\left<f,x\right>$ is used for
$f\in\text{Hom}_{\mathcal{RO}}(M,M^{\prime})$. We shall state our results for
left $\mathbb{O}$-modules, while the similar results for right
$\mathbb{O}$-modules shall be collected in subsection 3.8.

Let $M$ be a left $\mathbb{O}$-module and $M^{\prime}$ an
$\mathbb{O}$-bimodule. For simplicity, we denote

$f_{\mathbb{R}}(x)=\text{Re}\,f(x)$

for any $x\in M$ and $f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$. In view of
(3.2), $f$ can be written as

$f(x)=f_{\mathbb{R}}(x)+\sum_{i=1}^{7}e_{i}f_{i}(x),$

where $f_{\mathbb{R}}(x)\in\text{Re}\,M^{\prime}$ and
$f_{i}(x)\in\text{Re}\,{M^{\prime}}\text{ for any }i=1,\dots,7$.

We now give a characterization of $\mathbb{O}$-para-linear maps in terms of
the real part operators. The proof runs in completely the same manner as in
our previous work [13].

######  Theorem 3.10.

Let $M$ be a left $\mathbb{O}$-module and $M^{\prime}$ an
$\mathbb{O}$-bimodule. Then the following are equivalent:

  1. (${\mathrm{i}}$)

$f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$;

  2. (${\mathrm{ii}}$)

$f_{i}(x)=f_{\mathbb{R}}(\overline{e_{i}}x)$ for any $i=0,\dots,7$;

  3. (${\mathrm{iii}}$)

$A_{p}(x,f)=\sum_{i=1}^{7}e_{i}f_{\mathbb{R}}([e_{i},p,x])$.

###### Proof.

We first prove that assertion $({\mathrm{i}})$ implies assertion
$({\mathrm{ii}})$. By definition, we have

$\displaystyle f_{\mathbb{R}}(\overline{e_{i}}x)$
$\displaystyle=\text{Re}\,f(\overline{e_{i}}x)$
$\displaystyle=\text{Re}\,(\overline{e_{i}}f(x))$
$\displaystyle=\sum_{j=0}^{7}\text{Re}\,\left(\overline{e_{i}}e_{j}f_{j}(x)\right).$

According to (3.5), we have

$\displaystyle f_{\mathbb{R}}(\overline{e_{i}}x)$
$\displaystyle=\sum_{j=0}^{7}\text{Re}\,\left(\overline{e_{i}}e_{j}\right)f_{j}(x)$
$\displaystyle=f_{i}(x).$

This proves that assertion $({\mathrm{i}})$ implies assertion
$({\mathrm{ii}})$. We next come to prove that assertion $({\mathrm{ii}})$
implies assertion $({\mathrm{iii}})$. Without loss of generality we can assume
$p=e_{i}$. From above computation and assertion $({\mathrm{ii}})$, we get

$\displaystyle A_{e_{i}}(x,f)$ $\displaystyle=f(e_{i}x)-e_{i}f(x)$
$\displaystyle=f_{\mathbb{R}}(e_{i}x)+\sum_{j=1}^{7}e_{j}f_{j}(e_{i}x)-\left(-f_{i}(x)+e_{i}f_{\mathbb{R}}(x)+\sum_{j,k=1}^{7}\epsilon_{ijk}e_{k}f_{j}(x)\right)$
$\displaystyle=f_{\mathbb{R}}(e_{i}x)+f_{i}(x)-\sum_{j=1}^{7}e_{j}f_{\mathbb{R}}(e_{j}(e_{i}x))-e_{i}f_{\mathbb{R}}(x)-\sum_{j,k=1}^{7}\epsilon_{ijk}e_{k}f_{j}(x)$
$\displaystyle=\left(-\sum_{j=1}^{7}e_{j}f_{\mathbb{R}}\left((e_{j}e_{i})x-[e_{j},e_{i},x]\right)\right)-e_{i}f_{\mathbb{R}}(x)+\sum_{j,k=1}^{7}\epsilon_{ijk}e_{k}f_{\mathbb{R}}(e_{j}x)$
$\displaystyle=\Big{(}-\sum_{j,k=1}^{7}e_{j}f_{\mathbb{R}}\left((\epsilon_{jik}e_{k}-\delta_{ij})x\right)\Big{)}+\sum_{j=1}^{7}e_{j}f_{\mathbb{R}}([e_{j},e_{i},x])-e_{i}f_{\mathbb{R}}(x)+\sum_{j,k=1}^{7}\epsilon_{ijk}e_{k}f_{\mathbb{R}}(e_{j}x)$
$\displaystyle=\sum_{j=1}^{7}e_{j}f_{\mathbb{R}}([e_{j},e_{i},x]).$

We used the fact that $\epsilon_{ijk}$ is alternative in the last line.

Finally we prove that assertion $({\mathrm{iii}})$ implies assertion
$({\mathrm{i}})$. This is trivial in view of Theorem 3.1. ∎

As an important consequence of Theorem 3.10, we find that the second
associator vanishes for associative elements and an $\mathbb{O}$-para-linear
map $f$ is uniquely determined by $f_{\mathbb{R}}$.

######  Corollary 3.11.

Assume that $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$.

  1. (${\mathrm{i}}$)

For each associative element $x\in\mathscr{A}(M)$, we have

$A_{p}(x,f)=0$

and hence $f(px)=pf(x)$ for all $p\in\mathbb{O}$.

  2. (${\mathrm{ii}}$)

$f_{\mathbb{R}}=0\iff f=0$.

######  Corollary 3.12.

Let $M$ be an $\mathbb{O}$-bimodule and
$f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$.

  1. (${\mathrm{i}}$)

If $f\lvert_{\text{Re}\,{M}}=0$, then $f=0$;

  2. (${\mathrm{ii}}$)

$f\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$ if and only if
$f\lvert_{\text{Re}\,{M}}\in\text{Hom}_{\mathbb{R}}(\text{Re}\,{M},\text{Re}\,{M^{\prime}})$.

###### Proof.

We first prove assertion $({\mathrm{i}})$. Since $M$ is an
$\mathbb{O}$-bimodule, each $x\in M$ can be expressed as

$x=x_{0}+\sum_{i=1}^{7}e_{i}x_{i},$

where $x_{j}\in\text{Re}\,M$ for any $j=0,\dots,7.$ According to assertion
$({\mathrm{i}})$ of Corollary 3.11, we obtain

$f(x)=f(x_{0})+\sum_{i=1}^{7}e_{i}f(x_{i}),$

so that the condition $f\lvert_{\text{Re}\,{M}}=0$ will yield $f=0$.

We come to prove assertion $({\mathrm{ii}})$. If
$f\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$, it follows from Lemma 3.3 that
$f(\text{Re}\,M)\subseteq\text{Re}\,M^{\prime}.$

Conversely, if $f(\text{Re}\,M)\subseteq\text{Re}\,M^{\prime}$, then by
assertion $({\mathrm{i}})$ of Corollary 3.11 again, for all $p\in\mathbb{O}$
we have

$\displaystyle f(px)$
$\displaystyle\xlongequal{}f(px_{0})+\sum_{i=1}^{7}f((pe_{i})x_{i})$
$\displaystyle\xlongequal{}pf(x_{0})+\sum_{i=1}^{7}(pe_{i})f(x_{i})$
$\displaystyle\xlongequal{}pf(x_{0})+\sum_{i=1}^{7}p(e_{i}f(x_{i}))$
$\displaystyle\xlongequal{}pf(x).$

∎

We now introduce two very useful bijections.

######  Lemma 3.13.

If $M$ is a left $\mathbb{O}$-module and $M^{\prime}$ an
$\mathbb{O}$-bimodule. Then we have a bijection of (left) para-linear lift
map:

$\displaystyle
l\text{-}\mathrm{lif}\,:\qquad\text{Hom}_{\mathbb{R}}(M,\text{Re}\,M^{\prime})$
$\displaystyle\to$ $\displaystyle\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$
(3.27) $\displaystyle g$ $\displaystyle\mapsto$
$\displaystyle(l\text{-}\mathrm{lif}\,g)(x):=g(x)-\sum_{i=1}^{7}e_{i}g(e_{i}x).$

Its inverse map is given by

$\displaystyle\text{Re}\,_{*}:\qquad\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$
$\displaystyle\to$
$\displaystyle\text{Hom}_{\mathbb{R}}(M,\text{Re}\,M^{\prime})$ (3.28)
$\displaystyle f$ $\displaystyle\mapsto$ $\displaystyle\text{Re}\,\circ
f=f_{\mathbb{R}}.$

###### Proof.

We first show that the $l\text{-}\mathrm{lif}\,$ map is well-defined. In view
of Theorem 3.10, $l\text{-}\mathrm{lif}\,(g)$ is a left para-linear map so
that $l\text{-}\mathrm{lif}\,(g)\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$.
It is direct to check that

$\text{Re}\,_{*}\circ l\text{-}\mathrm{lif}\,=\text{id},\quad
l\text{-}\mathrm{lif}\,\circ\text{Re}\,_{*}=\text{id}.$

∎

######  Lemma 3.14.

If $M$ and $M^{\prime}$ are two $\mathbb{O}$-bimodules. Then we have a
bijection of (left) para-linear extension:

$\displaystyle
l\text{-}\mathrm{ext}\,:\qquad\text{Hom}_{\mathbb{R}}(\text{Re}\,M,M^{\prime})$
$\displaystyle\to$ $\displaystyle\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$
(3.29) $\displaystyle g$ $\displaystyle\mapsto$ $\displaystyle
l\text{-}\mathrm{ext}\,g,$

which the $l\text{-}\mathrm{ext}\,$ map defined by

(3.30) $\displaystyle(l\text{-}\mathrm{ext}\,g)(px):=p(g(x))$

for all $p\in\mathbb{O}$ and all $x\in\text{Re}\,M.$ Its inverse map is given
by

$\displaystyle\text{Re}\,^{*}:\qquad\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$
$\displaystyle\to$
$\displaystyle\text{Hom}_{\mathbb{R}}(\text{Re}\,M,M^{\prime})$ (3.31)
$\displaystyle f$ $\displaystyle\mapsto$ $\displaystyle
f\circ\text{Re}\,=f|_{\text{Re}\,M}.$

###### Proof.

Based on Theorem 3.1, the para-linear extension is uniquely defined via
(3.30).

We need to prove that
$l\text{-}\mathrm{ext}\,g\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ for any
$g\in\text{Hom}_{\mathbb{R}}(\text{Re}\,M,M^{\prime})$. Let $p\in\mathbb{O}$
and

$x=x_{0}+\sum_{i=1}^{7}e_{i}x_{i}\in M.$

Then by definition, we have

$\displaystyle A_{p}(x,l\text{-}\mathrm{ext}\,g)$
$\displaystyle=(l\text{-}\mathrm{ext}\,g)(px)-p(l\text{-}\mathrm{ext}\,g)(x)$
$\displaystyle=(l\text{-}\mathrm{ext}\,g)(px_{0}+\sum_{i=1}^{7}(pe_{i})x_{i})-p(l\text{-}\mathrm{ext}\,g)(x_{0}+\sum_{i=1}^{7}e_{i}x_{i})$
$\displaystyle=pg(x_{0})+\sum_{i=1}^{7}(pe_{i})g(x_{i})-pg(x_{0})-\sum_{i=1}^{7}p(e_{i}g(x_{i}))$
$\displaystyle=\sum_{i=1}^{7}[p,e_{i},g(x_{i})].$

It follows that $\text{Re}\,A_{p}(x,l\text{-}\mathrm{ext}\,g)=0$ and thus
$l\text{-}\mathrm{ext}\,g\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$.

By definition, one can check that

$\text{Re}\,^{*}\circ l\text{-}\mathrm{ext}\,=\text{id},\quad
l\text{-}\mathrm{ext}\,\circ\text{Re}\,^{*}=\text{id}.$

This completes the proof. ∎

We shall denote

$\mathrm{lif}\,=l\text{-}\mathrm{lif}\,,\quad\mathrm{ext}\,=l\text{-}\mathrm{ext}\,$

for simple if no confusion arises.

###  3.4. Second associator of para-linear maps

In subsection 3.2, we have studied the second associator of real linear maps
over $\mathbb{O}$-modules. In this subsection, we focus on the properties of
the second associator related to $\mathbb{O}$-para-linear maps. Some
properties have appeared in [13] for $\mathbb{O}$-para-linear functions.

######  Proposition 3.15.

Let $M$ be a left $\mathbb{O}$-module and $M^{\prime}$ an
$\mathbb{O}$-bimodule. If $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, then
for any $p,q\in\mathbb{O}$ and $x\in M$ we have

(3.32) $\displaystyle pA_{p}(x,f)$ $\displaystyle=A_{p}(x,f)\overline{p},$
(3.33) $\displaystyle\text{Re}\,\left(A_{p}(x,f)q\right)$
$\displaystyle=-\text{Re}\,\left(A_{q}(x,f)p\right).$

###### Proof.

As usual, we write

$f(x)=f_{\mathbb{R}}(x)+\sum_{i=1}^{7}e_{i}f_{i}(x).$

We first prove (3.32). Without loss of generality we can assume
$p\in\mathit{Im}(\mathbb{O})$. Let $p=\sum_{i=1}^{7}p_{i}e_{i}$ with
$p_{i}\in\mathbb{R}$. Then by Theorem 3.10, we have

$pA_{p}(x,f)=\sum_{i,j=1}^{7}p_{i}e_{i}A_{p_{j}e_{j}}(x,f)=\sum_{i,j,k=1}^{7}p_{i}p_{j}e_{i}e_{k}f_{\mathbb{R}}([e_{k},e_{j},x]),$

and

$A_{p}(x,f)\overline{p}=\sum_{i,j=1}^{7}A_{p_{j}e_{j}}(x,f)\overline{(p_{i}e_{i})}=-\sum_{i,j,k=1}^{7}p_{i}p_{j}e_{k}e_{i}f_{\mathbb{R}}([e_{k},e_{j},x]).$

Since

$e_{i}e_{k}+e_{k}e_{i}=-2\delta_{ik},$

we conclude

$pA_{p}(x,f)-A_{p}(x,f)\overline{p}=-2\sum_{i,j=1}^{7}p_{i}p_{j}f_{\mathbb{R}}([e_{i},e_{j},x])=0.$

The last step used the skew symmetricity of the associator $[e_{i},e_{j},x]$
in $i,j$.

To prove (3.33), we first prove the following identity

(3.34)
$\displaystyle\text{Re}\,\left(f([p,q,x])\right)=\text{Re}\,\left(A_{p}(x,f){q}\right).$

It suffices to show the case where $q=e_{i}$ for each $i=1,\dots,7$. By direct
calculations, we have

$\displaystyle\text{Re}\,\left(A_{p}(x,f){q}\right)$
$\displaystyle=\text{Re}\,\left(\sum_{j=1}^{7}(e_{j}f_{\mathbb{R}}([e_{j},p,x]))e_{i}\right)$
by Theorem 3.10
$\displaystyle=\text{Re}\,\left(\sum_{j=1}^{7}f_{\mathbb{R}}([e_{j},p,x])e_{j}e_{i}\right)$
by (3.6) $\displaystyle=-f_{\mathbb{R}}([e_{i},p,x])$ by (3.5)
$\displaystyle=\text{Re}\,\left(f([p,q,x])\right).$

Due to (3.34), we conclude that (3.33) holds.

∎

######  Proposition 3.16.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules and
$f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$. Then for all $p$,
$q\in\mathbb{O}$, and $x\in M$ we have

(3.35) $\displaystyle\text{Re}\,\left(f([p,x,q])\right)$
$\displaystyle=\text{Re}\,\left(pA_{q}(x,f)\right).$

###### Proof.

As usual, we write

$f(x)=f_{\mathbb{R}}(x)+\sum_{i=1}^{7}e_{i}f_{i}(x),\qquad
x=x_{0}+\sum_{i=1}^{7}e_{i}x_{i}.$

Without loss of generality we can assume that $p=e_{i}$ with
$i\in\\{1,\dots,7\\}$. It follows from Theorem 3.10 that

$\displaystyle\text{Re}\,\left(e_{i}A_{q}(x,f)\right)$
$\displaystyle=\text{Re}\,\left(\sum_{j=1}^{7}e_{i}e_{j}f_{\mathbb{R}}([e_{j},q,x])\right)$
$\displaystyle=-f_{\mathbb{R}}([e_{i},q,x])$
$\displaystyle=\text{Re}\,\left(f([e_{i},x,q])\right).$

This proves (3.35).

∎

###  3.5. $\mathbb{O}$-module structure of
$\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$

Now we can provide $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ with an
$\mathbb{O}$-module structure.

######  Theorem 3.17.

Let $M$ be a left $\mathbb{O}$-module and $M^{\prime}$ an
$\mathbb{O}$-bimodule. Then $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ admits a
right $\mathbb{O}$-module structure with respect to the right scalar
multiplication defined by (3.12), i.e.,

$\left<x,f\odot r\right>:=\left<x,f\right>r-A_{r}(x,f)$

for all $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, $r\in\mathbb{O}$, and
$x\in M$.

###### Proof.

First we claim that $f\odot r\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ for
all $r\in\mathbb{O}$. Recall identity (3.13)

$\displaystyle A_{p}(x,f\odot r)$
$\displaystyle=[p,f(x),r]+A_{p}(x,f)r-A_{r}(px,f)+pA_{r}(x,f).$

In order to deduce $\text{Re}\,A_{p}(x,f\odot r)=0$, we only need to show that

$\text{Re}\,(A_{p}(x,f)r)=-\text{Re}\,(pA_{r}(x,f)).$

This follows from identity (3.33) and the fact

$\text{Re}\,(pA_{r}(x,f))=\text{Re}\,(A_{r}(x,f)p).$

Next we show $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ is a right
$\mathbb{O}$-module. It suffices to prove

$(f\odot r)\odot r=f\odot r^{2}$

for any $r\in\mathbb{O}$. By definition (3.12), we have

$\displaystyle\left<x,(f\odot r)\odot r\right>$ $\displaystyle=\left<x,f\odot
r\right>r-A_{r}(x,f\odot r)$ (3.36)
$\displaystyle=\left<x,f\right>r^{2}-A_{r}(x,f)r-A_{r}(x,f\odot r)$ and (3.37)
$\displaystyle\left<x,f\odot r^{2}\right>$
$\displaystyle=\left<x,f\right>r^{2}-A_{r^{2}}(x,f).$

Since $f,f\odot r\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, by Definition
3.9 we have

$\text{Re}\,A_{r}(x,f\odot r)=0,\quad\text{Re}\,A_{r^{2}}(x,f)=0.$

Combining (3.36) and (3.37), we get

$\displaystyle\text{Re}\,\left<x,(f\odot r)\odot r-f\odot r^{2}\right>$
$\displaystyle=-\text{Re}\,(A_{r}(x,f)r)$
$\displaystyle=\text{Re}\,(A_{\overline{r}}(x,f)r)$ assertion $({\mathrm{i}})$
of Proposition 3.6
$\displaystyle=\text{Re}\,(\overline{r}A_{\overline{r}}(x,f))$ by identity
(3.32) $\displaystyle=\text{Re}\,A_{\overline{r}}(rx,f)$ assertion
$({\mathrm{iii}})$ of Proposition 3.6 $\displaystyle=0.$

Note that

$(f\odot r)\odot r-f\odot r^{2}\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime}).$

It follows from assertion $({\mathrm{ii}})$ of Corollary 3.11 that $(f\odot
r)\odot r=f\odot r^{2}$. This completes the proof. ∎

######  Theorem 3.18.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules. Then
$\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ admits a left $\mathbb{O}$-module
structure with the left scalar multiplication defined by (3.15), i.e.,

$\left<x,r\odot f\right>:=\left<xr,f\right>+A_{r}(x,f)$

for all $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, $r\in\mathbb{O}$, and
$x\in M$.

###### Proof.

First we claim that

$r\odot f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$

for all $r\in\mathbb{O}$ and $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$.
Indeed, using identities (3.35) and (3.16), we obtain

$\text{Re}\,\left(A_{p}(x,r\odot
f)\right)=\text{Re}\,(f([p,x,r]))-\text{Re}\,(pA_{r}(x,f))=0.$

Next we show that $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ is a left
$\mathbb{O}$-module. It suffices to prove $r\odot(r\odot f)=r^{2}\odot f$ for
all $r\in\mathbb{O}$. For any $x\in M$, by definition (3.15) we have

(3.38) $\displaystyle\left<x,r\odot(r\odot f)\right>=\left<xr,r\odot
f\right>+A_{r}(x,r\odot f)=\left<xr^{2},f\right>+A_{r}(xr,f)+A_{r}(x,r\odot
f).$

Since $f,r\odot f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, we have

$\displaystyle\text{Re}\,\left<x,r\odot(r\odot f)-r^{2}\odot f\right>$
$\displaystyle=\text{Re}\,\left(\left<xr^{2},f\right>+A_{r}(xr,f)+A_{r}(x,r\odot
f)\right)-\text{Re}\,\left(\left<xr^{2},f\right>+A_{r^{2}}(x,f)\right)$
$\displaystyle=\text{Re}\,(A_{r}(xr,f)+A_{r}(x,r\odot f)-A_{r^{2}}(x,f))$
$\displaystyle=0.$

It follows from Corollary 3.11 that $r\odot(r\odot f)=r^{2}\odot f$. This
finishes the proof. ∎

As a consequence, we have the following property about the second associators.

######  Proposition 3.19.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules and
$f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$. Then for all , $x\in M$ and
$r\in\mathbb{O}$ we have

(3.39) $\displaystyle A_{r}(xr,f)$ $\displaystyle=rA_{r}(x,f);$ (3.40)
$\displaystyle A_{r}(x,f)\overline{r}$ $\displaystyle=A_{r}(x{r},f).$

###### Proof.

Let $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, $x\in M$ and
$r\in\mathbb{O}$. Combining (3.38) with (3.16), we obtain

$\left<x,r\odot(r\odot
f)\right>=\left<xr^{2},f\right>+2A_{r}(xr,f)+A_{r}(rx,f)-rA_{r}(x,f).$

In view of assertion $({\mathrm{ii}})$ of Proposition 3.6, we get

$\displaystyle\left<x,r^{2}\odot f\right>$
$\displaystyle=\left<xr^{2},f\right>+A_{r^{2}}(x,f)$
$\displaystyle=\left<xr^{2},f\right>+rA_{r}(x,f)+A_{r}(rx,f).$

From Theorem 3.18, we know that $r^{2}\odot f=r\odot(r\odot f)$. Therefore we
have

$2A_{r}(xr,f)-rA_{r}(x,f)=rA_{r}(x,f).$

This implies identity (3.39) holds. Combining (3.32) with (3.39), we conclude

$A_{p}(x,f)\overline{p}={p}A_{p}(x,f)=A_{p}(x{p},f)$

as desired. ∎

######  Remark 3.20.

Propositions 3.6, 3.15, 3.16 and 3.19 generalize Lemma 3.9 in [9] from the
specific $\mathbb{O}$-module $\mathbb{O}$ itself to the general
$\mathbb{O}$-modules.

By Theorems 3.17 and 3.18, $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ admits a
right $\mathbb{O}$-module structure as well as a left $\mathbb{O}$-module
structure. Then a natural question arises whether the left and right scalar
multiplications in $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ are compatible so
that $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ admits an $\mathbb{O}$-bimodule
structure. We shall give an affirmative answer.

######  Lemma 3.21.

Let $M,$ $M^{\prime}$ be two $\mathbb{O}$-bimodules and
$f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$. Then with respect to the scalar
multiplications (3.12) and (3.15), we have

$[p,q,f]=[q,f,p]=[f,p,q]$

for all $p,q\in\mathbb{O}$.

###### Proof.

In view of Corollary 3.12, we only need to show that

$[p,q,f](x)=[q,f,p](x)=[f,p,q](x)$

for all $x\in\text{Re}\,{M}$.

Let $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ and fix an arbitrary
$x\in\text{Re}\,M$. In view of assertion $({\mathrm{i}})$ of Corollary 3.11,
we immediately conclude from identity (3.14) that

$[f,p,q](x)=[f(x),p,q].$

Similarly, from (3.17) we have

$[p,q,f](x)=-A_{q}(xp,f).$

Since $x\in\text{Re}\,M$, it follows from Theorem 3.1 that

$\displaystyle A_{p}(xq,f)$
$\displaystyle=\left<p(xq),f\right>-p\left<xq,f\right>$
$\displaystyle=\left<(pq)x,f\right>-p\left<qx,f\right>$
$\displaystyle=(pq)f(x)-p(qf(x))$ $\displaystyle=[p,q,f(x)].$

This implies that

${[p,q,f]}(x)=[p,q,f(x)]=[f(x),p,q]=[f,p,q](x).$

It remains to show

$[q,f,p](x)=[p,q,f](x).$

We conclude from identity (3.24) that

${[q,f,p]}(x)=A_{p}(xq,f).$

Thus we obtain $[q,f,p](x)=[p,q,f](x)$ as desired. ∎

As a consequence, we have

######  Theorem 3.22.

Let $M,$ $M^{\prime}$ be two $\mathbb{O}$-bimodules. Then
$\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ admits a canonical
$\mathbb{O}$-bimodule structure.

###  3.6. Real part of $\mathbb{O}$-para-linear maps

We assume that $M$, $M^{\prime}$ are two $\mathbb{O}$-bimodules in this
subsection. We come to study the real part structure of the
$\mathbb{O}$-bimodule $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$.

######  Theorem 3.23.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules. Then

$\text{Re}\,(\text{Hom}_{\mathcal{LO}}(M,M^{\prime}))=\text{Hom}_{\mathbb{O}}(M,M^{\prime}).$

More precisely, for all $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, we have

(3.41)
$\displaystyle\text{Re}\,f=\mathrm{ext}\,(f_{\mathbb{R}}|_{\text{Re}\,M}).$

###### Proof.

We first prove that

(3.42)
$\displaystyle\text{Hom}_{\mathbb{O}}(M,M^{\prime})\subseteq\text{Re}\,{\text{Hom}_{\mathcal{LO}}(M,M^{\prime})}.$

For any $f\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$, by definition we have
$A_{p}(x,f)=0$ for all $p\in\mathbb{O}$ and $x\in M$. Hence we conclude from
identity (3.14) that

$\displaystyle[f,p,q](x)$ $\displaystyle=[f(x),p,q]-A_{q}(x,f\odot p).$

This yields

$\text{Re}\,([f,p,q](x))=0$

for all $x\in M$. Notice that
$[f,p,q]\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ is a para-linear map due
to Theorem 3.17. It follows from Corollary 3.11 that $[f,p,q]=0$. This means

$f\in\mathscr{A}(\text{Hom}_{\mathcal{LO}}(M,M^{\prime})).$

Since $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ is an $\mathbb{O}$-bimodule,
it follows from (3.3) that

$f\in\text{Re}\,{\text{Hom}_{\mathcal{LO}}(M,M^{\prime})}.$

Conversely, if $f\in\text{Re}\,{\text{Hom}_{\mathcal{LO}}(M,M^{\prime})}$,
then for all $x\in M$,

$[f,p,q](x)=0.$

Fix $x\in\text{Re}\,{M}$. It follows from identity (3.14) and Corollary 3.11
that

$[f(x),p,q]=A_{p}(x,f)q+A_{q}(x,f\odot p)-A_{pq}(x,f)=0$

for all $p,q\in\mathbb{O}$. This means $f(x)\in\text{Re}\,{M^{\prime}}$ and
hence we obtain

$f(\text{Re}\,{M})\subseteq\text{Re}\,{M^{\prime}}.$

By assertion $({\mathrm{ii}})$ of Corollary 3.12, we conclude that
$f\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$. This proves

(3.43)
$\displaystyle\text{Re}\,{\text{Hom}_{\mathcal{LO}}(M,M^{\prime})}\subseteq\text{Hom}_{\mathbb{O}}(M,M^{\prime}).$

Combining (3.42) and (3.43), we conclude

$\text{Re}\,{\text{Hom}_{\mathcal{LO}}(M,M^{\prime})}=\text{Hom}_{\mathbb{O}}(M,M^{\prime}).$

We now prove (3.41). For any $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, we
get an $\mathbb{R}$-linear map

$f_{\mathbb{R}}|_{\text{Re}\,M}:\text{Re}\,M\to M^{\prime}.$

By the definition of the $\mathrm{ext}\,$ map, we have

$\mathrm{ext}\,(f_{\mathbb{R}}|_{\text{Re}\,M})\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime}).$

Because of Corollary 3.12, to show (3.41) it suffices to show that

$\left<x,\text{Re}\,f\right>=\left<x,f_{\mathbb{R}}|_{\text{Re}\,M}\right>$

for all $x\in\text{Re}\,M$. Fix an arbitrary $x\in\text{Re}\,M$. Since
$\text{Re}\,f$ is $\mathbb{O}$-linear, we conclude that

$(\text{Re}\,f)(x)\in\text{Re}\,M^{\prime}.$

Therefore,

$\displaystyle\left<x,\text{Re}\,f\right>$
$\displaystyle=\text{Re}\,\left<x,\text{Re}\,f\right>$ Re is a projection
operator
$\displaystyle=\text{Re}\,\Big{(}\langle{x},{\frac{5}{12}f}\rangle-\langle{x},{\frac{1}{12}\sum_{i=1}^{7}e_{i}\odot
f\odot e_{i}}\rangle\Big{)}$ by (3.4)
$\displaystyle=\frac{5}{12}f_{\mathbb{R}}(x)-\frac{1}{12}\text{Re}\,\sum_{i=1}^{7}f(xe_{i})e_{i}$
using definitions (3.12) and (3.15)
$\displaystyle=\frac{5}{12}f_{\mathbb{R}}(x)-\frac{1}{12}\text{Re}\,\sum_{i=1}^{7}e_{i}f(xe_{i})$
using (3.6)
$\displaystyle=\frac{5}{12}f_{\mathbb{R}}(x)-\frac{1}{12}\sum_{i=1}^{7}f_{\mathbb{R}}(e_{i}xe_{i})$
using $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$
$\displaystyle=f_{\mathbb{R}}(\text{Re}\,x)$ $\displaystyle=f_{\mathbb{R}}(x)$
$\displaystyle=f_{\mathbb{R}}|_{\text{Re}\,M}(x).$

This completes the proof. ∎

######  Remark 3.24.

It follows from assertion $({\mathrm{iii}})$ of Theorem 3.1 that for any
$f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, we have

$f=\sum_{i=0}^{7}e_{i}\odot\text{Re}\,(\overline{e_{i}}\odot
f)=:\sum_{i=0}^{7}e_{i}\odot f_{(i)}.$

It is worth pointing out that the following seemingly natural equality

$(e_{i}\odot f_{(i)})(x)=e_{i}f_{(i)}(x)$

is not correct since the map under considered is left para-linear. It should
be modified as

(3.44) $\displaystyle(e_{i}\odot f_{(i)})(x)=f_{(i)}(x)e_{i}.$

In fact, since $f_{(i)}\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$ is
$\mathbb{O}$-linear, we have

$\displaystyle(e_{i}\odot f_{(i)})(x)$
$\displaystyle=\left<xe_{i},f_{(i)}\right>+A_{e_{i}}(x,f_{(i)})$ by definition
(3.18) $\displaystyle=f_{(i)}(xe_{i})$ using $A_{e_{i}}(x,f_{(i)})=0$
$\displaystyle=f_{(i)}(x)e_{i}$ using (3.8).

Theorem 3.23 leads to an important and useful uniqueness result.

######  Corollary 3.25.

Under the assumptions above, we have

$\text{Re}\,f=0\mbox{\ if \ and \ only \ if \ }\text{Re}\,f(x)=0\text{ for all
}x\in\text{Re}\,{M}.$

###### Proof.

The condition in the right hand side above is equivalent to

(3.45) $\displaystyle f_{\mathbb{R}}|_{\text{Re}\,M}=0.$

It follows from Corollary 3.12 and (3.41) that the condition $\text{Re}\,f=0$
is also equivalent to (3.45). This completes the proof. ∎

###  3.7. Regular composition of $\mathbb{O}$-para-linear maps

Since the para-linearity does not preserve under the ordinary composition of
two para-linear maps, we need to introduce a new composition.

In this subsection, we always assume that $M,\ M^{\prime}$ are two left
$\mathbb{O}$-modules and $\ M^{\prime\prime}$ is an $\mathbb{O}$-bimodule.

######  Definition 3.26.

Let $f\in\text{Hom}_{\mathbb{R}}(M^{\prime},M^{\prime\prime})$ and
$g\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$. The (left) regular composition of
$f$ and $g$ is defined as

$f\circledcirc_{l}g(x):=f\circ g(x)+[f,g,x],$

where

(3.46)
$\displaystyle[f,g,x]:=-\sum_{j=1}^{7}e_{j}\text{Re}\,\big{(}f(A_{e_{j}}(x,g))\big{)}.$

######  Remark 3.27.

We shall just denote $f\circledcirc_{l}g=f\circledcirc g$ if there is no
confusion. By direct calculation, we have an identity about associators
concerning five terms:

(3.47) $A_{p}(x,f\circledcirc
g)=A_{p}(g(x),f)+f(A_{p}(x,g))+[f,g,px]-p[f,g,x].$

We shall show in Proposition 3.29 that the para-linearity preserves under the
regular compositions. This fact justifies our Definition 3.26. The point is
that the regular composition is non-associative in contrast to the classical
composition. This new composition plays a central role in the theory of non-
associative categories.

We give a sufficient condition for the regular composition becoming the
ordinary composition.

######  Lemma 3.28.

Let $f\in\text{Hom}_{\mathbb{R}}(M^{\prime},M^{\prime\prime})$ and
$g\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$. Suppose one of the following
conditions holds

  1. (${\mathrm{i}}$)

$g\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$;

  2. (${\mathrm{ii}}$)

$M^{\prime}$ is an $\mathbb{O}$-bimodule, $x\in\text{Re}\,{M}$ and
$g\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$;

  3. (${\mathrm{iii}}$)

$M^{\prime}$ is an $\mathbb{O}$-bimodule,
$f\in\text{Hom}_{\mathbb{O}}(M^{\prime},M^{\prime\prime})$ and
$g\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$.

We then have

$[f,g,x]=0$

for all $x\in M$ and

$f\circledcirc g=f\circ g.$

###### Proof.

It suffices to show that $[f,g,x]=0$ for all $x\in M$ in each case.

If $g\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$ then $[f,g,x]=0$ due to the
fact that $A_{e_{j}}(x,g)=0$.

Let $M^{\prime}$ be an $\mathbb{O}$-bimodule. If $x\in\text{Re}\,{M}$ and
$g\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, by Corollary 3.11 we conclude

$A_{e_{j}}(x,g)=0.$

This yields $[f,g,x]=0$.

If $f\in\text{Hom}_{\mathbb{O}}(M^{\prime},M^{\prime\prime})$ and
$g\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, we then conclude from Lemma 3.3
that

$\text{Re}\,f(A_{e_{j}}(x,g))=f(\text{Re}\,A_{e_{j}}(x,g))$

and from Definition 3.9 that

$\text{Re}\,A_{e_{j}}(x,g)=0.$

Therefore,

$\text{Re}\,f(A_{e_{j}}(x,g))=f(\text{Re}\,A_{e_{j}}(x,g))=0.$

Thus by (3.46), we have $[f,g,x]=0$. This proves the lemma. ∎

######  Proposition 3.29.

If $f\in\text{Hom}_{\mathcal{LO}}(M^{\prime},M^{\prime\prime})$ and
$g\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$, then

$f\circledcirc g\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime\prime}).$

###### Proof.

We need to show that

$\text{Re}\,A_{p}(x,f\circledcirc g)=0$

for all $p\in\mathbb{O}$ and $x\in M$. In view of identity (3.47), it is
sufficient to prove that

$\text{Re}\,(f(A_{p}(x,g))-p[f,g,x])=0.$

Without loss of generality we can assume $p=e_{k}$. In view of (3.1),

$\displaystyle\text{Re}\,\big{(}f(A_{e_{k}}(x,g))-e_{k}[f,g,x]\big{)}=\text{Re}\,f(A_{e_{k}}(x,g))+\text{Re}\,\Bigl{(}e_{k}\sum_{j=1}^{7}e_{j}\text{Re}\,\big{(}f(A_{e_{j}}(x,g))\big{)}\Bigl{)}=0.$

This completes the proof. ∎

For later use, we give some relations among various associators.

######  Lemma 3.30.

Let $M,M^{\prime},M^{\prime\prime},M^{\prime\prime\prime}$ be
$\mathbb{O}$-bimodules and let

$h\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime}),\quad
g\in\text{Hom}_{\mathcal{LO}}(M^{\prime},M^{\prime\prime}),\quad
f\in\text{Hom}_{\mathcal{LO}}(M^{\prime\prime},M^{\prime\prime\prime}).$

Then for all $x\in M$, we have

(3.48) $\displaystyle[f,g,h](x)+f[g,h,x]=[f,g,h(x)]-[f,g\circledcirc
h,x]+[f\circledcirc g,h,x],$

where

$[f,g,h]:=(f\circledcirc g)\circledcirc h-f\circledcirc(g\circledcirc h).$

Moreover, we have the following properties.

  * •

Let $f,g,h$ be as above. Then

$\text{Re}\,[f,g,h]=0;$

  * •

If one of $f,g,h$ is $\mathbb{O}$-linear, then

$[f,g,h]=0.$

###### Proof.

Identity (3.48) follows by direct calculation. Combining (3.48) with assertion
$({\mathrm{ii}})$ of Lemma 3.28, we deduce that

$\text{Re}\,([f,g,h](x))=\text{Re}\,[f,g,h(x)]=0\text{ for all
}x\in\text{Re}\,{M}.$

It then follows from Corollary 3.25 that $\text{Re}\,[f,g,h]=0$.

Suppose $f$ is an $\mathbb{O}$-linear map. For any $x\in M$, by identity
(3.48) and assertion $({\mathrm{iii}})$ of Lemma 3.28, we have

(3.49) $\displaystyle[f,g,h](x)=[f\circ g,h,x]-f([g,h,x]).$

Then by definition (3.46), the right hand side above becomes

$\sum_{i=1}^{7}e_{i}\text{Re}\,f\circ
g(A_{e_{i}}(x,h))-f\left(\sum_{i=1}^{7}e_{i}\text{Re}\,g(A_{e_{i}}(x,h))\right).$

In view of assertion $({\mathrm{i}})$ of Corollary 3.11, the second term above
becomes

(3.50) $\displaystyle
f\left(\sum_{i=1}^{7}e_{i}\text{Re}\,g(A_{e_{i}}(x,h))\right)=\sum_{i=1}^{7}e_{i}f(\text{Re}\,g(A_{e_{i}}(x,h))).$

By Lemma 3.3, we conclude

(3.51) $\displaystyle\sum_{i=1}^{7}e_{i}\text{Re}\,f\circ
g(A_{e_{i}}(x,h))=\sum_{i=1}^{7}e_{i}f(\text{Re}\,g(A_{e_{i}}(x,h))).$

Combining (3.49), (3.50) and (3.51), we obtain

$[f,g,h](x)=0.$

We have proved the identity above when $f$ is $\mathbb{O}$-linear. ∎

Finally, we establish some properties of the right multiplication operators in
an $\mathbb{O}$-bimodule $M$

$\displaystyle R_{p}:M$ $\displaystyle\to$ $\displaystyle M$ $\displaystyle x$
$\displaystyle\mapsto$ $\displaystyle xp$

for the use in the sequel.

######  Lemma 3.31.

Let $M$ be a left $\mathbb{O}$-module and $M^{\prime}$ an
$\mathbb{O}$-bimodule. Then for all $p\in\mathbb{O}$, $x\in M$ and
$f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, we have

  1. (${\mathrm{i}}$)

$[R_{p},f,x]=-A_{p}(x,f)$;

  2. (${\mathrm{ii}}$)

$R_{p}\circledcirc f=f\odot p$.

###### Proof.

We first prove assertion $({\mathrm{i}})$. From (3.33), we have

$\displaystyle[R_{p},f,x]$
$\displaystyle=-\sum_{j=1}^{7}e_{j}\text{Re}\,\big{(}A_{e_{j}}(x,f)p\big{)}$
$\displaystyle=\sum_{j=1}^{7}e_{j}\text{Re}\,\big{(}A_{p}(x,f)e_{j}\big{)}$
$\displaystyle=-A_{p}(x,f).$

Now we prove assertion $({\mathrm{ii}})$. By definition (3.12), we get

$\displaystyle R_{p}\circledcirc
f(x)=f(x)p+[R_{p},f,x]=f(x)p-A_{p}(x,f)=\left<x,f\odot p\right>.$

This completes the proof. ∎

######  Lemma 3.32.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules. Then for all
$p\in\mathbb{O},\,x\in M$ and $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$,
we have

  1. (${\mathrm{i}}$)

$[f,R_{p},x]=A_{p}(x,f)$;

  2. (${\mathrm{ii}}$)

$f\circledcirc R_{p}=p\odot f$.

###### Proof.

We first prove assertion $({\mathrm{i}})$. Notice that

$A_{e_{j}}(x,R_{p})=[e_{j},x,p]$

for each $j=1,\dots,7$. Combining this with (3.35), we get

$\displaystyle[f,R_{p},x]$
$\displaystyle=-\sum_{j=1}^{7}e_{j}\text{Re}\,\big{(}f(A_{e_{j}}(x,R_{p}))\big{)}$
$\displaystyle=-\sum_{j=1}^{7}e_{j}\text{Re}\,\big{(}f([e_{j},x,p])\big{)}$
$\displaystyle=-\sum_{j=1}^{7}e_{j}\text{Re}\,\big{(}e_{j}A_{p}(x,f)\big{)}$
$\displaystyle=A_{p}(x,f).$

We come to prove assertion $({\mathrm{ii}})$. By definition (3.15), we have

$\displaystyle f\circledcirc
R_{p}=f(xp)+[f,R_{p},x]=f(xp)+A_{p}(x,f)=\left<x,p\odot f\right>.$

This finishes the proof. ∎

###  3.8. Right para-linear maps

The results of right para-linear maps are all parallel with the left cases.
Given a right $\mathbb{O}$-module, we can endow it with a natural left
$\mathbb{O}$-module structure so that we can interpret a right para-linear map
as a left para-linear map. Thus we can establish these results via duality.

Let $M$ be a right $\mathbb{O}$-module. It induces a left scalar
multiplication on $M$ as follows:

$p\cdot x:=x\overline{p},\quad\text{for all $x\in M$, $p\in\mathbb{O}$}.$

One can check that this is indeed a left $\mathbb{O}$-module structure on $M$.
Similarly, a left $\mathbb{O}$-module also can induce a right
$\mathbb{O}$-module structure. And an $\mathbb{O}$-bimodule can induce a new
$\mathbb{O}$-bimodule structure. The induced module is denoted by $M^{C}$ for
all cases.

Let $M$ and $N$ be two right $\mathbb{O}$-modules. For any

$f\in\text{Hom}_{\mathbb{R}}(M,N),$

we can regard $f$ as a map between the left $\mathbb{O}$-modules $M^{C}$ and
$N^{C}$, denoted by

$f^{C}\in\text{Hom}_{\mathbb{R}}(M^{C},N^{C}).$

Moreover, their associators have the following relation

$B_{p}(f,x)=A_{p}(x,f^{C})$

for any $x\in M$ and $p\in\mathbb{O}$. This can be checked by direct
calculation

$\displaystyle f^{C}(p\cdot
x)=f(x\overline{p})=f(x)\overline{p}-B_{\overline{p}}(f,x)=p\cdot
f^{C}(x)+B_{p}(f,x).$

Now we can state the dual version of Theorem 3.10.

######  Theorem 3.33.

Let $M$ be a right $\mathbb{O}$-module and $M^{\prime}$ be an
$\mathbb{O}$-bimodule. Suppose that
$f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$ and write

$f(x)=f_{\mathbb{R}}(x)+\sum_{i=1}^{7}f_{i}(x)e_{i}.$

Then the following are equivalent:

  1. (${\mathrm{i}}$)

$f\in\text{Hom}_{\mathcal{RO}}(M,M^{\prime})$;

  2. (${\mathrm{ii}}$)

$f_{i}(x)=-f_{\mathbb{R}}(xe_{i}),\quad i=1,\dots,7$;

  3. (${\mathrm{iii}}$)

$B_{p}(f,x)=\sum_{i=1}^{7}f_{\mathbb{R}}([x,p,e_{i}])e_{i}$.

Similarly, we can also define the related bijections of right para-linear lift
and right para-linear extension. For a real linear map
$f\in\text{Hom}_{\mathbb{R}}(M,N)$, which can be viewed as
$f^{C}\in\text{Hom}_{\mathbb{R}}(M^{C},N^{C})$ as well, it follows from
definitions (3.12) and (3.18) that

$\left<x,f^{C}\odot r\right>=f^{C}(x)r-A_{r}(x,f^{C})=\overline{r}\cdot
f(x)+B_{\overline{r}}(f,x)=\left<\overline{r}\odot f,x\right>.$

Hence it holds

$f^{C}\odot r=\overline{r}\odot f.$

Similarly, we also have

$r\odot f^{C}=f\odot\overline{r}.$

Therefore, we have the following dual versions of Theorems 3.17, 3.22 and
3.23.

######  Theorem 3.34.

Let $M$ be a right $\mathbb{O}$-module and $M^{\prime}$ an
$\mathbb{O}$-bimodule. Then $\text{Hom}_{\mathcal{RO}}(M,M^{\prime})$ admits a
left $\mathbb{O}$-module structure with the left scalar multiplication defined
by (3.18).

######  Theorem 3.35.

Let $M,$ $M^{\prime}$ be two $\mathbb{O}$-bimodules. Then
$\text{Hom}_{\mathcal{RO}}(M,M^{\prime})$ admits an $\mathbb{O}$-bimodule
structure with the left scalar multiplication defined by (3.18) and the right
scalar multiplication defined by (3.21).

######  Theorem 3.36.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules. Then

(3.52) $\displaystyle\text{Re}\,{\text{Hom}_{\mathcal{RO}}(M,M^{\prime})}$
$\displaystyle=\text{Hom}_{\mathbb{O}}(M,M^{\prime})$

We now define right regular compositions also in a dual manner.

######  Definition 3.37.

Let $M,\ M^{\prime}$ be two right $\mathbb{O}$-modules and $\
M^{\prime\prime}$ an $\mathbb{O}$-bimodule. If
$f\in\text{Hom}_{\mathbb{R}}(M^{\prime},M^{\prime\prime})$ and
$g\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$, we define their (right) regular
composition as

$(f\circledcirc_{r}g)\,(x):=(f\circ g)(x)+[x,f,g],$

where

$[x,f,g]:=\sum_{j=1}^{7}\Big{(}\text{Re}\,f\big{(}B_{e_{j}}(g,x)\big{)}\Big{)}e_{j}.$

######  Remark 3.38.

By direct calculation, for right regular compositions we have

(3.53)
$B_{p}(f\circledcirc_{r}g,x)=B_{p}(f,g(x))+f(B_{p}(g,x))-[xp,f,g]+[x,f,g]p.$

We point out that the associator defined in the regular composition of right
para-linear maps are the same with the left one essentially. Indeed, it holds

$\displaystyle[x,f,g]=[f^{C},g^{C},x].$

This can be checked by direct calculations.

We shall omit the subscripts $``l"$ and $``r"$ in regular compositions for
simplicity, if the module is understood well.

We now discuss the relations between left and right para-linear maps over
$\mathbb{O}$-bimodules. We first introduce a notion of transpose of para-
linear maps, which makes us transform a left para-linear map into a right one.
Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules from now on.

######  Definition 3.39.

Let $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ be a left para-linear map. A
right para-linear map $g\in\text{Hom}_{\mathcal{RO}}(M,M^{\prime})$ is called
a (right) transpose of $f$ if

$\text{Re}\,\big{(}f(x)-g(x)\big{)}=0\text{ \quad for\ all }x\in M.$

One can define (left) transpose similarly.

######  Remark 3.40.

By Theorems 3.10 and 3.33, there exists a unique (right) transpose for every
left para-linear map $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, denoted by
$\hat{f}$. More precisely,

(3.54) $\displaystyle\hat{f}=r\text{-}\mathrm{lif}\,(f_{\mathbb{R}}).$

Similar results hold for left transpose. The left transpose of a right para-
linear map $f$ will still be denoted by $\hat{f}$.

The transpose map turns out to be an isomorphism.

######  Theorem 3.41.

Let $M,M^{\prime}$ be two $\mathbb{O}$-bimodules. Then

$\displaystyle\varphi:\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$
$\displaystyle\to$ $\displaystyle\text{Hom}_{\mathcal{RO}}(M,M^{\prime})$
$\displaystyle f$ $\displaystyle\mapsto$ $\displaystyle\hat{f}$

is an isomorphism of $\mathbb{O}$-bimodules.

###### Proof.

It is easy to see that $\hat{\hat{f}}=f$. It remains to show that $\varphi$ is
an $\mathbb{O}$-homomorphism. Let
$f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ be a left para-linear map. For
any $x\in\text{Re}\,M$, it follows from Theorem 3.10 that

$\displaystyle\left<\varphi(e_{i}\odot f),x\right>$
$\displaystyle=\text{Re}\,\left<x,e_{i}\odot
f\right>-\sum_{j=1}^{7}e_{j}\text{Re}\,\left<xe_{j},e_{i}\odot f\right>$
$\displaystyle=f_{\mathbb{R}}(xe_{i})-\sum_{j,k=1}^{7}e_{j}f_{\mathbb{R}}(xe_{k}\epsilon_{jik}-\delta_{ij}x)$
$\displaystyle=f_{\mathbb{R}}(xe_{i})+e_{i}f_{\mathbb{R}}(x)-\sum_{j,k=1}^{7}\epsilon_{jik}e_{j}f_{\mathbb{R}}(xe_{k}),$
and similarly, $\displaystyle\left<e_{i}\odot\varphi(f),x\right>$
$\displaystyle=e_{i}\big{(}f_{\mathbb{R}}(x)-\sum_{j=1}^{7}e_{j}f_{\mathbb{R}}(xe_{j})\big{)}$
$\displaystyle=e_{i}f_{\mathbb{R}}(x)-\sum_{j,k=1}^{7}(\epsilon_{ijk}e_{k}-\delta_{ij})f_{\mathbb{R}}(xe_{j})$
$\displaystyle=f_{\mathbb{R}}(xe_{i})+e_{i}f_{\mathbb{R}}(x)-\sum_{j,k=1}^{7}\epsilon_{ijk}e_{k}f_{\mathbb{R}}(xe_{j}).$

Symmetry in $i,\,j$ ensures that $\left<\varphi(e_{i}\odot
f),x\right>=\left<e_{i}\odot\varphi(f),x\right>$ for $x\in\text{Re}\,M$ and
thus

$\varphi(e_{i}\odot f)=e_{i}\odot\varphi(f)$

for $i=1,\dots,7$. This shows that $\varphi$ is an $\mathbb{O}$-homomorphism
by (3.8). ∎

The common subset of $\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$ and
$\text{Hom}_{\mathcal{RO}}(M,M^{\prime})$ is exactly the set of
$\mathbb{O}$-linear maps.

######  Theorem 3.42.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules. Then

(3.55)
$\displaystyle\text{Hom}_{\mathcal{LO}}(M,M^{\prime})\cap\text{Hom}_{\mathcal{RO}}(M,M^{\prime})=\text{Hom}_{\mathbb{O}}(M,M^{\prime})$

###### Proof.

According to identity (3.8), we have

(3.56)
$\displaystyle\text{Hom}_{\mathcal{LO}}(M,M^{\prime})\cap\text{Hom}_{\mathcal{RO}}(M,M^{\prime})\supseteq\text{Hom}_{\mathbb{O}}(M,M^{\prime}).$

Conversely, let
$f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})\cap\text{Hom}_{\mathcal{RO}}(M,M^{\prime})$
and choose $x\in\text{Re}\,{M}$. Then for any $p\in\mathbb{O}$, we have

$\displaystyle[f(x),p]$ $\displaystyle=f(x)p-pf(x)$
$\displaystyle=f(xp)+B_{p}(f,x)-f(px)+A_{p}(x,f)$
$\displaystyle=B_{p}(f,x)+A_{p}(x,f)+f([x,p])$
$\displaystyle=B_{p}(f,x)+A_{p}(x,f).$

The last step follows from (3.3). By Corollary 3.11, we conclude from
$x\in\text{Re}\,M$ that $A_{p}(x,f)=0$, and similarly $B_{p}(f,x)=0$. Hence we
obtain $[f(x),p]=0$ for all $p\in\mathbb{O}$. This yields
$f(x)\in\text{Re}\,M^{\prime}$ by (3.3). Then Corollary 3.12 ensures that
$f\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$. This completes the proof. ∎

######  Remark 3.43.

Recall definitions (3.12), (3.15) (3.18) and (3.21). Starting from the
relation (3.55), we observe that an $\mathbb{O}$-linear function $f$ between
$\mathbb{O}$-bimodules can be considered as either types of left or right
para-linear map, so that it admits four different scalar multiplications. More
precisely, if we regard $f\in\text{Hom}_{\mathcal{LO}}(M,M^{\prime})$, then
for any $r\in\mathbb{O}$,

$\displaystyle\left<x,f\odot r\right>=\left<x,f\right>r,\quad\left<x,r\odot
f\right>=\left<xr,f\right>;$

and if we regard $f\in\text{Hom}_{\mathcal{RO}}(M,M^{\prime})$, then for any
$r\in\mathbb{O}$,

$\displaystyle\left<r\odot f,x\right>=r\left<f,x\right>,\quad\left<f\odot
r,x\right>=\left<f,rx\right>.$

Note that all the related associators in definitions vanish since $f$ is
$\mathbb{O}$-linear.

###  3.9. Semi-reflexivity

In this subsection, we shall show that octonionic modules are semi-reflexive.
We denote by $M^{*_{l}}$ ($M^{*_{r}}$) the dual module
$\text{Hom}_{\mathcal{LO}}(M,\mathbb{O})$
($\text{Hom}_{\mathcal{RO}}(M,\mathbb{O})$) of an $\mathbb{O}$-module $M$
respectively. And we shall omit the subscript ‘l’, ‘r’ when the module is
understood well.

Let $M$ be a left $\mathbb{O}$-module and $x_{0}$ be an arbitrary element of
$M$. We define

$\displaystyle x_{0}^{\prime\prime}:\quad M^{*}$ $\displaystyle\to$
$\displaystyle\mathbb{O}$ $\displaystyle f$ $\displaystyle\mapsto$
$\displaystyle f(x_{0}).$

It follows from Theorem 3.17 that $M^{*}$ is a right $\mathbb{O}$-module and
similarly $M^{**}$ is a left $\mathbb{O}$-module. We claim that
$x_{0}^{\prime\prime}\in M^{**}$ is a right para-linear map on $M^{*}$. To
show this, we first compute the right associator

(3.57) $\displaystyle
B_{p}(x_{0}^{\prime\prime},f)=x_{0}^{\prime\prime}(f)p-x_{0}^{\prime\prime}(f\odot
p)=f(x_{0})p-\left<x_{0},f\odot p\right>=A_{p}(x_{0},f)$

for any $f\in M^{*}$ and any $p\in\mathbb{O}$. We used definition (3.12) in
the last equality. This shows that

$\text{Re}\,(B_{p}(x_{0}^{\prime\prime},f))=0$

and hence $x_{0}^{\prime\prime}$ is a right $\mathbb{O}$-para-linear map.

We define

(3.58) $\displaystyle\tau:\quad M$ $\displaystyle\rightarrow$ $\displaystyle
M^{**}$ $\displaystyle x$ $\displaystyle\mapsto$ $\displaystyle
x^{\prime\prime}.$

Then $\tau$ is an embedding $\mathbb{O}$-homomorphism. Indeed, by definition,
we have

$\displaystyle\left<\tau(rx),f\right>=\left<rx,f\right>=rf(x)+A_{r}(x,f)$

for any $r\in\mathbb{O}$, any $x\in M$ and any $f\in M^{*}$. Next, by
definition (3.18) and identity (3.57), we obtain

$\displaystyle\left<r\odot\tau(x),f\right>=r\left<\tau(x),f\right>+B_{r}(\tau(x),f)=rf(x)+A_{r}(x,f).$

Thus $\tau(rx)=r\odot\tau(x)$. This shows that $\tau$ is an
$\mathbb{O}$-homomorphism. Thus each left $\mathbb{O}$-module $M$ is a semi-
reflexive module.

######  Remark 3.44.

Let $\tau_{\mathbb{R}}$ denote the canonical embedding

$\tau_{\mathbb{R}}:M\rightarrow M^{*_{\mathbb{R}}*_{\mathbb{R}}}.$

Then for all $f\in M^{*}$ and $x\in M$ we have

(3.59)
$\displaystyle\left<\tau(x)_{\mathbb{R}},f\right>=\text{Re}\,(f(x))=f_{\mathbb{R}}(x)=\left<\tau_{\mathbb{R}}(x),f_{\mathbb{R}}\right>.$

##  4\. Non-associative categories of para-linear maps

In this section, we aim to adopt the view of non-associative categories to
study para-linear maps on $\mathbb{O}$-modules. The category $\mathcal{LO}$,
which will be introduced later, is a typical example of non-associative
categories, whose objects in this category are $\mathbb{O}$-bimodules,
morphisms are left para-linear maps, and compositions are left regular
compositions. This motivates us to introduce the notion of non-associative
categories and functors between non-associative categories and establish some
results on $\mathcal{LO}$ similar to the classical case.

###  4.1. Non-associative categories and weak functors

For the sake of completeness, we give the definition of non-associative
categories and functors in this subsection.

######  Definition 4.1.

A non-associative category $\mathcal{C}$ consists of three ingredients: a
class $\text{obj}\,(\mathcal{C})$ of objects, a set of morphisms
$\text{Hom}_{\mathcal{C}}(X,Y)$ for every ordered pair $(X,Y)$ of objects, and
composition

$\displaystyle\text{Hom}_{\mathcal{C}}(X,Y)\times\text{Hom}_{\mathcal{C}}(Y,Z)$
$\displaystyle\to$ $\displaystyle\text{Hom}_{\mathcal{C}}(X,Z)$
$\displaystyle(f,g)$ $\displaystyle\mapsto$ $\displaystyle gf,$

for every ordered triple $X,Y,Z$ of objects. These ingredients are subject to
the following axioms:

  1. (${\mathrm{i}}$)

Each $f\in\text{Hom}_{\mathcal{C}}(X,Y)$ has a unique domain $X$ and a unique
target $Y$;

  2. (${\mathrm{ii}}$)

For each object $X$, there is an identity morphism
$1_{X}\in\text{Hom}_{\mathcal{C}}(X,X)$ such that

$f\,1_{X}=f,\qquad 1_{Y}\,f=f$

for all $f\in\text{Hom}_{\mathcal{C}}(X,Y)$.

######  Definition 4.2.

Let $\mathcal{C}$ be a non-associative category. A morphism
$f\in\text{Hom}_{\mathcal{C}}(X,Y)$ is called an associative morphism if it
satisfies:

$\displaystyle(fg)h=f(gh),$ $\forall X^{\prime\prime}\stackrel{{\scriptstyle
h}}{{\to}}{X^{\prime}}\stackrel{{\scriptstyle
g}}{{\to}}{X}\stackrel{{\scriptstyle f}}{{\to}}{Y}$;
$\displaystyle(gf)h=g(fh),$ $\forall X^{\prime}\stackrel{{\scriptstyle
g}}{{\to}}{X}\stackrel{{\scriptstyle f}}{{\to}}{Y}\stackrel{{\scriptstyle
h}}{{\to}}{X^{\prime\prime}}$; $\displaystyle(gh)f=g(hf),$ $\forall
X\stackrel{{\scriptstyle f}}{{\to}}{Y}\stackrel{{\scriptstyle
g}}{{\to}}{X^{\prime}}\stackrel{{\scriptstyle h}}{{\to}}{X^{\prime\prime}}$.

Associated to the non-associative category $\mathcal{C}$ is an associative
category $\mathscr{A}(\mathcal{C})$. In $\mathscr{A}(\mathcal{C})$, the
objects coincide with $\mathcal{C}$, the morphisms are associative morphisms
from category $\mathcal{C}$, and composition is inherited from $\mathcal{C}$.
We call $\mathscr{A}(\mathcal{C})$ the nucleus subcategory of ${\mathcal{C}}$.

######  Example 4.3.

We define a category $\mathcal{LO}$. Its objects are $\mathbb{O}$-bimodules,
morphisms are left para-linear maps, and compositions are left regular
compositions. Clearly, $\mathcal{LO}$ is non-associative. we claim that the
nucleus subcategory of $\mathcal{LO}$ is the associative category of
$\mathbb{O}$-bimodules with $\mathbb{O}$-linear morphisms and usual
compositions.

Indeed, it follows from Lemma 3.30 that every $\mathbb{O}$-linear map between
$\mathbb{O}$-bimodules is an associative morphism. Conversely, if
$f\in\text{Hom}_{\mathcal{LO}}(X,Y)$ is an associative morphism, then for any
$p,q\in\mathbb{O}$, by definition we have

(4.1) $\displaystyle(f\circledcirc R_{q})\circledcirc
R_{p}=f\circledcirc(R_{q}\circledcirc R_{p}).$

It can be checked that

$R_{q}\circledcirc R_{p}=R_{pq}.$

Combining this with Lemma 3.32 and (4.1), we conclude that

$p\odot(q\odot f)=(pq)\odot f$

for all $p,q\in\mathbb{O}$. This means that

$f\in\mathscr{A}(\text{Hom}_{\mathcal{LO}}(X,Y))=\text{Hom}_{\mathbb{O}}(X,Y)$

by Theorem 3.23. This proves the claim.

Similarly we can also define a category $\mathcal{RO}$. The objects are also
$\mathbb{O}$-bimodules, morphisms are right para-linear maps, and compositions
are right regular compositions. Similar result holds for category
$\mathcal{RO}$.

We introduce the notion of functors and weak functors between non-associative
categories.

######  Definition 4.4.

Let $\mathcal{C}$ and $\mathcal{D}$ be two non-associative categories.

  * •

A covariant functor $T:\mathcal{C}\to\mathcal{D}$ consists of data:

    1. (${\mathrm{i}}$)

for each $X\in\text{obj}\,(\mathcal{C})$, a function

$\displaystyle\text{obj}\,(\mathcal{C})$ $\displaystyle\to$
$\displaystyle\text{obj}\,(\mathcal{D})$ $\displaystyle X$
$\displaystyle\mapsto$ $\displaystyle T(X);$

    2. (${\mathrm{ii}}$)

for each $X,\,X^{\prime}\in\text{obj}\,(\mathcal{C})$, a function

$\displaystyle\text{Hom}_{\mathcal{C}}(X,X^{\prime})$ $\displaystyle\to$
$\displaystyle\text{Hom}_{\mathcal{D}}(T(X),T(X^{\prime}))$ $\displaystyle f$
$\displaystyle\mapsto$ $\displaystyle T(f)$

satisfying the following axioms:

    1. (${\mathrm{i}}$)

for any $X\stackrel{{\scriptstyle
f}}{{\to}}{X^{\prime}}\stackrel{{\scriptstyle g}}{{\to}}{X^{\prime\prime}}$ in
$\mathcal{C}$, $T(gf)=T(g)T(f)$ in $\mathcal{D}$,

    2. (${\mathrm{ii}}$)

$T(1_{X})=1_{T(X)}$ for every $X\in\text{obj}\,(\mathcal{C})$.

  * •

A covariant weak functor $T:\mathcal{C}\to\mathcal{D}$ consists of data:

    1. (${\mathrm{i}}$)

for each $X\in\text{obj}\,(\mathcal{C})$, a function

$\displaystyle\text{obj}\,(\mathcal{C})$ $\displaystyle\to$
$\displaystyle\text{obj}\,(\mathcal{D})$ $\displaystyle X$
$\displaystyle\mapsto$ $\displaystyle T(X);$

    2. (${\mathrm{ii}}$)

for each $X,\,X^{\prime}\in\text{obj}\,(\mathcal{C})$, a function

$\displaystyle\text{Hom}_{\mathcal{C}}(X,X^{\prime})$ $\displaystyle\to$
$\displaystyle\text{Hom}_{\mathcal{D}}(T(X),T(X^{\prime}))$ $\displaystyle f$
$\displaystyle\mapsto$ $\displaystyle T(f)$

satisfying the following axioms:

    1. (${\mathrm{i}}$)

for any $X\stackrel{{\scriptstyle
f}}{{\to}}{X^{\prime}}\stackrel{{\scriptstyle g}}{{\to}}{X^{\prime\prime}}$ in
$\mathcal{C}$, if one of $f$ and $g$ is an associative morphism, then
$T(gf)=T(g)T(f)$ in $\mathcal{D}$,

    2. (${\mathrm{ii}}$)

$T(1_{X})=1_{T(X)}$ for every $X\in\text{obj}\,(\mathcal{C})$.

Similar definitions for contravariant functors and weak contravariant
functors.

######  Definition 4.5.

Let $\mathcal{C}$ and $\mathcal{D}$ be two non-associative categories.

  * •

A contravariant functor $T:\mathcal{C}\to\mathcal{D}$ consists of data:

    1. (${\mathrm{i}}$)

for each $X\in\text{obj}\,(\mathcal{C})$, a function

$\displaystyle\text{obj}\,(\mathcal{C})$ $\displaystyle\to$
$\displaystyle\text{obj}\,(\mathcal{D})$ $\displaystyle X$
$\displaystyle\mapsto$ $\displaystyle T(X);$

    2. (${\mathrm{ii}}$)

for each $X,\,X^{\prime}\in\text{obj}\,(\mathcal{C})$, a function

$\displaystyle\text{Hom}_{\mathcal{C}}(X,X^{\prime})$ $\displaystyle\to$
$\displaystyle\text{Hom}_{\mathcal{D}}(T(X^{\prime}),T(X))$ $\displaystyle f$
$\displaystyle\mapsto$ $\displaystyle T(f)$

satisfying the following axioms:

    1. (${\mathrm{i}}$)

for any $X\stackrel{{\scriptstyle
f}}{{\to}}{X^{\prime}}\stackrel{{\scriptstyle g}}{{\to}}{X^{\prime\prime}}$ in
$\mathcal{C}$, $T(gf)=T(f)T(g)$ in $\mathcal{D}$,

    2. (${\mathrm{ii}}$)

$T(1_{X})=1_{T(X)}$ for every $X\in\text{obj}\,(\mathcal{C})$.

  * •

A contravariant weak functor $T:\mathcal{C}\to\mathcal{D}$ consists of data:

    1. (${\mathrm{i}}$)

for each $X\in\text{obj}\,(\mathcal{C})$, a function

$\displaystyle\text{obj}\,(\mathcal{C})$ $\displaystyle\to$
$\displaystyle\text{obj}\,(\mathcal{D})$ $\displaystyle X$
$\displaystyle\mapsto$ $\displaystyle T(X);$

    2. (${\mathrm{ii}}$)

for each $X,\,X^{\prime}\in\text{obj}\,(\mathcal{C})$, a function

$\displaystyle\text{Hom}_{\mathcal{C}}(X,X^{\prime})$ $\displaystyle\to$
$\displaystyle\text{Hom}_{\mathcal{D}}(T(X^{\prime}),T(X))$ $\displaystyle f$
$\displaystyle\mapsto$ $\displaystyle T(f)$

satisfying the following axioms:

    1. (${\mathrm{i}}$)

for any $X\stackrel{{\scriptstyle
f}}{{\to}}{X^{\prime}}\stackrel{{\scriptstyle g}}{{\to}}{X^{\prime\prime}}$ in
$\mathcal{C}$, if one of $f$ and $g$ is an associative morphism, then
$T(gf)=T(f)T(g)$ in $\mathcal{D}$,

    2. (${\mathrm{ii}}$)

$T(1_{X})=1_{T(X)}$ for every $X\in\text{obj}\,(\mathcal{C})$.

######  Remark 4.6.

By definition, we know that every functor is automatically a weak functor and
the two notions of weak functor and functor coincide in the case of
associative categories. Generally, the composition of two weak functors may be
not a weak functor any more. However, it is easy to see that the composition
of a weak functor and a functor continues to be a weak functor.

######  Example 4.7.

We denote by Sets the associative category of sets. Let $\mathcal{C}$ be a
non-associative category and $B\in\text{obj}\,(\mathcal{\mathcal{C}})$. Then
there is a natural covariant weak Hom functor

$w-\text{Hom}_{\mathcal{C}}(B,-):\mathcal{C}\to\text{\bf{Sets}}$

and a contravariant weak Hom functor

$w-\text{Hom}_{\mathcal{C}}(-,B):\mathcal{C}\to\text{\bf{Sets}}.$

The covariant weak Hom functor is defined, for all
$C\in\text{obj}\,(\mathcal{\mathcal{C}})$, by

$w-\text{Hom}_{\mathcal{C}}(B,-)(C)=\text{Hom}_{\mathcal{C}}(B,C)$

and if $f:C\to C^{\prime}\in\mathcal{C}$, then
$w-\text{Hom}_{\mathcal{C}}(B,-)(f):\text{Hom}_{\mathcal{C}}(B,C)\to\text{Hom}_{\mathcal{C}}(B,C^{\prime})$
is given by

$w-\text{Hom}_{\mathcal{C}}(B,-)(f):h\mapsto fh.$

The contravariant weak Hom functor is defined, for all
$C\in\text{obj}\,(\mathcal{\mathcal{C}})$, by

$w-\text{Hom}_{\mathcal{C}}(-,B)(C)=\text{Hom}_{\mathcal{C}}(C,B)$

and if $f:C\to C^{\prime}\in\mathcal{C}$, then
$w-\text{Hom}_{\mathcal{C}}(-,B)(f):\text{Hom}_{\mathcal{C}}(C^{\prime},B)\to\text{Hom}_{\mathcal{C}}(C,B)$
is given by

$w-\text{Hom}_{\mathcal{C}}(-,B)(f):h\mapsto hf.$

We emphasize that the Hom functors defined above are all weak functors rather
than genuine functors. This is because that the category Sets is associative.
For the special case of non-associative categories of $\mathcal{LO}$ and
$\mathcal{RO}$, we shall introduce Hom functors whose target categories are
non-associative in Subsection 4.3.

######  Example 4.8.

We denote by $\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}$ the associative
category whose objects are $\mathbb{O}$-bimodules and morphisms are
$\mathbb{R}$-linear maps and compositions are usual compositions. Note that
the associative morphisms in $\mathcal{LO}$ are $\mathbb{O}$-linear maps. Then
in view of Lemma 3.28, we have a covariant weak functor

$\displaystyle\iota:\qquad\quad\mathcal{LO}\quad$
$\displaystyle\xlongrightarrow[]{\hskip
28.45274pt}\quad\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}$ $\displaystyle
M\quad$ $\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\quad
M$ $\displaystyle f:M\to N\quad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\quad f.$

######  Example 4.9.

Let

$U:\qquad\quad\mathcal{LO}\quad\xlongrightarrow[]{\hskip
28.45274pt}\quad\text{\bf{Sets}}$

be the forgetful functor which assigns to each $\mathbb{O}$-bimodule $M$ its
underlying set and views each left para-linear map as a mere function. Then by
the similar argument, we conclude that $U$ is covariant weak functor.

######  Example 4.10.

Let $\mathbb{R}\text{-}\textbf{Vec}$ denote the category of real vector
spaces. We define a weak functor

$\displaystyle\mathfrak{Re}:\qquad\quad\mathcal{LO}\quad$
$\displaystyle\xlongrightarrow[]{\hskip
28.45274pt}\quad\mathbb{R}\text{-}\textbf{Vec}$ $\displaystyle M\quad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\quad\text{Re}\,M$ $\displaystyle f:M\to N\quad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\quad(\text{Re}\,f)|_{\text{Re}\,M}.$

By (3.41), we know that

$\mathfrak{Re}(f)=(\text{Re}\,f)|_{\text{Re}\,M}=f_{\mathbb{R}}|_{\text{Re}\,M}:\text{Re}\,M\to\text{Re}\,M^{\prime}.$

This means for any $x\in\text{Re}\,M$, we have

(4.2) $\displaystyle(\mathfrak{Re}(f))(x)=f_{\mathbb{R}}(x)=\text{Re}\,f(x).$

Let $X\stackrel{{\scriptstyle f}}{{\to}}{X^{\prime}}\stackrel{{\scriptstyle
g}}{{\to}}{X^{\prime\prime}}$. Then for all $x\in\text{Re}\,X$,

$(\mathfrak{Re}(g\circledcirc f))(x)=\text{Re}\,((g\circledcirc f)(x))$

and

$(\mathfrak{Re}(g))\circ(\mathfrak{Re}(f))(x)=\text{Re}\,g(\text{Re}\,f(x)).$

If one of $f$ and $g$ is $\mathbb{O}$-linear, then it follows from Lemma 3.28
that

$\text{Re}\,((g\circledcirc f)(x))=\text{Re}\,((g\circ f)(x)).$

If $f$ is $\mathbb{O}$-linear, then

$(\mathfrak{Re}(g))\circ(\mathfrak{Re}(f))(x)=\text{Re}\,g(f(x)).$

If $g$ is $\mathbb{O}$-linear, then it follows from Lemma (3.3) that

$\text{Re}\,((g\circ f)(x))=g(\text{Re}\,f(x))=\text{Re}\,g(\text{Re}\,f(x)).$

In both cases, we obtain that

$(\mathfrak{Re}(g))\circ(\mathfrak{Re}(f))=\text{Re}\,((g\circledcirc f).$

This proves that $\mathfrak{Re}$ is a weak functor.

######  Definition 4.11.

Let $\mathcal{C}$, $\mathcal{D}$ be two non-associative categories. Suppose
$S,T:\mathcal{C}\rightarrow\mathcal{D}$ are two covariant weak functors. A
natural transformation $\tau:S\rightarrow T$ is a one-parameter family of
morphisms in $\mathcal{{D}}$,

$\tau=\left(\tau_{X}:S(X)\rightarrow
T(X)\right)_{X\in\text{obj}\,(\mathcal{\mathcal{C}})}$

making the following diagram commute in $\mathcal{{D}}$:

$\begin{array}[c]{ccc}S(X)&\stackrel{{\scriptstyle\tau_{X}}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}&T(X)\\\
\,\Biggm{\downarrow}\scriptstyle{S(f)}&&\Biggm{\downarrow}\scriptstyle{T(f)}\\\
S(Y)&\stackrel{{\scriptstyle\tau_{Y}}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}&T(Y).\end{array}$

for all $X\stackrel{{\scriptstyle f}}{{\longrightarrow}}Y$ in ${\mathcal{C}}$.

A natural isomorphism is a natural transformation $\tau$ for which each
$\tau_{X}$ is an isomorphism. Natural transformation between contravariant
functors can be defined similarly.

######  Remark 4.12.

We point out that the usual composition of two natural transformations may be
not a natural transformation generally.

We make the following notation. If $F,G:\mathcal{C}\to\mathcal{{D}}$ are weak
functors of the same variance, then we denote

$\text{Nat}(F,G):=\\{\text{natural transformations }F\to G\\}.$

######  Theorem 4.13 (Yoneda lemma).

Let $\mathcal{C}$ be a category, let
$A\in\text{obj}\,(\mathcal{\mathcal{C}})$, and let
$G:\mathcal{C}\to\text{\bf{Sets}}$ be a covariant functor. Then there is an
bijection

$\displaystyle y:\text{Nat}(w-\text{Hom}_{\mathcal{C}}(A,-),G)$
$\displaystyle\to$ $\displaystyle G(A)$ $\displaystyle\tau$
$\displaystyle\mapsto$ $\displaystyle\tau_{A}(1_{A}).$

###### Proof.

If $\tau:w-\text{Hom}_{\mathcal{C}}(A,-)\to G$ is a natural transformation,
then $\tau_{A}(1_{A})$ lies in the set $G(A)$, for
$\tau_{A}:\text{Hom}_{\mathcal{C}}(A,A)\to G(A).$ Thus, $y$ is well-defined.

We prove that $y$ is an injection. For each
$B\in\text{obj}\,(\mathcal{\mathcal{C}})$ and
$f\in\text{Hom}_{\mathcal{C}}(A,B)$, there is a commutative diagram

$\begin{array}[c]{ccc}\text{Hom}_{\mathcal{C}}(A,A)&\stackrel{{\scriptstyle\tau_{A}}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}&G(A)\\\
\quad\Biggm{\downarrow}\scriptstyle{w-\text{Hom}_{\mathcal{C}}(A,-)(f)}&&\Biggm{\downarrow}\scriptstyle{G(f)}\\\
\text{Hom}_{\mathcal{C}}(A,B)&\stackrel{{\scriptstyle\tau_{B}}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}&G(B),\end{array}$

so that

$(G(f))\tau_{A}(1_{A})=\tau_{B}(w-\text{Hom}_{\mathcal{C}}(A,-)(f)(1_{A}))=\tau_{B}(f1_{A})=\tau_{B}(f).$

If $\sigma:w-\text{Hom}_{\mathcal{C}}(A,-)\to G$ is another natural
transformation, then

$\sigma_{B}(f)=(G(f))\sigma_{A}(1_{A}).$

Hence, if $\sigma_{A}(1_{A})=\tau_{A}(1_{A})$, then $\sigma_{B}=\tau_{B}$ for
all $B\in\text{obj}\,(\mathcal{\mathcal{C}})$ and, hence, $\sigma=\tau$.
Therefore, $y$ is an injection.

To see that $y$ is a surjection, take $x\in G(A)$. For
$B\in\text{obj}\,(\mathcal{\mathcal{C}})$ and
$g\in\text{Hom}_{\mathcal{C}}(A,B)$, define

$\tau_{B}(g)=(G(g))(x).$

We claim that $\tau$ is a natural transformation; that is, if $h:B\to C$ is a
morphism in $\mathcal{C}$, then we have

$(G(h)\circ\tau_{B})(g)=G(h)\circ G(g)(x)$

and

$\tau_{C}(w-\text{Hom}_{\mathcal{C}}(A,-)(h)(g))=\tau_{C}(hg)=G(hg)(x).$

Since $G$ is a covariant functor, it follows that

$G(h)\circ G(g)=G(hg).$

This implies that $\tau$ is a natural transformation and

$y(\tau)=\tau_{A}(1_{A})=G(1_{A})(x)=x.$

Hence $y$ is a surjection. This completes the proof. ∎

######  Remark 4.14.

If $G$ is a weak functor, then we shall only get an injection in the Yoneda
lemma.

These discussions show that the notion of weak functors in the setting of non-
associative categories is natural. However, to study functors between non-
associative categories is difficult and important. We shall discuss functors
on the specific non-associative category $\mathcal{LO}$ in the following
subsections.

###  4.2. Conjugate functor

The duality in subsection 3.8 can be interpreted as a covariant functor
between two non-associative categories $\mathcal{LO}$ and $\mathcal{RO}$. We
call it the conjugate functor. It is defined by

$\displaystyle C:\qquad\quad\mathcal{LO}\quad$
$\displaystyle\xlongrightarrow[]{\hskip 28.45274pt}\quad\mathcal{RO}$
$\displaystyle M\quad$ $\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\quad M^{C}$ $\displaystyle
f\in\text{Hom}_{\mathcal{LO}}(M,N)\quad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\quad
C(f):=f^{C}.$

We recall the notation $M^{C}$ of an $\mathbb{O}$-bimodule $M$ and the
notation $f^{C}$ in subsection 3.8, where the bimodule structure on $M^{C}$ is
defined as:

$p\cdot x=x\overline{p},\quad x\cdot p=\overline{p}x.$

For each $f\in\text{Hom}_{\mathcal{LO}}(M,N)$, the morphism $C(f)$ is just
defined to be $f$ as a map of sets (which is denoted by $f^{C}$ in subsection
3.8).

######  Theorem 4.15.

The conjugate functor $C:\mathcal{LO}\longrightarrow\mathcal{RO}$ is a
covariant functor.

###### Proof.

We need to verify two assertions.

  * •

Assertion 1:

$\text{For any }f\in\text{Hom}_{\mathcal{LO}}(M,N),\quad\mbox{ we have
}C(f)\in\text{Hom}_{\mathcal{RO}}(M^{C},N^{C}).$

In fact, we have proved this in subsection 3.8. More precisely, we have

$\text{Re}\,B_{p}(C(f),x)=\text{Re}\,A_{p}(x,f)=0,$

which means that $C(f)\in\text{Hom}_{\mathcal{RO}}(M^{C},N^{C})$.

  * •

Assertion 2:

$C(f\circledcirc_{l}g)=C(f)\circledcirc_{r}C(g).$

Here the subscripts indicate which type of the regular composition is used.

Indeed, in view of Remark 3.38 we have

$\displaystyle C(f\circledcirc_{l}g)(x)$ $\displaystyle=$
$\displaystyle(f\circledcirc_{l}g)(x)$ $\displaystyle=$ $\displaystyle f\circ
g(x)+[f,g,x]$ $\displaystyle=$ $\displaystyle f^{C}\circ
g^{C}(x)+[x,f^{C},g^{C}]$ $\displaystyle=$
$\displaystyle(C(f)\circledcirc_{r}C(g))(x).$

This proves that $C$ is a covariant functor. ∎

Similarly, we also have a covariant functor from $\mathcal{RO}$ to
$\mathcal{LO}$, also denoted by $C$,

$\displaystyle C:\qquad\quad\mathcal{RO}\quad$
$\displaystyle\xlongrightarrow[]{\hskip 28.45274pt}\quad\mathcal{LO}$
$\displaystyle M\quad$ $\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\quad M^{C}$ $\displaystyle
f\in\text{Hom}_{\mathcal{RO}}(M,N)\quad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\quad
C(f):=f^{C}.$

For each $f\in\text{Hom}_{\mathcal{RO}}(M,N)$, the morphism $C(f)$ is just
defined to be $f$ as a map of sets.

Obviously, $C$ is an isomorphism between categories $\mathcal{LO}$ and
$\mathcal{RO}$.

###  4.3. Hom functors

We introduce the Hom functors between the non-associative categories
$\mathcal{LO}$ and $\mathcal{RO}$.

######  Theorem 4.16.

For any $\mathbb{O}$-bimodule $M$, we have a covariant functor

$\text{Hom}_{\mathcal{LO}}(M,-):\mathcal{LO}\to\mathcal{LO}$

and a contravariant functor

$\text{Hom}_{\mathcal{LO}}(-,M):\mathcal{LO}\to\mathcal{RO}.$

Moreover, both functors are $\mathbb{O}$-linear.

###### Proof.

For short, we denote the two functors above as

$\displaystyle T$ $\displaystyle=$
$\displaystyle\text{Hom}_{\mathcal{LO}}(M,-),$ $\displaystyle S$
$\displaystyle=$ $\displaystyle\text{Hom}_{\mathcal{LO}}(-,M).$

The definition of $T$ and $S$ is canonical. More precisely,

$\displaystyle T(X)$ $\displaystyle=$
$\displaystyle\text{Hom}_{\mathcal{LO}}(M,X),$ $\displaystyle S(X)$
$\displaystyle=$ $\displaystyle\text{Hom}_{\mathcal{LO}}(X,M)$

for any $X\in\text{obj}\,(\mathcal{LO})$. For each morphism
$f\in\text{Hom}_{\mathcal{LO}}(X,Y),$ we define for every
$g\in\text{Hom}_{\mathcal{LO}}(M,X)$

$\displaystyle\left<g,T(f)\right>$ $\displaystyle:=$ $\displaystyle
f\circledcirc_{l}g,$

and for every $g\in\text{Hom}_{\mathcal{LO}}(Y,M)$

$\displaystyle\left<S(f),g\right>$ $\displaystyle:=$ $\displaystyle
g\circledcirc_{l}f.$

For simplicity, we omit the subscript ‘l’ since the regular composition is
understood well.

We show that $T$ is a covariant functor. For any $p\in\mathbb{O}$,
$f\in\text{Hom}_{\mathcal{LO}}(X,Y)$, and
$g\in\text{Hom}_{\mathcal{LO}}(M,X)$, it follows from Lemma 3.32 that

$\displaystyle A_{p}(g,T(f))$ $\displaystyle=f\circledcirc(p\odot
g)-p\odot(f\circledcirc g)$ $\displaystyle=f\circledcirc(g\circledcirc
R_{p})-(f\circledcirc g)\circledcirc R_{p}$ $\displaystyle=-[f,g,R_{p}].$

In view of Lemma 3.30, we have

$\text{Re}\,A_{p}(g,T(f))=0,$

which means

$T(f)\in\text{Hom}_{\mathcal{LO}}(T(X),T(Y)).$

For a pair of morphisms

$X\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}{X^{\prime}}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}{X^{\prime\prime}}$

in $\mathcal{LO}$, we need to verify

(4.3) $\displaystyle T(f_{1}\circledcirc f_{2})=T(f_{1})\circledcirc
T(f_{2}).$

For any

$g\in\text{Hom}_{\mathbb{O}}(M,X),$

we have

$\displaystyle{\ \ \ \,}\left<g,T(f_{1}\circledcirc
f_{2})-T(f_{1})\circledcirc T(f_{2})\right>$ $\displaystyle=(f_{1}\circledcirc
f_{2})\circledcirc g-\big{(}T(f_{1})\circ
T(f_{2})\big{)}(g)-[T(f_{1}),T(f_{2}),g]$ using Lemma 3.28
$\displaystyle=(f_{1}\circledcirc f_{2})\circledcirc
g-T(f_{1})(f_{2}\circledcirc g)$ $\displaystyle=(f_{1}\circledcirc
f_{2})\circledcirc g-f_{1}\circledcirc(f_{2}\circledcirc g)$
$\displaystyle=[f_{1},f_{2},g]$ using Lemma 3.30 $\displaystyle=0.$

Notice that

$\text{Re}\,\text{Hom}_{\mathcal{LO}}(M,X)=\text{Hom}_{\mathbb{O}}(M,X)$

due to Theorem 3.23. Since both sides of (4.3) are left para-linear maps, it
follows from Corollary 3.11 that (4.3) holds.

Next we come to show that $T$ is an $\mathbb{O}$-linear functor, i.e., for any
$p\in\mathbb{O}$ and $f\in\text{Hom}_{\mathcal{LO}}(X,Y)$,

$T(p\odot f)=p\odot T(f).$

If
$g\in\text{Re}\,\text{Hom}_{\mathcal{LO}}(M,X)=\text{Hom}_{\mathbb{O}}(M,X)$
(by Theorem 3.23), then again from Lemma 3.30 we have

$\displaystyle\left<g,T(p\odot f)-p\odot T(f)\right>$ $\displaystyle=$
$\displaystyle(p\odot f)\circledcirc g-f\circledcirc(g\odot p)$
$\displaystyle=$ $\displaystyle(f\circledcirc R_{p})\circledcirc
g-f\circledcirc(R_{p}\circledcirc g)$ $\displaystyle=$
$\displaystyle[f,R_{p},g]$ $\displaystyle=$ $\displaystyle 0.$

It follows from Corollary 3.11 that $T$ is $\mathbb{O}$-linear.

Secondly, we shall prove that $S$ is a contravariant functor. For any
$p\in\mathbb{O}$, $f\in\text{Hom}_{\mathcal{LO}}(X,Y)$, and
$g\in\text{Hom}_{\mathcal{LO}}(Y,M)$, it follows from Lemma 3.31 that

$\displaystyle B_{p}(S(f),g)$ $\displaystyle=(g\circledcirc f)\odot p-(g\odot
p)\circledcirc f$ $\displaystyle=R_{p}\circledcirc(g\circledcirc
f)-(R_{p}\circledcirc g)\circledcirc f$ $\displaystyle=-[R_{p},g,f].$

Hence by Lemma 3.30, we have

$\text{Re}\,B_{p}(S(f),g)=0,$

which means

$S(f)\in\text{Hom}_{\mathcal{RO}}(S(Y),S(X)).$

For a pair of morphisms

$X\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}{X^{\prime}}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}{X^{\prime\prime}}$

in $\mathcal{LO}$, we need to show

$S(f_{1}\circledcirc f_{2})=S(f_{2})\circledcirc S(f_{1}).$

In fact, for any

$g\in\text{Re}\,\text{Hom}_{\mathcal{LO}}(Y,M)=\text{Hom}_{\mathbb{O}}(Y,M),$

we have

$\displaystyle\left<S(f_{1}\circledcirc f_{2})-S(f_{2})\circledcirc
S(f_{1}),g\right>$ $\displaystyle=g\circledcirc(f_{1}\circledcirc
f_{2})-(g\circledcirc f_{1})\circledcirc f_{2}$
$\displaystyle=-[g,f_{1},f_{2}]$ $\displaystyle=0.$

Then it follows from Corollary 3.11 that $S$ is a contravariant functor.

Finally, it remains to show that $S$ is an $\mathbb{O}$-linear functor, i.e.,
for any $p\in\mathbb{O}$ and $f\in\text{Hom}_{\mathcal{LO}}(X,Y)$,

$S(p\odot f)=p\odot S(f).$

Let $g\in\text{Re}\,\text{Hom}_{\mathcal{LO}}(M,Y)$. Then

$\displaystyle\left<S(p\odot f)-p\odot S(f),g\right>$ $\displaystyle=$
$\displaystyle g\circledcirc(p\odot f)-p\odot(g\circledcirc f)$
$\displaystyle=$ $\displaystyle g\circledcirc(f\circledcirc
R_{p})-(g\circledcirc f)\circledcirc R_{p}$ $\displaystyle=$
$\displaystyle-[g,f,R_{p}]$ $\displaystyle=$ $\displaystyle 0.$

It follows from Corollary 3.11 again that $S$ is $\mathbb{O}$-linear. This
completes the proof. ∎

######  Remark 4.17.

Similarly, we also have a covariant functor

$\text{Hom}_{\mathcal{RO}}(M,-):\mathcal{RO}\to\mathcal{RO}$

and a contravariant functor

$\text{Hom}_{\mathcal{RO}}(-,M):\mathcal{RO}\to\mathcal{LO}.$

In contrast to the classical associative case where the target category of a
Hom functor is always the Sets category, in the non-associative case the Hom
functors admit a non-associative target category. Moreover, the weak Hom
functors introduced in Example 4.7 have a close relation with Hom functors.
For example, we have

$w-\text{Hom}_{\mathcal{LO}}(M,-)=U\circ\text{Hom}_{\mathcal{LO}}(M,-),$

where $U$ is the forgetful functor introduced in Example 4.9. This provides an
alternative viewpoint to see that $w-\text{Hom}_{\mathcal{LO}}(M,-)$ is a weak
functor.

We shall always denote

$f_{*_{l}}:=\text{Hom}_{\mathcal{LO}}(M,-)(f):\text{Hom}_{\mathcal{LO}}(M,X)\to\text{Hom}_{\mathcal{LO}}(M,Y),$
$f^{*_{l}}:=\text{Hom}_{\mathcal{LO}}(-,M)(f):\text{Hom}_{\mathcal{LO}}(Y,M)\to\text{Hom}_{\mathcal{LO}}(X,M)$

for any $X\stackrel{{\scriptstyle f}}{{\longrightarrow}}{Y}$ in
$\mathcal{LO}$. Similar notations can be defined for $\mathcal{RO}$.

Theorem 4.16 provides a fundamental fact in the study of octonionic functional
analysis. The Hom functors can be used to define the Banach dual operator and
Hilbert dual operator. This answers that why the seemly right method to define
the Hilbert dual of an operator $T$ as

$\left<Tx,y\right>=\left<x,Ty\right>$

does not work in an octonionic Hilbert space [8]. As a direct consequence of
Theorem 4.16, we have the basic properties of dual operators:

$\displaystyle(p\odot f)^{*_{l}}=p\odot f^{*_{l}},$ $\displaystyle(f\odot
p)^{*_{l}}=f^{*_{l}}\odot p,$ $\displaystyle(f\circledcirc
g)^{*_{l}}=g^{*_{l}}\circledcirc f^{*_{l}}$

for any $p\in\mathbb{O}$ and $X\stackrel{{\scriptstyle
g}}{{\longrightarrow}}{Y}\stackrel{{\scriptstyle f}}{{\longrightarrow}}{Z}$.

As a consequence, we have a covariant functor

$\text{Hom}_{\mathcal{RO}}(\text{Hom}_{\mathcal{LO}}(-,\mathbb{O}),\mathbb{O}):\mathcal{LO}\to\mathcal{RO}\to\mathcal{LO}.$

We denote by $1_{\mathcal{LO}}$ the identity functor. Then the embedding
$\mathbb{O}$-homomorphism

$\tau_{X}:X\to X^{**}$

introduced in Subsection 3.9 is a natural transformation of functors. This can
be seen from the following lemma.

######  Lemma 4.18.

Let $X$ and $Y$ be two $\mathbb{O}$-bimodules. Then for any
$\phi\in\text{Hom}_{\mathcal{LO}}(X,Y)$, we have

(4.4)
$\displaystyle\phi^{*_{l}*_{r}}\circledcirc_{l}\tau_{X}=\tau_{Y}\circledcirc_{l}\phi.$

###### Proof.

Note that we have shown in Subsection 3.9 that $\tau_{X}$ is an
$\mathbb{O}$-linear map. Hence (4.4) is equivalent to

$\phi^{*_{l}*_{r}}\circ\tau_{X}=\tau_{Y}\circ\phi.$

For any $x\in X$ and any $f\in Y^{*_{l}}$, we denote
$x^{\prime\prime}=\tau_{X}(x)$ as usual, then

$\displaystyle\left<\phi^{*_{l}*_{r}}(x^{\prime\prime}),f\right>$
$\displaystyle=$
$\displaystyle(x^{\prime\prime}\circledcirc_{r}\phi^{*_{l}})(f)$
$\displaystyle=$ $\displaystyle
x^{\prime\prime}(\phi^{*_{l}}(f))+[f,x^{\prime\prime},\phi^{*_{l}}]$
$\displaystyle=$
$\displaystyle\left<x,\phi^{*_{l}}(f)\right>+[f,x^{\prime\prime},\phi^{*_{l}}].$

By Definition 3.37, we get

$\displaystyle[f,x^{\prime\prime},\phi^{*_{l}}]$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{7}\big{(}\text{Re}\,\left<x^{\prime\prime},B_{e_{j}}(\phi^{*_{l}},f)\right>\big{)}e_{j}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{7}\big{(}\text{Re}\,\left<x,B_{e_{j}}(\phi^{*_{l}},f)\right>\big{)}e_{j}.$

It follows from Lemma 3.31 that for all $p\in\mathbb{O}$,

$\displaystyle B_{p}(\phi^{*_{l}},f)$ $\displaystyle=$
$\displaystyle\phi^{*_{l}}(f)\odot p-\phi^{*_{l}}(f\odot p)$ $\displaystyle=$
$\displaystyle(f\circledcirc\phi)\odot p-(f\odot p)\circledcirc\phi$
$\displaystyle=$ $\displaystyle
R_{p}\circledcirc(f\circledcirc\phi)-(R_{p}\circledcirc f)\circledcirc\phi$
$\displaystyle=$ $\displaystyle-[R_{p},f,\phi].$

In view of (3.48), we thus obtain

$\displaystyle\sum_{j=1}^{7}\big{(}\text{Re}\,\left<x,B_{e_{j}}(\phi^{*_{l}},f)\right>\big{)}e_{j}$
$\displaystyle=$
$\displaystyle-\sum_{j=1}^{7}\big{(}\text{Re}\,\left<x,[R_{e_{j}},f,\phi]\right>\big{)}e_{j}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{7}\big{(}\text{Re}\,R_{e_{j}}([f,\phi,x])\big{)}e_{j}$
$\displaystyle=$ $\displaystyle-[f,\phi,x].$

We used the fact that $[f,\phi,x]\in\mathit{Im}(\mathbb{O})$ in the last
equality. Therefore
$\phi^{*_{l}*_{r}}(x^{\prime\prime})=(\phi(x))^{\prime\prime}$ for all $x\in
X$. That is, $\phi^{*_{l}*_{r}}\circ\tau_{X}=\tau_{Y}\circ\phi$. ∎

###  4.4. Tensor functors

In this subsection, we introduce the tensor functors between non-associative
categories $\mathcal{LO}$ and $\mathcal{RO}$.

We define the tensor product of two $\mathbb{O}$-bimodules $M$ and
$M^{\prime}$ as

(4.5) $\displaystyle
M\otimes_{\mathbb{O}}M^{\prime}:=(\text{Re}\,M\otimes_{\mathbb{R}}\text{Re}\,M^{\prime})\otimes_{\mathbb{R}}\mathbb{O}.$

Then $M\otimes_{\mathbb{O}}M^{\prime}$ is an $\mathbb{O}$-bimodule and

$\text{Re}\,(M\otimes_{\mathbb{O}}M^{\prime})=\text{Re}\,M\otimes_{\mathbb{R}}\text{Re}\,M^{\prime}.$

For $m\in M$ and $m^{\prime}\in M^{\prime}$, we write

$m=\sum_{i=0}^{7}e_{i}m_{i},\qquad
m^{\prime}=\sum_{i=0}^{7}e_{i}m^{\prime}_{i},$

where $m_{i}\in\text{Re}\,M$ and $m^{\prime}_{i}\in\text{Re}\,M^{\prime}$. We
define

$\displaystyle
m\otimes_{\mathbb{O}}m^{\prime}:=\sum_{i,j=0}^{7}(m_{i}\otimes_{\mathbb{R}}m^{\prime}_{j})\otimes_{\mathbb{R}}(e_{i}e_{j}).$

We remark that in general for any $p\in\mathbb{O}$,

$mp\otimes_{\mathbb{O}}m^{\prime}\neq m\otimes_{\mathbb{O}}pm^{\prime}.$

Instead, by direct calculation, we have the following identity

(4.6) $\displaystyle
mp\otimes_{\mathbb{O}}m^{\prime}-m\otimes_{\mathbb{O}}pm^{\prime}=\sum_{i,j=0}^{7}(m_{i}\otimes_{\mathbb{R}}m^{\prime}_{j})\otimes_{\mathbb{R}}[e_{i},p,e_{j}].$

We shall use the specific case of (4.6) and its variants as follows.

######  Lemma 4.19.

Let $M$, $M^{\prime}$ be two $\mathbb{O}$-bimodules and $p\in\mathbb{O}$,
$m\in M$, $m^{\prime}\in M^{\prime}$. If one of the three elements $p$, $m$
and $m^{\prime}$ is an associative element, then we have

(4.7) $\displaystyle
p(m\otimes_{\mathbb{O}}m^{\prime})=pm\otimes_{\mathbb{O}}m^{\prime},$ (4.8)
$\displaystyle
mp\otimes_{\mathbb{O}}m^{\prime}=m\otimes_{\mathbb{O}}pm^{\prime},$ (4.9)
$\displaystyle(m\otimes_{\mathbb{O}}m^{\prime})p=m\otimes_{\mathbb{O}}m^{\prime}p.$

###### Proof.

Assertion (4.8) follows from (4.6) directly. The proof of (4.7) and (4.9) is
similar. ∎

Now we come to define tensor functors. Fix an $\mathbb{O}$-bimodule $M$. For
any morphism $X\stackrel{{\scriptstyle f}}{{\longrightarrow}}X^{\prime}$, we
introduce a map

$\displaystyle f_{M}:\text{Re}\,M\otimes_{\mathbb{R}}\text{Re}\,X$
$\displaystyle\to$ $\displaystyle M\otimes_{\mathbb{O}}X^{\prime}$
$\displaystyle m\otimes_{\mathbb{R}}x$ $\displaystyle\mapsto$ $\displaystyle
m\otimes_{\mathbb{O}}f(x).$

By the left para-linear extension defined in Lemma 3.14, $f_{M}$ induces a
left para-linear map:

$l\text{-}\mathrm{ext}\,f_{M}\in\text{Hom}_{\mathcal{LO}}(M\otimes_{\mathbb{O}}X,M\otimes_{\mathbb{O}}X^{\prime}).$

In view of identities (4.7) and (4.8), by direct calculations we obtain that

(4.10) $\displaystyle
l\text{-}\mathrm{ext}\,f_{M}(m\otimes_{\mathbb{O}}x)=m\otimes_{\mathbb{O}}f(x)$

for all $m\in\text{Re}\,M$ and all $x\in X$. We thus have a natural tensor
functor

$\displaystyle M\otimes^{ll}_{\mathbb{O}}-:\qquad\mathcal{LO}\qquad$
$\displaystyle\xlongrightarrow[]{\hskip 28.45274pt}\qquad\mathcal{LO}$
$\displaystyle X\qquad$ $\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\qquad M\otimes_{\mathbb{O}}X$ $\displaystyle
X\stackrel{{\scriptstyle f}}{{\to}}X^{\prime}\qquad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\qquad
l\text{-}\mathrm{ext}\,f_{M}.$

######  Theorem 4.20.

For any $\mathbb{O}$-bimodule $M$, the tensor functor

$M\otimes^{ll}_{\mathbb{O}}-:\mathcal{LO}\xlongrightarrow[]{\hskip
28.45274pt}\mathcal{LO}$

is a covariant functor.

###### Proof.

To show $M\otimes^{ll}_{\mathbb{O}}-$ is a covariant functor, it only needs to
verify that

(4.11) $\displaystyle l\text{-}\mathrm{ext}\,f_{M}\circledcirc
l\text{-}\mathrm{ext}\,g_{M}=l\text{-}\mathrm{ext}\,(f\circledcirc g)_{M}$

for any morphisms

$X\stackrel{{\scriptstyle
g}}{{\longrightarrow}}X^{\prime}\stackrel{{\scriptstyle
f}}{{\longrightarrow}}X^{\prime\prime}$

in $\mathcal{LO}$.

Let $m\in\text{Re}\,M$ and $x\in\text{Re}\,X$. It follows that
$m\otimes_{\mathbb{O}}x\in\text{Re}\,(M\otimes_{\mathbb{O}}X)$. Then by Lemma
3.28, we have

(4.12) $\displaystyle(l\text{-}\mathrm{ext}\,f_{M}\circledcirc
l\text{-}\mathrm{ext}\,g_{M})(m\otimes_{\mathbb{O}}x)$ $\displaystyle=$
$\displaystyle(l\text{-}\mathrm{ext}\,f_{M}\circ
l\text{-}\mathrm{ext}\,g_{M})(m\otimes_{\mathbb{R}}x)$ $\displaystyle=$
$\displaystyle l\text{-}\mathrm{ext}\,f_{M}\,(m\otimes_{\mathbb{O}}g(x)).$

Combining (4.12) with (4.10), we obtain

(4.13) $\displaystyle(l\text{-}\mathrm{ext}\,f_{M}\circledcirc
l\text{-}\mathrm{ext}\,g_{M})(m\otimes_{\mathbb{O}}x)$ $\displaystyle=$
$\displaystyle m\otimes_{\mathbb{O}}f\circ g(x).$

Using (4.10) again, we get

(4.14) $\displaystyle l\text{-}\mathrm{ext}\,(f\circledcirc
g)_{M}(m\otimes_{\mathbb{O}}x)=m\otimes_{\mathbb{O}}(f\circledcirc
g)(x)=m\otimes_{\mathbb{O}}f\circ g(x).$

It follows from (4.13)and (4.14) that

$(l\text{-}\mathrm{ext}\,f_{M}\circledcirc
l\text{-}\mathrm{ext}\,g_{M})(m\otimes_{\mathbb{O}}x)=l\text{-}\mathrm{ext}\,(f\circledcirc
g)_{M}(m\otimes_{\mathbb{O}}x).$

Note that

$\text{Re}\,(M\otimes_{\mathbb{O}}X)=\text{Re}\,M\otimes_{\mathbb{R}}\text{Re}\,X.$

Hence (4.11) follows from Corollary 3.12. ∎

Similarly, we can also define three tensor functors
$M\otimes^{lr}_{\mathbb{O}}-$, $M\otimes^{rr}_{\mathbb{O}}-$ and
$M\otimes^{rl}_{\mathbb{O}}-$. They are defined as follows

$\displaystyle M\otimes^{lr}_{\mathbb{O}}-:\qquad\mathcal{LO}\qquad$
$\displaystyle\xlongrightarrow[]{\hskip 28.45274pt}\qquad\mathcal{RO}$
$\displaystyle X\qquad$ $\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\qquad M\otimes_{\mathbb{O}}X$ $\displaystyle
X\stackrel{{\scriptstyle f}}{{\to}}X^{\prime}\qquad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\qquad
r\text{-}\mathrm{ext}\,f_{M};$ $\displaystyle
M\otimes^{rr}_{\mathbb{O}}-:\qquad\mathcal{RO}\qquad$
$\displaystyle\xlongrightarrow[]{\hskip 28.45274pt}\qquad\mathcal{RO}$
$\displaystyle X\qquad$ $\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\qquad M\otimes_{\mathbb{O}}X$ $\displaystyle
X\stackrel{{\scriptstyle f}}{{\to}}X^{\prime}\qquad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\qquad
r\text{-}\mathrm{ext}\,f_{M};$ $\displaystyle
M\otimes^{rl}_{\mathbb{O}}-:\qquad\mathcal{RO}\qquad$
$\displaystyle\xlongrightarrow[]{\hskip 28.45274pt}\qquad\mathcal{LO}$
$\displaystyle X\qquad$ $\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\qquad M\otimes_{\mathbb{O}}X$ $\displaystyle
X\stackrel{{\scriptstyle f}}{{\to}}X^{\prime}\qquad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\qquad
l\text{-}\mathrm{ext}\,f_{M}.$

We shall denote

$(M\otimes^{ll}_{\mathbb{O}}-)(f):=1_{M}\otimes^{ll}f:\quad
M\otimes_{\mathbb{O}}X\to M\otimes_{\mathbb{O}}X^{\prime}$

for $X\stackrel{{\scriptstyle f}}{{\longrightarrow}}X^{\prime}$ in category
$\mathcal{LO}$ and denote the similar notations for other tensor functors.

###  4.5. Adjoint functor theorem

It turns out that the two functors

$(M\otimes^{ll}_{\mathbb{O}}-,\quad\text{Hom}_{\mathcal{LO}}(M,-))$

constitute an adjoint pair.

######  Definition 4.21.

Let $\mathcal{C}$ and $\mathcal{D}$ be two non-associative categories. Let
$F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{D}\to\mathcal{C}$ be covariant
functors. The ordered pair $(F,G)$ is an adjoint pair if, for each
$C\in\text{obj}\,(\mathcal{C})$ and $D\in\text{obj}\,(\mathcal{D})$, there are
bijections

$\tau_{C,D}:w-\text{Hom}_{\mathcal{D}}(FC,D)\to
w-\text{Hom}_{\mathcal{C}}(C,GD)$

that are natural transformations in ${C}$ and in ${D}$.

We get the adjoint functor theorem for $\mathbb{O}$-bimodules as follows:

######  Theorem 4.22.

Let $M$ be an $\mathbb{O}$-bimodule. Then
$(M\otimes^{ll}_{\mathbb{O}}-,\text{Hom}_{\mathcal{LO}}(M,-))$ is an adjoint
pair.

More precisely, we have an isomorphism of $\mathbb{O}$-bimodules for any
$\mathbb{O}$-bimodules $X,\,Y$:

$\tau_{X,Y}:\text{Hom}_{\mathcal{LO}}(M\otimes_{\mathbb{O}}X,Y)\to\text{Hom}_{\mathcal{LO}}(X,\text{Hom}_{\mathcal{LO}}(M,Y)).$

Moreover, the following two diagrams commute for all $f:X^{\prime}\to X$ in
$\mathcal{LO}$ and $g:Y\to Y^{\prime}$ in $\mathcal{LO}$:

$\begin{array}[c]{ccc}\text{Hom}_{\mathcal{LO}}(M\otimes_{\mathbb{O}}X,Y)&\stackrel{{\scriptstyle\tau_{X,Y}}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}&\text{Hom}_{\mathcal{LO}}(X,\text{Hom}_{\mathcal{LO}}(M,Y))\\\
\Biggm{\downarrow}\scriptstyle{(1_{M}\otimes^{ll}f)^{*_{l}}}&&\Biggm{\downarrow}\scriptstyle{f^{*_{l}}}\\\
\text{Hom}_{\mathcal{LO}}(M\otimes_{\mathbb{O}}X^{\prime},Y)&\stackrel{{\scriptstyle\tau_{X^{\prime},Y}}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}&\text{Hom}_{\mathcal{LO}}(X^{\prime},\text{Hom}_{\mathcal{LO}}(M,Y));\end{array}$

$\begin{array}[c]{ccc}\text{Hom}_{\mathcal{LO}}(M\otimes_{\mathbb{O}}X,Y)&\stackrel{{\scriptstyle\tau_{X,Y}}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}&\text{Hom}_{\mathcal{LO}}(X,\text{Hom}_{\mathcal{LO}}(M,Y))\\\
\Biggm{\downarrow}\scriptstyle{g_{*_{l}}}&&\Biggm{\downarrow}\scriptstyle{(\text{Hom}_{\mathcal{LO}}(M,g)_{*_{l}}}\\\
\text{Hom}_{\mathcal{LO}}(M\otimes_{\mathbb{O}}X,Y^{\prime})&\stackrel{{\scriptstyle\tau_{X,Y^{\prime}}}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}&\text{Hom}_{\mathcal{LO}}(X,\text{Hom}_{\mathcal{LO}}(M,Y^{\prime})).\end{array}$

###### Proof.

We first define the map $\tau_{X,Y}$ by extension. For any
$\alpha\in\text{Hom}_{\mathcal{LO}}(M\otimes_{\mathbb{O}}X,Y)$, and
$x\in\text{Re}\,X$, we define

$\left<m,\tau_{X,Y}(\alpha)(x)\right>:=\alpha(m\otimes_{\mathbb{O}}x)$

for all $m\in\text{Re}\,M$. Then by the left para-extension defined in Lemma
3.14, we get

$\tau_{X,Y}(\alpha)(x)\in\text{Hom}_{\mathcal{LO}}(M,Y)$

for all $x\in\text{Re}\,X$. Then by the left para-extension again, we obtain

$\tau_{X,Y}(\alpha)\in\text{Hom}_{\mathcal{LO}}(X,\text{Hom}_{\mathcal{LO}}(M,Y)).$

By construction, if $m\in\text{Re}\,M$, then for all $x\in X$ we have

(4.15)
$\displaystyle\left<m,\tau_{X,Y}(\alpha)(x)\right>=\alpha(m\otimes_{\mathbb{O}}x).$

Indeed, if we write $x=\sum_{i=0}^{7}e_{i}x_{i}$ as usual, then by left para-
extension in construction, for any $m\in\text{Re}\,M$ we have

$\displaystyle\left<m,\tau_{X,Y}(\alpha)(x)\right>$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{7}\left<m,e_{i}\odot(\tau_{X,Y}(\alpha)(x_{i}))\right>$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{7}\left<e_{i}m,\tau_{X,Y}(\alpha)(x_{i})\right>$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{7}e_{i}\left<m,\tau_{X,Y}(\alpha)(x_{i})\right>$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{7}e_{i}\alpha(m\otimes_{\mathbb{O}}x_{i}).$

And it follows from Lemma 4.19 that

$\alpha(m\otimes_{\mathbb{O}}x)=\sum_{i=0}^{7}\alpha(m\otimes_{\mathbb{O}}e_{i}x_{i})=\sum_{i=0}^{7}e_{i}\alpha(m\otimes_{\mathbb{O}}x_{i}).$

This proves (4.15).

We come to show that $\tau_{X,Y}$ is $\mathbb{O}$-linear. Bearing Corollary
3.12 in mind, for all $r\in\mathbb{O}$ and
$\alpha\in\text{Hom}_{\mathcal{LO}}(M\otimes_{\mathbb{O}}X,Y)$ we get

$\displaystyle\tau_{X,Y}(r\odot\alpha)=r\odot\tau_{X,Y}(\alpha)$
$\displaystyle\iff\left<x,\tau_{X,Y}(r\odot\alpha)\right>=\left<x,r\odot\tau_{X,Y}(\alpha)\right>,\quad$
$\displaystyle\text{for all }x\in\text{Re}\,X;$
$\displaystyle\iff\tau_{X,Y}(r\odot\alpha)(x)=\tau_{X,Y}(\alpha)(xr),\quad$
$\displaystyle\text{for all }x\in\text{Re}\,X;$
$\displaystyle\iff\left<m,\tau_{X,Y}(r\odot\alpha)(x)\right>=\left<m,\tau_{X,Y}(\alpha)(xr)\right>,\quad$
$\displaystyle\text{for all }x\in\text{Re}\,X,\text{for all
}m\in\text{Re}\,M;$
$\displaystyle\stackrel{{\scriptstyle\eqref{eqpf:tau}}}{{\iff}}\left<m\otimes_{\mathbb{O}}x,r\odot\alpha\right>=\left<m\otimes_{\mathbb{O}}xr,\alpha\right>,\quad$
$\displaystyle\text{for all }x\in\text{Re}\,X,\text{for all
}m\in\text{Re}\,M;$
$\displaystyle\iff\left<(m\otimes_{\mathbb{O}}x)r,\alpha\right>=\left<m\otimes_{\mathbb{O}}xr,\alpha\right>,\quad$
$\displaystyle\text{for all }x\in\text{Re}\,X,\text{for all
}m\in\text{Re}\,M.$

Then (4.8) implies that $\tau_{X,Y}$ is $\mathbb{O}$-linear. It is trivial to
see that $\tau_{X,Y}$ is a bijection. Hence we conclude that $\tau_{X,Y}$ is
an isomorphism of $\mathbb{O}$-bimodules as desired.

Similarly, to show

$f^{*_{l}}\circ\tau_{X,Y}=\tau_{X^{\prime},Y}\circ(1_{M}\otimes^{ll}f)^{*_{l}},$

it suffices to show

(4.16)
$\displaystyle\left<m,\left<x^{\prime},f^{*_{l}}(\tau_{X,Y}(\alpha))\right>\right>=\left<m,\left<x^{\prime},\tau_{X^{\prime},Y}((1_{M}\otimes^{ll}f)^{*_{l}}(\alpha))\right>\right>$

for all $\alpha\in\text{Hom}_{\mathcal{LO}}(M\otimes_{\mathbb{O}}X,Y)$ and all
$x^{\prime}\in X^{\prime}$, $m\in\text{Re}\,M$. By direct calculations, we
have

$\displaystyle\left<m,\left<x^{\prime},f^{*_{l}}(\tau_{X,Y}(\alpha))\right>\right>$
$\displaystyle=\left<m,\left<x^{\prime},\tau_{X,Y}(\alpha)\circledcirc
f\right>\right>$
$\displaystyle=\left<m,(\tau_{X,Y}(\alpha)(f(x^{\prime})\right>$ since
$x^{\prime}\in\text{Re}\,X^{\prime}$
$\displaystyle=\alpha(m\otimes_{\mathbb{O}}f(x^{\prime}))$ by (4.15).

Since
$m\otimes_{\mathbb{O}}x^{\prime}\in\text{Re}\,(M\otimes_{\mathbb{O}}X^{\prime})$,
we have

$\displaystyle\left<m,\left<x^{\prime},\tau_{X^{\prime},Y}((1_{M}\otimes^{ll}f)^{*_{l}}(\alpha))\right>\right>$
$\displaystyle=\left<m,\left<x^{\prime},\tau_{X^{\prime},Y}(\alpha\circledcirc(1_{M}\otimes^{ll}f))\right>\right>$
$\displaystyle=(\alpha\circledcirc(1_{M}\otimes^{ll}f))(m\otimes_{\mathbb{O}}x^{\prime})$
$\displaystyle=\alpha(m\otimes_{\mathbb{O}}f(x^{\prime})).$

This proves (4.16). ∎

###  4.6. Enveloping categories

Now we introduce a new notion called the enveloping category. The point is
that it becomes a bridge between the non-associative category and the
associative category. To do this, we have to focus our attention on additive
categories.

######  Definition 4.23.

A non-associative additive category is a non-associative category
$\mathcal{C}$ satisfying the following axioms:

  1. (${\mathrm{i}}$)

Every set $\operatorname{Hom}_{\mathcal{C}}(X,Y)$ is equipped with a structure
of an abelian group (written additively) such that composition of morphisms is
biadditive with respect to this structure.

  2. (${\mathrm{ii}}$)

There exists a zero object $0\in\text{obj}\,(\mathcal{\mathcal{C}})$ such that
$\operatorname{Hom}_{\mathcal{C}}(0,0)=0$.

  3. (${\mathrm{iii}}$)

(Existence of direct sums.) For any objects
$X_{1},X_{2}\in\text{obj}\,(\mathcal{\mathcal{C}})$ there exists an object
$Y\in\text{obj}\,(\mathcal{\mathcal{C}})$ and morphisms $p_{1}:Y\rightarrow
X_{1}$, $p_{2}:Y\rightarrow X_{2}$, $i_{1}:X_{1}\rightarrow Y$,
$i_{2}:X_{2}\rightarrow Y$ such that $p_{1}i_{1}=1_{X_{1}}$,
$p_{2}i_{2}=1_{X_{2}}$, and $i_{1}p_{1}+i_{2}p_{2}=1_{Y}$.

######  Definition 4.24.

Let $\mathcal{C}$ be a non-associative additive category.

  1. (${\mathrm{i}}$)

$(\mathcal{D},\iota)$ is called an associative cover category of $\mathcal{C}$
if $\mathcal{D}$ is an associative additive category and
$\iota:\mathcal{C}\to\mathcal{D}$ is a faithful weak additive functor.

  2. (${\mathrm{ii}}$)

An associative cover category $(\widetilde{\mathcal{C}},\iota)$ is called the
enveloping category of $\mathcal{C}$ if for any associative cover category
$({\mathcal{D}},\eta)$, there exists a unique additive functor
$T:\widetilde{\mathcal{C}}\to\mathcal{D}$ such that the following diagram
commutes:

${\widetilde{\mathcal{C}}}$${\mathcal{D}}$${\mathcal{C}}$$\exists!\
T$$\iota$$\eta$

######  Example 4.25.

We now provide some typical examples of enveloping categories.

  1. (${\mathrm{i}}$)

If $\mathcal{C}$ is an associative additive category, then one can check that
($\mathcal{C}$, $1_{\mathcal{C}}$) is the enveloping category of
$\mathcal{C}$.

  2. (${\mathrm{ii}}$)

The non-associative category $\mathcal{LO}$ takes
$(\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}},\iota)$ as its enveloping
category. Here we recall that the associative category
$\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}$ and the weak functor
$\iota:\mathcal{LO}\to\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}$ are defined
in Example 4.8. We delay its proof to Theorem 4.26.

  3. (${\mathrm{iii}}$)

If $\mathcal{U}$ be a unital non-associative algebra, then it takes
$(\mathfrak{M}(\mathcal{U}),\iota)$ as its enveloping category, where
$\mathfrak{M}(\mathcal{U})$ is the associative subalgebra generated by left
multiplications of $\mathcal{U}$ and $\iota$ is defined as

$\iota:\mathcal{U}\to\mathfrak{M}(\mathcal{U}),\quad x\mapsto L_{x}.$

Here $\mathcal{U}$ has been viewed as a specific sort of non-associative
additive category with a single object, whose morphisms are represented by the
elements of $\mathcal{U}$ and whose composition is the multiplication of
$\mathcal{U}$. Similarly, $\mathfrak{M}(\mathcal{U})$ is interpreted as an
associative category.

######  Theorem 4.26.

The enveloping category of $\mathcal{LO}$ is
$(\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}},\iota)$ and so is
$\mathcal{RO}$.

To prove this theorem, we need some lemmas. Let $M$ be an
$\mathbb{O}$-bimodule and $\alpha\in\text{End}_{\mathbb{R}}(\mathbb{O})$. Then
there associates a map $\alpha_{M}\in\text{End}_{\mathbb{R}}(M)$ defined by

$\alpha_{M}(px)=\alpha(p)x$

for all $p\in\mathbb{O}$ and $x\in\text{Re}\,M$.

######  Lemma 4.27.

Let $f\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$ and
$\alpha\in\text{End}_{\mathbb{R}}(\mathbb{O})$. Then

(4.17) $\displaystyle f\circ\alpha_{M}=\alpha_{M^{\prime}}\circ f.$

###### Proof.

This follows from direct calculations. For any $x=\sum_{i=0}^{7}e_{i}x_{i}\in
M$,

$\displaystyle(f\circ\alpha_{M})(x)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{7}f(\alpha(e_{i})x_{i})$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{7}\alpha(e_{i})f(x_{i})$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{7}\alpha_{M^{\prime}}(e_{i}f(x_{i}))$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{7}\alpha_{M^{\prime}}(f(e_{i}x_{i}))$
$\displaystyle=$ $\displaystyle(\alpha_{M^{\prime}}\circ f)(x).$

This proves the lemma. ∎

######  Lemma 4.28.

For any $f\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})$, there is a unique
decomposition

(4.18) $\displaystyle f=\sum_{i,j=0}^{7}f_{ij}\circ{\alpha^{ij}}_{M}$

for which $f_{ij}\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$,
$\alpha^{ij}\in\text{End}_{\mathbb{R}}(\mathbb{O})$, and

$\alpha^{ij}(e_{k})=\delta_{jk}e_{i}.$

###### Proof.

Let $f(x)=\sum_{i=0}^{7}e_{i}f_{i}(x)$, where
$f_{i}(x)\in\text{Re}\,(M^{\prime})$. We define

$f_{ij}(x)=f_{i}(e_{j}x)$

for all $x\in\text{Re}\,M$ and then we obtain
$f_{ij}\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$ by para-linear extension. One
can check that for all $x=\sum_{i=0}^{7}e_{i}x_{i}\in M$, we have

$\displaystyle\sum_{i,j=0}^{7}(f_{ij}\circ{\alpha^{ij}}_{M})(x)$
$\displaystyle=$
$\displaystyle\sum_{i,j,k=0}^{7}f_{ij}({\alpha^{ij}}_{M}(e_{k}x_{k}))$
$\displaystyle=$
$\displaystyle\sum_{i,j,k=0}^{7}f_{ij}({\alpha^{ij}}(e_{k})x_{k})$
$\displaystyle=$ $\displaystyle\sum_{i,j=0}^{7}f_{ij}(e_{i}x_{j})$
$\displaystyle=$ $\displaystyle\sum_{i,j=0}^{7}e_{i}f_{ij}(x_{j})$
$\displaystyle=$ $\displaystyle\sum_{i,j=0}^{7}e_{i}f_{i}(e_{j}x_{j})$
$\displaystyle=$ $\displaystyle\sum_{i=0}^{7}e_{i}f_{i}(x)$ $\displaystyle=$
$\displaystyle f(x).$

The uniqueness is trivial. ∎

Now we come to prove Theorem 4.26.

###### Proof of Theorem 4.26..

Let $(\mathcal{D},\eta)$ be an associative cover category of $\mathcal{LO}$.
We need to define a functor
$T:\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}\to\mathcal{D}$. Note that
$\text{End}_{\mathbb{R}}(\mathbb{O})$ can be generated by $\\{R_{p}\mid
p\in\mathbb{O}\\}$ algebraically. We define

$T(R_{p})=\eta(R_{p})$

for any $R_{p}\in\text{End}_{\mathcal{LO}}(\mathbb{O})$ and extend it to
$\text{End}_{\mathbb{R}}(\mathbb{O})$ by preserving the composition and linear
combination. This induces the definition of $T(\alpha_{M})$ for any
$\alpha\in\text{End}_{\mathbb{R}}(\mathbb{O})$ and $\mathbb{O}$-bimodule $M$.
We thus define

$\displaystyle
T:\qquad\qquad\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}\qquad$
$\displaystyle\xlongrightarrow[]{\hskip 28.45274pt}\qquad\mathcal{D}$
$\displaystyle M\qquad$ $\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip
28.45274pt}\qquad\eta(M)$ $\displaystyle
f=\sum_{i,j=0}^{7}f_{ij}\circ{\alpha^{ij}}_{M}\in\text{Hom}_{\mathbb{R}}(M,M^{\prime})\qquad$
$\displaystyle\shortmid\\!\xlongrightarrow[]{\hskip 28.45274pt}\qquad
T(f):=\sum_{i,j=0}^{7}\eta({f_{ij}})T({\alpha^{ij}}_{M}).$

We first claim that

$T(f)T(\alpha_{M})=T(\alpha_{M^{\prime}})T(f)$

for all $f\in\text{Hom}_{\mathbb{O}}(M,M^{\prime})$ and
$\alpha\in\text{End}_{\mathbb{R}}(\mathbb{O})$. Without loss of generality we
can assume $\alpha_{M}=R_{p}$ for some $p\in\mathbb{O}$. Since $\eta$ is a
weak functor, it follows that

$T(f)T(\alpha_{M})=\eta(f)\eta(R_{p})=\eta(f\circ R_{p})=\eta(R_{p}\circ
f)=\eta(R_{p})\eta(f).$

This proves the claim. For any

$M\stackrel{{\scriptstyle
f}}{{\longrightarrow}}M^{\prime}\stackrel{{\scriptstyle
g}}{{\longrightarrow}}M^{\prime\prime}$

in $\mathbb{O}\text{-}\textbf{Mod}_{\mathbb{R}}$, it follows from Lemma 4.27
that

$\displaystyle T(g\circ f)$ $\displaystyle=$ $\displaystyle
T(g_{mn}\circ{\alpha^{mn}}_{M^{\prime}}\circ f_{ij}\circ{\alpha^{ij}}_{M})$
$\displaystyle=$ $\displaystyle T(g_{mn}\circ
f_{ij}\circ{\alpha^{mn}}_{M}\circ{\alpha^{ij}}_{M})$ $\displaystyle=$
$\displaystyle\eta(g_{mn}\circ
f_{ij})T({\alpha^{mn}}_{M}\circ{\alpha^{ij}}_{M})$ $\displaystyle=$
$\displaystyle T(g_{mn})T(f_{ij})T({\alpha^{mn}}_{M})T({\alpha^{ij}}_{M})$
$\displaystyle=$ $\displaystyle
T(g_{mn})T({\alpha^{mn}}_{M^{\prime}})T(f_{ij})T({\alpha^{ij}}_{M})$
$\displaystyle=$ $\displaystyle T(g)T(f).$

The uniqueness is obvious. ∎

###  4.7. Natural associated functor

For a given functor between non-associative categories, it is natural to ask
what is its natural counterpart functor in associative categories.

######  Definition 4.29.

Let $\mathcal{C}$ and $\mathcal{D}$ be two non-associative additive
categories. Assume $(\mathcal{\widetilde{C}},\iota_{1})$ and
$(\mathcal{\widetilde{D}},\iota_{2})$ are their enveloping categories,
respectively. Suppose
$S:\mathcal{\widetilde{C}}\rightarrow\mathcal{\widetilde{D}}$,
$T:\mathcal{C}\rightarrow\mathcal{D}$ are covariant functors. A natural lift
transformation $\tau:S\rightarrow T$ is a one-parameter family of morphisms in
$\mathcal{\widetilde{D}}$,

$\tau=\left(\tau_{X}:S(\iota_{1}(X))\rightarrow\iota_{2}(T(X))\right)_{X\in\text{obj}\,(\mathcal{\mathcal{C}})}$

making the following diagram commute in $\mathcal{\widetilde{D}}$:

$\begin{array}[c]{ccc}S(\iota_{1}(X))&\stackrel{{\scriptstyle\tau_{X}}}{{\xlongrightarrow[]{\hskip
28.45274pt}}}&\iota_{2}(T(X))\\\