On split regular BiHom-Poisson superalgebras

TL;DR

This paper introduces split regular BiHom-Poisson superalgebras, generalizing previous algebraic structures, and develops root connection techniques to analyze their structure and simplicity conditions.

Contribution

It defines split regular BiHom-Poisson superalgebras and characterizes their structure and simplicity, extending existing algebraic frameworks.

Findings

A decomposition of the algebra into subspaces and ideals.

Conditions for algebra simplicity in maximal length cases.

Development of root connection techniques for these algebras.

Abstract

The paper introduces the class of split regular BiHom-Poisson superalgebras, which is a natural generalization of split regular Hom-Poisson algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Poisson superalgebras $A$ is of the form $A=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $A$, satisfying $[I_\a, I_\b]+I_\a I_\b = 0$ if $[\a]\neq [\b]$. Under certain conditions, in the case of $A$ being of maximal length, the simplicity of the algebra is characterized.

Equations

$$\varphi (x)(yz)=(xy)\varphi (y),$$

$$[xy,\varphi (z)]=\varphi (x)[y,z]+(-1{)}^{\overline{j}\overline{k}}[x,z]\varphi (y)$$

$$\varphi \circ \psi =\psi \circ \varphi ,$$

$$\varphi (a{a}^{\prime })=\varphi (a)\varphi (a^{\prime }),\psi (a{a}^{\prime })=\psi (a)\psi (a^{\prime }),$$

$$\varphi (a)(a^{\prime }{a}^{\prime \prime })=(a{a}^{\prime })\psi (a^{\prime \prime }).$$

$$[x,y]\subset L_{\overline{i}+\overline{j}},$$

$$\varphi \circ \psi =\psi \circ \varphi ,$$

$$[\psi (x),\varphi (y)]=-(-1{)}^{\overline{i}\overline{j}}[\psi (y),\varphi (x)],$$

$$(-1{)}^{\overline{k}\overline{i}}[{\psi }^2(x),[\psi (y),\varphi (z)]]+(-1{)}^{\overline{i}\overline{j}}[{\psi }^2(y),[\psi (z),\varphi (x)]]+(-1{)}^{\overline{j}\overline{k}}[{\psi }^2(z),[\psi (x),\varphi (y)]]=0.$$

$$\varphi (x)(yz)=(xy)\psi (y),$$

$$[xy,\varphi \psi (z)]=\varphi (x)[y,\varphi (z)]+(-1{)}^{|j||k|}[x,\psi (z)]\varphi (y)$$

$$\varphi (e_1)=a{e}_1\varphi (e_2)=a{e}_2\psi (e_1)=b{e}_1\psi (e_2)=b{e}_2,a,b\ne 0,1,$$

$$e_1\cdot e_1=ab{e}_1,e_2\cdot e_2=ab{e}_1,e_1\cdot e_2=e_2\cdot e_1=ab{e}_2,[e_2,e_2]=2ab{e}_2.$$

$$\varphi (e_1)=a{e}_1\varphi (e_2)=e_1+e_2\varphi (e_3)=0,a\ne 0,1,$$

$$\psi (e_1)=b{e}_1\psi (e_2)=e_1+e_2,\psi (e_3)=0,b\ne 0,1,$$

$$e_1\cdot e_2=a{e}_1,e_2\cdot e_2=e_1+e_2,[e_1,e_2]=a{e}_1.$$

$$\alpha :H_{\overline{0}}\to \mathbb{K},$$

$$\displaystyle A_{\alpha}:=\{v_{\alpha}\in A:[h_{\overline{0}},\phi(v_{\alpha})]=\alpha(h)\phi\psi(v_{\alpha}),\mbox{for any $h_{\overline{0}}\in H_{\overline{0}}$}\}.$$

$$A=H\oplus \underset{\alpha \in \Lambda }{⨁}{A}_{\alpha }.$$

$$[h_{\overline{0}},\varphi (v_{\alpha }{v}_{\beta })]=[\psi {\psi }^{-1}(h_{\overline{0}}),\varphi (v_{\alpha }{v}_{\beta })],$$

$$[\psi {\psi }^{-1}(h_{\overline{0}}),\varphi (v_{\alpha }{v}_{\beta })]$$

$$A_{\alpha ,\overline{i}}{A}_{\beta ,\overline{j}}\subset A_{\alpha {\varphi }^{-1}+\beta {\psi }^{-1},\overline{i}+\overline{j}},\overline{i},\overline{j}\in {\mathbb{Z}}_2.$$

$$A=H\oplus (\underset{\alpha \in \Lambda }{⨁}{A}_{\alpha })$$

$$-\Lambda :=\{-\alpha :\alpha \in \Lambda \}\text{mbox}and\pm \Lambda :\Lambda \cup (-\Lambda ).$$

$$A=H\oplus (\underset{\alpha \in \Gamma }{⨁}{A}_{\alpha {\psi }^{-1}})$$

$$\displaystyle\beta=\epsilon\alpha\phi^{z_{1}}\psi^{z_{2}}~{}~{}\mbox{for some $z_{1},z_{2}\in\mathbb{Z}$ and $\epsilon\in\{-1,1\}$}$$

$${\alpha }_1{\varphi }^{-k+1}+{\alpha }_2{\varphi }^{-k+2}{\psi }^{-1}+{\alpha }_3{\varphi }^{-k+3}{\psi }^{-1}+\cdot \cdot \cdot +{\alpha }_i{\varphi }^{-k+i}{\psi }^{-1}+\cdot \cdot \cdot +{\alpha }_k{\psi }^{-1}=ϵ\beta {\varphi }^{-m}{\psi }^{-s},$$

$$\{\beta {\varphi }^{-m}{\psi }^{-s},-ϵ{\alpha }_k{\varphi }^{-1},-ϵ{\alpha }_{k-1}{\varphi }^{-3},-ϵ{\alpha }_{k-2}{\varphi }^{-5},\cdot \cdot \cdot ,-ϵ{\alpha }_2{\varphi }^{-2k+3}\}$$

$${\alpha }_1{\varphi }^{-k+1}+{\alpha }_2{\varphi }^{-k+2}{\psi }^{-1}+\cdot \cdot \cdot +{\alpha }_k{\psi }^{-1}=ϵ\beta {\varphi }^{-m}{\psi }^{-s}$$

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

$$I_{H,[\alpha ]}=spa{n}_{\mathbb{K}}\{[A_{\beta {\psi }^{-1}},A_{-\beta {\varphi }^{-1}}]+A_{\beta {\psi }^{-1}}{A}_{-\beta {\varphi }^{-1}}:\beta \in [\alpha ]\}.$$

$$V_{[\alpha ]}:=\underset{\beta \in [\alpha ]}{⨁}{A}_{\beta }.$$

$$I_{[\alpha ]}:=I_{H,[\alpha ]}\oplus V_{[\alpha ]}.$$

$$[I_{[\alpha ]},I_{[\alpha ]}]$$

Full text

On split regular BiHom-Poisson superalgebras

**Shuangjian Guo1, Yuanyuan Ke2

**1. School of Mathematics and Statistics, Guizhou University of Finance and Economics

Guiyang 550025, P. R. China

2. School of Mathematics and Computer Science, Jianghan University

Wuhan, 430056, P. R. China **Corresponding author, Email: keyy086@126.com **

ABSTRACT

The paper introduces the class of split regular BiHom-Poisson superalgebras, which is a natural generalization of split regular Hom-Poisson algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Poisson superalgebras $A$ is of the form $A=U+\sum_{\alpha}I_{\alpha}$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\alpha}$, a well described ideal of $A$, satisfying $[I_{\alpha},I_{\beta}]+I_{\alpha}I_{\beta}=0$ if $[\alpha]\neq[\beta]$. Under certain conditions, in the case of $A$ being of maximal length, the simplicity of the algebra is characterized.

Key words: BiHom-Lie superalgebra, BiHom-associative superalgebra, BiHom-Poisson superalgebra, root, structure theory.

2010 Mathematics Subject Classification: 17A30, 17B63

INTRODUCTION

The notion of Hom-Lie algebras was first introduced by Hartwig, Larsson and Silvestrov in [15], who developed an approach to deformations of the Witt and Virasoro algebras basing on $\sigma$-deformations. In fact, Hom-Lie algebras include Lie algebras as a subclass, but the deformation of Lie algebras twisted by a homomorphism.

The twisting parts of the defining identities can transfer one algebra to the other algebraic structures. In [17, 19], Makhlouf and Silvestrov introduced the notions of Hom-associative algebras, Hom-coassociative coalgebras, Hom-bialgebras and Hom-Hopf algebras. The original definition of a Hom-bialgebra involved two linear maps, one was twisting the associativity condition and the other was twisting the coassociativity condition. In the case of Hom-Lie algebras, the relevant structure for a tensor theory is a Hom-Poisson algebra structure. A Hom-Poisson algebra has simultaneously a Hom-Lie algebra structure and a Hom-associative algebra structure, satisfying the Hom-Leibniz identity in [18]. In [20], Wang, Zhang and Wei characterized Hom-Leibniz superalgebras and Hom-Leibniz Poisson superalgebras, and presented the methods to construct these superalgebras.

A BiHom-algebra is an algebra in such a way that the identities defining the structure are twisted by two homomorphisms $\phi$ and $\psi$. This class of algebras was introduced from a categorical approach in [13] which as an extension of the class of Hom-algebras. If the two linear maps are the same automorphisms, BiHom-algebras will be return to Hom-algebras. These algebraic structures include BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras. The representation theory of BiHom-Lie algebras was introduced by Cheng and Qi in [12], in which, BiHom-cochain complexes, derivation, central extension, derivation extension, trivial representation and adjoint representation of BiHom-Lie algebras were studied. More applications of BiHom-algebras, BiHom-Lie superalgebras, BiHom-Lie colour algebras and BiHom-Novikov algebras can be found in ([16], [21], [1], [14]).

The class of the split algebras is specially related to addition quantum numbers, graded contractions and deformations. For instance, for a physical system which displays a symmetry, it is interesting to know the detailed structure of the split decomposition, since its roots can be seen as certain eigenvalues which are the additive quantum numbers characterizing the state of such system. Determining the structure of split algebras will become more and more meaningful in the area of research in mathematical physics. Recently, in ([2]-[8], [9]-[11], [22]-[23]), the structure of different classes of split algebras have been determined by the techniques of connections of roots. The purpose of this paper is to consider the class of split regular BiHom-Poisson superalgebras, which is a natural extension of split regular BiHom-Lie superalgebras and split Hom-Lie superalgebras.

In Section 2, we prove that such an arbitrary split regular BiHom-Poisson superalgebras $A$ is of the form $A=U+\sum_{\alpha}I_{\alpha}$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\alpha}$, a well described ideal of $A$, satisfying $[I_{\alpha},I_{\beta}]+I_{\alpha}I_{\beta}=0$ if $[\alpha]\neq[\beta]$.

In Section 3, we present that under certain conditions, in the case of $A$ being of maximal length, the simplicity of the algebra is characterized.

1 Preliminaries

Throughout this paper, we will denote by $\mathbb{N}$ the set of all nonnegative integers and by $\mathbb{Z}$ the set of all integers. Split regular BiHom-Poisson superalgebras are considered of arbitrary dimension and over an arbitrary base field $\mathbb{K}$. Any unexplained definitions and notations can be found in [13] and [18], and we recall some basic definitions and results related to our paper.

**1.1. Hom-Poisson superalgebra ** A Hom-Poisson superalgebra is a Hom-Lie superalgebra $(A,[\cdot,\cdot],\phi)$ endowed with a Hom-associative superproduct,that is, a bilinear product denoted by juxtaposition such that

[TABLE]

for all $x,y,z\in A$, and such that the Hom-Leibniz superidentity

[TABLE]

holds for any $x\in A_{\overline{i}},y\in A_{\overline{j}},z\in A_{\overline{k}}$ and $\overline{i},\overline{j},\overline{k}\in\mathbb{Z}_{2}$.

If $\phi$ is furthermore a Poisson automorphism, that is, a linear bijective on such that $\phi([x,y])=[\phi(x),\phi(y)]$ and $\phi(xy)=\phi(x)\phi(y)$ for any $x,y\in A$, then $A$ is called a regular Hom-Poisson superalgebra.

1.2. BiHom-associative superalgebra A BiHom-associative superalgebra is a 4-tuple $(A,\mu,\phi,\psi)$, where $A$ is a superspace, $\phi:A\rightarrow A$, $\psi:A\rightarrow A$ and $\mu:A\otimes A\rightarrow A$ are linear maps, with notation $\mu(a\otimes b)=ab$, satisfying the following conditions, for all $a,a^{\prime},a^{\prime\prime}\in A$:

[TABLE]

And the maps $\phi$ and $\psi$ are called the structure maps of $A$.

Clearly, a Hom-associative algebra $(A,\mu,\phi)$ can be regarded as the BiHom-associative algebra $(A,\mu,\phi,\phi)$.

**1.3. BiHom-Lie superalgebra ** A BiHom-Lie superalgebra $L$ is a $\mathbb{Z}_{2}$-graded algebra $L=L_{\overline{0}}\oplus L_{\overline{1}}$, endowed with an even bilinear mapping $[\cdot,\cdot]:L\times L\rightarrow L$ and two homomorphisms $\phi,\psi:L\rightarrow L$ satisfying the following conditions, for all $x\in L_{\overline{i}},y\in L_{\overline{j}},z\in L_{\overline{k}}$ and $\overline{i},\overline{j},\overline{k}\in\mathbb{Z}_{2}$:

[TABLE]

When $\phi$ and $\psi$ are algebra automorphisms, it is said that $L$ is a regular BiHom-Lie superalgebra.

2 Decomposition

Definition 2.1**.**

A BiHom-Poisson superalgebra $A$ is a BiHom-Lie superalgebra $(A,[\cdot,\cdot],\phi,\psi)$ endowed with a BiHom-associative superproduct, that is, a bilinear product denoted by juxtaposition such that

[TABLE]

for all $x,y,z\in A$, and such that the BiHom-Leibniz superidentity

[TABLE]

holds for any $x\in A_{\overline{i}},y\in A_{\overline{j}},z\in A_{\overline{k}}$ and $\overline{i},\overline{j},\overline{k}\in\mathbb{Z}_{2}$.

Furthermore, if $\phi$ and $\psi$ are Poisson automorphisms, it is said that $A$ is a regular BiHom-Poisson superalgebra.

Example 2.2**.**

Let $(A,\mu,[\cdot,\cdot])$ be a Poisson superalgebra and $\phi,\psi:A\rightarrow A$ two commuting Possion superalgebras automorphism. If we endow the underlying linear space $A$ with new products $[\cdot,\cdot]^{\prime}:A\times A\rightarrow A,\mu^{\prime}=\mu\circ(\phi\otimes\psi)$ defined by $[x,y]^{\prime}=[\phi(x),\psi(y)],\mu^{\prime}(x\otimes y)=\phi(x)\psi(y)$ for any $x,y\in A$, we know that $(A,\mu^{\prime},[\cdot,\cdot]^{\prime},\phi,\psi)$ becomes a regular BiHom-Poisson superalgebra.

Example 2.3**.**

Let $A=A_{\overline{0}}\oplus A_{\overline{1}}$ be a 2-dimensional superspace, where $A_{\overline{0}}$ is generated by $e_{1}$ and $A_{\overline{1}}$ is generated by $e_{2}$ and nonzero products and $\phi,\psi:A\rightarrow A$ are given by

[TABLE]

Then $(A,\cdot,[\cdot,\cdot],\phi,\psi)$ is a regular BiHom-Poisson superalgebra.

Example 2.4**.**

Let $A=A_{\overline{0}}\oplus A_{\overline{1}}$ be a 3-dimensional superspace, where $A_{\overline{0}}$ is generated by $e_{1},e_{2}$ and $A_{\overline{1}}$ is generated by $e_{3}$ and nonzero products and $\phi,\psi:A\rightarrow A$ are given by

[TABLE]

Then $(A,\cdot,[\cdot,\cdot],\phi,\psi)$ is a regular BiHom-Poisson superalgebra.

Note that $A_{\overline{0}}$ is a BiHom-Poisson algebra called the even or bosonic part of $A$, while $A_{\overline{1}}$ is called the odd or fermonic part of $A$. The usual regularity concepts will be understood in the graded sense. That is, a subalgebra $H=H_{\overline{0}}\oplus H_{\overline{1}}$ of $A$ is a graded subspace such that $[H,H]+HH\subset A$ and $\phi(H)=\psi(H)=H$. A graded subspace $I=I_{\overline{0}}\oplus I_{\overline{1}}$ of $A$ is called an ideal if $[I,A]+IA+AI\subset I$ and $\phi(I)=\psi(I)=I$. A BiHom-Poisson superalgebra $A$ will be called simple if $[A,A]+AA\neq 0$ and its only ideals are $\{0\}$ and $A$.

We recall from [22] that a BiHom-Lie superalgebra $(A,[\cdot,\cdot],\phi,\psi)$ and a maximal abelian sualgebra $H$ of $A$, for a linear functional

[TABLE]

we define the root space of $A$ associated to $\alpha$ as the subspace

[TABLE]

The elements $\alpha:H_{\overline{0}}\rightarrow\mathbb{K}$ satisfying $A_{\alpha}\neq 0$ are called roots of $A$ with respect to $H$ and we denote $\Lambda:=\{\alpha\in(H_{\overline{0}})^{\ast}/\{0\}:A_{\alpha}\neq 0\}$. We call that $A$ is a split regular BiHom-Lie superalgebra with respect to $H$ if

[TABLE]

We also say that $\Lambda$ is the root system of $A$.

To simplify notation, the mappings $\phi|_{H},\psi|_{H},\phi^{-1}|_{H},\psi^{-1}|_{H}:H\rightarrow H$ will be denoted by $\phi,\phi^{-1}$, $\psi$ and $\psi^{-1}$ respectively.

We recall some properties of split regular BiHom-Lie superalgebras that can be found in [22].

Lemma 2.5**.**

Let $(A,[\cdot,\cdot],\phi,\psi)$ be a split regular BiHom-Lie superalgebra. Then, for any $\alpha,\beta\in A\cup\{0\}$,

(1)$\phi(A_{\alpha})=A_{\alpha\phi^{-1}}$ and $\psi(A_{\alpha})=A_{\alpha\psi^{-1}}$,

(2) $\phi^{-1}(A_{\alpha})=A_{\alpha\phi}$ and $\psi^{-1}(A_{\alpha})=A_{\alpha\psi}$,

(3) $[A_{\alpha},A_{\beta}]\subset A_{\alpha\phi^{-1}+\beta\psi^{-1}}$,

(4) If $\alpha\in\Lambda$, then $\alpha\phi^{-z_{1}}\psi^{-z_{2}}\in\Lambda$ for any $z_{1},z_{2}\in\mathbb{Z}$,

(5) $A_{0}=H$.

Lemma 2.6**.**

Let $A$ be a split regular BiHom-Poisson superalgebra. Then for any $\alpha,\beta\in A\cup\{0\}$, we have $A_{\alpha}A_{\beta}\subset A_{\alpha\phi^{-1}+\beta\psi^{-1}}$.

Proof Let $h_{\overline{0}}\in H_{\overline{0}},v_{\alpha}\in A_{\alpha,{\overline{i}}}$ and $v_{\beta}\in A_{\beta,{\overline{j}}}$, we can write

[TABLE]

and denote $h^{\prime}_{\overline{0}}=\psi^{-1}(h_{\overline{0}})$. By applying the BiHom-Leibniz superidentity, we get

[TABLE]

That is $A_{\alpha}A_{\beta}\subset A_{\alpha\phi^{-1}+\beta\psi^{-1}}$. $\square$

By Lemma 2.6, we can assert that

[TABLE]

In what follows, $A$ denotes a split regular BiHom-Poisson superalgebra and

[TABLE]

the corresponding root spaces decomposition. Given a linear functional $\alpha:H_{\overline{0}}\rightarrow\mathbb{K}$, we denote by $-\alpha:H_{\overline{0}}\rightarrow\mathbb{K}$ the element in $H^{\ast}_{\overline{0}}$ defined by $(-\alpha)(h_{\overline{0}}):=-\alpha(h_{\overline{0}})$. We write

[TABLE]

Example 2.7**.**

Let $A=H\oplus(\bigoplus_{\alpha\in\Gamma}A_{\alpha})$ be a split Possion superalgebra, $\phi,\psi:A\rightarrow A$ two automorphism such that $\phi(H)=\psi(H)=H$ and $\phi\circ\psi=\psi\circ\phi$. By Example 2.2, we know that $(A,\mu^{\prime},[\cdot,\cdot]^{\prime},\phi,\psi)$ is a regular BiHom-Possion superalgebra. Then we have

[TABLE]

makes of the regular BiHom-Possion superalgebra $(A,\mu^{\prime},[\cdot,\cdot]^{\prime},\phi,\psi)$ being the roots system $\Lambda=\{\alpha\psi^{-1}:\alpha\in\Gamma\}$.

Definition 2.8**.**

Let $\alpha,\beta\in\Lambda$. We will say that $\alpha$ is connected to $\beta$ if either

[TABLE]

*or there exists $\{\alpha_{1},\cdot\cdot\cdot,\alpha_{k}\}\subset\pm\Lambda$ with $k\geq 2$, such that

1. $\alpha_{1}\in\{\alpha\phi^{-n}\psi^{-r}:~{}~{}n,r\in\mathbb{N}\}$.

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

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

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

$\cdot\cdot\cdot\cdot\cdot$**

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

$\alpha_{1}\phi^{-k+2}+\alpha_{2}\phi^{-k+3}\psi^{-1}+\alpha_{3}\phi^{-k+4}\psi^{-1}+\cdot\cdot\cdot+\alpha_{k-2}\phi^{-1}\psi^{-1}+\alpha_{k-1}\psi^{-1}\in\pm\Lambda$*.

3. $\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+2}\psi^{-1}+\alpha_{3}\phi^{-k+3}\psi^{-1}+\cdot\cdot\cdot+\alpha_{i}\phi^{-k+i}\psi^{-1}+\cdot\cdot\cdot+\alpha_{k-1}\phi^{-1}\psi^{-1}+\alpha_{k}\psi^{-1}\in\{\pm\beta\phi^{-m}\psi^{-s}:m,s\in\mathbb{N}\}$.*

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

The proof of the next result is analogous to the one of [22]. For the sake of completeness, we give a sketch of the proof.

Proposition 2.9**.**

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

Proof. If $\alpha\sim\beta$, then either $\beta=\epsilon\alpha\phi^{z_{1}}\psi^{z_{2}}$ for some $z_{1},z_{2}\in\mathbb{Z}$ and $\epsilon\in\{-1,1\}$, and so $\beta$ is connected to $\alpha$; or there exists $\{\alpha_{1},\cdot\cdot\cdot,\alpha_{k}\}\subset\pm\Lambda$ with $k\geq 2$, from $\alpha$ to $\beta$ with

[TABLE]

for some $m,s\in\mathbb{N}$, $\epsilon\in\{-1,1\}$. Then we can verify that

[TABLE]

is a connection from $\beta$ to $\alpha$ and the relation $\sim$ is symmetric.

Finally, suppose $\alpha\sim\beta$ and $\beta\sim\gamma$. If $\beta=\epsilon\alpha\phi^{z_{1}}\psi^{z_{2}}$ for some $z_{1},z_{2}\in\mathbb{Z}$, $\epsilon\in\{-1,1\}$ and $\gamma=\epsilon^{\prime}\alpha\phi^{z_{3}}\psi^{z_{4}}$ for some $z_{3},z_{4}\in\mathbb{Z}$, $\epsilon^{\prime}\in\{-1,1\}$, it is clear that $\alpha\sim\gamma$. Hence suppose $\{\alpha_{1},\cdot\cdot\cdot,\alpha_{k}\}$ with $k\geq 2$ is a connection from $\alpha$ to $\beta$ which satisfies

[TABLE]

for some $m,s\in\mathbb{N}$, $\epsilon\in\{-1,1\}$, and $\{h_{1},\cdot\cdot\cdot,h_{p}\}$ is a connection from $\beta$ to $\gamma$. Then $\{\alpha_{1},\cdot\cdot\cdot,\alpha_{k},\epsilon h_{2},\cdot\cdot\cdot,\epsilon h_{p}\}$ is connection from $\alpha$ to $\gamma$, so the connection relation is also transitive. $\square$

By Proposition 2.9 we can consider the quotient set

[TABLE]

with $[\alpha]$ being the set of nozero roots which are connected to $\alpha$. Our next goal is to associate an ideal $I_{[\alpha]}$ to $[\alpha]$. Fix $[\alpha]\in\Lambda/\sim$, we start by defining

[TABLE]

Now we define

[TABLE]

Finally, we denote by $I_{[}\alpha]$ the direct sum of the two subspaces above:

[TABLE]

Proposition 2.10**.**

For any $[\alpha]\in\Lambda/\sim$, the following assertions hold.

(1). $[I_{[\alpha]},I_{[\alpha]}]+I_{[\alpha]}I_{[\alpha]}\subset I_{[\alpha]}$,

(2). $\phi(I_{[\alpha]})=I_{[\alpha]}$ and $\psi(I_{[\alpha]})=I_{[\alpha]}$,

(3). For any $[\beta]\neq[\alpha]$, we have $[I_{[\alpha]},I_{[\beta]}]+I_{[\alpha]}I_{[\beta]}=0$.

Proof. (1) First we check that $[I_{[\alpha]},I_{[\alpha]}]\subset I_{[\alpha]}$, we can write

[TABLE]

Given $\beta\in[\alpha]$, we have $[I_{H,[\alpha]}]\subset A_{\beta\psi^{-1}}$. Since $\beta\psi^{-1}\in[\alpha]$, it follows that $[I_{H,[\alpha]},A_{\beta}]\subset V_{[\alpha]}$.

By a similar argument, we get $[A_{\beta},I_{H,\alpha}]\subset V_{[\alpha]}$.

Next we consider $[V_{[\alpha]},V_{[\alpha]}]$. If we take $\beta,\gamma\in[\alpha]$ such that $[A_{\beta},A_{\gamma}]\neq 0$, then $[A_{\beta},A_{\gamma}]\subset A_{\beta\phi^{-1}+\gamma\psi^{-1}}$. If $\beta\phi^{-1}+\gamma\psi^{-1}=0$, we get $[A_{\beta},A_{-\gamma}]\subset H$ and so $[A_{\beta},A_{-\gamma}]\subset I_{H,[\alpha]}$. Suppose that $\beta\phi^{-1}+\gamma\psi^{-1}\in\Lambda$. We infer that $\{\beta,\gamma\}$ is connection from $\beta$ to $\beta\phi^{-1}+\gamma\psi^{-1}$. The transitivity of $\sim$ now gives that $\beta\phi^{-1}+\gamma\psi^{-1}\in[\alpha]$ and so $[A_{\beta},A_{\gamma}]\subset V_{[\alpha]}$. Hence

[TABLE]

From (2.1) and (2.2), we get $[I_{[\alpha]},I_{[\alpha]}]\subset I_{[\alpha]}$.

Second, we will check that $I_{[\alpha]}I_{[\alpha]}\subset I_{[\alpha]}$. We have

[TABLE]

Similar considerations, we have

[TABLE]

Hence, it just remains to check that $I_{H,[\alpha]}I_{H,[\alpha]}$, observe that

[TABLE]

Consider the first summand on the right hand side of (2.4). By BiHom-Leibniz superidentity, we have

[TABLE]

Next we consider the last summand on the right hand side of (2.4). By BiHom-associativity, we know

[TABLE]

(2) It is easy to check that $\phi(I_{[\alpha]})=I_{[\alpha]}$ and $\psi(I_{[\alpha]})=I_{[\alpha]}$.

(3) We will study the expression $[I_{[\alpha]},I_{[\beta]}]+I_{[\alpha]}I_{[\beta]}$. Observe that

[TABLE]

and

[TABLE]

First we consider $[V_{[\alpha]},V_{[\beta]}]+V_{[\alpha]}V_{[\beta]}$ and suppose that there exist $\alpha_{1}\in[\alpha]$ and $\beta_{1}\in[\beta]$ such that $[A_{\alpha_{1}},A_{\beta_{1}}]+A_{\alpha_{1}}A_{\beta_{1}}\neq 0$. As necessarily $\alpha_{1}\phi^{-1}\neq-\beta_{1}\psi^{-1}$, then $\alpha_{1}\phi^{-1}+\beta_{1}\psi^{-1}\in\Lambda$. So $\{\alpha_{1},\beta_{1},-\alpha_{1}\phi^{-1}\}$ is a connection between $\alpha_{1}$ and $\beta_{1}$. By the transitivity of the connection relation we see $\alpha\in[\beta]$, a contradicition. Hence $[A_{\alpha_{1}},A_{\beta_{1}}]+A_{\alpha_{1}}A_{\beta_{1}}=0$, and so

[TABLE]

Next we consider the first summand $[I_{H,[\alpha]},V_{[\beta]}]$ on the right hand side of (2.5) and the second one $I_{H,[\alpha]}V_{[\beta]}$ of (2.6), and suppose that there exist $\alpha_{1}\in[\alpha]$ and $\beta_{1}\in[\beta]$ such that

[TABLE]

Then some of the four sunmands are different from zero.

If

[TABLE]

then BiHom-Leibniz identity gives

[TABLE]

Hence

[TABLE]

which contradicts (2.6). Therefore, $[[A_{\alpha_{1}},A_{-\alpha_{1}}],A_{\beta_{1}}]=0$.

If the second, third or fouth summand were nonzero, we can argue as above but using the BiHom-Leibniz or BiHom-associativity superidentites to show that these products are zero. Consequently,

[TABLE]

In a similar way we can prove that the remaining summands in (2.5) and (2.6) are zero, and the proof is complete. $\square$

Proposition 2.11**.**

For any $[\alpha]\in\Lambda/\sim$, we have

[TABLE]

Proof. Fix any $\beta\in[\alpha]$. On the one hand, by the BiHom-Leibniz superidentity, we get

[TABLE]

And on the other hand, by BiHom-associativity we know

[TABLE]

Theorem 2.12**.**

(1) For any $[\alpha]\in\Lambda/\sim$, the linear space $I_{[\alpha]}=I_{H,[\alpha]}+V_{[\alpha]}$ of $A$ associated to $[\alpha]$ is an ideal of $A$.

(2) If $A$ is simple, then there exists a connection from $\alpha$ to $\beta$ for any $\alpha,\beta\in\Lambda$ and $H=\sum_{\alpha\in\Lambda}([A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}})$.

Proof. (1) Since $[I_{[\alpha]},H]\subset I_{[\alpha]}$, by (2.4) and (2.5) we have

[TABLE]

According to Proposition 2.9 and Proposition 2.10, we have

[TABLE]

As we also have $\phi(I_{[\alpha]})=I_{[\alpha]}$ and $\psi(I_{[\alpha]})=I_{[\alpha]}$. So we conclude that $I_{[\alpha]}$ is an ideal of $A$.

(2) The simplicity of $A$ implies $I_{[\alpha]}=A$. From here, it is clear that $[\alpha]=\Lambda$ and $H=\sum_{\alpha\in\Lambda}([A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}})$. $\square$

Theorem 2.13**.**

We have

[TABLE]

where $U$ is a linear complement in $H$ of $span_{\mathbb{K}}\{[A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}}:\alpha\in\Lambda\}$ and any $I_{[\alpha]}$ is one of the ideals of $A$ described in Theorem 2.12, satisfying $[I_{[\alpha]},I_{[\beta]}]+I_{[\alpha]}I_{[\beta]}=0$ if $[\alpha]\neq[\beta]$.

Proof. $I_{[\alpha]}$ is well defined and is an ideal of $A$, being clear that

[TABLE]

Finally, Proposition 2.10 gives us $[I_{[\alpha]},I_{[\beta]}]+I_{[\alpha]}I_{[\beta]}=0$ if $[\alpha]\neq[\beta]$. $\square$

Let us denote by $Z(A):=\{v\in A:[v,A]+vA+Av=0\}$ the center of $A$.

Corollary 2.14**.**

If $Z(A)=0$ and $H=\sum_{\alpha\in\Lambda}([A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}})$. Then $A$ is the direct sum of the ideals given in Theorem 2.12,

[TABLE]

Furthermore, $[I_{[\alpha]},I_{[\beta]}]+I_{[\alpha]}I_{[\beta]}=0$ if $[\alpha]\neq[\beta]$. $\square$.

Proof. Since $H=\sum_{\alpha\in\Lambda}([A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}})$, it follows that $A=\sum_{[\alpha]\in\Lambda/\sim}I_{[\alpha]}$. To verify the direct character of the sum, take some $v\in I_{[\alpha]}\cap(\sum_{[\beta]\in\Lambda/\sim,[\beta]\neq[\alpha]}I_{[\beta]})$. Since $v\in I_{[\alpha]}$, the fact $[I_{[\alpha]},I_{[\beta]}]+I_{[\alpha]}I_{[\beta]}=0$ when $[\alpha]\neq[\beta]$ gives us

[TABLE]

In a similar way, $v\in\sum_{[\beta]\in\Lambda/\sim,[\beta]\neq[\alpha]}I_{[\beta]}$ implies $[v,I_{[\alpha]}]+vI_{[\alpha]}+I_{[\alpha]}v=0$. That is $v\in Z(A)$ and so $v=0$. $\square$

3 The simple components

In this section we focus on the simplicity of split regular BiHom-Poisson superalgebras by centering our attention on those of maximal length, we recall that a roots system $\Lambda$ of a split regular BiHom-Poisson superalgebra $A$ is called symmetric if it is satisfies that $\alpha\in\Lambda$ implies $-\alpha\in\Lambda$. From now on we will suppose that $\Lambda$ is symmetric.

For the grading of $I$, we have

[TABLE]

Lemma 3.1**.**

Suppose $H=\sum_{\alpha\in\Lambda}([A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}})$. If $I$ is an ideal of $A$ such that $I\subset H$, then $I\subset Z(A)$.

Proof. Observe that $[I,H]\subset[H,H]=0$ and

[TABLE]

Since $H=\sum_{\alpha\in\Lambda}([A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}})$, by the BiHom-Leibniz superidentity and the above observation, we obtain $HI+IH=0$. So $I\subset Z(A)$. $\square$

We use the same notions of [8] and [22], denote by $\Lambda_{\overline{0}}:=\{\alpha\in\Lambda:L_{\alpha,\overline{0}}\neq 0\}$ and $\Lambda_{\overline{1}}:=\{\alpha\in\Lambda:L_{\alpha,\overline{1}}\neq 0\}$, so $\Lambda=\Lambda_{\overline{0}}\cup\Lambda_{\overline{1}}$.

Definition 3.2**.**

A split regular BiHom-Poisson superalgebra $A$ is root multiplicative if $\alpha\in\Lambda_{\overline{i}},\beta\in\Lambda_{\overline{i}},\overline{i},\overline{j}\in\mathbb{Z}_{2}$ such that $\alpha\psi^{-1}+\beta\phi^{-1}\in\Lambda_{\overline{i}+\overline{j}}$, then $[A_{\alpha,\overline{i}},A_{\beta,\overline{j}}]+A_{\alpha,\overline{i}}A_{\beta,\overline{j}}\neq 0$.

Definition 3.3**.**

A split regular BiHom-Poisson superalgebra $A$ is of maximal length if dim$A_{\alpha,\overline{i}}=1$ for any $\alpha\in\Lambda$ and $\overline{i}\in\mathbb{Z}_{2}$.

Observe that if $A$ is of maximal lenth, then we have

[TABLE]

where $\Lambda^{\prime}_{\overline{i}}=\{\alpha\in\Lambda_{\overline{i}}:I_{\overline{i}}\cap L_{\alpha,\overline{i}}\neq 0\},\overline{i}\in\mathbb{Z}_{2}$.

Theorem 3.4**.**

Let $A$ be a split regular BiHom-Poisson superalgebra of maximal length and root multiplicative. Then $A$ is simple if and only if $Z(A)=0$, $H=\sum_{\alpha\in\Lambda}([A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}})$ and $\Lambda$ has all of its elements connected.

Proof. Suppose $A$ is simple. Since $Z(A)$ is an ideal of $A$, we have $Z(A)=0$. Now Theorem 2.12(2) completes the proof of the direct implication.

To prove the converse, consider a nonzero ideal of $A$. By (3.1), we can write $I=((I_{\overline{0}}\cap H_{\overline{0}})\oplus(\oplus_{\alpha\in\Lambda^{\prime}_{\overline{0}}}L_{\alpha,\overline{0}}))\oplus((I_{\overline{1}}\cap H_{\overline{1}})\oplus(\oplus_{\alpha\in\Lambda^{\prime}_{\overline{1}}}L_{\alpha,\overline{1}}))$, where $\Lambda^{\prime}_{\overline{i}}\subset\Lambda_{\overline{i}},\overline{i}\in\mathbb{Z}_{2}$, and some $\Lambda^{\prime}_{\overline{i}}\neq\emptyset$ as consequence of Lemma 3.1. Let us fix some $\alpha_{0}\in\Lambda^{\prime}_{\overline{i}}$ with $0\neq A_{\alpha_{0},\overline{i}}\subset I$. Since $\phi(I)=I$ and $\psi(I)=I$, it follows that

[TABLE]

In particular,

[TABLE]

Now, let us take any $\beta\in\Lambda$ satisfying $\beta\notin\{\alpha_{0}\phi^{z_{1}}\psi^{z_{2}}:z_{1},z_{2}\in\mathbb{Z}\}$. Since $\alpha_{0}$ and $\beta$ are connected, we have a connection $\{\alpha_{1},\cdot\cdot\cdot,\alpha_{k}\},k\geq 2$, from $\alpha_{0}$ to $\beta$ satisfying:

$\alpha_{1}=\alpha_{0}\phi^{-n}\psi^{-r}:~{}~{}n,r\in\mathbb{N}$.

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

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

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

$\cdot\cdot\cdot\cdot\cdot$

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

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

$\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+2}\psi^{-1}+\alpha_{3}\phi^{-k+3}\psi^{-1}+\cdot\cdot\cdot+\alpha_{i}\phi^{-k+i}\psi^{-1}+\cdot\cdot\cdot+\alpha_{k-1}\phi^{-1}\psi^{-1}+\alpha_{k}\psi^{-1}=\epsilon\beta\phi^{-m}\psi^{-s}:m,s\in\mathbb{N}$.

Taking into account that $\alpha_{1},\alpha_{2}\in\Lambda$ and $\alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1}\in\Lambda$, we have $\alpha_{1}\in\Lambda_{\overline{i}}$ and $\alpha_{1}\in\Lambda_{\overline{j}}$ such that $\alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1}\in\Lambda_{\overline{i}+\overline{j}}$. The root multiplicativity and maximal length of $A$ allow us to assert that either $0\neq[A_{\alpha_{1},\overline{i}},A_{\alpha_{2},\overline{j}}]=A_{\alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1},\overline{i}+\overline{j}}$ or $0\neq A_{\alpha_{1},\overline{i}}A_{\alpha_{2},\overline{j}}+A_{\alpha_{2},\overline{j}}A_{\alpha_{1},\overline{i}}=A_{\alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1},\overline{i}+\overline{j}}$.

Since $0\neq A_{\alpha_{1},\overline{i}}\subset I$ as a consequence of (3.3) we get

[TABLE]

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

[TABLE]

gives us $0\neq A_{\alpha_{1}\phi^{-2}+\alpha_{2}\phi^{-1}\psi^{-1}+\alpha_{3}\psi^{-1},\overline{k}}\subset I$. We can follow this process with the connection $\{\alpha_{1},\cdot\cdot\cdot,\alpha_{k}\}$ to get

[TABLE]

and then

[TABLE]

From (3.2) and (3.3), we have

[TABLE]

This can be reformulated by saying that for any $\alpha\in\Lambda$, either $\{\alpha\phi^{-z_{1}}\psi^{-z_{2}}\}$ or $\{-\alpha\phi^{-z_{1}}\psi^{-z_{2}}\}$ is contained in $\Lambda_{I}$. Taking now into account $H=\sum_{\alpha\in\Lambda}([A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}})$, we have

[TABLE]

Now for any $\alpha\in\Lambda$, since $A_{\alpha}=[H,A_{\alpha\psi}]$ by the maximal length of $A$, (3.4) gives us $A_{\alpha}\subset I$ and so $A=I$. That is, $A$ is simple. $\square$

Theorem 3.5**.**

Let $A$ be a split regular BiHom-Poisson superalgebra of maximal length and root multiplicative with $Z(A)=0$ and satisfying $H=\sum_{\alpha\in\Lambda}([A_{\alpha\psi^{-1}},A_{-\alpha\phi^{-1}}]+A_{\alpha\psi^{-1}}A_{-\alpha\phi^{-1}})$. Then

[TABLE]

where any $I_{[\alpha]}$ is a simple split ideal having its roots system $\Lambda_{I_{[\alpha]}}$, with all of its elements $\Lambda_{I_{[\alpha]}}$-connected.

Proof. By Corollary 2.11, we can write $A$ as the direct sum $\bigoplus_{[\alpha]\in\Lambda/\sim}I_{[\alpha]}$ of the family of ideals

[TABLE]

where each $I_{[\alpha]}$ is a split regular BiHom-Poisson superalgebra with root system $A_{I_{[\alpha]}}=[\alpha]$. To make use of Theorem 3.4 in each $I_{[\alpha]}$, we observe that the root multiplicativity of $A$ and Proposition 2.10 show that $A_{I_{[\alpha]}}$ has all of its elements $A_{I_{[\alpha]}}$ connected, that is, connected through connections contained in $A_{I_{[\alpha]}}$. Moreover, each $I_{[\alpha]}$ is root multiplicative by the root multiplicativity of $A$. So we infer that $I_{[\alpha]}$ is of maximal length, and finally its center $Z(I_{[\alpha]})=0$. As consequence $[I_{\alpha},I_{\beta}]+I_{\alpha}I_{\beta}=0$ if $[\alpha]\neq[\beta]$. Applying Theorem 3.4, we conclude that $I_{[\alpha]}$ is simple and $A=\bigoplus_{[\alpha]\in\Lambda/\sim}I_{[\alpha]}$. $\square$

ACKNOWLEDGEMENT

The paper is supported by the NSF of China (No. 11761017) and the Youth Project for Natural Science Foundation of Guizhou provincial department of education (No. KY[2018]155).