# On split regular Hom-Lie superalgebras

Helena Albuquerque  Helena Albuquerque, CMUC, Departamento de Matemática,
Universidade de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal. E-mail
address<EMAIL_ADDRESS>,  Elisabete Barreiro  Elisabete Barreiro, CMUC,
Departamento de Matemática, Universidade de Coimbra, Apartado 3008, 3001-454
Coimbra, Portugal. E-mail address<EMAIL_ADDRESS>,  A.J. Calderón  A.J.
Calderón, Departamento de Matemáticas, Universidad de Cádiz, Campus de Puerto
Real, 11510, Puerto Real, Cádiz, España. E-mail address:
<EMAIL_ADDRESS>and  José M. Sánchez  José M. Sánchez, CMUC,
Departamento de Matemática, Universidade de Coimbra, Apartado 3008, 3001-454
Coimbra, Portugal. E-mail address<EMAIL_ADDRESS>
###### Abstract.

We introduce the class of split regular Hom-Lie superalgebras as the natural
extension of the one of split Hom-Lie algebras and Lie superalgebras, and
study its structure by showing that an arbitrary split regular Hom-Lie
superalgebra ${\mathfrak{L}}$ is of the form ${\mathfrak{L}}=U+\sum_{j}I_{j}$
with $U$ a linear subspace of a maximal abelian graded subalgebra $H$ and any
$I_{j}$ a well described (split) ideal of ${\mathfrak{L}}$ satisfying
$[I_{j},I_{k}]=0$ if $j\neq k$. Under certain conditions, the simplicity of
${\mathfrak{L}}$ is characterized and it is shown that ${\mathfrak{L}}$ is the
direct sum of the family of its simple ideals.

Keywords: Hom-Lie superalgebra, Root, Root space, Structure theory.

2000 MSC: 17A60, 17A70, 17B22.

The first, second and fourth authors acknowledge financial assistance by the
Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013,
funded by the Portuguese Government through FCT/MEC and co-funded by the
European Regional Development Fund through the Partnership Agreement PT2020.
Third and fourth authors are supported by the PCI of the UCA ‘Teoría de Lie y
Teoría de Espacios de Banach’, by the PAI with project numbers FQM298, FQM7156
and by the project of the Spanish Ministerio de Educación y Ciencia
MTM2013-41208P. The fourth author acknowledges the Fundação para a Ciência e a
Tecnologia for the grant with reference SFRH/BPD/101675/2014.

##  1\. Introduction and first definitions

The motivation to study Hom-Lie structures are related to physics and to
deformations of Lie algebras (see for instance [1, 3, 13, 14, 19, 21, 22]). A
Hom-Lie superalgebra is a $\mathbb{Z}_{2}$-graded space with a bracket for
which the super Jacobi identity is twisted by a homomorphism. This class was
introduced by Ammar and Makhlouf in [1], where they have shown that the
supercommutator bracket defined using the multiplication in a Hom-associative
superalgebra leads naturally to a Hom-Lie superalgebra. Hom-Lie superalgebras
are generalizations of Lie superalgebras, Lie algebras as well as Hom-Lie
algebras, and also they are a particular case of $\Gamma$-graded quasi-Lie
algebras introduced by Larsson and Silvestrov in [15], who also have shown its
relation with discrete and deformed vector fields and differential calculus.
We also note that an analogous notion has been considered in other categories
like Hom-associative algebras, Hom-alternative algebras, Hom-Leibniz algebras,
etc. (see [1, 12, 16, 17, 18, 20, 23]).

In the present paper we introduce the class of split regular Hom-Lie
superalgebras as the natural extension of the one of split regular Hom-Lie
algebras (see [2]) and study its structure presenting them as a direct sum of
adequate ideals. We also characterize a simple split regular split Hom-Lie
superalgebra which together a relation of equivalence defined in its root
system allow us to obtain a final expression as direct sum of simple ideals.
Here it is interesting to mention the recent references [9, 10, 11, 24] where
different classes of split Hom-algebras are also studied from the viewpoint of
their inner structures.

We briefly describe the content of this paper. In Section 2 we develop
connection of roots techniques which becomes the main tool in our study. In
Section 3 we apply all of the machinary introduced in the previous section to
show that a split regular Hom-Lie superalgebra ${\mathfrak{L}}$ is of the form
${\mathfrak{L}}=U+\sum_{j}I_{j}$ with $U$ a linear subspace of a maximal
abelian graded subalgebra $H$ and any $I_{j}$ a well described ideal of
${\mathfrak{L}}$ satisfying $[I_{j},I_{k}]=0$ if $j\neq k$. In the final
section, under certain conditions, the simplicity of ${\mathfrak{L}}$ is
characterized and it is shown that ${\mathfrak{L}}$ is the direct sum of the
family of its simple ideals.

######  Definition 1.1.

A Hom-Lie superalgebra ${\mathfrak{L}}$ is a ${\hbox{\hueca Z}}_{2}$-graded
algebra
${\mathfrak{L}}={\mathfrak{L}}_{\bar{0}}\oplus{\mathfrak{L}}_{\bar{1}}$ over a
base field 𝕂 endowed with a bilinear product

$[\cdot,\cdot]:{\mathfrak{L}}\times{\mathfrak{L}}\to{\mathfrak{L}}$

and an even superspace homomorphism $\phi:{\mathfrak{L}}\to{\mathfrak{L}},$
satisfying

$[x,y]=-(-1)^{{\bar{i}}{\bar{j}}}[y,x]$ (1)
$(-1)^{{\bar{i}}{\bar{k}}}[[x,y],\phi(z)]+(-1)^{{\bar{i}}{\bar{j}}}[[y,z],\phi(x)]+(-1)^{{\bar{j}}{\bar{k}}}[[z,x],\phi(y)]=0,$
(2)

for all homogeneous elements
$x\in{\mathfrak{L}}_{{\bar{i}}},y\in{\mathfrak{L}}_{\bar{j}},z\in{\mathfrak{L}}_{\bar{k}},$
with ${\bar{i}},{\bar{j}},{\bar{k}}\in{\hbox{\hueca Z}}_{2},$ called skew-
supersymmetry and super Hom-Jacobi identity, respectively. Furthermore, if
$\phi$ is an superalgebra automorphism ${\mathfrak{L}}$ is called a regular
Hom-Lie superalgebra.

Note that ${\mathfrak{L}}_{\bar{0}}$ is a Hom-Lie algebra called the even or
bosonic part of $\mathfrak{L}$ while ${\mathfrak{L}}_{\bar{1}}$ is called the
odd or fermonic part of $\mathfrak{L}$. The usual regularity concepts will be
understood in the graded sense. A graded subalgebra $A$ of ${\mathfrak{L}}$ is
a graded subspace $A=A_{\bar{0}}\oplus A_{\bar{1}}$ of ${\mathfrak{L}}$ such
that $[A,A]\subset A$ and $\phi(A)=A$. An ideal $I$ of ${\mathfrak{L}}$ is a
graded subalgebra $I=I_{\bar{0}}\oplus I_{\bar{1}}$ of ${\mathfrak{L}}$
satisfying $[I,{\mathfrak{L}}]\subset I.$ A Hom-Lie superalgebra
$\mathfrak{L}$ is called simple if $[{\mathfrak{L}},{\mathfrak{L}}]\neq 0$ and
its only (graded) ideals are $\\{0\\}$ and ${\mathfrak{L}}$. Throughout this
paper we consider regular Hom-Lie superalgebras and denote by $\mathbb{N}_{0}$
the set of all non-negative integers and by $\mathbb{Z}$ the set of all
integers. Finally, we emphasize that the regular Hom-Lie superalgebras
${\mathfrak{L}}$ are considered of arbitrary dimension and over an arbitrary
base field 𝕂. Let us introduce the class of split algebras in the framework of
regular Hom-Lie superalgebras in a similar way to the cases of Hom-Lie
algebras, Lie algebras and Lie superalgebras, among other classes of algebras.
We begin by considering a maximal abelian graded subalgebra
$H=H_{\bar{0}}\oplus H_{\bar{1}}$ among the abelian graded subalgebras of
$\mathfrak{L}.$

######  Definition 1.2.

Let $\mathfrak{L}$ be a regular Hom-Lie superalgebra and denote by
$H=H_{\bar{0}}\oplus H_{\bar{1}}$ a maximal abelian graded subalgebra (MAGSA)
of $\mathfrak{L}$. For a linear functional $\alpha:H_{\bar{0}}\to\hbox{\hueca
K},$ we define the root space of $\mathfrak{L}$ (with respect to $H$)
associated to $\alpha$ as the subspace

${\mathfrak{L}}_{\alpha}:=\bigl{\\{}v_{\alpha}\in{\mathfrak{L}}:[h_{\bar{0}},v_{\alpha}]=\alpha(h_{\bar{0}})\phi(v_{\alpha})\hskip
2.84544pt{\it for}\hskip 2.84544pt{\it any}\hskip 2.84544pth_{\bar{0}}\in
H_{\bar{0}}\bigr{\\}}.$

The elements $\alpha\in(H_{\bar{0}})^{*}$ satisfying
${\mathfrak{L}}_{\alpha}\neq 0$ are called roots of ${\mathfrak{L}}$ (with
respect to $H$) and we denote
$\Lambda:=\\{\alpha\in(H_{\bar{0}})^{*}\setminus\\{0\\}:{\mathfrak{L}}_{\alpha}\neq
0\\}$ the root system of ${\mathfrak{L}}.$ We say that ${\mathfrak{L}}$ is a
split regular Hom-Lie superalgebra (with respect to $H$) if

${\mathfrak{L}}=H\oplus\Bigl{(}\bigoplus\limits_{\alpha\in\Lambda}{\mathfrak{L}}_{\alpha}\Bigr{)}.$

Split regular Hom-Lie algebras, split Lie superalgebras and split Lie algebras
are examples of split regular Hom-Lie superalgebras. Hence, the present paper
extends the results in [2, 4, 8]. Let us now show a proper example.

######  Example 1.1.

Consider the real ${\mathbb{Z}}_{2}$-graded vector space
${\mathfrak{L}}={\mathfrak{L}}_{\bar{0}}\oplus{\mathfrak{L}}_{\bar{1}}$, with
basis

$\\{e_{1},e_{2}\\}\cup\\{h_{n},x_{n},y_{n}:n\in{\mathbb{N}},n\geq 2\\}$

of ${\mathfrak{L}}_{\bar{0}}$ and

$\\{e_{3}\\}\cup\\{f_{n},g_{n}:n\in{\mathbb{N}},n\geq 2\\}$

of ${\mathfrak{L}}_{\bar{1}};$ and where the nonzero products on these
elements are induced by the relations:

$[e_{2},e_{1}]=e_{1},\hskip 8.5359pt[h_{n},x_{n}]=2n^{2}x_{n},\hskip
8.5359pt[h_{n},y_{n}]=-\frac{2}{n^{2}}y_{n},$ $[x_{n},y_{n}]=h_{n},\hskip
8.5359pt[y_{n},g_{n}]=\frac{1}{n}f_{n},\hskip 8.5359pt[x_{n},f_{n}]={n}g_{n},$
$[h_{n},f_{n}]=-\frac{1}{n}f_{n},\hskip 8.5359pt[h_{n},g_{n}]={n}g_{n},\hskip
8.5359pt[g_{n},f_{n}]=h_{n},\hskip 8.5359pt[g_{n},g_{n}]=-2n^{2}x_{n},$
$[f_{n},f_{n}]=\frac{2}{n^{2}}y_{n}.$

Then by considering the superspace homomorphism

$\phi:{\mathfrak{L}}\to{\mathfrak{L}}$

defined by

$\phi(e_{i})=e_{i}$, $i\in\\{1,2,3\\}$ and  $\phi(h_{n})=h_{n}$,
$\phi(x_{n})=n^{2}x_{n}$, $\phi(y_{n})=\frac{1}{n^{2}}y_{n}$,
$\phi(f_{n})=\frac{1}{n}f_{n}$, $\phi(g_{n})=ng_{n}$

for any $n\geq 2$, we have that
${\mathfrak{L}}={\mathfrak{L}}_{\bar{0}}\oplus{\mathfrak{L}}_{\bar{1}}$
becomes a, (non-Hom-Lie algebra and non-Lie superalgebra), split regular Hom-
Lie superalgebra

${\mathfrak{L}}=H\oplus{\mathfrak{L}}_{\alpha}\oplus(\bigoplus\limits_{n\in{\mathbb{N}},n\geq
2}{\mathfrak{L}}_{\beta_{n}})\oplus(\bigoplus\limits_{n\in{\mathbb{N}},n\geq
2}{\mathfrak{L}}_{-\beta_{n}})\oplus(\bigoplus\limits_{n\in{\mathbb{N}},n\geq
2}{\mathfrak{L}}_{\gamma_{n}})\oplus(\bigoplus\limits_{n\in{\mathbb{N}},n\geq
2}{\mathfrak{L}}_{-\gamma_{n}})$ (3)

where

$H=\langle e_{2},h_{2},h_{3},...,h_{n},...\rangle\oplus\langle e_{3}\rangle,$
$\hbox{ ${\mathfrak{L}}_{\alpha}=\langle e_{1}\rangle,$
${\mathfrak{L}}_{\beta_{n}}=\langle x_{n}\rangle$,
${\mathfrak{L}}_{-\beta_{n}}=\langle y_{n}\rangle$,
${\mathfrak{L}}_{\gamma_{n}}=\langle f_{n}\rangle$ and
${\mathfrak{L}}_{-\gamma_{n}}=\langle g_{n}\rangle$},$

being

$\alpha,\beta_{n},\gamma_{n}:H\to{\mathbb{R}},$

$n\in{\mathbb{N}},n\geq 2,$ defined by

$\hbox{$\alpha(e_{2})=1,$ and $\alpha(e_{3})=\alpha(h_{n})=0,$ for any $n\geq
2$},$ $\hbox{$\beta_{n}(e_{2})=\beta_{n}(e_{3})=0,$ $\beta_{n}(h_{n})=2,$ and
$\beta_{n}(h_{m})=0$ when $n\neq m$},$
$\hbox{$\gamma_{n}(e_{2})=\gamma_{n}(e_{3})=0,$ $\gamma_{n}(h_{n})=-1,$ and
$\gamma_{n}(h_{m})=0$ when $n\neq m$}.$

It is clear that the root space associated to the zero root satisfies
$H\subset{\mathfrak{L}}_{0}.$ Conversely, given any
$v_{0}\in{\mathfrak{L}}_{0}$ we can write
$v_{0}=h+\sum_{i=1}^{n}v_{\alpha_{i}}$ with $h\in H$ and
$v_{\alpha_{i}}\in{\mathfrak{L}}_{\alpha_{i}}$ for $i\in\\{1,\dots,n\\}$,
being $\alpha_{i}\in\Lambda$ with $\alpha_{i}\neq\alpha_{j}$ if $i\neq j$.
Hence
$0=[h_{\bar{0}},h+\sum_{i=1}^{n}v_{\alpha_{i}}]=\sum_{i=1}^{n}\alpha_{i}(h_{\bar{0}})\phi(v_{\alpha_{i}})$
for any $h_{\bar{0}}\in H_{\bar{0}}$. So, taking into account the direct
character of the sum, that $\alpha_{i}\neq 0$ and $\phi$ is an automorphism,
we have that any $v_{\alpha_{i}}=0$ and then $v_{0}\in H$. Consequently

$H={\mathfrak{L}}_{0}.$

By the grading of ${\mathfrak{L}},$ any
$v_{\alpha}\in{\mathfrak{L}}_{\alpha},\alpha\in\Lambda\cup\\{0\\}$ may be
expressed in the form $v_{\alpha}=v_{\alpha,{\bar{0}}}+v_{{\alpha},{\bar{1}}}$
with
$v_{\alpha,{\bar{i}}}\in{\mathfrak{L}}_{\bar{i}},{\bar{i}}\in{\hbox{\hueca
Z}}_{2}$, then

$[h_{\bar{0}},v_{\alpha,{\bar{i}}}]=\alpha(h_{\bar{0}})\phi(v_{\alpha,{\bar{i}}})$

for any $h_{\bar{0}}\in H_{\bar{0}}$. From here,
${\mathfrak{L}}_{\alpha}=({\mathfrak{L}}_{\alpha}\cap{\mathfrak{L}}_{\bar{0}})\oplus({\mathfrak{L}}_{\alpha}\cap{\mathfrak{L}}_{\bar{1}})$.
Hence, by denoting
${\mathfrak{L}}_{\alpha,{\bar{i}}}:={\mathfrak{L}}_{\alpha}\cap{\mathfrak{L}}_{\bar{i}},$
for $\bar{i}\in\mathbb{Z}_{2},$ we can write

${\mathfrak{L}}_{\alpha}={\mathfrak{L}}_{\alpha,{\bar{0}}}\oplus{\mathfrak{L}}_{\alpha,{\bar{1}}}$
(4)

for any $\alpha\in\Lambda\cup\\{0\\}$. From the above,

$H_{\bar{0}}={\mathfrak{L}}_{0,{\bar{0}}}$ and
$H_{\bar{1}}={\mathfrak{L}}_{0,{\bar{1}}}$,

and also

${\mathfrak{L}}_{\bar{0}}=H_{\bar{0}}\oplus(\bigoplus\limits_{\alpha\in\Lambda}{\mathfrak{L}}_{\alpha,{\bar{0}}})$
and
${\mathfrak{L}}_{\bar{1}}=H_{\bar{1}}\oplus(\bigoplus\limits_{\alpha\in\Lambda}{\mathfrak{L}}_{\alpha,{\bar{1}}})$.

Taking into account this expression of ${\mathfrak{L}}_{\bar{0}}$, the direct
character of the sum and the fact that $\alpha\neq 0$ for any
$\alpha\in\Lambda$, we have that $H_{\bar{0}}$ is a MASA of the Hom-Lie
algebra ${\mathfrak{L}}_{\bar{0}}$. Hence ${\mathfrak{L}}_{\bar{0}}$ is a
split regular Hom-Lie algebra (with respect to $H_{\bar{0}}$) (cf. [2]) and
${\mathfrak{L}}_{\bar{1}}$ is (in a sense) a split anti-Lie triple system
(with respect to $H_{\bar{0}}$) (cf. [5]). For an easier notation, the
mappings $\phi|_{H_{\bar{0}}},\phi|_{H_{\bar{0}}}^{-1}:H_{\bar{0}}\to
H_{\bar{0}}$ will be denoted by $\phi$ and $\phi^{-1},$ respectively, when
there is not possible confusion. The below lemma in an immediate consequence
of super Hom-Jacobi identity.

######  Lemma 1.1.

For any $\alpha,\beta\in\Lambda\cup\\{0\\}$ the following assertions hold.

  1. 1.

If $\phi({\mathfrak{L}}_{\alpha})\neq 0$ then
${\alpha}\phi^{-1}\in\Lambda\cup\\{0\\}$ and
$\phi({\mathfrak{L}}_{\alpha})={\mathfrak{L}}_{{\alpha}\phi^{-1}}.$ In a
similar way, if $\phi^{-1}({\mathfrak{L}}_{\alpha})\neq 0$ then
${\alpha}\phi\in\Lambda\cup\\{0\\}$ and
$\phi^{-1}({\mathfrak{L}}_{\alpha})={\mathfrak{L}}_{{\alpha}\phi}.$

  2. 2.

If $[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{\beta}]\neq 0$ then
$\alpha\phi^{-1}+\beta\phi^{-1}\in\Lambda\cup\\{0\\}$ and
$[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{\beta}]\subset{\mathfrak{L}}_{\alpha\phi^{-1}+\beta\phi^{-1}}.$

###### Proof.

1\. For any $h_{\bar{0}}\in H_{\bar{0}}$ and $0\neq
v_{\alpha}\in{\mathfrak{L}}_{\alpha},$ since
$[h_{\bar{0}},v_{\alpha}]=\alpha(h_{\bar{0}})\phi(v_{\alpha}),$ by writing
$h^{\prime}:=\phi(h_{\bar{0}})$ we have

$[h^{\prime},\phi(v_{\alpha})]=\phi([h_{\bar{0}},v_{\alpha}])=\alpha(h_{\bar{0}})\phi^{2}(v_{\alpha})=\alpha\phi^{-1}(h^{\prime})\phi(\phi(v_{\alpha})).$

That is,
$0\neq\phi(v_{\alpha})\in{\mathfrak{L}}_{\alpha\phi^{-1}},\alpha\phi^{-1}\in\Lambda\cup\\{0\\}$
and also

$\phi({\mathfrak{L}}_{\alpha})\subset{\mathfrak{L}}_{\alpha\phi^{-1}}.$ (5)

Now, for any $h_{\bar{0}}\in H_{\bar{0}}$ and $0\neq
v_{\alpha}\in{\mathfrak{L}}_{\alpha},$ as in the previous inclusion, for
$h^{\prime}:=\phi^{-1}(h_{\bar{0}})$ we get

$[h^{\prime},\phi^{-1}(v_{\alpha})]=\phi^{-1}([h_{\bar{0}},v_{\alpha}])=\alpha(h_{\bar{0}})\phi^{-1}(\phi(v_{\alpha}))=\alpha\phi(h^{\prime})\phi(\phi^{-1}(v_{\alpha})).$

That is,
$0\neq\phi^{-1}(v_{\alpha})\in{\mathfrak{L}}_{\alpha\phi},\alpha\phi\in\Lambda\cup\\{0\\}$
and conclude

$\phi^{-1}({\mathfrak{L}}_{\alpha})\subset{\mathfrak{L}}_{\alpha\phi}.$ (6)

Now for any $x\in{\mathfrak{L}}_{\alpha\phi^{-1}},$ we can write
$x=\phi(\phi^{-1}(x))$ and since $\alpha\phi^{-1}\in\Lambda\cup\\{0\\}$ by
Equation (6) we have $\phi^{-1}(x)\in{\mathfrak{L}}_{\alpha}$, and we conclude
${\mathfrak{L}}_{\alpha\phi^{-1}}\subset\phi({\mathfrak{L}}_{\alpha}).$ This
fact together with Equation (5) show
$\phi({\mathfrak{L}}_{\alpha})={\mathfrak{L}}_{\alpha\phi^{-1}}$. In a similar
way we also prove
$\phi^{-1}({\mathfrak{L}}_{{\alpha}})={\mathfrak{L}}_{{\alpha}\phi}.$

2\. For any $h_{\bar{0}}\in H_{\bar{0}},v_{\alpha}\in{\mathfrak{L}}_{\alpha}$
and $v_{\beta}\in{\mathfrak{L}}_{\beta}$ such that $[v_{\alpha},v_{\beta}]\neq
0,$ by denoting $h^{\prime}:=\phi(h_{\bar{0}}),$ we get

$\displaystyle\begin{split}[h^{\prime},[v_{\alpha},v_{\beta}]]&=-[[h_{\bar{0}},v_{\beta}],\phi(v_{\alpha})]+[[h_{\bar{0}},v_{\alpha}],\phi(v_{\beta})]\\\
&=(\alpha+\beta)(h_{\bar{0}})\phi([v_{\alpha},v_{\beta}])\\\
&=(\alpha+\beta)\phi^{-1}(h^{\prime})\phi([v_{\alpha},v_{\beta}]).\end{split}$

That is,
$0\neq[v_{\alpha},v_{\beta}]\in{\mathfrak{L}}_{\alpha\phi^{-1}+\beta\phi^{-1}}$
and $\alpha\phi^{-1}+\beta\phi^{-1}\in\Lambda\cup\\{0\\}$. ∎

######  Remark 1.1.

From Lemma 1.1 and Equation (4) we can assert that for any
$\alpha,\beta\in\Lambda\cup\\{0\\}$ and $\bar{i},\bar{j}\in\mathbb{Z}_{2}$:

  1. 1.

If $\phi({\mathfrak{L}}_{{\alpha},\bar{i}})\neq 0$ then
${\alpha}\phi^{-1}\in\Lambda\cup\\{0\\}$ and
$\phi({\mathfrak{L}}_{{\alpha},\bar{i}})={\mathfrak{L}}_{{\alpha}\phi^{-1},\bar{i}},$

  2. 2.

If $\phi^{-1}({\mathfrak{L}}_{{\alpha},\bar{i}})\neq 0$ then
${\alpha}\phi\in\Lambda\cup\\{0\\}$ and
$\phi^{-1}({\mathfrak{L}}_{{\alpha},\bar{i}})={\mathfrak{L}}_{{\alpha}\phi,\bar{i}},$

  3. 3.

If $[{\mathfrak{L}}_{\alpha,\bar{i}},{\mathfrak{L}}_{\beta,\bar{j}}]\neq 0$
then $\alpha\phi^{-1}+\beta\phi^{-1}\in\Lambda\cup\\{0\\}$ and
$[{\mathfrak{L}}_{\alpha,\bar{i}},{\mathfrak{L}}_{\beta,\bar{j}}]\subset{\mathfrak{L}}_{\alpha\phi^{-1}+\beta\phi^{-1},\bar{i}+\bar{j}},$

######  Lemma 1.2.

If $\alpha\in\Lambda$ then $\alpha\phi^{z}\in\Lambda$ for any
$z\in{\mathbb{Z}}$.

###### Proof.

Consequence of Lemma 1.1-1. ∎

######  Remark 1.2.

From Lemma 1.2, Remark 1.1 and Equation (4) we can assert if
${\mathfrak{L}}_{\alpha,\bar{i}}\neq 0,$ with ${\bar{i}}\in\mathbb{Z}_{2},$
then ${\mathfrak{L}}_{\alpha\phi^{z},\bar{i}}\neq 0$ for any
$z\in{\mathbb{Z}}.$

##  2\. Connections of roots techniques

In the following, ${\mathfrak{L}}$ denotes a split regular Hom-Lie
superalgebra and

${\mathfrak{L}}={\mathfrak{L}}_{0}\oplus\Bigl{(}\bigoplus\limits_{\alpha\in\Lambda}{\mathfrak{L}}_{\alpha}\Bigr{)}$

the corresponding root spaces decomposition. Given a linear functional
$\alpha:H_{\bar{0}}\to{\hbox{\hueca K}}$, we denote by
$-\alpha:H_{\bar{0}}\to{\hbox{\hueca K}}$ the element in $(H_{\bar{0}})^{*}$
defined by $(-\alpha)(h_{\bar{0}}):=-\alpha(h_{\bar{0}})$ for all
$h_{\bar{0}}\in H_{\bar{0}}$. We also denote by
$-\Lambda:=\\{-\alpha:\alpha\in\Lambda\\}$ and

$\pm\Lambda:=\Lambda\cup-\Lambda.$

######  Definition 2.1.

Let $\alpha$ and $\beta$ be two elements in $\Lambda$. We shall say that
$\alpha$ is connected to $\beta$ if there exists
$\\{\alpha_{1},\alpha_{2},\dots,\alpha_{k}\\}\subset\pm\Lambda$ such that:

  1. If $k=1$,

    1. 1.

$\alpha_{1}\in\\{\alpha\phi^{-n}:n\in{\mathbb{N}}_{0}\\}\cap\\{\pm\beta\phi^{-m}:m\in{\mathbb{N}}_{0}\\}.$

  2. If $k\geq 2$,

    1. 1.

$\alpha_{1}\in\\{\alpha\phi^{-n}:n\in{\mathbb{N}}_{0}\\}$.

    2. 2.

$\alpha_{1}\phi^{-1}+\alpha_{2}\phi^{-1}\in\pm\Lambda,$

$\alpha_{1}\phi^{-2}+\alpha_{2}\phi^{-2}+\alpha_{3}\phi^{-1}\in\pm\Lambda,$

$\alpha_{1}\phi^{-3}+\alpha_{2}\phi^{-3}+\alpha_{3}\phi^{-2}+\alpha_{4}\phi^{-1}\in\pm\Lambda,$

$\vdots$

$\alpha_{1}\phi^{-i}+\alpha_{2}\phi^{-i}+\alpha_{3}\phi^{-i+1}+\cdots+\alpha_{i+1}\phi^{-1}\in\pm\Lambda,$

$\vdots$

$\alpha_{1}\phi^{-k+2}+\alpha_{2}\phi^{-k+2}+\alpha_{3}\phi^{-k+3}+\cdots+\alpha_{i}\phi^{-k+i}+\cdots+\alpha_{k-1}\phi^{-1}\in\pm\Lambda.$

    3. 3.

$\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+1}+\alpha_{3}\phi^{-k+2}+\cdots+\alpha_{i}\phi^{-k+i-1}+\cdots+\alpha_{k}\phi^{-1}\in\\{\pm\beta\phi^{-m}:m\in{\mathbb{N}}_{0}\\}.$

We shall also say that $\\{\alpha_{1},\dots,\alpha_{k}\\}$ is a connection
from $\alpha$ to $\beta$.

Observe that the above concept of connection of nonzero roots from $\alpha$ to
$\beta$, given in Definition 2.1, for the case $k=1$ is equivalent to the fact
$\beta=\epsilon\alpha\phi^{z}$ for some $z\in{\mathbb{Z}}$ and
$\epsilon\in\\{1,-1\\}$. Our next goal is to show that the connection relation
is of equivalence. We begin with a couple of previous lemmas being their proof
analogous to the ones of Lemma 2.1 and Lemma 2.2 in [2].

######  Lemma 2.1.

For any $\alpha\in\Lambda,\alpha\phi^{z_{1}}$ is connected to
$\alpha\phi^{z_{2}}$ for any $z_{1},z_{2}\in{\mathbb{Z}},$ and also to
$-\alpha\phi^{z_{2}}$ in case $-\alpha\in\Lambda$.

######  Lemma 2.2.

Let $\\{\alpha_{1},\dots,\alpha_{k}\\}$ be a connection from $\alpha$ to
$\beta$. The following assertions hold.

  1. 1.

If $\alpha_{1}=\alpha\phi^{-n},n\in{\mathbb{N}}_{0},$ then for any
$r\in{\mathbb{N}}_{0}$ such that $r\geq n$, there exists a connection
$\\{\bar{\alpha}_{1},\dots,\bar{\alpha}_{k}\\}$ from $\alpha$ to $\beta$ such
that $\bar{\alpha}_{1}=\alpha\phi^{-r}$.

  2. 2.

If $\alpha_{1}=\epsilon\beta\phi^{-m}$ in case $k=1$ or

$\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+1}+\alpha_{3}\phi^{-k+2}+\cdots+\alpha_{k}\phi^{-1}=\epsilon\beta\phi^{-m}$

in case $k\geq 2$, with $m\in{\mathbb{N}}_{0}$ and $\epsilon\in\\{1,-1\\},$
then for any $r\in{\mathbb{N}}_{0}$ such that $r\geq m$, there exists a
connection $\\{\bar{\alpha}_{1},\dots,\bar{\alpha}_{k}\\}$ from $\alpha$ to
$\beta$ such that $\bar{\alpha}_{1}=\epsilon\beta\phi^{-r}$ in case $k=1$ or

$\bar{\alpha}_{1}\phi^{-k+1}+\bar{\alpha}_{2}\phi^{-k+1}+\bar{\alpha}_{3}\phi^{-k+2}+\cdots+\bar{\alpha}_{k}\phi^{-1}=\epsilon\beta\phi^{-r}$

in case $k\geq 2$.

The proof of the next result follows the ideas of [2] for split Hom-Lie
algebras. For the sake of completeness we add an sketch of the same.

######  Proposition 2.1.

The relation $\sim$ in $\Lambda$ defined by $\alpha\sim\beta$ if and only if
$\alpha$ is connected to $\beta$ is an equivalence relation.

###### Proof.

By Lemma 2.1 the relation $\sim$ is reflexive. The symmetric character of
$\sim$ is consequence of consider $\alpha\sim\beta$ with a connection
$\\{\alpha_{1},\alpha_{2},\alpha_{3},\dots,\alpha_{k-1},\alpha_{k}\\}\subset\pm\Lambda$
from $\alpha$ to $\beta$. If $k=1$ we get $\beta\sim\alpha$ with the
connection $\\{\epsilon\alpha_{1}\\}.$ If $k\geq 2$, we conclude
$\beta\sim\alpha$ taking the connection

$\\{\beta\phi^{-m},-\alpha_{k}\phi^{-1},-\alpha_{k-1}\phi^{-3},-\alpha_{k-2}\phi^{-5},\dots,-\alpha_{k-i}\phi^{-2i-1},\dots,-\alpha_{2}\phi^{-2k+3}\\}\subset\pm\Lambda$

in case
$\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+1}+\alpha_{3}\phi^{-k+2}+\cdots+\alpha_{k-i+1}\phi^{-i}+\cdots+\alpha_{k}\phi^{-1}=\beta\phi^{-m},$
and the connection

$\bigl{\\{}\beta\phi^{-m},\alpha_{k}\phi^{-1},\alpha_{k-1}\phi^{-3},\alpha_{k-2}\phi^{-5},\dots,\alpha_{k-i}\phi^{-2i-1},\dots,\alpha_{2}\phi^{-2k+3}\bigr{\\}}\pm\Lambda$

in case
$\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+1}+\alpha_{3}\phi^{-k+2}+\cdots+\alpha_{k-i+1}\phi^{-i}+\cdots+\alpha_{k}\phi^{-1}=-\beta\phi^{-m},$
for some $m\in{\mathbb{N}}_{0}.$

Finally, to verify that $\sim$ is transitive we suppose $\alpha\sim\beta$ with
connection $\\{\alpha_{1},\dots,\alpha_{k}\\}$ such that

$\alpha_{1}=\epsilon\beta\phi^{-m}$ if $k=1$ (7)

for some $m\in{\mathbb{N}}_{0}$, $\epsilon\in\\{1,-1\\},$ or

$\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+1}+\alpha_{3}\phi^{-k+2}+\cdots+\alpha_{k}\phi^{-1}=\epsilon\beta\phi^{-m}$
if $k\geq 2$, (8)

for some $r\in{\mathbb{N}}_{0},$ and suppose $\beta\sim\gamma$ with connection
$\\{h_{1},\dots,h_{p}\\}.$ If $p=1$, $\\{\alpha_{1},\dots,\alpha_{k}\\}$ is
also a connection from $\alpha$ to $\gamma$. If $p\geq 2$,
$\\{\alpha_{1},\dots,\alpha_{k},h_{2},\dots,h_{p}\\}$ is a connection from
$\alpha$ to $\gamma$ if $\epsilon=1$ in Equations (7) or (8), and
$\\{\alpha_{1},\dots,\alpha_{k},-h_{2},\dots,-h_{p}\\}$ is the referred
connection if $\epsilon=-1$ in Equations (7) or (8). Summarizing, the
connection relation is also transitive and so it is an equivalence relation. ∎

##  3\. Decompositions

By Proposition 2.1 the connection relation is an equivalence relation in
$\Lambda$ and so we can consider the quotient set

$\Lambda/\sim:=\\{[\alpha]:\alpha\in\Lambda\\},$

becoming $[\alpha]$ the set of nonzero roots ${\mathfrak{L}}$ which are
connected to $\alpha$. Our next goal in this section is to associate an
(adequate) ideal ${I}_{{[\alpha]}}$ of the split regular Hom-Lie superalgebra
${\mathfrak{L}}$ to any $[\alpha]$ of $\Lambda/\sim$. For $[\alpha]$, with
$\alpha\in\Lambda$, we define

$H_{[\alpha]}:=span_{\hbox{\hueca
K}}\bigl{\\{}[{\mathfrak{L}}_{\beta},{\mathfrak{L}}_{\delta}]:\beta,\delta\in[\alpha]\bigr{\\}}\cap{\mathfrak{L}}_{0}.$

Applying Lemma 1.1-2 we obtain

$H_{[\alpha]}:=span_{\hbox{\hueca
K}}\bigl{\\{}[{\mathfrak{L}}_{\beta},{\mathfrak{L}}_{-\beta}]:\beta\in[\alpha]\bigr{\\}}=$
$\Bigl{(}\sum\limits_{\beta\in[\alpha]}\bigl{(}[{\mathfrak{L}}_{\beta,{\bar{0}}},{\mathfrak{L}}_{-\beta,{\bar{0}}}]+[{\mathfrak{L}}_{\beta,{\bar{1}}},{\mathfrak{L}}_{-\beta,{\bar{1}}}]\bigr{)}\Bigr{)}\oplus\Bigl{(}\sum\limits_{\beta\in[\alpha]}\bigl{(}[{\mathfrak{L}}_{\beta,{\bar{0}}},{\mathfrak{L}}_{-\beta,{\bar{1}}}]+[{\mathfrak{L}}_{\beta,{\bar{1}}},{\mathfrak{L}}_{-\beta,{\bar{0}}}]\bigr{)}\Bigl{)}\subset
H_{\bar{0}}\oplus H_{\bar{1}},$

last equality being consequence of Equation (4), and we also define

$V_{[\alpha]}:=\bigoplus\limits_{\beta\in[\alpha]}{\mathfrak{L}}_{\beta}=(\bigoplus\limits_{\beta\in[\alpha]}{\mathfrak{L}}_{\beta,{\bar{0}}})\oplus(\bigoplus\limits_{\beta\in[\alpha]}{\mathfrak{L}}_{\beta,{\bar{1}}}).$

We denote by ${\mathfrak{L}}_{[\alpha]}$ the following (graded) subspace of
${\mathfrak{L}}$,

${\mathfrak{L}}_{[\alpha]}:=H_{[\alpha]}\oplus V_{[\alpha]}.$

######  Proposition 3.1.

For any $[\alpha]\in\Lambda/\sim$, the linear subspace
${\mathfrak{L}}_{[\alpha]}$ is a graded subalgebra of ${\mathfrak{L}}$.

###### Proof.

First, we have to check that ${\mathfrak{L}}_{{[\alpha]}}$ satisfies
$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[\alpha]}]\subset{\mathfrak{L}}_{[\alpha]}$.
Since $H_{[\alpha]}\subset{\mathfrak{L}}_{0}=H$, then
$[H_{[\alpha]},H_{[\alpha]}]=0$ and we have

$\bigl{[}H_{[\alpha]}\oplus V_{[\alpha]},H_{[\alpha]}\oplus
V_{[\alpha]}\bigr{]}\subset[H_{[\alpha]},V_{[\alpha]}]+[V_{[\alpha]},V_{[\alpha]}].$
(9)

Let us consider the first summand in Equation (9). Given $\beta\in[\alpha]$ we
have
$[H_{[\alpha]},{\mathfrak{L}}_{\beta}]\subset{\mathfrak{L}}_{\beta\phi^{-1}}$,
being $\beta\phi^{-1}\in[\alpha]$ by Lemma 2.1. Hence

$[H_{[\alpha]},{\mathfrak{L}}_{\beta}]\subset V_{[\alpha]}.$ (10)

Consider now the second summand in Equation (9), that is,
$[V_{[\alpha]},V_{[\alpha]}]$. Given $\beta,\gamma\in[\alpha]$ such that
$[{\mathfrak{L}}_{\beta},{\mathfrak{L}}_{\gamma}]\neq 0$, if $\gamma=-\beta$
then clearly
$[{\mathfrak{L}}_{\beta},{\mathfrak{L}}_{\gamma}]=[{\mathfrak{L}}_{\beta},{\mathfrak{L}}_{-\beta}]\subset
H_{[\alpha]}.$ Suppose $\gamma\neq-\beta$. Taking into account that the fact
$[{{\mathfrak{L}}}_{\beta},{{\mathfrak{L}}}_{\gamma}]\neq 0$ together with
Lemma 1.1-2 ensure $\beta\phi^{-1}+\gamma\phi^{-1}\in\Lambda$, we have that
$\\{\beta,\gamma\\}$ is a connection from $\beta$ to
$\beta\phi^{-1}+\gamma\phi^{-1}$. The transitivity of $\sim$ gives now that
$\beta\phi^{-1}+\gamma\phi^{-1}\in[\alpha]$ and so
$[{\mathfrak{L}}_{\beta},{\mathfrak{L}}_{\gamma}]\subset{\mathfrak{L}}_{\beta\phi^{-1}+\gamma\phi^{-1}}\subset
V_{[\alpha]}$. Hence
$[\bigoplus\limits_{\beta\in[\alpha]}{\mathfrak{L}}_{\beta},\bigoplus\limits_{\beta\in[\alpha]}{\mathfrak{L}}_{\beta}]\subset
H_{[\alpha]}\oplus V_{[\alpha]}$. That is,

$[V_{[\alpha]},V_{[\alpha]}]\subset{\mathfrak{L}}_{[\alpha]}.$ (11)

From Equations (9)-(11) we get
$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[\alpha]}]=[H_{[\alpha]}\oplus
V_{[\alpha]},H_{[\alpha]}\oplus
V_{[\alpha]}]\subset{\mathfrak{L}}_{[\alpha]}.$

Second, we have to verify that
$\phi({\mathfrak{L}}_{[\alpha]})={\mathfrak{L}}_{[\alpha]}$. But this is a
direct consequence of Lemma 1.1-1 and Lemma 2.1. ∎

######  Proposition 3.2.

If $[\alpha]\neq[\gamma]$ then
$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[{\gamma}]}]=0$.

###### Proof.

We have

$\bigl{[}H_{[\alpha]}\oplus V_{[\alpha]},H_{[\gamma]}\oplus
V_{[\gamma]}\bigr{]}\subset[H_{[\alpha]},V_{[\gamma]}]+[V_{[\alpha]},H_{[\gamma]}]+[V_{[\alpha]},V_{[\gamma]}].$
(12)

Consider the above third summand $[V_{[\alpha]},V_{[\gamma]}]$ and suppose
there exist $\alpha_{1}\in[\alpha]$ and $\gamma_{1}\in[{\gamma}]$ such that
$[{\mathfrak{L}}_{\alpha_{1}},{\mathfrak{L}}_{\gamma_{1}}]\neq 0$. As
necessarily $\alpha_{1}\neq-\gamma_{1}$, then
$\alpha_{1}\phi^{-1}+\gamma_{1}\phi^{-1}\in\Lambda$. So
$\\{\alpha_{1},\gamma_{1},-\alpha_{1}\phi^{-1}\\}$ is a connection between
$\alpha_{1}$ and $\gamma_{1}$. By the transitivity of the connection relation
we have $\alpha\in[\gamma]$, a contradiction. Hence
$[{\mathfrak{L}}_{\alpha_{1}},{\mathfrak{L}}_{\gamma_{1}}]=0$ and so

$[V_{[\alpha]},V_{[\gamma]}]=0.$ (13)

Consider now the first summand $[H_{[\alpha]},V_{[\gamma]}]$ in Equation (12)
and suppose there exist $\alpha_{1}\in[\alpha]$ and $\gamma_{1}\in[\gamma]$
such that
$[[{\mathfrak{L}}_{\alpha_{1}},{\mathfrak{L}}_{-\alpha_{1}}],{\mathfrak{L}}_{\gamma_{1}}]\neq
0$. By Lemma 1.1-1 we have
${\mathfrak{L}}_{\gamma_{1}}=\phi({{\mathfrak{L}}}_{\gamma_{1}\phi})$, so we
obtain
$\bigl{[}[{\mathfrak{L}}_{\alpha_{1}},{\mathfrak{L}}_{-\alpha_{1}}],\phi({\mathfrak{L}}_{\gamma_{1}\phi})\bigr{]}\neq
0.$ Using the grading of ${\mathfrak{L}}_{\alpha_{1}}$ we get

$\bigl{[}[{\mathfrak{L}}_{\alpha_{1},\bar{0}},{\mathfrak{L}}_{-\alpha_{1}}],\phi({{\mathfrak{L}}}_{\gamma_{1}\phi})\bigr{]}+\bigl{[}[{\mathfrak{L}}_{\alpha_{1},\bar{1}},{\mathfrak{L}}_{-\alpha_{1}}],\phi({{\mathfrak{L}}}_{\gamma_{1}\phi})\bigr{]}\neq
0.$

Therefore
$\bigl{[}[{\mathfrak{L}}_{\alpha_{1},\bar{0}},{\mathfrak{L}}_{-\alpha_{1}}],\phi({{\mathfrak{L}}}_{\gamma_{1}\phi})\bigr{]}\neq
0$ or
$\bigl{[}[{\mathfrak{L}}_{\alpha_{1},\bar{1}},{\mathfrak{L}}_{-\alpha_{1}}],\phi({{\mathfrak{L}}}_{\gamma_{1}\phi})\bigr{]}\neq
0.$ In the first case, by super Hom-Jacobi identity we get either
$[{\mathfrak{L}}_{-\alpha_{1}},{\mathfrak{L}}_{\gamma_{1}\phi}]\neq 0$ or
$[{\mathfrak{L}}_{\gamma_{1}\phi},{\mathfrak{L}}_{\alpha_{1},{\bar{0}}}]\neq
0$. From here $[V_{[\alpha]},V_{[\gamma]}]\neq 0$ in any case, what
contradicts Equation (13). In the second case we have
$0\bigl{[}[{\mathfrak{L}}_{\alpha_{1},{\bar{1}}},{\mathfrak{L}}_{-\alpha_{1},{\bar{0}}}],{\mathfrak{L}}_{\gamma_{1}\phi}\bigr{]}+\bigl{[}[{\mathfrak{L}}_{\alpha_{1},{\bar{1}}},{\mathfrak{L}}_{-\alpha_{1},{\bar{1}}}],{\mathfrak{L}}_{\gamma_{1}\phi}\bigr{]}\neq
0$ and so either
$\bigl{[}[{\mathfrak{L}}_{\alpha_{1},{\bar{1}}},{\mathfrak{L}}_{-\alpha_{1},{\bar{0}}}],{\mathfrak{L}}_{\gamma_{1}\phi}\bigr{]}\neq
0$ or
$\bigl{[}[{\mathfrak{L}}_{\alpha_{1},{\bar{1}}},{\mathfrak{L}}_{-\alpha_{1},{\bar{1}}}],{\mathfrak{L}}_{\gamma_{1}\phi}\bigr{]}\neq
0.$ By applying super Hom-Jacobi identity to any of these summands we have as
above that necessarily $[V_{[\alpha]},V_{[\gamma]}]\neq 0$ what contradicts
again Equation (13). From here $[H_{[\alpha]},V_{[\gamma]}]=0$.

In a similar way we get $[V_{[\alpha]},H_{[\gamma]}]=0$ and we conclude,
together with Equations (12) and (13) that
$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[\beta]}]=0.$ ∎

######  Theorem 3.1.

Let ${\mathfrak{L}}$ be a split regular Hom-Lie superalgebra. The following
assertions hold:

  1. 1.

For any $[\alpha]\in\Lambda/\sim$, the (graded) subalgebra

${\mathfrak{L}}_{[\alpha]}=H_{[\alpha]}\oplus V_{[\alpha]}$

of ${\mathfrak{L}}$ associated to $[\alpha]$ is an ideal of ${\mathfrak{L}}$.

  2. 2.

If ${\mathfrak{L}}$ is simple then there exists a connection from $\alpha$ to
$\beta$ for any $\alpha,\beta\in\Lambda$ and
$H=\sum\limits_{\alpha\in\Lambda}[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{-\alpha}]$.

###### Proof.

1\. Since $[{\mathfrak{L}}_{[\alpha]},H]\subset{\mathfrak{L}}_{[\alpha]}$ we
have by Proposition 3.1 and Proposition 3.2 that

$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}]=[{\mathfrak{L}}_{[\alpha]},H\oplus(\bigoplus\limits_{\beta\in[\alpha]}{\mathfrak{L}}_{{\beta}})\oplus(\bigoplus\limits_{{\gamma}\notin[\alpha]}{{\mathfrak{L}}}_{{\gamma}})]\subset{\mathfrak{L}}_{[\alpha]}.$

We also have by Proposition 3.1 that
$\phi({\mathfrak{L}}_{[\alpha]})={\mathfrak{L}}_{[\alpha]}$, we conclude
${\mathfrak{L}}_{[\alpha]}$ is an ideal of ${\mathfrak{L}}$.

2\. The simplicity of ${\mathfrak{L}}$ implies
${\mathfrak{L}}_{[\alpha]}={\mathfrak{L}}$. From here, it is clear that
$[\alpha]=\Lambda$ and
$H=\sum\limits_{\alpha\in\Lambda}[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{-\alpha}]$.
∎

######  Theorem 3.2.

A split regular Hom-Lie superalgebra ${\mathfrak{L}}$ can be represented as

${\mathfrak{L}}=U+\sum\limits_{[\alpha]\in\Lambda/\sim}{\mathfrak{L}}_{[\alpha]},$

where $U$ is a linear complement in $H$ of
$span_{\mathbb{K}}\\{[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{-\alpha}]:\alpha\in\Lambda\\}$
and any ${\mathfrak{L}}_{[\alpha]}$ is one of the ideals of ${\mathfrak{L}}$
described in Theorem 3.1-1, satisfying
$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[\gamma]}]=0$ if
$[\alpha]\neq[\gamma].$

###### Proof.

We have ${\mathfrak{L}}_{[\alpha]}$ is well defined and, by Theorem 3.1-1, an
ideal of $\mathfrak{L}$, being clear that

${\mathfrak{L}}=H\oplus(\bigoplus\limits_{\alpha\in\Lambda}{\mathfrak{L}}_{\alpha})=U+\sum\limits_{[\alpha]\in\Lambda/\sim}{\mathfrak{L}}_{[\alpha]}.$

Finally from Proposition 3.2 we obtain
$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[\gamma]}]=0$ if
$[\alpha]\neq[\gamma].$ ∎

Let us denote by
${\mathcal{Z}}({\mathfrak{L}}):=\bigl{\\{}v\in{\mathfrak{L}}:[v,{\mathfrak{L}}]=0\bigr{\\}}$
the center of ${\mathfrak{L}}$.

######  Corollary 3.1.

If ${\mathcal{Z}}({\mathfrak{L}})=0$ and
$H=\sum_{\alpha\in\Lambda}[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{-\alpha}]$
then ${\mathfrak{L}}$ is the direct sum of the ideals given in Theorem 3.1,

${\mathfrak{L}}=\bigoplus\limits_{[\alpha]\in\Lambda/\sim}{\mathfrak{L}}_{[\alpha]},$

being $[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[\gamma]}]=0$ if
$[\alpha]\neq[\gamma].$

###### Proof.

Since
$H=\sum_{\alpha\in\Lambda}[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{-\alpha}]$
we get
${\mathfrak{L}}=\sum_{[\alpha]\in\Lambda/\sim}{\mathfrak{L}}_{[\alpha]}$.
Finally, the direct character of the sum can be followed from the facts
$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[\gamma]}]=0$, if
$[\alpha]\neq[\gamma]$, and ${\mathcal{Z}}({\mathfrak{L}})=0$. ∎

######  Example 3.1.

Let us consider the split Hom-Lie superalgebra ${{\mathfrak{L}}}$ in Example
1.1. We have that

$\Lambda=\\{\alpha\\}\cup\\{\pm\beta_{n},\pm\gamma_{n};n\in{\mathbb{N}},n\geq
2\\}.$

It is straightforward to verify that

$\Lambda/\sim=\\{[\alpha]\\}\cup\\{[\beta_{n}]:n\in{\mathbb{N}},n\geq 2\\},$

where $[\alpha]=\\{\alpha\\}$ and any
$[\beta_{n}]=\\{\pm\beta_{n},\pm\gamma_{n}\\}$. Indeed, for checking for
instance $\beta_{n}\sim\gamma_{n}$, observe that the fact
$\beta_{n}\phi^{-1}+\gamma_{n}\phi^{-1}=-\gamma_{n}$ gives us that the set
$\\{\beta_{n},\gamma_{n}\\}$ is a connection from $\beta_{n}$ to $\gamma_{n}.$

From here, the results in this section allow us to consider the family of
ideals

${\mathfrak{L}}_{[\alpha]}:={\mathfrak{L}}_{\alpha},$ and
$\hbox{${\mathfrak{L}}_{[\beta_{n}]}:=\langle
h_{n}\rangle\oplus{\mathfrak{L}}_{\beta_{n}}\oplus{\mathfrak{L}}_{-\beta_{n}}\oplus{\mathfrak{L}}_{\gamma_{n}}\oplus{\mathfrak{L}}_{-\gamma_{n}}$,
for $n\geq 2$};$

and the decomposition of ${{\mathfrak{L}}}$ as the sum of the ideals

${{\mathfrak{L}}}=\langle
e_{2},e_{3}\rangle\oplus{\mathfrak{L}}_{[\alpha]}\oplus(\bigoplus\limits_{n\in{\mathbb{N}},n\geq
2}{\mathfrak{L}}_{[\beta_{n}]}).$ (14)

##  4\. The simple components

In this section we are interested in studying under which conditions
${\mathfrak{L}}$ decomposes as the direct sum of the family of its simple
ideals. We recall that a roots system $\Lambda$ of a split regular Hom-Lie
superalgebra ${\mathfrak{L}}$ is called symmetric if it satisfies that
$\alpha\in\Lambda$ implies $-\alpha\in\Lambda$. From now on we will suppose
$\Lambda$ is symmetric.

######  Lemma 4.1.

If $I$ is an ideal of ${{\mathfrak{L}}}$ such that $I\subset H$, then
$I\subset{\mathcal{Z}}({{\mathfrak{L}}}).$

###### Proof.

Consequence of $[I,H]\subset[H,H]=0$ and
$[I,\bigoplus\limits_{\alpha\in{\Lambda}}{{\mathfrak{L}}}_{\alpha}]\subset(\bigoplus\limits_{\alpha\in{\Lambda}}{{\mathfrak{L}}}_{\alpha})\cap
H=0.$ ∎

Observe that the grading of $I$ and Equation (4) let us write

$I=I_{\bar{0}}\oplus I_{\bar{1}}=\Bigl{(}\bigl{(}I_{\bar{0}}\cap
H_{\bar{0}}\bigr{)}\oplus\bigl{(}\bigoplus\limits_{\alpha\in\Lambda}(I_{\bar{0}}\cap{\mathfrak{L}}_{\alpha,{\bar{0}}})\bigr{)}\Bigr{)}\oplus\Bigl{(}\bigl{(}I_{\bar{1}}\cap
H_{\bar{1}}\bigr{)}\oplus\bigl{(}\bigoplus\limits_{\alpha\in\Lambda}(I_{\bar{1}}\cap{\mathfrak{L}}_{\alpha,{\bar{1}}})\bigr{)}\Bigr{)}.$
(15)

Next lemmas have a proof similar to the ones of Lemma 4.2 and Lemma 4.3 in
[2].

######  Lemma 4.2.

For any $\alpha,\beta\in\Lambda$ with $\alpha\neq\beta$ there exists $h_{0}\in
H_{\bar{0}}$ such that $\alpha(h_{0})\neq 0$ and
$\alpha(h_{0})\neq\beta(h_{0})$.

######  Lemma 4.3.

If $I$ is an ideal of ${\mathfrak{L}}$ and
$x=h+\sum_{j=1}^{n}v_{\alpha_{j}}\in I$, with $h\in H$,
$v_{\alpha_{j}}\in{\mathfrak{L}}_{\alpha_{j}}$ and
${\alpha_{j}}\neq\alpha_{k}$ if $j\neq k$. Then any $v_{\alpha_{j}}\in I$.

Let us introduce the concepts of root-multiplicativity and maximal length in
the framework of split Hom-Lie superalgebras in a similar way to the ones for
split Hom-Lie algebras, split Lie algebras, split color algebras, split
Leibniz algebras and split Lie superalgebras (see [2, 4, 6, 7, 8] for these
notions and examples). We denote by
$\Lambda_{\bar{0}}:=\\{\alpha\in\Lambda:{\mathfrak{L}}_{\alpha,{\bar{0}}}\neq
0\\}$ and by
$\Lambda_{\bar{1}}:=\\{\alpha\in\Lambda:{\mathfrak{L}}_{\alpha,{\bar{1}}}\neq
0\\}$ (see Equation (4)). So $\Lambda=\Lambda_{\bar{0}}\cup\Lambda_{\bar{1}}$,
being a non necessarily disjoint union.

######  Definition 4.1.

We say that a split regular Hom-Lie superalgebra ${\mathfrak{L}}$ is root-
multiplicative if given $\alpha\in\Lambda_{\bar{i}},$
$\beta\in\Lambda_{\bar{j}},$ for $\bar{i},\bar{j}\in\mathbb{Z}_{2},$ such that
$\alpha\phi^{-1}+\beta\phi^{-1}\in\Lambda_{\bar{i}+\bar{j}},$ then
$[{\mathfrak{L}}_{\alpha,\bar{i}},{\mathfrak{L}}_{\beta,\bar{j}}]\neq 0.$

######  Definition 4.2.

It is said that a split regular Hom-Lie superalgebra ${\mathfrak{L}}$ is of
maximal length if $\dim{\mathfrak{L}}_{\alpha,\bar{i}}\in\\{0,1\\}$ for any
$\alpha\in\Lambda$ with $\bar{i}\in\mathbb{Z}_{2}.$

Observe that if ${\mathfrak{L}}$ is of maximal length, then the grading of $I$
and Equation (4) allow us to assert that given any nonzero ideal $I$ of
${\mathfrak{L}}$ then

$I=\Bigl{(}\bigl{(}I_{\bar{0}}\cap
H_{\bar{0}}\bigr{)}\oplus\bigl{(}\bigoplus\limits_{\alpha\in\Lambda^{I}_{\bar{0}}}{\mathfrak{L}}_{\alpha,{\bar{0}}}\bigr{)}\Bigr{)}\oplus\Bigl{(}\bigl{(}I_{\bar{1}}\cap
H_{\bar{1}}\bigr{)}\oplus\bigl{(}\bigoplus\limits_{\alpha\in\Lambda^{I}_{\bar{1}}}{\mathfrak{L}}_{\alpha,{\bar{1}}}\bigr{)}\Bigr{)}$
(16)

${\rm where}\hskip
2.84544pt\Lambda_{\bar{i}}^{I}:=\\{\alpha\in\Lambda_{\bar{i}}:I_{\bar{i}}\cap{\mathfrak{L}}_{\alpha,{\bar{i}}}\neq
0\\},$ for ${\bar{i}}\in{\hbox{\hueca Z}}_{2}.$

######  Theorem 4.1.

Let ${\mathfrak{L}}$ be a split regular Hom-Lie superalgebra of maximal
length, root multiplicative and with $\mathcal{Z}({\mathfrak{L}})=0$. It holds
that ${\mathfrak{L}}$ is simple if and only if it has all its nonzero roots
connected and
$H=\sum\limits_{\alpha\in\Lambda}[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{-\alpha}]$.

###### Proof.

The necessary implication is Theorem 3.1-2. To prove the converse, consider
$I$ a nonzero ideal of ${\mathfrak{L}}$. By Lemmas 4.2, 4.3 and Equation (16)
we can write

$I=\Bigl{(}\bigl{(}I_{\bar{0}}\cap
H_{\bar{0}}\bigr{)}\oplus\bigl{(}\bigoplus\limits_{\alpha\in\Lambda^{I}_{\bar{0}}}{\mathfrak{L}}_{\alpha,{\bar{0}}}\bigr{)}\Bigr{)}\oplus\Bigl{(}\bigl{(}I_{\bar{1}}\cap
H_{\bar{1}}\bigr{)}\oplus\bigl{(}\bigoplus\limits_{\alpha\in\Lambda^{I}_{\bar{1}}}{\mathfrak{L}}_{\alpha,{\bar{1}}}\bigr{)}\Bigr{)}$

with $\Lambda_{\bar{i}}^{I}\subset\Lambda_{\bar{i}},$ for
${\bar{i}}\in{\hbox{\hueca Z}}_{2}$ and some
$\Lambda_{\bar{i}}^{I}\neq\emptyset$ by Lemma 4.1. Hence, we can take
$\alpha_{0}\in\Lambda_{\bar{i}}^{I}$ being so

$0\neq{\mathfrak{L}}_{\alpha_{0},{\bar{i}}}\subset I,$ (17)

thus

$\\{{\mathfrak{L}}_{\alpha_{0}\phi^{z},\bar{i}}:z\in{\hbox{\hueca
Z}}\\}\subset I$ (18)

because $\phi(I)=I$ together with Lemma 1.1 and Remark 1.1 allow us to assert
that

$\hbox{if $\alpha\in\Lambda^{I}_{\bar{i}}$ then
$\\{\alpha\phi^{z}:z\in{\hbox{\hueca Z}}\\}\subset\Lambda^{I}_{\bar{i}}$}.$
(19)

Now, let us take any
$\beta\in\Lambda\setminus\\{\alpha_{0}\phi^{z},-\alpha_{0}\phi^{z}:z\in{\hbox{\hueca
Z}}\\}$. Since $\alpha_{0}$ and $\beta$ are connected we have a connection
$\\{\alpha_{1},\alpha_{2},\dots,\alpha_{k}\\}\subset\Lambda,k\geq 2$, from
$\alpha_{0}$ to $\beta$ satisfying:

  1. 1.

$\alpha_{1}=\alpha_{0}\phi^{-n}$ for some $n\in{\mathbb{N}}_{0}$.

  2. 2.

$\alpha_{1}\phi^{-1}+\alpha_{2}\phi^{-1}\in\Lambda$
$\alpha_{1}\phi^{-2}+\alpha_{2}\phi^{-2}+\alpha_{3}\phi^{-1}\in\Lambda$
$\hskip 56.9055pt\vdots$
$\alpha_{1}\phi^{-i}+\alpha_{2}\phi^{-i}+\alpha_{3}\phi^{-i+1}+\cdots+\alpha_{i+1}\phi^{-1}\in\Lambda$
$\hskip 56.9055pt\vdots$
$\alpha_{1}\phi^{-k+2}+\alpha_{2}\phi^{-k+2}+\alpha_{3}\phi^{-k+3}+\cdots+\alpha_{i}\phi^{-k+i}+\cdots+\alpha_{k-1}\phi^{-1}\in\Lambda$

  3. 3.

$\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+1}+\alpha_{3}\phi^{-k+2}+\cdots+\alpha_{i}\phi^{-k+i-1}+\cdots+\alpha_{k}\phi^{-1}=\epsilon\beta\phi^{-m}$
for some $m\in{\mathbb{N}}_{0}$ and $\epsilon\in\\{1,-1\\}.$

Consider $\alpha_{1}=\alpha_{0}\in\Lambda_{\bar{i}}$. Since
$\alpha_{2}\in\Lambda$ it follows ${\mathfrak{L}}_{\alpha_{2},\bar{j}}\neq 0,$
for some ${\bar{j}}\in{\hbox{\hueca Z}}_{2}$, and so
$\alpha_{2}\in\Lambda_{{\bar{j}}}$. We have $\alpha_{1}\in\Lambda_{{\bar{i}}}$
and $\alpha_{2}\in\Lambda_{{\bar{j}}}$ such that
$\alpha_{1}\phi^{-1}+\alpha_{2}\phi^{-1}\in\Lambda_{{\bar{i}}+\bar{j}}$ by
Remark 1.1. The root-multiplicativity and maximal length of ${\mathfrak{L}}$
allow us to assert
$0\neq[{\mathfrak{L}}_{\alpha_{1},\bar{i}},{\mathfrak{L}}_{\alpha_{2},\bar{j}}]={\mathfrak{L}}_{\alpha_{1}\phi^{-1}+\alpha_{2}\phi^{-1},\bar{i}+\bar{j}}.$
As consequence of Equation (17) we get

$0\neq{\mathfrak{L}}_{\alpha_{1}\phi^{-1}+\alpha_{2}\phi^{-1},\bar{i}+\bar{j}}\subset
I.$

A similar argument applied to
$\alpha_{1}\phi^{-1}+\alpha_{2}\phi^{-1},\alpha_{3}$ and

$(\alpha_{1}\phi^{-1}+\alpha_{2}\phi^{-1})\phi^{-1}+\alpha_{3}\phi^{-1}=\alpha_{1}\phi^{-2}+\alpha_{2}\phi^{-2}+\alpha_{3}\phi^{-1}$

gives us
$0\neq{\mathfrak{L}}_{\alpha_{1}\phi^{-2}+\alpha_{2}\phi^{-2}+\alpha_{3}\phi^{-1},\bar{l}}\subset
I,$ for certain $\bar{l}\in\mathbb{Z}_{2}.$ We can follow this process with
the connection $\\{\alpha_{1},\alpha_{2},\dots,\alpha_{k}\\}$ to get

$0\neq{\mathfrak{L}}_{\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+1}+\alpha_{3}\phi^{-k+2}+\cdots+\alpha_{i}\phi^{-k+i-1}+\cdots+\alpha_{k}\phi^{-1},\bar{h}}\subset
I,$

for some $\bar{h}\in\mathbb{Z}_{2},$ and then

$0\neq{\mathfrak{L}}_{\epsilon\beta\phi^{-m},\bar{h}}\subset I,$ for certain
$\epsilon\in\\{1,-1\\}$ (20)

for any
$\beta\in\Lambda\setminus\\{\alpha_{0}\phi^{z},-\alpha_{0}\phi^{z}:z\in{\hbox{\hueca
Z}}\\}$ and some $m\in\mathbb{N}_{0}.$ In general, from Equations (18) and
(20) we get for any $\alpha\in\Lambda$

either $\\{{\mathfrak{L}}_{\alpha\phi^{z},\bar{i}}:z\in{\hbox{\hueca
Z}}\\}\subset I$ or
$\\{{\mathfrak{L}}_{-\alpha\phi^{z},\bar{i}}:z\in{\hbox{\hueca Z}}\\}\subset
I,$ for a fixed $\bar{i}\in\mathbb{Z}_{2}$.

which can be reformulated by given any $\alpha\in\Lambda$:

either $\\{\alpha\phi^{z}:z\in\mathbb{Z}\\}$ or
$\\{-\alpha\phi^{z}:z\in\mathbb{Z}\\}$ is contained in
$\Lambda^{I}_{\bar{i}},$ for some $\bar{i}\in\mathbb{Z}_{2}$. (21)

From here, if we have

$\Lambda_{\bar{0}}\cap\Lambda_{\bar{1}}=\emptyset,$

Equation (21) shows that for any $\alpha\in\Lambda$ we have

${\mathfrak{L}}_{\epsilon\alpha}\subset I$ (22)

for some $\epsilon\in\\{1,-1\\}$. Since the expression of $H$ we get

$H\subset I.$ (23)

Now, for $-\epsilon\alpha\in\Lambda,$ the facts $-\epsilon\alpha\neq
0,H\subset I$ and the maximal length of ${\mathfrak{L}}$ show

$[H_{\bar{0}},{\mathfrak{L}}_{-\epsilon\alpha}]={\mathfrak{L}}_{-\epsilon\alpha}\subset
I.$ (24)

From Equations (22)-(24) we conclude $I={\mathfrak{L}}$.

Therefore, it just remains to study the case in which there exists
$\alpha\in\Lambda$ such that

$\alpha\in\Lambda_{\bar{0}}\cap\Lambda_{\bar{1}}.$

We begin with the following observation.

If there exists some $\alpha\in\Lambda_{\bar{0}}\cap\Lambda_{\bar{1}}$
satisfying
${\mathfrak{L}}_{\alpha,{\bar{0}}}\oplus{\mathfrak{L}}_{\alpha,{\bar{1}}}\subset
I$ then $I={\mathfrak{L}}$. (25)

Indeed, by arguing as above with the fact that ${\mathfrak{L}}$ has all its
nonzero roots connected from ${\mathfrak{L}}_{\alpha,{\bar{0}}}$, we get that
for any
$\beta\in\Lambda\setminus\\{\alpha,-\alpha\\},0\neq{\mathfrak{L}}_{\epsilon\beta,{\bar{h}}}\subset
I$ for some $\epsilon\in\\{1,-1\\}$ and some ${\bar{h}}\in{\hbox{\hueca
Z}}_{2}$. If we argue similarly from ${\mathfrak{L}}_{\alpha,{\bar{1}}}$ we
obtain $0\neq{\mathfrak{L}}_{\epsilon\beta,{\bar{h}}+{\bar{1}}}\subset I$. So
${\mathfrak{L}}_{\epsilon\beta}\subset I.$ From here, as
$H=\sum\limits_{\alpha\in\Lambda}[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{-\alpha}]$
we have

$H\subset I.$ (26)

Given now any $\gamma\in\Lambda$, the facts $\gamma\neq 0$, $H\subset I$ and
the maximal length of ${\mathfrak{L}}$ show

$[H_{\bar{0}},{\mathfrak{L}}_{\gamma,\bar{i}}]={\mathfrak{L}}_{\gamma,\bar{i}}\subset
I$

for any $\bar{i}\in\mathbb{Z}_{2}.$ Therefore

$[H_{\bar{0}},{\mathfrak{L}}_{\gamma}]={\mathfrak{L}}_{\gamma}\subset I.$ (27)

From Equations (26) and (27) we conclude $I={\mathfrak{L}}$.

Let us return to our $\alpha\in\Lambda$ such that
$\alpha\in\Lambda_{\bar{0}}\cap\Lambda_{\bar{1}}$. Taking into account
Equation (21) $\epsilon\alpha\phi^{-m}\in\Lambda,$ for certain
$\epsilon\in\\{1,-1\\}$ and some $m\in\mathbb{N}_{0},$ then
$[H_{\bar{0}},{\mathfrak{L}}_{\epsilon\alpha\phi^{-m},{\bar{h}}}]\neq 0$ for
fixed $\bar{h}\in\mathbb{Z}_{2}$. Since
$H_{\bar{0}}=\sum_{\gamma\in\Lambda}([{\mathfrak{L}}_{\gamma,{\bar{0}}},{\mathfrak{L}}_{-\gamma,{\bar{0}}}]+[{\mathfrak{L}}_{\gamma,{\bar{1}}},{\mathfrak{L}}_{-\gamma,{\bar{1}}}])$,
there exists $\gamma\in\Lambda$ such that either
$[[{\mathfrak{L}}_{\gamma,{\bar{0}}},{\mathfrak{L}}_{-\gamma,{\bar{0}}}],{\mathfrak{L}}_{\epsilon\alpha\phi^{-m},{\bar{h}}}]\neq
0$ or
$[[{\mathfrak{L}}_{\gamma,{\bar{1}}},{\mathfrak{L}}_{-\gamma,{\bar{1}}}],{\mathfrak{L}}_{\epsilon\alpha\phi^{-m},{\bar{h}}}]\neq
0.$ That is, by Lemma 1.1 and Remark 1.2 is
$\phi({\mathfrak{L}}_{\epsilon\beta\phi^{-m+1},{\bar{h}}})={\mathfrak{L}}_{\epsilon\alpha\phi^{-m},{\bar{h}}},$
and so

$[[{\mathfrak{L}}_{\gamma,{\bar{n}}},{\mathfrak{L}}_{-\gamma,{\bar{n}}}],\phi({\mathfrak{L}}_{\epsilon\alpha\phi^{-m+1},{\bar{h}}})]\neq
0$ (28)

for some ${\bar{n}}\in{\hbox{\hueca Z}}_{2}$ and fixed
$\bar{h}\in{\hbox{\hueca Z}}_{2},m\in\mathbb{N}_{0}$. By super Hom-Jacobi
identity

either
$[{\mathfrak{L}}_{\epsilon\alpha\phi^{-m+1},{\bar{h}}},{\mathfrak{L}}_{\gamma,{\bar{n}}}]\neq
0$ or
$[{\mathfrak{L}}_{-\gamma,{\bar{n}}},{\mathfrak{L}}_{\epsilon\alpha\phi^{-m+1},{\bar{h}}}]\neq
0$

and so
${\mathfrak{L}}_{\gamma+\epsilon\alpha\phi^{-m+1},{\bar{n}}+{\bar{h}}}\neq 0$
or ${\mathfrak{L}}_{-\gamma+\epsilon\alpha\phi^{-m+1},{\bar{n}}+{\bar{h}}}\neq
0$. That is, by Equation (20),

$0\neq{\mathfrak{L}}_{\kappa\gamma+\epsilon\alpha\phi^{-m+1},{\bar{n}}+{\bar{h}}}\subset
I$ (29)

for some $\kappa\in\\{1,-1\\}$.

Let us distinguish two possibilities, if
${\mathfrak{L}}_{-\kappa\gamma,{\bar{n}}+{\bar{1}}}\neq 0$, we get by the
root-multiplicativity and maximal length of ${\mathfrak{L}}$ that

$0\neq[{\mathfrak{L}}_{\kappa\gamma+\epsilon\alpha\phi^{-m+1},{\bar{n}}+{\bar{h}}},{\mathfrak{L}}_{-\kappa\gamma,{\bar{n}}+{\bar{1}}}]={\mathfrak{L}}_{\epsilon\alpha\phi^{-m+1},{\bar{h}}+{\bar{1}}}\subset
I.$

From here, Equations (21) and (25) give us $I={\mathfrak{L}}$.

If ${\mathfrak{L}}_{-\kappa\gamma,{\bar{n}}+{\bar{1}}}=0$, as by symmetry of
$\Lambda$ is $-\epsilon\alpha\phi^{-m}\in\Lambda,$ and by Lemma 1.2 we also
obtain $-\epsilon\alpha\phi^{-m+1}\in\Lambda,$ so
${\mathfrak{L}}_{-\epsilon\alpha\phi^{-m+1},{\bar{m}}}\neq 0$ for some
${\bar{m}}\in{\hbox{\hueca Z}}_{2}$. By Equation (29) and the root-
multiplicativity and maximal length of ${\mathfrak{L}}$ we obtain

$0\neq[{\mathfrak{L}}_{\kappa\gamma+\epsilon\alpha\phi^{-m+1},{\bar{n}}+{\bar{h}}},{\mathfrak{L}}_{-\epsilon\alpha\phi^{-m+1},{\bar{m}}}]={\mathfrak{L}}_{\kappa\gamma,{\bar{n}}+{\bar{h}}+{\bar{m}}}\subset
I.$ (30)

If ${\bar{h}}+{\bar{m}}={\bar{0}}$, taking into account that Equation (28)
gives us
$[[{\mathfrak{L}}_{\gamma,{\bar{n}}},{\mathfrak{L}}_{-\gamma,{\bar{n}}}],{\mathfrak{L}}_{\epsilon\alpha\phi^{-m},{\bar{h}}}]\neq
0,$ that is,

$\alpha([{\mathfrak{L}}_{\gamma,{\bar{n}}},{\mathfrak{L}}_{-\gamma,{\bar{n}}}])\neq
0,$

Equation (30) and the fact $\alpha\in\Lambda_{\bar{0}}\cap\Lambda_{\bar{1}}$
let us assert
$0\neq[[{\mathfrak{L}}_{\gamma,{\bar{n}}},{\mathfrak{L}}_{-\gamma,{\bar{n}}}],{\mathfrak{L}}_{\alpha,{\bar{p}}}]={\mathfrak{L}}_{\alpha,{\bar{p}}}\subset
I$ for any ${\bar{p}}\in{\hbox{\hueca Z}}_{2}$. From here, Equation (25) gives
us $I={\mathfrak{L}}$.

If ${\bar{h}}+{\bar{m}}={\bar{1}}$ we have by Equation (30) and the root-
multiplicativity and maximal length of ${\mathfrak{L}}$ that
$0\neq[{\mathfrak{L}}_{\kappa\gamma,{\bar{n}}+{\bar{1}}},{\mathfrak{L}}_{\epsilon\alpha\phi^{-m+1},{\bar{h}}}]={\mathfrak{L}}_{\kappa\gamma+\epsilon\alpha\phi^{-m+1},{\bar{n}}+{\bar{h}}+{\bar{1}}}\subset
I$. From here, Equations (29) and (25) give us $I={\mathfrak{L}}$.
Consequently ${\mathfrak{L}}$ is simple. ∎

######  Theorem 4.2.

Let ${\mathfrak{L}}$ be a split regular Hom-Lie superalgebra of maximal
length, root multiplicative satisfying $\mathcal{Z}({\mathfrak{L}})=0$ and
$H=\sum_{\alpha\in\Lambda}[{\mathfrak{L}}_{\alpha},{\mathfrak{L}}_{-\alpha}].$
Then ${\mathfrak{L}}$ is the direct sum of the family of its minimal ideals,
each one being a simple split regular Hom-Lie superalgebra having all its
nonzero roots connected.

###### Proof.

By Corollary 3.1,
${\mathfrak{L}}=\bigoplus_{[\alpha]\in\Lambda/\sim}{\mathfrak{L}}_{[\alpha]}$
is the direct sum of the ideals

${\mathfrak{L}}_{[\alpha]}=H_{[\alpha]}\oplus
V_{[\alpha]}=\Bigl{(}\sum\limits_{\beta\in[\alpha]}[{\mathfrak{L}}_{\beta},{\mathfrak{L}}_{-\beta}]\Bigr{)}\oplus\Bigl{(}\bigoplus\limits_{\beta\in[\alpha]}{\mathfrak{L}}_{\beta}\Bigr{)}$

having any ${\mathfrak{L}}_{[\alpha]}$ its root system $[\alpha]$ with all of
its roots connected. Taking into account the facts $[\alpha]=-[\alpha]$ and
$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[\alpha]}]\subset{\mathfrak{L}}_{[\alpha]}$
(see Proposition 3.1-1) we easily deduce that $[\alpha]$ has all of its roots
connected through roots in $[\alpha]$. We also have that any of the
${\mathfrak{L}}_{[\alpha]}$ is root-multiplicative as consequence of the root-
multiplicativity of ${\mathfrak{L}}$ and trivially is regular. Finally, is
clear that ${\mathfrak{L}}_{[\alpha]}$ is of maximal length and
${\mathcal{Z}}_{{\mathfrak{L}}_{[\alpha]}}({\mathfrak{L}}_{[\alpha]})=0$
(where ${\mathcal{Z}}_{{\mathfrak{L}}_{[\alpha]}}({\mathfrak{L}}_{[\alpha]})$
denotes de center of ${\mathfrak{L}}_{[\alpha]}$ in
${\mathfrak{L}}_{[\alpha]}$) as consequence of
$[{\mathfrak{L}}_{[\alpha]},{\mathfrak{L}}_{[\gamma]}]=0$ if
$[\alpha]\neq[\gamma]$ (Theorem 3.2) and ${\mathcal{Z}}({\mathfrak{L}})=0$. We
can apply Theorem 4.1 to any ${\mathfrak{L}}_{[\alpha]}$ so as to conclude
${\mathfrak{L}}_{[\alpha]}$ is simple. It is clear that the decomposition
${\mathfrak{L}}=\bigoplus_{[\alpha]\in\Lambda/\sim}{\mathfrak{L}}_{[\alpha]}$
satisfies the assertions of the theorem. ∎

######  Remark 4.1.

We finally note that the results in the present paper also hold for non
regular Hom-Lie superalgebras ${\mathfrak{L}}$ if we introduce the class of
split non regular Hom-Lie superalgebras by asking for $H$ not only the
condition of being a maximal abelian subalgebra of ${\mathfrak{L}}$ but also
the one of satisfying that $\phi|_{H}$ is a linear bijection from $H$ onto
$H.$

######  Example 4.1.

Let us consider the real ${\mathbb{Z}}_{2}$-graded vector space
${\mathfrak{L}}={\mathfrak{L}}_{\bar{0}}\oplus{\mathfrak{L}}_{\bar{1}}$, with
basis

$\\{e_{1},e_{2}\\}\cup\\{h_{n},x_{n},y_{n}:n\in{\mathbb{N}},n\geq 2\\}$

of ${\mathfrak{L}}_{\bar{0}}$ and

$\\{e_{3}\\}\cup\\{f_{n},g_{n}:n\in{\mathbb{N}},n\geq 2\\}$

of ${\mathfrak{L}}_{\bar{1}}$ given in Example 1.1. Here the nonzero products
on these elements were induced by the relations:

$[e_{2},e_{1}]=e_{1},\hskip 8.5359pt[h_{n},x_{n}]=2n^{2}x_{n},\hskip
8.5359pt[h_{n},y_{n}]=-\frac{2}{n^{2}}y_{n},$ $[x_{n},y_{n}]=h_{n},\hskip
8.5359pt[y_{n},g_{n}]=\frac{1}{n}f_{n},\hskip 8.5359pt[x_{n},f_{n}]={n}g_{n},$
$[h_{n},f_{n}]=-\frac{1}{n}f_{n},\hskip 8.5359pt[h_{n},g_{n}]={n}g_{n},\hskip
8.5359pt[g_{n},f_{n}]=h_{n},\hskip 8.5359pt[g_{n},g_{n}]=-2n^{2}x_{n},$
$[f_{n},f_{n}]=\frac{2}{n^{2}}y_{n}.$

If we now consider the superspace homomorphism

$\phi^{\prime}:{\mathfrak{L}}\to{\mathfrak{L}}$

defined by

$\phi^{\prime}(e_{1})=e_{1}$, $\phi^{\prime}(e_{2})=e_{2}$,
$\phi^{\prime}(e_{3})=0$ and  $\phi^{\prime}(h_{n})=h_{n}$,
$\phi^{\prime}(x_{n})=n^{2}x_{n}$,
$\phi^{\prime}(y_{n})=\frac{1}{n^{2}}y_{n}$,
$\phi^{\prime}(f_{n})=\frac{1}{n}f_{n}$, $\phi^{\prime}(g_{n})=ng_{n}$

for any $n\geq 2$, we have that
${\mathfrak{L}}={\mathfrak{L}}_{\bar{0}}\oplus{\mathfrak{L}}_{\bar{1}}$ also
becomes a split Hom-Lie superalgebra, with a root spaces decomposition as in
Equation (3), which is not regular since $\phi^{\prime}$ is not an
automorphism of $({\mathfrak{L}},[\cdot,\cdot])$. However,
$\phi^{\prime}|_{H}$ is a linear bijection from $H$ onto $H.$ From here,
Remark 4.1 allows us to assert that ${\mathfrak{L}}$ admits an analogous
decomposition as sum of ideals as the one given in Equation (14).

Acknowledgment. We would like to thank the referee for the detailed reading of
this work and for the suggestions which have improved the final version of the
same.

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,,

#  Bicriteria Optimization in Designing Magnetic Composites

Krzysztof Z. Sokalski<EMAIL_ADDRESS>Institute of Computer Science,
Czȩstochowa University of Technology, Al. Armii Krajowej 17, 42-200
Czȩstochowa, Poland  Barbara Ślusarek<EMAIL_ADDRESS>Bartosz
Jankowski<EMAIL_ADDRESS>Marek Przybylski
<EMAIL_ADDRESS>Tele and Radio Research Institute, Ratuszowa 11
Street, 03-450 Warszawa, Poland

###### Abstract

A novel algorithm for designing values of technological parameters for
production of soft magnetic composites (SMC) has been created. These
parameters are hardening temperature $T$ and compaction pressure $p$. They
enable us to optimize power losses and induction. The advantage of the
presented algorithm consists in bicriteria optimization. The crucial role
played by the presented algorithm is the scaling and pseudo-state equation. On
this basis mathematical models of power losses and induction have been
created. The model parameters have been calculated on the basis of the power
loss characteristics and hysteresis loops. The created optimization system has
been applied to specimens of Somaloy 500. The obtained output consists of a
finite set of feasible solutions. In order to select a unique solution an
additional criterion has been formulated. Keywords: soft magnetic composites;
power losses; induction; bicriteria optimization

###### keywords:

A.Laminates; B.Magnetic properties; C.Analytical modelling; Scaling

††journal: Journal of LaTeX Templates

##  1 Introduction

Soft magnetic composites (SMC) have physical properties which are used for
adapting these materials to specific applications [1],[2]. Very often the
functionality of these materials depends on more than one feature. This leads
to multi-criteria optimization problems, which has not been applied yet in the
design of SMC. However, there are papers which treat more than one physical
property of SMC but these are not considered as target functions in an
optimization procedure [3],[4]. Recently an algorithm for designing values of
the hardening temperature and the compaction pressure in the production
process of soft magnetic composites (SMC) has been derived by using the
concept of the pseudo-state equation [5]. In equilibrium thermodynamics the
equation of state relates thermodynamic parameters. For instance, in the case
of gas-liquid system they are the temperature, pressure and volume of the
considered material. By analogy with the equation of state we consider a
phenomenological relation between the technological parameters and physical
properties of the material. Such an approach for SMC is possible thanks to the
topology of the completed set of scaled power loss characteristics. The most
important features of this topology are the following,[5]:

Set of characteristics consists of one variable smooth function

$\frac{P_{tot}}{(B_{m})^{\beta}}=F\left(\frac{f}{B_{m}^{\alpha}}\right),$ (1)

where $P_{tot}$ is density of power loss, $B_{m}$ is peak of induction, $f$ is
frequency of electromagnetic field wave, $F(\cdot)$ is a function of the
following form [5]:

$F\left(\frac{f}{B_{m}^{\alpha}}\right)=(f/B_{m}^{\alpha}\cdot\left(\Gamma_{1}+f/B_{m}^{\alpha}\cdot\left(\Gamma_{2}+f/B_{m}^{\alpha}\cdot\left(\Gamma_{3}+f/B_{m}^{\alpha}\cdot\Gamma_{4}\right)\right)\right),$
(2)

where $\Gamma_{i},\alpha$ and $\beta$ have to be determined from experimental
data. The form (1) has been derived from the assumption about the power losses
as a homogeneous function in a general sense. Each characteristic is
determined by the values of $\Gamma_{i}$ coefficients and $\alpha$ as well as
$\beta$ exponents. These are functions of the technological parameters $T$ and
$p$

$\Gamma_{i}=\Gamma_{i}(T,p),\hskip 5.69054pt\alpha=\alpha(T,p),\hskip
5.69054pt\beta=\beta(T,p),$ (3)

where $T$ and $p$ are hardening temperature and compaction pressure,
respectively. (3) reveals that characteristics (1), (2) of samples composed at
different $T,p$ conditions possess different dimensions, whereas all
disentangled characteristics

$P_{tot}=B_{m}^{\beta}(f/B_{m}^{\alpha}\cdot\left(\Gamma_{1}+f/B_{m}^{\alpha}\cdot\left(\Gamma_{2}+f/B_{m}^{\alpha}\cdot\left(\Gamma_{3}+f/B_{m}^{\alpha}\cdot\Gamma_{4}\right)\right)\right),$
(4)

possess a common physical dimension. Why do we use the implicit form (1),(2)?
Note that the right-hand side of this equation depends only on one effective
variable $\frac{f}{B_{m}^{\alpha}}$. Therefore, calculations performed with
(1),(2) are represented by one curve for all values of $f$ and $B_{m}$,
whereas results of calculations performed with (4) are split into many curves.
For instance, if one needs $P_{tot}$ as a function of $f$ then the number of
generated characteristics is equal to the required number of different values
of $B_{m}$. Because of the different dimensions of the different
characteristics they do not cross each other except at origin point
$\frac{f}{B_{m}^{\alpha}}=0$, for which dimension is not very important.
According to the Egenhofer theorem [6] the relations between characteristics
are invariant with respect to scaling, translation and rotation. Just the
conservation of the relations with respect to the scaling enables us to use
the implicit form of characteristics. All the characteristics are monotonic
increasing functions of $\frac{f}{B_{m}^{\alpha}}$ According to (3) the power
loss characteristics are parameterized by pressure and temperature. This
dependence enables us to introduce a measure of distance in the space of
characteristics. Let $(p_{1},T_{1})$ and $(p_{2},T_{2})$ be labels of the
characteristics of the two composites which have been composed under
conditions corresponding to these pressures and temperatures, respectively.
Then the distance between these characteristics has the following general
form:

$\rho(p_{1},T_{1},p_{2},T_{2})=R(|p_{1}-p_{2}|,|T_{1}-T_{2}|),$ (5)

where $R(\cdot,\cdot)$ satisfies axioms of the distance function. Therefore,
the set of all characteristics constitutes metric space.We have shown in [5]
that this space consists of two subspaces. Therefore, by introducing the
distance measure in the space of all characteristics we make each of these
subspaces compact.

Each compact set corresponds to physical phase which is defined by
characteristic values of the physical parameters. For instance, in [5] we
considered the low and high losses phases of SOMALOY 500. All the properties
mentioned above are presented in Fig. 1; however, the compactness of
characteristics’ subsets is ensured by the existence of (5). These properties
have enabled us to introduce a measure of power loss $V(T,p)$ which was the
average of characteristics with respect to $\frac{f}{B_{m}^{\alpha}}$ [5]:

$V({T},{p})=\frac{1}{\phi_{max}-\phi_{min}}\int_{\phi_{min}}^{\phi_{max}}\frac{P_{tot}(\frac{f}{B_{m}^{\alpha}})}{B_{m}^{\beta}}\,d(\frac{f}{B_{m}^{\alpha}}).$
(6)

Note that the dimension of the denominator in front of the integral and the
dimension of integration limits cancel themselves out.

Figure 1: The five loss characteristics for $T\leq 550^{o}C$ corresponding to
the low losses phase and the one characteristic for $T=600^{o}C$ corresponding
to the high losses phase.

From the topological properties of the characteristics’ set and on (3) as well
as on (6) the pseudo-state equation for soft magnetic composites has been
derived. This equation has enabled us to determine the optimal values of the
technological parameters [5]. However, the described optimization relates only
to the power losses and in design processes optimization of induction is also
important. Therefore, the goal of this paper is to create a new algorithm for
optimization of both the power losses and the induction within the frame of a
bicriteria problem.

##  2 Experimental Data

Specimens were produced by cold pressing under pressure of 500$\dots$900 MPa.
The specimens made of Somaloy 500 powder were cured at a temperature of
400$\dots$600oC for 30 minutes in air atmosphere. The specimens used in
experiments were ring-shaped with a square cross-section. The specimens had
the following dimensions: external diameter 55 mm, internal diameter 45 mm and
thickness 5 mm. Total power loss density $P_{tot}$, expressed in watts per
kilogram (W/ kg), was obtained from measurements of the AC hysteresis cycle
according to IEC Standard 60404-6 using the system AMH-20K-HS produced by
Laboratorio Elettrofisico Walker LDJ Scientific. Total power losses$P_{tot}$
were measured at maximum flux density $B_{m}$ = 0.1…1.3 T over a frequency
range of 10 to 5000 Hz. During measurements of the total power losses
$P_{tot}$, the shape factor of the secondary voltage was equal to
1.111$\pm$1.5 %. Maximum measurement error of the total energy losses was
equal to 3%. In order to optimize the magnetic properties, the magnetic
inductions $B$ at fixed magnetic field $H$ equal to 1000 A/m were determined.
These values were obtained from measurements of the DC magnetization curve
according to IEC Standard 60404-4 using the same measuring system.

##  3 Power Losses and Induction Pseudo-State Equations

Optimization of the power losses was based on the topological properties of
the characteristics. However, in the case of magnetic properties the situation
is much simpler. For optimization of magnetic properties we selected induction
$B_{1000}$ for the fixed magnetic field $H$=1000$($A/m$)$. We chose this value
because the magnetic permeability of the soft magnetic composites reaches a
maximum value around this magnetic field. We expected that the pseudo-state
equation would properly describe induction at this point as a function of $T$
and $p$. In the previous paper [5] it was assumed and confirmed that the loss
measure $V$ obeys the scaling. Here, this assumption was extended to
induction. In order to justify this assumption we referred to two phenomena:
invariance of power losses (area of the hysteresis loop) with respect to
scaling and invariance of the hysteresis loop with respect to scaling [7].
Therefore, for the bicriteria optimization problem, minimization of the power
losses and maximization of the induction for a fixed magnetic field, we used
the following pseudo-state equations of general form:

$V\left(T,p\right)=\left(\frac{p}{p_{c}}\right)^{\gamma}\,\cdot\,\Phi(X),$ (7)
$B_{1000}\left(T,p\right)=\left(\frac{p}{p^{\prime}_{c}}\right)^{\gamma^{\prime}}\,\cdot\,\Lambda(X^{\prime}),$
(8)

where

$\displaystyle X=\frac{\frac{T}{T_{c}}}{(\frac{p}{p_{c}})^{\delta}},$ (9)
$\displaystyle
X^{\prime}=\frac{\frac{T}{T^{\prime}_{c}}}{(\frac{p}{p^{\prime}_{c}})^{\delta^{\prime}}},$
(10)

where $\Phi(\cdot)$ and $\Lambda(\cdot)$ were arbitrary functions to be
determined. $\gamma$, $\delta$ , $\gamma^{\prime}$, $\delta^{\prime}$ and
$T_{c}$, $p_{c}$, $T^{\prime}_{c}$, $p^{\prime}_{c}$ are scaling exponents and
scaling parameters respectively, and were to be determined. In the case of the
power losses’ pseudo-state equation all calculations concerning modelling of
$\Phi(\cdot)$ and fitting of scaling exponents as well as model parameters
were done in [5]. The most important result was the derivation of an infinite
set of solutions for the technological parameters which minimized the power
losses:

$\frac{\frac{T}{T_{c}}}{(\frac{p}{p_{c}})^{\delta}}=19,75.$ (11)

##  4  Induction Pseudo-State Equation

In this Section we derive a pseudo-state equation for induction $B_{1000}$
which will constitute a function of the two variables $p$ and $T$. This
function and the power losses’ pseudo-state equation (7) will enable us to
optimize induction and losses together. The optimization criteria are the
following: find $V=V_{min}$ and $B_{1000}=B_{1000\,max}$ with respect to $p$
and $T$. Deriving in [5] the form for $\Phi(\cdot)$ we reveal two phases of
Somaloy 500: low losses and high losses. Therefore, in terms of the induction
pseudo-state equation we have to take into account this phase separation.
Measurement data of $B_{1000}$ vs. $T$ and $p$ are separated into these two
phases in Table 1.

Table 1: Somaloy 500. Measure of induction $B_{1000}$ vs. hardening temperature $T$ and compaction pressure $p$ for magnetic field $H$=1000 $($A/m$)$. Temperature | Pressure | Induction
---|---|---
$($K$)$ |  $($MPa$)$ |  $($T$)$
723,15 | 800 | 0,378
773,15 | 900 | 0,496
773,15 | 700 | 0,483
673,15 | 800 | 0,335
773,15 | 600 | 0,467
823,15 | 800 | 0,546
773,15 | 500 | 0,414
741,15 | 764 | 0,425
773,15 | 750 | 0,489
773,15 | 800 | 0,504
773,15 | 650 | 0,469
773,15 | 725 | 0,467
873,15 | 800 | 0,568

The horizontal line between $B_{1000}=0,414($T$)$ and $B_{1000}=0,425($T$)$
indicates the crossover between the low losses phase and the high losses
phase. This transition is clearly visible in the jump of the $V(T,p)$ function
around the separation line [5]. For each phase we assume an independent branch
of the pseudo-state equation in the form of the Padé approximant. In order to
simplify notations we introduce the following abbreviations:

$\pi=\frac{p}{p_{c}},\hskip 5.69054pt\pi^{\prime}=\frac{p}{p^{\prime}_{c}},$
(12)

Expressing $\Lambda(\cdot)$ in (8) by the Padé approximant we get the
following form for the induction pseudo-equation of state:

$B_{1000}(T,p)=\pi^{\prime\gamma^{\prime}}\frac{\tilde{G}_{0}+\tilde{G}_{1}\,X^{\prime}+\tilde{G}_{2}\,X^{\prime
2}+\tilde{G}_{3}\,X^{\prime 3}+\tilde{G}_{4}\,X^{\prime
4}}{1+\tilde{D}_{1}\,X^{\prime}+\tilde{D}_{2}\,X^{\prime
2}+\tilde{D}_{3}\,X^{\prime 3}+\tilde{D}_{4}\,X^{\prime 4}},$ (13)

where $\tilde{G}_{0},\dots,\tilde{G}_{4},\tilde{D}_{1},\dots,\tilde{D}_{4}$
are parameters of the Padé approximant. All parameters have to be determined
from the data presented in Table 1. The corresponding pseudo-state equation
for the power losses has been derived in [5]:

$V(T,p)=\pi^{\gamma}\frac{G_{0}+G_{1}\,X+G_{2}\,X^{2}+G_{3}\,X^{3}+G_{4}\,X^{4}}{1+D_{1}\,X+D_{2}\,X^{2}+D_{3}\,X^{3}+D_{4}\,X^{4}}.$
(14)

##  5  Estimation of Parameters for Induction Pseudo-State Equation

The above-mentioned crossover between low-loss and high-loss phases is
observed as A sudden change of $V$ between two points: $[773,15;500,0]$ and
$[742,15;764,0]$ Table 1. However, this effect is not seen in the induction
magnitude. Therefore in order to have a compact description of the power
losses and the induction we take that into account and we divide the data of
Table 1 into two subsets corresponding to the two respective phases.
Minimizations of $\chi^{2}$ for both phases have been performed with Microsoft
Excel 2010, where

$\chi^{2}=\sum_{i=1}^{N}\left(B_{1000}(\tau^{\prime}_{i},\pi^{\prime}_{i})-{\pi^{\prime}_{i}}^{\gamma^{\prime}}\frac{\tilde{G}_{0}+\tilde{G}_{1}\,{X^{\prime}}_{i}+\tilde{G}_{2}\,{X^{\prime}}_{i}^{2}+\tilde{G}_{3}\,{X^{\prime}}_{i}^{3}+\tilde{G}_{4}\,{X^{\prime}}_{i}^{4}}{1+\tilde{D}_{1}\,{X^{\prime}}_{i}+\tilde{D}_{2}\,{X^{\prime}}_{i}^{2}+\tilde{D}_{3}\,{X^{\prime}}_{i}^{3}+\tilde{D}_{4}\,{X^{\prime}}_{i}^{4}}\right)^{2},$
(15)

where $N=7$ and $N=6$ for the low-losses and high-losses phases, respectively.
Table 2 and Table 3 present estimated values of the model parameters for the
low-loss and high-loss phases, respectively.

Table 2: Somaloy 500, low-loss phase. Values of the induction pseudo-state equation’s parameters and the Padé approximant’s coefficients of (13).  $\gamma^{\prime}$ | $\delta^{\prime}$ | $T^{\prime}_{c}$ | $p^{\prime}_{c}$ | $\tilde{G}_{0}$ | $\tilde{G}_{1}$ | $\tilde{G}_{2}$
---|---|---|---|---|---|---
1,114 | 0,499 | 32,186 | 25,849 | -784,41 | 764,05 | -276,06
$\tilde{G}_{3}$ | $\tilde{G}_{4}$ | $\tilde{D}_{1}$ | $\tilde{D}_{2}$ | $\tilde{D}_{3}$ | $\tilde{D}_{4}$ | -
43,412 | -2,4315 | 3,6486 | 2,9005 | 2,8373 | 3,975 | -
Table 3: Somaloy 500, high-losses phase. Values of the induction pseudo-state equation’s parameters and the Padé approximant’s coefficients of (13).  $\gamma^{\prime}$ | $\delta^{\prime}$ | $T^{\prime}_{c}$ | $p^{\prime}_{c}$ | $\tilde{G}_{0}$ | $\tilde{G}_{1}$ | $\tilde{G}_{2}$
---|---|---|---|---|---|---
1,1146 | 0,4992 | 32,19 | 25,83 | -808,91 | 747,44 | -266,95
$\tilde{G}_{3}$ | $\tilde{G}_{4}$ | $\tilde{D}_{1}$ | $\tilde{D}_{2}$ | $\tilde{D}_{3}$ | $\tilde{D}_{4}$ | -
42,852 | -2,5142 | 0,5846 | -3,001 | -9,968 | 7,1432 | -

##  6 Optimization of Induction and Power Losses

Figure 2: Scaled $B_{1000}$ vs. scaled temperature in the low loss and high
loss phases.

In the optimization of the power loss problem [5] we have applied low loss
phase solutions and high loss phase solutions have not been considered.
However, it is not clear whether this simplification excludes important
solutions for the induction. The binary relations are invariant with respect
to scaling [6],[5]. This enables us to present all scaled characteristics in
the one picture Fig.2 and draw the following conclusion. All considered
pressure characteristics of the high losses phase are covered by the set of
the low losses phase characteristics. Therefore for further investigations we
limit our searching to the low losses phase. To this end we draw part of the
phase diagram of Somaloy 500 corresponding to the low losses phase Fig. 3 and
we deliver values of $G_{i},D_{i},p_{c},\gamma$ which are displayed in Table
4.

Figure 3: Scaled $V$ vs. scaled temperature in the low losses phase. According to (9) $\tau\,\pi^{-\delta}=X$. Table 4: Somaloy 500, low-losses phase. Values of the $V$ pseudo-state equation’s parameters and the Padé approximant’s coefficients of (13) [5]. This table is an amended version of Table 3 in [5] $\gamma$ | $\delta$ | $T_{c}$ | $p_{c}$ | $G_{0}$ | $G_{1}$ | $G_{2}$
---|---|---|---|---|---|---
1,2812 | 0,1715 | 21,622 | 37,729 | 370315315 | -47752251 | 1734952
$G_{3}$ | $G_{4}$ | $D_{1}$ | $D_{2}$ | $D_{3}$ | $D_{4}$ | -
-1,3764 | -678,26 | 170,80 | 6243,8 | 386,96 | -28,699 | -
Table 5: Somaloy 500, low-losses phase. Optimum solutions in technological and in physical spaces.  $p\textrm{(MPa)}$ | $T(^{o}\textrm{C})$ | $V(\textrm{W/kgT}^{-\beta})$ | $B_{1000}\textrm{(T)}$
---|---|---|---
389 | 370 | 14,1 | 0,300
492 | 407 | 20,0 | 0,356
584 | 440 | 27,3 | 0,400
683 | 478 | 40,0 | 0,449
733 | 499 | 50,0 | 0,479
764 | 515 | 58,5 | 0,500
800 | 532 | 70,0 | 0,525
838 | 549 | 82,9 | 0,550
906 | 570 | 101 | 0,580
979 | 584 | 116 | 0,600
Figure 4: Technological optimum curve presenting dependence of optimum
temperature vs optimum pressure. Figure 5: Physical optimum curve. Induction
$B_{1000}$ vs.losses measure $V$.

##  7 Details of Bicriteria Problem

All calculations in this section have to satisfy the following conditions:
$18,4<X<22,9$, which results from limitation of the presented calculations to
the Low Losses Phase presented in Fig. 3. The considered bicriteria problem is
formulated by the initial value of $V=V_{1}$, feasible set of $(p,T)$ and the
two criteria : $V(p,T)=V_{min}$ whereas $B_{1000}(p,T)=B_{1000\,max}$. Since
increase of $B_{1000}$ causes increase of $V$ these conditions are in
contradiction. Therefore looked for solving criterion should lead to self-
consistency between $V_{min}$ and $B_{1000\,max}$. Such consistency will be
achieved as a fixed point of the following recurrence procedure. Let
maximiztion and minimization procedures be reprezented by the operators
$\hat{O}_{max}$ and $\hat{O}_{min}$, respectively. Let $B_{1000}(T,p)$ and
$V(T,p)$ be functions defined by $(\ref{eq8a})$ and $(\ref{eq8aa})$
respectively. Then the one step of independent optimizations of
$B_{1000}(T,p)$ and $V(T,p)$ can be writen in the following form:

$\displaystyle\hat{O}_{max}B_{1000}=B_{1000,max}\hskip 5.69054ptfor\hskip
5.69054ptT=T_{1},\hskip 5.69054ptp=p_{1}$ (16)
$\displaystyle\hat{O}_{min}V=V_{min}\hskip 5.69054ptfor\hskip
5.69054ptT=T_{2},\hskip 5.69054ptp=p_{2}.$ (17)

The obtained result consits of the two points $(T_{1},p_{1}),(T_{2},p_{2})$
and the corresponding values of magnitudes to be optimized
$B_{1000}(T_{1},p_{1})$ and $V(T_{2},p_{2})$. Therefore, any further
optimization is not possible and the bicriteria problem is not solved. In
order to mesh $B_{1000,max}$ and $V_{min}$ we introduce constrain
$V(T,p)=V_{0}$ which protects (16) and (17) against collapse, where $V_{0}$ is
an initial value of loss. Then (16) get the following form which coupled
$B_{1000,max}$ and $V_{min}$ as well as leaved some space for further
optimization:

$V(T,p)=V_{0},\hskip
5.69054pt\hat{O}_{max}B_{1000}(T,p)=B_{1000}(T_{1},p_{1}).$ (18)

Having $B_{1000}(T_{1},p_{1})$ we protect (17) against collapse:

$B_{1000}(T,p)=B_{1000}(T_{1},p_{1}),\hskip
5.69054pt\hat{O}_{min}V=V(T_{2},p_{2}).$ (19)

Therefore, after $n$ steps we obtain:

$\displaystyle V(T,p)=V(T_{2\,n},p_{2\,n}),\hskip
5.69054pt\hat{O}_{max}B_{1000}(T,p)=B_{1000}(T_{2\,n+1},p_{2\,n+1}),$ (20)
$\displaystyle B_{1000}(T,p)=B_{1000}(T_{2\,n+1},p_{2\,n+1}),\hskip
5.69054pt\hat{O}_{min}V=V(T_{2\,n+2},p_{2\,n+2}).$ (21)

(18)-(21) generate the two converging series: $T_{1},T_{2},\cdots,T_{2\,n+2}$
and $p_{1},p_{2},\cdots,p_{2\,n+2}$:

$\displaystyle\lim\limits_{k\to\infty}T_{k}=T^{*}$ (22)
$\displaystyle\lim\limits_{k\to\infty}p_{k}=p^{*}.$ (23)

Substituting $T^{*}$ and $p^{*}$ to $(\ref{eq8a})$ and $(\ref{eq8aa})$ we
derive the meshed values of $V$ and $B_{1000}$:

$V^{*}=V(T^{*},p^{*}),\hskip 5.69054ptB_{1000}^{*}=B_{1000}(T^{*},p^{*}).$
(24)

The found solutions are not unique. Selecting set of initial values for
$V_{o}$ we derive the set of final solutions. Optimization has been done by
SOLVER routine of EXCEL2010 program. Obtained output is presented in TABLE 5.
Fig.4 and Fig.5 present these results in technological and in physical spaces,
respectively. The obtained results represented by markers are fixed points of
the proposed procedure. There is one to one correspondence between these
points in physical and technological spaces (TABLE 5). In order to select an
unique solution one must provide an additional criterion resulting from a
relation between importance of losses and induction. For instance, assuming
the deepest minimum for the scaled measure of losses $V\pi^{-\gamma}$ we apply
condition given by (11). Intersection of two curves presented in Fig.6 leads
to the following single solution $p=382(\textrm{MPa})$ and
$T=363(^{o}\textrm{C})$. In the physical space this point corresponds to
$V=13,64(\textrm{W}\,\textrm{kg}^{-1}\textrm{T}^{-\beta})$ and
$B_{1000}=0,29(\textrm{T})$.

Figure 6: Reduction of the feasible set of solutions for the technological
parameters $\\{p,T\\}$ to the single point
$(p=382\textrm{MPa},T=368^{o}\textrm{C})$.

At the end we pay some attention to the power losses measure $V$. This is an
auxiliary magnitude which help us to derive values of designing technological
parameters due to the following features:

  * 1.

$V$ is pseudo-thermodynamic average with respect to magnitude created with the
peak of induction and the frequency of electromagnetic field wave. Therefore,
this includes information about both independent variables.

  * 2.

$V$ depends on the technological parameters.

  * 3.

Physical dimension of $V$ is unknown yet due to a dummy exponent $\beta$.
However, the value of $V$ is well determined together with values of $p$ and
$T$ which enables us to compose SMC specimen and to perform measurements of
its characteristics. Finally, applying (1),(2) we are able to calculate
$\beta$ and to determine the physical dimension of the current $V$.

##  8 Conclusions

We have presented method for the bicriteria optimization of the chosen
physical properties of Soft Magnetic Composites. By this way we have solved
the problem mentioned in [5] which concerns optimization of losses and
induction. Achievement of the fixed point is interpreted as revelation of an
equilibrium between the both assumed criteria.The crucial roles in the
presented method play scaling and the notion of pseudo-state equation. The
created system is as good as the experimental data which have been used for
the estimations of model parameters. Therefore, presented here the first
version will be improved by forthcoming new experimental data. The presented
example in this paper is a minimum nontrivial case of Multiphysics problem and
shows that this approach suits for designing Magnetic Composites. Therefore,
the presented algorithm is going to be extended for more than two physical
features of the composing material. For instance, the designing of magnetic
composites requires also optimization of mechanical properties, since the
susceptibility of such materials to cracking in service is of fundamental
concern [8]. We address the derived algorithm to designers of SMCs.

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