On $\mathcal{L}$-simulation mappings in partial metric spaces

Hassen Aydi; M. A. Barakat; Erdal Karapinar; Zoran D. Mitrović; Tawseef Rashid; Hassen Aydi; M. A. Barakat; Erdal Karapinar; Zoran D. Mitrović; Tawseef Rashid

doi:10.3934/math.2019.4.1034

AIMS Mathematics

2019, Volume 4, Issue 4: 1034-1045. doi: 10.3934/math.2019.4.1034

Previous Article Next Article

Research article

On $\mathcal{L}$ -simulation mappings in partial metric spaces

1.
Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia
2.
Department of Medical Research, China Medical University Hospital, China Medical University, 40402, Taichung, Taiwan
3.
Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524, Egypt
4.
Department of Mathematics, College of Al Wajh, University of Tabuk, Saudi Arabia
5.
University of Banja Luka, Faculty of Electrical Engineering, 78 000 Banja Luka, Bosnia and Herzegovina
6.
Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India

The class of L-contractive mappings was introduced by Cho ^[12]. In this paper, we provide some fixed point results for such mappings via a control function introduced by Jleli and Samet ^[14] in the class of partial metric spaces. Some illustrating examples are given.

Keywords:

Citation: Hassen Aydi, M. A. Barakat, Erdal Karapinar, Zoran D. Mitrović, Tawseef Rashid. On $\mathcal{L}$ -simulation mappings in partial metric spaces[J]. AIMS Mathematics, 2019, 4(4): 1034-1045. doi: 10.3934/math.2019.4.1034

Related Papers:

[1]	Madeeha Tahir, Ayesha Naz, Muhammad Imran, Hasan Waqas, Ali Akgül, Hussein Shanak, Rabab Jarrar, Jihad Asad . Activation energy impact on unsteady Bio-convection nanomaterial flow over porous surface. AIMS Mathematics, 2022, 7(11): 19822-19845. doi: 10.3934/math.20221086
[2]	Khalil Ur Rehman, Wasfi Shatanawi, Zeeshan Asghar, Haitham M. S. Bahaidarah . Neural networking analysis for MHD mixed convection Casson flow past a multiple surfaces: A numerical solution. AIMS Mathematics, 2023, 8(7): 15805-15823. doi: 10.3934/math.2023807
[3]	Taqi A. M. Shatnawi, Nadeem Abbas, Wasfi Shatanawi . Comparative study of Casson hybrid nanofluid models with induced magnetic radiative flow over a vertical permeable exponentially stretching sheet. AIMS Mathematics, 2022, 7(12): 20545-20564. doi: 10.3934/math.20221126
[4]	Nadeem Abbas, Wasfi Shatanawi, Taqi A. M. Shatnawi . Innovation of prescribe conditions for radiative Casson micropolar hybrid nanofluid flow with inclined MHD over a stretching sheet/cylinder. AIMS Mathematics, 2025, 10(2): 3561-3580. doi: 10.3934/math.2025164
[5]	Muhammad Asif Zahoor Raja, Kottakkaran Sooppy Nisar, Muhammad Shoaib, Ajed Akbar, Hakeem Ullah, Saeed Islam . A predictive neuro-computing approach for micro-polar nanofluid flow along rotating disk in the presence of magnetic field and partial slip. AIMS Mathematics, 2023, 8(5): 12062-12092. doi: 10.3934/math.2023608
[6]	M. S. Alqarni . Thermo-bioconvection flow of Walter's B nanofluid over a Riga plate involving swimming motile microorganisms. AIMS Mathematics, 2022, 7(9): 16231-16248. doi: 10.3934/math.2022886
[7]	Kiran Sajjan, N. Ameer Ahammad, C. S. K. Raju, M. Karuna Prasad, Nehad Ali Shah, Thongchai Botmart . Study of nonlinear thermal convection of ternary nanofluid within Darcy-Brinkman porous structure with time dependent heat source/sink. AIMS Mathematics, 2023, 8(2): 4237-4260. doi: 10.3934/math.2023211
[8]	J. Kayalvizhi, A. G. Vijaya Kumar, Ndolane Sene, Ali Akgül, Mustafa Inc, Hanaa Abu-Zinadah, S. Abdel-Khalek . An exact solution of heat and mass transfer analysis on hydrodynamic magneto nanofluid over an infinite inclined plate using Caputo fractional derivative model. AIMS Mathematics, 2023, 8(2): 3542-3560. doi: 10.3934/math.2023180
[9]	Ali Raza, Umair Khan, Aurang Zaib, Wajaree Weera, Ahmed M. Galal . A comparative study for fractional simulations of Casson nanofluid flow with sinusoidal and slipping boundary conditions via a fractional approach. AIMS Mathematics, 2022, 7(11): 19954-19974. doi: 10.3934/math.20221092
[10]	Latifa I. Khayyat, Abdullah A. Abdullah . The onset of Marangoni bio-thermal convection in a layer of fluid containing gyrotactic microorganisms. AIMS Mathematics, 2021, 6(12): 13552-13565. doi: 10.3934/math.2021787

Abstract

1. Introduction

A Poisson algebra is a triple, $(L, \cdot, \left[{-, - } \right])$ , where $(L, \cdot)$ is a commutative associative algebra and $(L, \left[{-, - } \right])$ is a Lie algebra that satisfies the following Leibniz rule:

$\left[ {x, y\cdot z} \right] = \left[ {x, y} \right]\cdot z + y\cdot \left[ {x, z} \right], \forall x, y, z \in L.$

Poisson algebras appear naturally in the study of Hamiltonian mechanics and play a significant role in mathematics and physics, such as in applications of Poisson manifolds, integral systems, algebraic geometry, quantum groups, and quantum field theory (see ^[7,11,24,25]). Poisson algebras can be viewed as the algebraic counterpart of Poisson manifolds. With the development of Poisson algebras, many other algebraic structures have been found, such as Jacobi algebras ^[1,9], Poisson bialgebras ^[20,23], Gerstenhaber algebras, Lie-Rinehart algebras ^[16,17,26], $F$ -manifold algebras ^[12], Novikov-Poisson algebras ^[28], quasi-Poisson algebras ^[8] and Poisson $n$ -Lie algebras ^[10].

As a dual notion of a Poisson algebra, the concept of a transposed Poisson algebra was recently introduced by Bai et al. ^[2]. A transposed Poisson algebra $(L, \cdot, \left[{-, - } \right])$ is defined by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra:

$2z \cdot \left[ {x, y} \right] = \left[ {z \cdot x, y} \right] + \left[ {x, z \cdot y} \right], \forall x, y, z \in L,$

where $(L, \cdot)$ is a commutative associative algebra and $(L, \left[{-, - } \right])$ is a Lie algebra.

It is shown that a transposed Poisson algebra possesses many important identities and properties and can be naturally obtained by taking the commutator in the Novikov-Poisson algebra ^[2]. There are many results on transposed Poisson algebras, such as those on transposed Hom-Poisson algebras ^[18], transposed BiHom-Poisson algebras ^[21], a bialgebra theory for transposed Poisson algebras ^[19], the relation between $\frac{1}{2}$ -derivations of Lie algebras and transposed Poisson algebras ^[14], the relation between $\frac{1}{2}$ -biderivations and transposed Poisson algebras ^[29], and the transposed Poisson structures with fixed Lie algebras (see ^[6] for more details).

The notion of an $n$ -Lie algebra (see Definition 2.1), as introduced by Filippov ^[15], has found use in many fields in mathematics and physics ^[4,5,22,27]. The explicit construction of $n$ -Lie algebras has become one of the important problems in this theory. In ^[3], Bai et al. gave a construction of $(n+1)$ -Lie algebras through the use of $n$ -Lie algebras and some linear functions. In ^[13], Dzhumadil′daev introduced the notion of a Poisson $n$ -Lie algebra which can be used to construct an ( $n+1$ )-Lie algebra under an additional strong condition. In ^[2], Bai et al. showed that this strong condition for $n = 2$ holds automatically for a transposed Poisson algebra, and they gave a construction of 3-Lie algebras from transposed Poisson algebras with derivations. They also found that this constructed $3$ -Lie algebra and the commutative associative algebra satisfy the analog of the compatibility condition for transposed Poisson algebras, which is called a transposed Poisson $3$ -Lie algebra. This motivated them to introduce the concept of a transposed Poisson $n$ -Lie algebra (see Definition 2.2) and propose the following conjecture:

Conjecture 1.1. ^[2] Let $n\geq 2$ be an integer and $(L, \cdot, \mu_n)$ a transposed Poisson $n$ -Lie algebra. Let $D$ be a derivation of $(L, \cdot)$ and $(L, \mu_n)$ . Define an $(n+1)$ -ary operation:

$\mu_{n+1}(x_1, \cdots, x_{n+1}): = \sum\limits_{i = 1}^{n+1} (-1)^{i-1}D(x_i) \mu_n(x_1, \cdots, \hat{x}_i, \cdots, x_{n+1}), \quad \forall x_1, \cdots, x_{n+1}\in L,$

where $\hat{x}_i$ means that the $i$ -th entry is omitted. Then, $(L, \cdot, \mu_{n+1})$ is a transposed Poisson $(n+1)$ -Lie algebra.

In this paper, based on the identities for transposed Poisson $n$ -Lie algebras given in Section 2, we prove that Conjecture 1.1 holds under a certain strong condition described in Section 3 (see Definition 2.3 and Theorem 3.2).

Throughout the paper, all vector spaces are taken over a field of characteristic zero. To simplify notations, the commutative associative multiplication $(\cdot)$ will be omitted unless the emphasis is needed.

2. Identities in transposed Poisson $n$ -Lie algebras

In this section, we first recall some definitions, and then we exhibit a class of identities for transposed Poisson $n$ -Lie algebras.

Definition 2.1. ^[15] Let $n\geq 2$ be an integer. An $n$ -Lie algebra is a vector space $L$ , together with a skew-symmetric linear map $\left[{-, \cdots, - } \right]:{ \otimes ^n}L \to L$ , such that, for any $x_i, y_j \in L, 1 \le i \le n-1, 1 \le j \le n,$ the following identity holds:

$\begin{equation} {\left[ {{{\left[ {{y_1}, \cdots , {y_n}} \right]}}, {x_1}, \cdots , {x_{n - 1}}} \right]} = \sum\limits_{i = 1}^n {{{( - 1)}^{i - 1}}{{\left[ {{{\left[ {{y_i}, {x_1}, \cdots , {x_{n - 1}}} \right]}}, {y_1}, \cdots , {{\hat y}_i}, \cdots , {y_n}} \right]}}}. \end{equation}$

(2.1)

Definition 2.2. ^[2] Let $n\geq 2$ be an integer and $L$ a vector space. The triple $(L, \cdot, \left[{-, \cdots, - } \right])$ is called a transposed Poisson $n$ -Lie algebra if $(L, \cdot)$ is a commutative associative algebra and $(L, \left[{-, \cdots, - } \right])$ is an $n$ -Lie algebra such that, for any $h, {x_i} \in L, 1 \le i \le n$ , the following identity holds:

$\begin{equation} nh\left[ {{x_1}, \cdots , {x_n}} \right] = \sum\limits_{i = 1}^n {\left[ {{x_1}, \cdots , h{x_i}, \cdots , {x_n}} \right]}. \end{equation}$

(2.2)

Some identities for transposed Poisson algebras in ^[2] can be extended to the following theorem for transposed Poisson $n$ -Lie algebras.

Theorem 2.1. Let $(L, \cdot, \left[{-, \cdots, - } \right])$ be a transposed Poisson $n$ -Lie algebra. Then, the following identities hold:

(1) For any $x_i\in L, 1\leq i\leq n+1$ , we have

$\begin{align} \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}{x_i}\left[ {{x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n + 1}}} \right]} = 0; \end{align}$

(2.3)

(2) For any $h, x_i, y_j\in L, 1\leq i\leq n-1, 1\leq j\leq n$ , we have

$\begin{align} \sum\limits_{i = 1}^n {{{( - 1)}^{i - 1}}\left[ {h\left[ {{y_i}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {y_n}} \right] = } \left[ {h\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right]; \end{align}$

(2.4)

(3) For any $x_i, y_j\in L, 1\leq i\leq n-1, 1\leq j\leq n+1$ , we have

$\begin{align} \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}\left[{y_i}, {{x_1}, \cdots , {x_{n-1}}} \right]\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} = 0; \end{align}$

(2.5)

(4) For any $x_1, x_2, y_i\in L, 1\leq i\leq n$ , we have

$\begin{align} \sum\limits_{i = 1}^n {\sum\limits_{j = 1, j \ne i}^n {\left[ {{y_1}, \cdots , {y_i}{x_1}, \cdots , {y_j}{x_2}, \cdots , {y_n}} \right]} } = n(n - 1){x_1}{x_2}\left[ {{y_1}, {y_2}, \cdots , {y_n}} \right]. \end{align}$

(2.6)

Proof. (1) By Eq (2.2), for any $1\leq i\leq n+1$ , we have

$n{x_i}\left[ {{x_1}, \cdots , {x_{i - 1}}, {x_{i + 1}}, \cdots , {x_{n+1}}} \right] = \sum\limits_{j \ne i}^{} {\left[ {{x_1}, \cdots , {x_{i - 1}}, {x_{i + 1}}, \cdots , {x_i}{x_j}, \cdots , {x_{n+1}}} \right]}.$

Thus, we obtain

$\sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}n{x_i}\left[ {{x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n + 1}}} \right]} = \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {{{( - 1)}^{i - 1}}\left[ {{x_1}, \cdots , {{\hat x}_i}, \cdots , {x_i}{x_j}, \cdots , {x_{n + 1}}} \right]} }.$

Note that, for any $i > j$ , we have

$\begin{eqnarray*} &&{( - 1)^{i - 1}}\left[ {{x_1}, \cdots , {x_{j - 1}}, {x_i}{x_j}, {x_{j + 1}}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right] \\ &&+ {( - 1)^{j - 1}}\left[ {{x_1}, \cdots , {{\hat x}_j}, \cdots , {x_{i - 1}}, {x_j}{x_i}, {x_{i + 1}}, \cdots , {x_n}} \right]\\ & = & {( - 1)^{i - 1 + (i - j - 1)}}\left[ {{x_1}, \cdots , {x_{j - 1}}, {x_{j + 1}}, \cdots , {x_{i - 1}}, {x_i}{x_j}, {x_{i + 1}}, \cdots , {x_n}} \right] \\ && + {( - 1)^{j - 1}}\left[ {{x_1}, \cdots , {{\hat x}_j}, \cdots , {x_{i - 1}}, {x_j}{x_i}, {x_{i + 1}}, \cdots , {x_n}} \right]\\ & = & \big( {{{( - 1)}^{ - j - 2}} + {{( - 1)}^{j - 1}}} \big)\left[ {{x_1}, \cdots , {x_{j - 1}}, {x_{j + 1}}, \cdots , {x_{i - 1}}, {x_i}{x_j}, {x_{i + 1}}, \cdots , {x_n}} \right]\\ & = & 0, \end{eqnarray*}$

which gives $\sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {{{(- 1)}^{i - 1}}\left[{{x_1}, \cdots, {{\hat x}_i}, \cdots, {x_i}{x_j}, \cdots, {x_{n + 1}}} \right]} } = 0.$

Hence, we get

$\sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}n{x_i}\left[ {{x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n + 1}}} \right]} = 0.$

(2) By Eq (2.2), we have

and, for any $1\leq j\leq n$ ,

$\begin{eqnarray*} &&{( - 1)^{j - 1}}(\left[ {h\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {\hat{y}_j}, \cdots , {y_{n - 1}}} \right] \\ &&+\sum\limits_{i = 1, i\neq j}^{n } {\left[ {\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , h{y_i}, \cdots , \hat{y}_j, \cdots , {y_{n - 1}}} \right]}) \\ & = &{( - 1)^{j - 1}}nh\left[ {\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {\hat{y}_j}, \cdots , {y_{n - 1}}} \right]. \end{eqnarray*}$

By taking the sum of the above $n+1$ identities and applying Eq (2.1), we get

$\begin{eqnarray*} &&- \left[ {h\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right]- \sum\limits_{i = 1}^{n - 1} {\left[ {\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , h{x_i}, \cdots , {x_{n - 1}}} \right]}\\ &&+\sum\limits_{j = 1}^{n} {( - 1)^{j - 1}}(\left[ {h\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {\hat{y}_j}, \cdots , {y_{n - 1}}} \right] \\ &&+\sum\limits_{i = 1, i\neq j}^{n } {\left[ {\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , h{y_i}, \cdots , \hat{y}_j, \cdots , {y_{n - 1}}} \right]}) \\ & = & -nh\left[ {\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right]+\\ &&nh\sum\limits_{j = 1}^{n} {( - 1)^{j - 1}}\left[ {\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {\hat{y}_j}, \cdots , {y_{n - 1}}} \right]\\ & = & 0. \end{eqnarray*}$

We denote

$\begin{eqnarray*} {{A_j}} & : = &\sum\limits_{i = 1, i \ne j}^n {{{( - 1)}^{i - 1}}\left[ {\left[ {{y_i}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , h{y_j}, \cdots , {{\hat y}_i}, \cdots , {y_n}} \right]}, 1\leq j\leq {n}, \\ {B_i} &: = & \left[ {\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , h{x_i}, \cdots , {x_{n - 1}}} \right], 1\leq i\leq {n-1}. \end{eqnarray*}$

Then, the above equation can be rewritten as

$\begin{equation} \begin{array}{l} \sum\limits_{i = 1}^n {{{( - 1)}^{i - 1}}\left[ {h\left[ {{y_i}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {y_n}} \right]} - \left[ {h\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right]\\ + \sum\limits_{j = 1}^n {{A_j}} - \sum\limits_{i = 1}^{n - 1} {{B_i}} = 0. \end{array} \end{equation}$

(2.7)

By applying Eq (2.1) to ${{A_j}}, 1\leq j\leq n$ , we have

$\begin{eqnarray*} {{A_j}} & = & \sum\limits_{i = 1, i \ne j}^n {{{( - 1)}^{i - 1}}\left[ {\left[ {{y_i}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , h{y_j}, \cdots , {{\hat y}_i}, \cdots , {y_n}} \right]}\\ & = & \left[ {\left[ {{y_1}, \cdots , h{y_j}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right] \\ &&+ {( - 1)^j}\left[ {\left[ {h{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_j}, \cdots , {y_n}} \right]. \end{eqnarray*}$

Thus, we get

$\begin{eqnarray*} \sum\limits_{j = 1}^n { {{A_j}} } & = &\sum\limits_{j = 1}^n {\left[ {\left[ {{y_1}, \cdots , h{y_j}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right]} \\ && + \sum\limits_{j = 1}^n {( - 1)^j}{\left[ {\left[ {h{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_j}, \cdots , {y_n}} \right]}\\ & = & n\left[ {h\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right]\\ && + \sum\limits_{j = 1}^n {( - 1)^j}{\left[ {\left[ {h{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_j}, \cdots , {y_n}} \right]}. \end{eqnarray*}$

By applying Eq (2.1) to ${{B_i}}, 1\leq i\leq n-1$ , we have

$\begin{eqnarray*} B_i & = &\left[ {\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , h{x_i}, \cdots , {x_{n - 1}}} \right] \\ & = & \sum\limits_{j = 1}^n {{{( - 1)}^{j - 1}}\left[ {\left[ {{y_j}, {x_1}, \cdots , h{x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right]}. \end{eqnarray*}$

Thus, we get

$\begin{eqnarray*} \sum\limits_{i = 1}^{n - 1} {{B_i}} & = & \sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = 1}^n {{{( - 1)}^{j - 1}}\left[ {\left[ {{y_j}, {x_1}, \cdots , h{x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right]} } \\ & = &\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^{n - 1} {{{( - 1)}^{j - 1}}\left[ {\left[ {{y_j}, {x_1}, \cdots , h{x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right]} }. \end{eqnarray*}$

Note that, by Eq (2.2), we have

$\begin{eqnarray*} &&\sum\limits_{i = 1}^{n - 1} {{{( - 1)}^{j-1}}\left[ {\left[ {{y_j}, {x_1}, \cdots , h{x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right]} \\ & = &{( - 1)^{j-1}}n\left[ {h\left[ {{y_j}, {x_1}, \cdots , {x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right] \\ &&+{( - 1)^{j}} \left[ {\left[ {h{y_j}, {x_1}, \cdots , {x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right]. \end{eqnarray*}$

Thus, we obtain

$\begin{eqnarray*} \sum\limits_{i = 1}^{n - 1} {{B_i}} & = &\sum\limits_{j = 1}^n {{{( - 1)}^{j - 1}}n\left[ {h\left[ {{y_j}, {x_1}, \cdots , {x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right]} \\ && + \sum\limits_{j = 1}^n {{{( - 1)}^j}\left[ {\left[ {h{y_j}, {x_1}, \cdots , {x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right]}. \end{eqnarray*}$

By substituting these equations into Eq (2.7), we have

$\begin{eqnarray*} &&\sum\limits_{i = 1}^n {{{( - 1)}^{i - 1}}\left[ {h\left[ {{y_i}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {y_n}} \right]} - \left[ {h\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right]\\ &&+ n\left[ {h\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right] + \sum\limits_{j = 1}^n {( - 1)^j}{\left[ {\left[ {h{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_j}, \cdots , {y_n}} \right]}\\ &&- \sum\limits_{j = 1}^n {{{( - 1)}^{j - 1}}n\left[ {h\left[ {{y_j}, {x_1}, \cdots , {x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right]} \\ &&-\sum\limits_{j = 1}^n {{{( - 1)}^j}\left[ {\left[ {h{y_j}, {x_1}, \cdots , {x_i}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots {{\hat y}_j}, \cdots , {y_n}} \right]} \\ & = &0, \end{eqnarray*}$

which implies that

$(n - 1)(\sum\limits_{i = 1}^n {{{( - 1)}^i}\left[ {h\left[ {{y_i}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {y_n}} \right]} + \left[ {h\left[ {{y_1}, \cdots , {y_n}} \right], {x_1}, \cdots , {x_{n - 1}}} \right]) = 0.$

Therefore, the proof of Eq (2.4) is completed.

(3) By Eq (2.2), for any $1\leq j\leq n+1$ , we have

$\begin{eqnarray*} &&{( - 1)^{j - 1}}n\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right]\left[ {{y_1}, \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]\\ & = & \sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}\left[ {{y_1}, \cdots , {y_i}\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]}. \end{eqnarray*}$

By taking the sum of the above $n+1$ identities, we obtain

$\begin{eqnarray*} &&\sum\limits_{j = 1}^{n + 1} {{{( - 1)}^{j - 1}}n\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right]\left[ {{y_1}, \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]} \\ & = & \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}\left[ {{y_1}, \cdots , {y_i}\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]} }. \end{eqnarray*}$

Thus, we only need to prove the following equation:

$\sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}\left[ {{y_1}, \cdots , {y_i}\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]} } = 0.$

Note that

$\begin{eqnarray*} &&\sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}\left[ {{y_1}, \cdots , {y_i}\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]} } \\ & = &\sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {{{( - 1)}^{j - 1}}\left[ {{y_1}, \cdots , {y_i}\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]} } \\ & = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1}^{i - 1} {{{( - 1)}^{i + j - 1}}\left[ {{y_i}\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_j}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } \\ && + \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = i + 1}^{n + 1} {{{( - 1)}^{i + j}}\left[ {{y_i}\left[ {{y_j}, {x_1}, \cdots , {x_{n - 1}}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]} } \\ &\mathop = \limits^{(2.4)}& \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^i}\left[ {{y_i}\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {x_{n - 1}}} \right]} \\ &\mathop = \limits^{(2.3)}& 0. \end{eqnarray*}$

Hence, the conclusion holds.

(4) By applying Eq (2.2), we have

$\begin{eqnarray*} &&{n^2}{x_1}{x_2}\left[ {{y_1}, {y_2}, \cdots , {y_n}} \right] = n{x_1}\sum\limits_{j = 1}^n {\left[ {{y_1}, \cdots , {y_j}{x_2}, \cdots , {y_n}} \right]} \\ & = & \sum\limits_{i = 1}^n {\sum\limits_{j = 1, j \ne i}^n {\left[ {{y_1}, \cdots , {y_i}{x_1}, \cdots , {y_j}{x_2}, \cdots , {y_n}} \right]} } + \sum\limits_{j = 1}^n {\left[ {{y_1}, \cdots , {y_j}{x_1}{x_2}, \cdots , {y_n}} \right]}\\ & = &\sum\limits_{i = 1}^n {\sum\limits_{j = 1, j \ne i}^n {\left[ {{y_1}, \cdots , {y_i}{x_1}, \cdots , {y_j}{x_2}, \cdots , {y_n}} \right]} } + n{x_1}{x_2}\left[ {{y_1}, \cdots , {y_n}} \right], \end{eqnarray*}$

which gives

$n(n - 1){x_1}{x_2}\left[ {{y_1}, {y_2}, \cdots , {y_n}} \right] = \sum\limits_{i = 1}^n {\sum\limits_{j = 1, j \ne i}^n {\left[ {{y_1}, \cdots , {y_i}{x_1}, \cdots , {y_j}{x_2}, \cdots , {y_n}} \right]} } .$

Hence, the proof is completed.

To prove Conjecture 1.1, we need the following extra condition.

Definition 2.3. A transposed Poisson $n$ -Lie algebra $(L, \cdot, \left[{-, \cdots, - } \right])$ is called strong if the following identity holds:

$\begin{equation} \begin{array}{l} {y_1}\left[ {h{y_2}, {x_1}, \cdots , {x_{n - 1}}} \right] - {y_2}\left[ {h{y_1}, {x_1}, \cdots , {x_{n - 1}}} \right] + \sum\limits_{i = 1}^{n - 1} {{{( - 1)}^{i - 1}}h{x_i}\left[ {{y_1}, {y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n - 1}}} \right]} \\ = 0 \end{array} \end{equation}$

(2.8)

for any ${y_1}, {y_2}, {x_i}\in L, 1\leq i\leq n - 1.$

Remark 2.1. When $n = 2$ , the identity is

${y_1}\left[ {h{y_2}, {x_1}} \right] + {y_2}\left[ {{x_1}, h{y_1}} \right] + h{x_1}\left[ {{y_1}, {y_2}} \right] = 0,$

which is exactly Theorem 2.5 (11) in ^[2]. Thus, in the case of a transposed Poisson algebra, the strong condition always holds. So far, we cannot prove that the strong condition fails to hold for $n\geq 3.$

Proposition 2.1. Let $(L, \cdot, \left[{-, \cdots, - } \right])$ be a strong transposed Poisson $n$ -Lie algebra. Then,

$\begin{equation} {y_1}\left[ {h{y_2}, {x_1}, \cdots , {x_{n - 1}}} \right] - h{y_1}\left[ {{y_2}, {x_1}, \cdots , {x_{n - 1}}} \right] = {y_2}\left[ {h{y_1}, {x_1}, \cdots , {x_{n - 1}}} \right] - h{y_2}\left[ {{y_1}, {x_1}, \cdots , {x_{n - 1}}} \right] \end{equation}$

(2.9)

for any ${y_1}, {y_2}, {x_i} \in L, 1\leq i\leq n - 1$ .

Proof. By Eq (2.3), we have

$- h{y_1}\left[ {{y_2}, {x_1}, \cdots , {x_{n - 1}}} \right] + h{y_2}\left[ {{y_1}, {x_1}, \cdots , {x_{n - 1}}} \right] = \sum\limits_{i = 1}^{n - 1} {{{( - 1)}^{i - 1}}h{x_i}\left[ {{y_1}, {y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n - 1}}} \right]}.$

Then, the statement follows from Eq (2.8).

3. Proof of the conjecture for strong transposed Poisson $n$ -Lie algebras

In this section, we will prove Conjecture 1.1 for strong transposed Poisson $n$ -Lie algebras. First, we recall the notion of derivations of transposed Poisson $n$ -Lie algebras.

Definition 3.1. Let $(L, \cdot, \left[{-, \cdots, - } \right])$ be a transposed Poisson $n$ -Lie algebra. The linear operation $D: L\rightarrow L$ is called a derivation of $(L, \cdot, \left[{-, \cdots, - } \right])$ if the following holds for any $u, v, x_{i}\in L, 1\leq i\leq n$ :

(1) $D$ is a derivation of $\left({L, \cdot} \right)$ , i.e., $D\left({uv} \right) = D\left(u \right)v + uD\left(v \right)$ ;

(2) $D$ is a derivation of $(L, \left[{-, \cdots, - } \right])$ , i.e.,

$D\left( \left[ {{x_1}, \cdots , x_n} \right] \right) = \sum\limits_{i = 1}^n {\left[ {{x_1}, \cdots , {x_{i - 1}}, D\left( {{x_i}} \right), {x_{i + 1}}, \cdots , {x_n}} \right]}.$

Lemma 3.1. Let $(L, \cdot, \left[{-, \cdots, - } \right])$ be a transposed Poisson $n$ -Lie algebra and $D$ a derivation of $(L, \cdot, \left[{-, \cdots, - } \right])$ . For any $y_i\in L, 1\leq i\leq n+1$ , we have the following:

(1)

$\begin{align} &\sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}D({y_i})} D\left( {\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} \right)\\ = &\sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {{{( - 1)}^{i - 1}}D({y_i})\left[ {{y_1}, \cdots , D({y_j}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} }; \end{align}$

(3.1)

(2)

$\begin{align} &\sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}D({y_i})} D\left( {\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} \right)\\ = &\sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}{y_i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } }, \end{align}$

(3.2)

where, for any $i > j$ , $\sum\limits_{i}^{j}$ denotes the empty sum, which is equal to zero.

Proof. (1) The statement follows immediately from Definition 3.1.

(2) By applying Eq (3.1), we need to prove the following equation:

$\begin{eqnarray*} & &\sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {{{( - 1)}^{i - 1}}nD({y_i})\left[ {{y_1}, \cdots , D({y_j}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } \\ & = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}n{y_i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } }. \end{eqnarray*}$

For any $1\leq i\leq n+1$ , denote $A_i : = n\sum\limits_{j = 1, j \ne i}^{n + 1} {{{(- 1)}^{i - 1}}D({y_i})\left[{{y_1}, \cdots, D({y_j}), \cdots, {{\hat y}_i}, \cdots, {y_{n + 1}}} \right]}$ . Then, we have

$\sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {{{( - 1)}^{i - 1}}nD({y_i})\left[ {{y_1}, \cdots , D({y_j}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } = \sum\limits_{i = 1}^{n + 1} { {{A_i}} }.$

Note that

$\begin{eqnarray*} &&{A_i} = {( - 1)^{i - 1}}(nD({y_i})\left[ {D({y_1}), {y_2}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right] + nD({y_i})\left[ {{y_1}, D({y_2}), {y_3}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\\ &&+ \cdots + nD({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_n}, D({y_{n + 1}})} \right])\\ & = & {( - 1)^{i - 1}}(\left[ {D({y_i})D({y_1}), {y_2}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right] \\ &&+ \sum\limits_{k = 2, k \ne i}^{n + 1} {\left[ {D({y_1}), {y_2}, \cdots , {y_k}D({y_i}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} \\ &&+ \left[ {{y_1}, D({y_i})D({y_2}), {y_3}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\\ && + \sum\limits_{k = 1, k \ne 2, i}^{n + 1} {\left[ {{y_1}, D({y_2}), {y_3}, \cdots , {y_k}D({y_i}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} \\ &&+ \cdots + \left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_n}, D({y_i})D({y_{n + 1}})} \right] \\ &&+ \sum\limits_{k = 1, k \ne i}^n {\left[ {{y_1}, \cdots , {y_k}D({y_i}), \cdots , {{\hat y}_i}, \cdots , {y_n}, D({y_{n + 1}})} \right]} )\\ & = & {( - 1)^{i - 1}}\sum\limits_{j = 1, j \ne i}^{n + 1} {\left[ {{y_1}, \cdots , D({y_i})D({y_j}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} \\ &&+ {( - 1)^{i - 1}}\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = 1, k \ne j, i}^{n + 1} {\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_k}D({y_i}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} }. \end{eqnarray*}$

Thus, we have

$\begin{eqnarray*} \sum\limits_{i = 1}^{n + 1} { {{A_i}} } & = &\sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}\left[ {{y_1}, \cdots , D({y_j})D({y_i}), \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]} } \\ &&+ \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = 1, k \ne i, j}^{n + 1} {{( - 1)^{i-1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_k}D({y_i}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } } \\ & = & {T_1} + {T_2}, \end{eqnarray*}$

where

$\begin{eqnarray*} {T_1} &: = &\sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}\left[ {{y_1}, \cdots , D({y_j})D({y_i}), \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]} }, \\ {T_2}& : = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = 1, k \ne i, j}^{n + 1} {{( - 1)^{i-1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_k}D({y_i}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } }. \\ \end{eqnarray*}$

Note that

$\begin{eqnarray*} {T_1} = \sum\limits_{j, i = 1}^{n + 1} {{B_{ji}}} , \end{eqnarray*}$

where ${B_{ji}} = {(- 1)^{j - 1}}\left[{{y_1}, \cdots, D({y_j})D({y_i}), \cdots, {{\hat y}_j}, \cdots, {y_{n + 1}}} \right]$ for any $1\leq j\neq i\leq n+1$ , and ${B_{ii}} = 0$ for any $1\leq i\leq n+1$ .

For any $1\leq i, j\leq n+1,$ without loss of generality, assume that $i < j$ ; then, we have

$\begin{eqnarray*} &&{B_{ji}} + {B_{ij}}\\ & = & {( - 1)^{j - 1}}\left[ {{y_1}, \cdots , D({y_j})D({y_i}), \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]\\ &&+ {( - 1)^{i - 1}}\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , D({y_i})D({y_j}), \cdots , {y_{n + 1}}} \right]\\ & = & {( - 1)^{j - 1}}\left[ {{y_1}, \cdots , D({y_j})D({y_i}), \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]\\ &&+ {( - 1)^{i - 1 + j - i + 1}}\left[ {{y_1}, \cdots , D({y_j})D({y_i}), \cdots , {{\hat y}_j}, \cdots , {y_{n + 1}}} \right]\\ & = & 0, \end{eqnarray*}$

which implies that ${T_1} = \sum\limits_{j, i = 1}^{n + 1} {{B_{ji}}} = 0.$

Thus, we get that $\sum\limits_{i = 1}^{n + 1} { {{A_i}} } = {T_2}$ .

We rewrite

$\begin{eqnarray*} &&\sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n+1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}n{y_i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } }\\ & = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n+1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {\sum\limits_{t = 1, t \ne j, k, i}^{n + 1} {{{( - 1)}^i}}} } }\\ &&\cdot\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {y_t}{y_i}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right] \\ &&+ \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n+1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , {y_i}D({y_j}), \cdots , D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } } \\ &&+ \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n+1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } } \\ & = & {M_1} + {M_2} + {M_3}, \end{eqnarray*}$

where

$\begin{eqnarray*} {M_1}& : = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {\sum\limits_{t = 1, t \ne j, k, i}^{n + 1} {{{( - 1)}^i}}} } }\\ &&\cdot\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {y_t}{y_i}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], \\ {M_2}& : = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , {y_i}D({y_j}), \cdots , D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } }, \\ {M_3} &: = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } }. \end{eqnarray*}$

Note that

$\begin{eqnarray*} {M_1}& = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n+1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {\sum\limits_{t = 1, t \ne j, k, i}^{n + 1} {{{( - 1)}^i}}} } }\\ &&\cdot\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {y_t}{y_i}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right] \\ & = &\sum\limits_{i, j, k, t = 1}^{n + 1} {{B_{ijkt}}}, \end{eqnarray*}$

where

$\begin{gather*} {B_{ijkt}} = \begin{cases} 0, { \quad } \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{if any two indices are equal or } k < j; \\ {( - 1)^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {y_t}{y_i}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], { \quad } \text{otherwise}. \end{cases} \end{gather*}$

For any $1\leq j, k\leq n+1,$ without loss of generality, assume that $t < i;$ then, we have

$\begin{eqnarray*} &&{B_{ijkt}} + {B_{tjki}}\\ & = & {( - 1)^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {y_t}{y_i}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\\ &&+ {( - 1)^t}\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {{\hat y}_t}, \cdots , {y_t}{y_i}, \cdots , {y_{n + 1}}} \right]\\ & = & {( - 1)^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {y_t}{y_i}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\\ &&+ {( - 1)^{t + i - t - 1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , D({y_k}), \cdots , {y_t}{y_i}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\\ & = & 0, \end{eqnarray*}$

which implies that ${M_1} = 0.$

Therefore, we only need to prove the following equation:

$\begin{eqnarray*} &&{M_2} + {M_3} \\ & = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = 1, k \ne i, j}^{n + 1} {{{( - 1)}^{i - 1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_k}D({y_i}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } }. \end{eqnarray*}$

First, we have

$\begin{eqnarray*} && \sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , {y_i}D({y_j}), \cdots , D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } \\ &&+ \sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } \\ & = & \sum\limits_{k = 1, k \ne i}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{k - 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , {y_i}D({y_j}), \cdots , D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } \\ &&+ \sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } \\ & = & \sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = 1, k \ne i}^{j - 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , {y_i}D({y_k}), \cdots , D({y_j}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } \\ &&+ \sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = j + 1, k \ne i}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} }\\ & = &\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = 1, k \ne i, j}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } . \end{eqnarray*}$

Thus,

$\begin{eqnarray*} &&M_2+M_3 \\ & = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = 1, k \ne i, j}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } } \\ & = & \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = 1, k \ne i, j}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } }\\ & = & \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = 1, k \ne j}^{i - 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } } \\ &&+ \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = i + 1, k \ne j}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {{\hat y}_i}, \cdots , {y_i}D({y_k}), \cdots , {y_{n + 1}}} \right]} } } . \end{eqnarray*}$

Note that, for any $1\leq j\leq n+1,$ we have

$\begin{eqnarray*} &&\sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = 1, k \ne j}^{i - 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } \\ & = & \sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = 1, k \ne j}^{i - 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_{k - 1}}, {y_i}D({y_k}), {y_{k + 1}}, \cdots , {{\hat y}_i} \cdots , {y_{n + 1}}} \right]} } \\ & = & \sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = 1, k \ne j}^{i - 1} {{{( - 1)}^{k - 1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {{\hat y}_k}, \cdots , {y_{i - 1}}, {y_i}D({y_k}), {y_{i + 1}}, \cdots , {y_{n + 1}}} \right]} } \\ & = & \sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = 1, k \ne j}^{i - 1} {{{( - 1)}^{k - 1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {{\hat y}_k}, \cdots , {y_i}D({y_k}), \cdots , {y_{n + 1}}} \right]} }. \end{eqnarray*}$

Similarly, we have

$\begin{eqnarray*} &&\sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = i + 1, k \ne j}^{n + 1} {{{( - 1)}^i}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {{\hat y}_i}, \cdots , {y_i}D({y_k}), \cdots , {y_{n + 1}}} \right]} } \\ & = & \sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = i + 1, k \ne j}^{n + 1} {{{( - 1)}^{k-1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} }. \end{eqnarray*}$

Thus,

$\begin{eqnarray*} &&{M_2} + {M_3}\\ & = & \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = 1, k \ne j}^{i - 1} {{{( - 1)}^{k - 1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {{\hat y}_k}, \cdots , {y_i}D({y_k}), \cdots , {y_{n + 1}}} \right]} } } \\ &&+ \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = i + 1, k \ne j}^{n + 1} {{{( - 1)}^{k-1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} } } \\ & = & \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {\sum\limits_{k = 1, k \ne i, j}^{n + 1} {{{( - 1)}^{k - 1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} } }\\ & = & \sum\limits_{k = 1}^{n + 1} {\sum\limits_{j = 1, j \ne k}^{n + 1} {\sum\limits_{i = 1, i \ne j, k}^{n + 1} {{{( - 1)}^{k - 1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_i}D({y_k}), \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} } }\\ & = &\sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {\sum\limits_{k = 1, k \ne j, i}^{n + 1} {{{( - 1)}^{i - 1}}\left[ {{y_1}, \cdots , D({y_j}), \cdots , {y_k}D({y_i}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } } . \end{eqnarray*}$

The proof is completed.

Theorem 3.1. Let $(L, \cdot, \left[{-, \cdots, - } \right])$ be a strong transposed Poisson $n$ -Lie algebra and $D$ a derivation of $(L, \cdot, \left[{-, \cdots, - } \right])$ . Define an $(n + 1)$ -ary operation:

$\begin{equation} {\mu _{n + 1}}({x_1}, \cdots , {x_{n + 1}}): = \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}}D({x_i})[ {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n + 1}}] \end{equation}$

(3.3)

for any ${x_i}\in L, 1\leq i\leq n+1.$ Then, $(L, \mu_{n+1})$ is an $(n+1)$ -Lie algebra.

Proof. For convenience, we denote

${\mu _{n + 1}}\left( {{x_1}, \cdots , {x_{n + 1}}} \right): = \left[ {{x_1}, \cdots , {x_{n + 1}}} \right].$

On one hand, we have

$\begin{eqnarray*} && {\left[ {\left[ {{y_1}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {x_n}} \right]}\\ &\mathop = \limits^{(3.3)}& \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}\left[ {D({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {x_n}} \right]} \\ &\mathop = \limits^{(3.3)}& \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}D(D({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right])\left[ {{x_1}, \cdots , {x_n}} \right]} \\ &&{ + \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1}^n {{{( - 1)}^{i + j - 1}}D({x_j})} } \left[ {D({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]}\\ & = &\sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}{D^2}({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]} \\ &&+ \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}D({y_i})D(\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right])\left[ {{x_1}, \cdots , {x_n}} \right]}\\ &&+ \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1}^n {{{( - 1)}^{i + j - 1}}D({x_j})} } \left[ {D({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]\\ &\mathop = \limits^{(3.1)}& \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}{D^2}({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]}\\ &&+ \sum\limits_{i = 1}^{n + 1} {\sum\limits_{k = 1, k \ne i}^{n + 1} {{{( - 1)}^{i - 1}}D({y_i})\left[ {{y_1}, \cdots , D({y_k}), \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]} } \\ &&+ \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1}^n {{{( - 1)}^{i + j - 1}}D({x_j})} } \left[ {D({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]\\ & = & \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}{D^2}({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]} \\ &&+ \sum\limits_{k = 1}^{n + 1} {\sum\limits_{i = 1}^{k - 1} {{{( - 1)}^{k + i - 1}}D({y_i})\left[ {D({y_k}), {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]} }\\ &&+ \sum\limits_{k = 1}^{n + 1} {\sum\limits_{i = k + 1}^{n + 1} {{{( - 1)}^{i + k}}D({y_i})\left[ {D({y_k}), {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]} } \\ &&+ \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1}^n {{{( - 1)}^{i + j - 1}}D({x_j})} } \left[ {D({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]. \end{eqnarray*}$

On the other hand, for any $1\leq k\leq n$ , we have

$\begin{eqnarray*} &&{( - 1)^{k - 1}}\left[ {\left[ {{y_k}, {x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\\ &\mathop = \limits^{(3.3)}& {( - 1)^{k - 1}}\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\\ &&+ \sum\limits_{j = 1}^n {{{( - 1)}^{j + k - 1}}\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} \\ &\mathop = \limits^{(3.3)}& {( - 1)^{k - 1}}D(D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right])\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\\ &&+ \sum\limits_{i = 1}^{k - 1} {{{( - 1)}^{i + k - 1}}D({y_i})\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} \\ && + \sum\limits_{i = k + 1}^{n + 1} {{{( - 1)}^{i + k}}D({y_i})\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} \\ &&+ \sum\limits_{j = 1}^n {{{( - 1)}^{j + k - 1}}D(D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right])\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} \\ && +\sum\limits_{j = 1}^n {\sum\limits_{i = k + 1}^{n + 1} {{{(( - 1)}^{ i + j}}D({y_i}) } }\\ &&\cdot\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right] )\\ &&+ \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^{k - 1} {{{(( - 1)}^{i + j - 1}}D({y_i})} }\\ &&\cdot\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right])\\ & = & {( - 1)^{k - 1}}{D^2}({y_k})\left[ {{x_1}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\\ &&+ \sum\limits_{i = 1}^{k - 1} {{{( - 1)}^{i + k - 1}}D({y_i})\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} \\ &&+ \sum\limits_{i = k + 1}^{n + 1} {{{( - 1)}^{i + k}}D({y_i})\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} \\ &&+ \sum\limits_{j = 1}^n {{{( - 1)}^{j + k - 1}}{D^2}({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} \\ &&+ \sum\limits_{j = 1}^n {\sum\limits_{i = k + 1}^{n + 1} {{{(( - 1)}^{ i + j}}D({y_i}) } }\\ &&\cdot\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right] )\\ &&+\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^{k - 1} {{{(( - 1)}^{ i + j - 1}}D({y_i}) } }\\ &&\cdot\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right])\\ &&+ {( - 1)^{k - 1}}D({y_k})D(\left[ {{x_1}, \cdots , {x_n}} \right])\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\\ &&+ \sum\limits_{j = 1}^n {{{( - 1)}^{j + k - 1}}D({x_j})D(\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right])\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} \\ &\mathop = \limits^{(3.2)}& {( - 1)^{k - 1}}{D^2}({y_k})\left[ {{x_1}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\\ &&+ \sum\limits_{i = 1}^{k - 1} {{{( - 1)}^{i + k - 1}}D({y_i})\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} \\ &&+ \sum\limits_{i = k + 1}^{n + 1} {{{( - 1)}^{i + k}}D({y_i})\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} \\ &&+ \sum\limits_{j = 1}^n {{{( - 1)}^{j + k - 1}}{D^2}({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} \\ && +\sum\limits_{j = 1}^n {\sum\limits_{i = k + 1}^{n + 1} {{{(( - 1)}^{ i + j}}D({y_i}) } }\\ &&\cdot\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right] )\\ &&+ \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^{k - 1} {{{(( - 1)}^{ i + j - 1}}D({y_i}) } }\\ &&\cdot\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right])\\ &&+ \sum\limits_{j = 1}^n {\sum\limits_{t = j + 1}^n {{{( - 1)}^k}{y_k}\left[ {{x_1}, \cdots , D({x_j}), \cdots , D({x_t}), \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} } \\ &&+ \sum\limits_{i = 1}^n {\sum\limits_{j = 1, j \ne i}^n {{{( - 1)}^{k + i}}{x_i}\left[ {D({y_k}), {x_1}, \cdots , D({x_j}), \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} } \\ &&+ \sum\limits_{i = 1}^n {\sum\limits_{j = 1, j \ne i}^n {\sum\limits_{t = j + 1, t \ne i}^n {{{( - 1)}^{k + i}}{x_i}\left[ {{y_k}, {x_1}, \cdots , D({x_j}), D({x_t}), \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} } } \\ &&\cdot \left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]. \end{eqnarray*}$

We denote

$\sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}\left[ {\left[ {{y_i}, {x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} = \sum\limits_{i = 1}^7 {{A_i}},$

where

$\begin{eqnarray*} {A_1} &: = & \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}{D^2}({y_i})\left[ {{x_1}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]}, \\ {A_2} &: = & \sum\limits_{k = 1}^{n + 1} {\sum\limits_{j = 1}^n {{{( - 1)}^{k + j - 1}}} {D^2}({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]}, \\ {A_3} &: = & \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1}^n {\sum\limits_{k = j + 1}^n {{{( - 1)}^i}{y_i}\left[ {{x_1}, \cdots , D({x_j}), \cdots , D({x_k}), \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} } }, \\ {A_4} &: = &\sum\limits_{k = 1}^{n + 1} {\sum\limits_{i = 1}^n {\sum\limits_{j = 1, j \ne i}^n {\sum\limits_{t = j + 1, t \ne i}^n {({{( - 1)}^{k + i}}{x_i}\left[ {{y_k}, {x_1}, \cdots , D({x_j}), D({x_t}), \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} } } } \\ && \cdot \left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]), \\ {A_5} &: = & \sum\limits_{k = 1}^{n + 1} {\sum\limits_{i = 1}^{k - 1} {{{( - 1)}^{k + i - 1}}D({y_i})\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} } \\ &&+ \sum\limits_{k = 1}^{n + 1} {\sum\limits_{i = k + 1}^{n + 1} {{{( - 1)}^{i + k}}D({y_i})\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} }, \\ {A_6}&: = & \sum\limits_{k = 1}^{n + 1} {\sum\limits_{i = 1}^n {\sum\limits_{j = 1, j \ne i}^n {{{(( - 1)}^{k + i}}{x_i}\left[ {D({y_k}), {x_1}, \cdots , D({x_j}), \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} } } \\ && \cdot\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right] ), \\ {A_7}&: = & \sum\limits_{k = 1}^{n + 1} {\sum\limits_{j = 1}^n {\sum\limits_{i = k + 1}^{n + 1} {{{(( - 1)}^{k + i + j}}D({y_i})} } }\\ &&\cdot\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right] )\\ &&+ \sum\limits_{k = 1}^{n + 1} {\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^{k - 1} {{{(( - 1)}^{k + i + j - 1}}D({y_i})} } }\\ &&\cdot\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]) . \end{eqnarray*}$

By Eq (2.5), for fixed $j$ , we have

$\sum\limits_{k = 1}^{n + 1} {{{( - 1)}^{k + j - 1}}{D^2}({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} = 0.$

So, we obtain that ${{A_2}} = 0.$

By Eq (2.3), for fixed $j$ and $k$ , we have

$\sum\limits_{i = 1}^{n + 1} {{{( - 1)}^i}{y_i}\left[ {{x_1}, \cdots , D({x_j}), \cdots , D({x_k}), \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} = 0.$

So, we obtain that ${{A_3}} = 0.$

By Eq (2.5), for fixed $j$ and $t$ , we have

$\sum\limits_{k = 1}^{n + 1} {{{( - 1)}^{k + i}}{x_i}\left[ {{y_k}, {x_1}, \cdots , D({x_j}), D({x_t}), \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} = 0.$

So, we obtain that ${{A_4}} = 0.$

By Eq (2.9), for fixed $i$ and $k$ , we have

$\begin{eqnarray*} &&{( - 1)^{k + i - 1}}D({y_i})\left[ {D({y_k})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\\ &&+ {( - 1)^{i + k}}D({y_k})\left[ {D({y_i})\left[ {{x_1}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\\ & = & {( - 1)^{k + i - 1}}D({y_i})\left[ {D({y_k}), {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]\\ &&+ {( - 1)^{i + k}}D({y_k})\left[ {D({y_i}), {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]. \end{eqnarray*}$

Thus, we obtain

$\begin{eqnarray*} {{A_5}} & = &\sum\limits_{k = 1}^{n + 1} {\sum\limits_{i = 1}^{k - 1} {{{( - 1)}^{k + i - 1}}D({y_i})\left[ {D({y_k}), {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]} } \\ && + \sum\limits_{k = 1}^{n + 1} {\sum\limits_{i = k + 1}^{n + 1} {{{( - 1)}^{i + k}}D({y_i})\left[ {D({y_k}), {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\left[ {{x_1}, \cdots , {x_n}} \right]} }. \end{eqnarray*}$

By Eq (2.3), for fixed $j$ and $k$ , we have

$\begin{eqnarray*} &&\sum\limits_{i = 1}^n {{{( - 1)}^{k + i}}{x_i}\left[ {D({y_k}), {x_1}, \cdots , D({x_j}), \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} \\ & = & {( - 1)^{k - 1}}D({y_k})\left[ {{x_1}, \cdots , D({x_j}), \cdots , {x_n}} \right] + {( - 1)^{k + j - 1}}D({x_j})\left[ {D({y_k}), {x_1}, \cdots , {x_n}} \right]\\ & = & {( - 1)^{k + j}}D({y_k})\left[ {D({x_j}), {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right] + {( - 1)^{k + j - 1}}D({x_j})\left[ {D({y_k}), {x_1}, \cdots , {x_n}} \right]. \end{eqnarray*}$

Thus, we get

$\begin{eqnarray*} {{A_6}} & = &\sum\limits_{k = 1}^{n + 1} {\sum\limits_{j = 1}^n {{{( - 1)}^{k + j}}D({y_k})\left[ {D({x_j}), {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} } \\ &&+ \sum\limits_{k = 1}^{n + 1} {\sum\limits_{j = 1}^n {{{( - 1)}^{k + j - 1}}D({x_j})\left[ {D({y_k}), {x_1}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} } . \end{eqnarray*}$

By Eq (2.4), for fixed $j$ and $i$ , we have

$\begin{eqnarray*} && \sum\limits_{k = i + 1}^{n + 1} {{{( - 1)}^{k + i + j - 1}}D({y_i})\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_i}, \cdots , {{\hat y}_k}, \cdots , {y_{n + 1}}} \right]} \\ && + \sum\limits_{k = 1}^{i - 1} {{{( - 1)}^{k + i + j}}D({y_i})\left[ {D({x_j})\left[ {{y_k}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right], {y_1}, \cdots , {{\hat y}_k}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]} \\ & = & {( - 1)^{j + i - 1}}D({y_i})\left[ {D({x_j})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]. \end{eqnarray*}$

So, we obtain

${{{A_7}}} = \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{j + i - 1}}D({y_i})\left[ {D({x_j})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]} } .$

By Eq (2.9), we have

$\begin{eqnarray*} &&{( - 1)^{i + j}}D({y_i})\left[ {D({x_j}), {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\\ &&+ {( - 1)^{i + j - 1}}D({x_j})\left[ {D({y_i}), {x_1}, \cdots , {x_n}} \right]\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right]\\ &&+ {( - 1)^{j + i - 1}}D({y_i})\left[ {D({x_j})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]\\ & = & {( - 1)^{j + i - 1}}D({x_j})\left[ {D({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]. \end{eqnarray*}$

So, we get

${{{A_6}}} + {{{A_7}}} = \sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1}^n {{{( - 1)}^{j + i - 1}}D({x_j})\left[ {D({y_i})\left[ {{y_1}, \cdots , {{\hat y}_i}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right].} }$

Thus, we have

$\sum\limits_{i = 1}^7 {{A_i}} = {{A_1}} +{{A_5}} + {{A_6}} + {{A_7}} = \left[ {\left[ {{y_1}, \cdots , {y_{n + 1}}} \right], {x_1}, \cdots , {x_n}} \right].$

Therefore, $(L, \mu_{n+1})$ is an $(n+1)$ -Lie algebra.

Now, we can prove Conjecture 1.1 for strong transposed Poisson $n$ -Lie algebras.

Theorem 3.2. With the notations in Theorem 3.1, $(L, \cdot, \mu_{n+1})$ is a strong transposed Poisson $(n + 1)$ -Lie algebra.

Proof. For convenience, we denote ${\mu _{n + 1}}\left({{x_1}, \cdots, {x_{n + 1}}} \right): = \left[{{x_1}, \cdots, {x_{n + 1}}} \right].$ According to Theorem 3.1, we only need to prove Eqs (2.2) and (2.8).

Proof of Eq (2.2). By Eq (3.3), we have

$\begin{eqnarray*} &&\sum\limits_{i = 1}^{n + 1} {\left[ {{x_1}, \cdots , h{x_i}, \cdots , {x_{n + 1}}} \right]} \\ & = & D(h{x_1})\left[ {{x_2}, \cdots , {x_{n + 1}}} \right] + \sum\limits_{j = 2}^{n + 1} {{{( - 1)}^{j - 1}}D({x_j})\left[ {h{x_1}, {x_2}, \cdots , {{\hat x}_j}, \cdots , {x_{n + 1}}} \right]} \\ &&- D(h{x_2})\left[ {{x_1}, {x_3}, \cdots , {x_{n + 1}}} \right] + \sum\limits_{j = 1, j \ne 2}^{n + 1} {{{( - 1)}^{j - 1}}D({x_j})\left[ {{x_1}, h{x_2}, {x_3}, \cdots , {{\hat x}_j}, \cdots , {x_{n + 1}}} \right]} \\ &&+ \cdots + {( - 1)^n}D(h{x_n})\left[ {{x_1}, \cdots , {x_n}} \right] + \sum\limits_{j = 1}^n {{{( - 1)}^{j - 1}}D({x_j})\left[ {{x_1}, \cdots , {{\hat x}_j}, \cdots , {x_n}, h{x_{n + 1}}} \right]}\\ & = & \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}D(h{x_i})\left[ {{x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} \\ &&+\sum\limits_{i = 1}^{n + 1} {\sum\limits_{j = 1, j \ne i}^{n + 1} {{{( - 1)}^{j - 1}}D({x_j})\left[ {{x_1}, , \cdots , h{x_i}, \cdots , {{\hat x}_j}, \cdots , {x_{n + 1}}} \right]} } \\ & = & \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}D(h{x_i})} \left[ {{x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n + 1}}} \right]\\ &&+ \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}D({x_j})\left[ {{x_1}, \cdots , h{x_i}, \cdots , {{\hat x}_j}, \cdots , {x_{n + 1}}} \right]} } \\ & = & \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}hD({x_i})} \left[ {{x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n + 1}}} \right] \\ &&+ \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}{x_i}D(h)} \left[ {{x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n + 1}}} \right]\\ && + \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}D({x_j})\left[ {{x_1}, \cdots , h{x_i}, \cdots , {{\hat x}_j}, \cdots , {x_{n + 1}}} \right]} } \\ & \mathop = \limits^{(2.3)}& \sum\limits_{i = 1}^{n + 1} {{{( - 1)}^{i - 1}}hD({x_i})} \left[ {{x_1}, \cdots , {{\hat x}_i}, \cdots , {x_{n + 1}}} \right] \\ &&+\sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}D({x_j})\left[ {{x_1}, \cdots , h{x_i}, \cdots , {{\hat x}_j}, \cdots , {x_{n + 1}}} \right]} } \\ &\mathop = \limits^{(3.3)}& h\left[ {{x_1}, \cdots , {x_{n + 1}}} \right] + \sum\limits_{j = 1}^{n + 1} {\sum\limits_{i = 1, i \ne j}^{n + 1} {{{( - 1)}^{j - 1}}D({x_j})\left[ {{x_1}, \cdots , h{x_i}, \cdots , {{\hat x}_j}, \cdots , {x_{n + 1}}} \right]} } \\ &\mathop = \limits^{(2.2)}& h\left[ {{x_1}, \cdots , {x_{n + 1}}} \right] + nh\sum\limits_{j = 1}^{n + 1} {{{( - 1)}^{j - 1}}D({x_j})\left[ {{x_1}, \cdots , {{\hat x}_j}, \cdots , {x_{n + 1}}} \right]} \\ &\mathop = \limits^{(3.3)}& h\left[ {{x_1}, \cdots , {x_{n + 1}}} \right] + nh\left[ {{x_1}, \cdots , {x_{n + 1}}} \right]\\ & = & (n + 1)h\left[ {{x_1}, \cdots , {x_{n + 1}}} \right]. \end{eqnarray*}$

Proof of Eq (2.8). By Eq (3.3), we have

$\begin{eqnarray*} &&{y_1}\left[ {h{y_2}, {x_1}, \cdots , {x_n}} \right] - {y_2}\left[ {h{y_1}, {x_1}, \cdots , {x_n}} \right] + \sum\limits_{i = 1}^n {{{( - 1)}^{i - 1}}h{x_i}\left[ {{y_1}, {y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} \\ & = &{y_1}{y_2}D(h)\left[ {{x_1}, \cdots , {x_n}} \right] + {y_1}hD({y_2})\left[ {{x_1}, \cdots , {x_n}} \right] - {y_1}D({x_1})\left[ {h{y_2}, {x_2}, \cdots , {x_n}} \right]\\ &&+ {y_1}D({x_2})\left[ {h{y_2}, {x_1}, {x_3}, \cdots , {x_n}} \right] + \cdots + {( - 1)^n}{y_1}D({x_n})\left[ {h{y_2}, {x_1}, \cdots , {x_{n - 1}}} \right]\\ &&- {y_2}{y_1}D(h)\left[ {{x_1}, \cdots , {x_n}} \right] - {y_2}hD({y_1})\left[ {{x_1}, \cdots , {x_n}} \right] + {y_2}D({x_1})\left[ {h{y_1}, {x_2}, \cdots , {x_n}} \right]\\ &&- {y_2}D({x_2})\left[ {h{y_1}, {x_1}, {x_3}, \cdots , {x_n}} \right] + \cdots + {( - 1)^{n - 1}}{y_2}D({x_n})\left[ {h{y_1}, {x_1}, \cdots , {x_{n - 1}}} \right]\\ && + h{x_1}D({y_1})\left[ {{y_2}, {x_2}, \cdots , {x_n}} \right] - h{x_1}D({y_2})\left[ {{y_1}, {x_2}, \cdots , {x_n}} \right]\\ &&+ h{x_1}D({x_2})\left[ {{y_1}, {y_2}, {x_3}, \cdots , {x_n}} \right]+ \cdots + {( - 1)^{n + 1}}h{x_1}D({x_n})\left[ {{y_1}, {y_2}, {x_2}, \cdots , {x_{n - 1}}} \right]\\ &&- h{x_2}D({y_1})\left[ {{y_2}, {x_1}, {x_3}, \cdots , {x_n}} \right] + h{x_2}D({y_2})\left[ {{y_1}, {x_1}, {x_3}, \cdots , {x_n}} \right] \\ &&- h{x_2}D({x_1})\left[ {{y_1}, {y_2}, {x_3}, \cdots , {x_n}} \right]+ \cdots + {( - 1)^{n + 2}}h{x_2}D({x_n})\left[ {{y_1}, {y_2}, {x_1}, {x_3}, \cdots , {x_{n - 1}}} \right]\\ &&+ \cdots + {( - 1)^{n - 1}}h{x_n}D({y_1})\left[ {{y_2}, {x_1}, \cdots , {x_{n - 1}}} \right] + {( - 1)^n}h{x_n}D({y_2})\left[ {{y_1}, {x_1}, \cdots , {x_{n - 1}}} \right]\\ &&+ {( - 1)^{n + 1}}h{x_n}D({x_1})\left[ {{y_1}, {y_2}, {x_2}, \cdots , {x_{n - 1}}} \right] \\ &&+ \cdots + {( - 1)^{2n - 1}}h{x_n}D({x_{n - 1}})\left[ {{y_1}, {y_2}, {x_1}, \cdots , {x_{n - 2}}} \right]\\ & = & - {y_2}hD({y_1})\left[ {{x_1}, \cdots , {x_n}} \right] + h{x_1}D({y_1})\left[ {{y_2}, {x_2}, \cdots , {x_n}} \right]\\ &&+ \sum\limits_{i = 2}^n {{{( - 1)}^{i - 1}}h{x_i}D({y_1})\left[ {{y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} \\ &&+ {y_1}hD({y_2})\left[ {{x_1}, \cdots , {x_n}} \right] - h{x_1}D({y_2})\left[ {{y_1}, {x_2}, \cdots , {x_n}} \right] \\ &&+ \sum\limits_{i = 2}^n {{{( - 1)}^i}h{x_i}D({y_2})\left[ {{y_1}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]}\\ &&- {y_1}D({x_1})\left[ {h{y_2}, {x_2}, \cdots , {x_n}} \right] + {y_2}D({x_1})\left[ {h{y_1}, {x_2}, \cdots , {x_n}} \right]\\ &&+ \sum\limits_{i = 2}^n {{{( - 1)}^{i - 1}}h{x_i}D({x_1})\left[ {{y_1}, {y_2}, {x_2}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} \\ &&+ {y_1}D({x_2})\left[ {h{y_2}, {x_1}, {x_3}, \cdots , {x_n}} \right] - {y_2}D({x_2})\left[ {h{y_1}, {x_1}, {x_3}, \cdots , {x_n}} \right]\\ &&+ h{x_1}D({x_2})\left[ {{y_1}, {y_2}, {x_3}, \cdots , {x_n}} \right] + \sum\limits_{i = 3}^n {{{( - 1)}^i}h{x_i}D({x_2})\left[ {{y_1}, {y_2}, {x_1}, {x_3}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} \\ &&\cdots+ {( - 1)^n}{y_1}D({x_n})\left[ {h{y_2}, {x_1}, \cdots , {x_{n - 1}}} \right] + {( - 1)^{n - 1}}{y_2}D({x_n})\left[ {h{y_1}, {x_1}, \cdots , {x_{n - 1}}} \right]\\ &&+ \sum\limits_{j = 1}^{n - 1} {{{( - 1)}^{n + j - 1}}h{x_j}D({x_n})\left[ {{y_1}, {y_2}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {x_{n - 1}}} \right]} \\ & = & {A_1} + {A_2} + \sum\limits_{i = 1}^n {{B_i}}, \end{eqnarray*}$

where

$\begin{eqnarray*} {A_1} &: = & - {y_2}hD({y_1})\left[ {{x_1}, \cdots , {x_n}} \right] + \sum\limits_{i = 1}^n {{{( - 1)}^{i -1}}h{x_i}D({y_1})\left[ {{y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]}, \\ {A_2} &: = & {y_1}hD({y_2})\left[ {{x_1}, \cdots , {x_n}} \right] + \sum\limits_{i = 1}^n {{{( -1)}^i}h{x_i}D({y_2})\left[ {{y_1}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]}, \\ \end{eqnarray*}$

and, for any $1\leq i\leq n,$

$\begin{eqnarray*} {B_i} &: = &{( - 1)^i}{y_1}D({x_i})\left[ {h{y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right] + {( - 1)^{i - 1}}{y_2}D({x_i})\left[ {h{y_1}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]\\ &&+ \sum\limits_{j = 1}^{i - 1} {{{( - 1)}^{i + j - 1}}h{x_j}D({x_i})\left[ {{y_1}, {y_2}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} \\ &&+ \sum\limits_{j = i + 1}^n {{{( - 1)}^{i + j}}h{x_j}D({x_i})\left[ {{y_1}, {y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]} . \end{eqnarray*}$

By Eq (2.3), we have

$\begin{eqnarray*} {A_1}& = & hD({y_1})\left( { - {y_2}\left[ {{x_1}, \cdots , {x_n}} \right] + \sum\limits_{i = 1}^n {{{( - 1)}^{i - 1}}{x_i}\left[ {{y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} } \right) = 0. \end{eqnarray*}$

Similarly, we have that ${A_2} = 0.$

By Eq (2.8), for any $1\leq i\leq n$ , we have

$\begin{eqnarray*} {B_i} & = &{( - 1)^i}D({x_i})({y_1}\left[ {h{y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right] - {y_2}\left[ {h{y_1}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]\\ &&+ \sum\limits_{j = 1}^{i - 1} {{{( - 1)}^{j - 1}}h{x_j}\left[ {{y_1}, {y_2}, {x_1}, \cdots , {{\hat x}_j}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} \\ &&+ \sum\limits_{j = i + 1}^n {{{( - 1)}^j}h{x_j}\left[ {{y_1}, {y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {{\hat x}_j}, \cdots , {x_n}} \right]} )\\ & = & 0. \end{eqnarray*}$

Thus, we get

${y_1}\left[ {h{y_2}, {x_1}, \cdots , {x_n}} \right] - {y_2}\left[ {h{y_1}, {x_1}, \cdots , {x_n}} \right] + \sum\limits_{i = 1}^n {{{( - 1)}^{i - 1}}h{x_i}\left[ {{y_1}, {y_2}, {x_1}, \cdots , {{\hat x}_i}, \cdots , {x_n}} \right]} = 0.$

The proof is completed.

Example 3.1. The commutative associative algebra $L = k[x_1, x_2, x_3]$ , together with the bracket

$\begin{eqnarray*} [x, y]: = x \cdot {D_1}(y) - y \cdot {D_1}(x), \forall x, y \in L. \end{eqnarray*}$

gives a transposed Poisson algebra $(L, \cdot, [-, -])$ , where ${D_1} = \partial_{x_1}$ ([, Proposition 2.2]). Note that the transposed Poisson algebra $(L, \cdot, [-, -])$ is strong according to Remark 2.5. Now, let ${D_2} = \partial_{x_2}$ ; one can check that $D_2$ is a derivation of $(L, \cdot, [-, -])$ . Then, there exists a strong transposed Poisson 3-Lie algebra defined by

$\begin{eqnarray*} [x, y, z]: = {D_2}(x)(y{D_1}(z) - z{D_1}(y)) + {D_2}(y)(z{D_1}(x) - x{D_1}(z))+{D_2}(z)(x{D_1}(y) - y{D_1}(x)), \ \forall x, y, z \in L. \end{eqnarray*}$

We note that $[{x_1}, {x_2}, {x_3}] = {x_3}$ , which is non-zero. The strong condition can be checked as follows:

For any $h, y_1, y_2, z_1, z_2\in L,$ by a direct calculation, we have

$\begin{eqnarray*} {y_1}[h{y_2}, {z_1}, {z_2}] & = & {y_1}{z_1}h{D_1}({z_2}){D_2}({y_2}) - {y_1}{z_2}h{D_1}({z_1}){D_2}({y_2})\\ &&+ {y_1}{y_2}{z_1}{D_1}({z_2}){D_2}({h}) - {y_1}{y_2}{z_2}{D_1}({z_1}){D_2}({h})\\ &&- {y_1}{y_2}h{D_1}({z_2}){D_2}({z_1}) + {y_1}{z_2}h{D_1}({y_2}){D_2}({z_1}) + {y_1}{y_2}{z_2}{D_1}(h){D_2}({z_1})\\ &&+ {y_1}{y_2}h{D_1}({z_1}){D_2}({z_2}) - {y_1}{z_1}h{D_1}({y_2}){D_2}({z_2}) - {y_1}{y_2}{z_1}{D_1}(h){D_2}({z_2}), \end{eqnarray*}$

$\begin{eqnarray*} - {y_2}[h{y_1}, {z_1}, {z_2}] & = & - {y_2}{z_1}h{D_1}({z_2}){D_2}({y_1}) + {y_2}{z_2}h{D_1}({z_1}){D_2}({y_1})\\ &&- {y_1}{y_2}{z_1}{D_1}({z_2}){D_2}(h) + {y_1}{y_2}{z_2}{D_1}({z_1}){D_2}(h)\\ &&+ {y_1}{y_2}h{D_1}({z_2}){D_2}({z_1}) - {y_2}{z_2}h{D_1}({y_1}){D_2}({z_1}) - {y_1}{y_2}{z_2}{D_1}(h){D_2}({z_1})\\ &&- {y_1}{y_2}h{D_1}({z_1}){D_2}({z_2}) + {y_2}{z_1}h{D_1}({y_1}){D_2}({z_2}) + {y_1}{y_2}{z_1}{D_1}(h){D_2}({z_2}), \end{eqnarray*}$

$\begin{eqnarray*} h{z_1}[{y_1}, {y_2}, {z_2}] & = & h{y_2}{z_1}{D_1}({z_2}){D_2}({y_1}) - h{z_1}{z_2}{D_1}({y_2}){D_2}({y_1})- h{y_1}{z_1}{D_1}({z_2}){D_2}({y_2})\\ &&+ h{z_1}{z_2}{D_1}({y_1}){D_2}({y_2})+ h{y_1}{z_1}{D_1}({y_2}){D_2}({z_2}) - h{y_2}{z_1}{D_1}({y_1}){D_2}({z_2}), \end{eqnarray*}$

$\begin{eqnarray*} - h{z_2}[{y_1}, {y_2}, {z_1}]& = & - h{y_2}{z_2}{D_1}({z_1}){D_2}({y_1}) + h{z_1}{z_2}{D_1}({y_2}){D_2}({y_1})+ h{y_1}{z_2}{D_1}({z_1}){D_2}({y_2}) \\ &&- h{z_1}{z_2}{D_1}({y_1}){D_2}({y_2})- h{y_1}{z_2}{D_1}({y_2}){D_2}({z_1}) + h{y_2}{z_2}{D_1}({y_1}){D_2}({z_1}). \end{eqnarray*}$

Thus, we get

${y_1}[h{y_2}, {z_1}, {z_2}] - {y_2}[h{y_1}, {z_1}, {z_2}] + h{z_1}[{y_1}, {y_2}, {z_2}] - h{z_2}[{y_1}, {y_2}, {z_1}] = 0.$

4. Conclusions

We have studied transposed Poisson $n$ -Lie algebras. We first established an important class of identities for transposed Poisson $n$ -Lie algebras, which were subsequently used throughout the paper. We believe that the identities developed here will be useful in investigations of the structure of transposed Poisson $n$ -Lie algebras in the future. Then, we introduced the notion of a strong transposed Poisson $n$ -Lie algebra and derived an $(n+1)$ -Lie algebra from a strong transposed Poisson $n$ -Lie algebra with a derivation. Finally, we proved the conjecture of Bai et al. ^[2] for strong transposed Poisson $n$ -Lie algebras.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Acknowledgments

Ming Ding was supported by the Guangdong Basic and Applied Basic Research Foundation (2023A1515011739) and the Basic Research Joint Funding Project of University and Guangzhou City under grant number 202201020103.

Conflict of interest

The authors declare that there is no conflict of interest.

References

[1]	T. Abdeljawad, H. Aydi, E. Karapinar, Coupled fixed points for Meir-Keeler contractions in ordered partial metric spaces, Math. Probl. Eng., 2012 (2012).
[2]	M. Abbas, B. Ali, C. Vetro, A Suzuki type fixed point theorem for a generalized multivalued mapping on partial Hausdorff metric spaces, Topol. Appl., 160 (2013), 553-563. doi: 10.1016/j.topol.2013.01.006
[3]	P. Agarwal, M. A. Alghamdi, N. Shahzad, Fixed point theory for cyclic generalized contractions in partial metric spaces, Fixed Point Theory A., 2012 (2012), 40.
[4]	I. Altun, A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory A., 2011 (2011), 508730.
[5]	E. Ameer, H. Aydi, M. Arshad, et al. Hybrid multivalued type contraction mappings in α_k-complete partial b-metric spaces and applications, Symmetry, 11 (2019), 86.
[6]	H. Aydi, M. A. Barakat, Z. D. Mitrović, et al. A Suzuki type multi-valued contraction on weak partial metric spaces and application, J. Inequal. Appl., 2018 (2018), 270.
[7]	H. Aydi, E. Karapinar, W. Shatanawi, Coupled fixed point results for (ψ,φ)-weakly contractive condition in ordered partial metric spaces, Comput. Math. Appl., 62 (2011), 4449-4460. doi: 10.1016/j.camwa.2011.10.021
[8]	H. Aydi, E. Karapinar, New Meir-Keeler type tripled fixed point theorems on ordered partial metric spaces, Math. Probl. Eng., 2012 (2012), 1-17.
[9]	H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler's fixed point theorem on partial metric spaces, Topol. Appl., 159 (2012), 3234-3242. doi: 10.1016/j.topol.2012.06.012
[10]	B. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1992), 133-181.
[11]	L. j. Ćirić, B. Samet, H. Aydi, et al. Common fixed points of generalized contractions on partial metric spaces and an application, Appl. Math. Comput., 218 (2011), 2398-2406.
[12]	S. H. Cho, Fixed point theorems for $\mathcal{L}$ -contractions in generalized metric spaces, Abstr. Appl. Anal., 2018 (2018), 1-6.
[13]	S. Gulyaz, E. Karapinar, A coupled fixed point result in partially ordered partial metric spaces through implicit function, Hacet. J. Math. Stat., 42 (2013), 347-357.
[14]	M. Jleli, B. Samet, new generalization of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 38.
[15]	E. Karapınar, R. P. Agarwal, H. Aydi, Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces, Mathematics, 6 (2018), 256.
[16]	E. Karapinar, W. Shatanawi, K. Tas, Fixed point theorem on partial metric spaces involving rational expressions, Miskolc Math. Notes, 14 (2013), 135-142. doi: 10.18514/MMN.2013.471
[17]	E. Karapinar, S. Romaguera, Nonunique fixed point theorems in partial metric spaces, Filomat, 27 (2013), 1305-1314. doi: 10.2298/FIL1307305K
[18]	E. Karapinar, I. Erhan, A. Ozturk, Fixed point theorems on quasi-partial metric spaces, Math. Comput. Model., 57 (2013), 2442-2448. doi: 10.1016/j.mcm.2012.06.036
[19]	W. Kirk, N. Shahzad, Fixed point theory in distance spaces, Springer International Publishing, Switzerland, 2014.
[20]	F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (2015), 1189-1194. doi: 10.2298/FIL1506189K
[21]	S. G. Matthews, Partial metric topology, Ann. NY Acad. Sci., 728 (1994), 183-197. doi: 10.1111/j.1749-6632.1994.tb44144.x
[22]	S. J. O'Neill, Partial metrics, valuations and domain theory, Ann. NY Acad. Sci., 806 (1996), 304-315. doi: 10.1111/j.1749-6632.1996.tb49177.x
[23]	L. Pasicki, Dislocated quasi-metric and generalized contractions, Fixed Point Theory, 19 (2018), 359-368. doi: 10.24193/fpt-ro.2018.1.27
[24]	S. Radenović, Classical fixed point results in 0-complete partial metric spaces via cyclic-type extension, The Allahabad Mathematical Society, 31 (2016), 39-55.
[25]	S. Radenović, Coincidence point results for generalized weakly (ψ,φ)-contractive mappings in ordered partial metric spaces, J. Indian Math. Soc., 3 (2014), 319-333.
[26]	S. Romaguera, Fixed point theorems for generalized contraction on partial metric spaces, Topol. Appl., 159 (2012), 194-199. doi: 10.1016/j.topol.2011.08.026
[27]	W. Shatanawi, M. Postolache, Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces, Fixed Point Theory A., 2013 (2013), 54.
[28]	W. Shatanawi, S. Manro, Fixed point results for cyclic (ψ,φ,A,B)-contraction in partial metric spaces, Fixed Point Theory A., 2012 (2012), 165.
[29]	W. Shatanawi, H. K. Nashine, N. Tahat, Generalization of some coupled fixed point results on partial metric spaces, International Journal of Mathematics and Mathematical Sciences, 2012 (2012), 1-10.
[30]	W. Shatanawi, B. Samet, M. Abbas, Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces, Math. Comput. Model., 55 (2012), 680-687. doi: 10.1016/j.mcm.2011.08.042
[31]	S. Shukla, S. Radenović, Some common fixed point theorems for $F$ -contraction type mappings in 0-complete partial metric spaces, Journal of Mathematics, 2013 (2013).

This article has been cited by:

Muhammad Sohail, Muhammad Hussain Ali, Kamaleldin Abodayeh, Syed Tehseen Abbas, Bio-convective boundary layer flow of Maxwell nanofluid via optimal homotopic procedure with radiation and Darcy-Forchheimer impacts over a stretched sheet, 2025, 46, 0143-0750, 10.1080/01430750.2025.2462583

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(4425) PDF downloads(637) Cited by(9)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

On $\mathcal{L}$ -simulation mappings in partial metric spaces

Related Papers:

Abstract

1. Introduction

2. Identities in transposed Poisson $n$ -Lie algebras

3. Proof of the conjecture for strong transposed Poisson $n$ -Lie algebras

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

On L\mathcal{L}-simulation mappings in partial metric spaces

Related Papers:

Abstract

1. Introduction

2. Identities in transposed Poisson n n -Lie algebras

3. Proof of the conjecture for strong transposed Poisson n n -Lie algebras

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

On $\mathcal{L}$ -simulation mappings in partial metric spaces

2. Identities in transposed Poisson $n$ -Lie algebras

3. Proof of the conjecture for strong transposed Poisson $n$ -Lie algebras