Results on generalized neutral fractional impulsive dynamic equation over time scales using nonlocal initial condition

1.   Introduction

The concept of local derivations was originally proposed by Kadison, Larson, and Sourour in 1990 for the study of Banach algebras (see^[6,7]). In 2016, Ayupov and Kudaybergenov studied the local derivations of a Lie algebra. They asserted that every local derivation of a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation (see ^[1]). Many researchers have focused on studying local derivations of Lie algebras (see ^[2,12,14]). Motivated by ^[1], Chen et al. introduced the definition of local superderivations for a Lie superalgebra in 2017 (see ^[5]). More and more scholars have begun to study local superderivations of Lie superalgebras. In ^[5,13], Chen, Wang, and Yuan et al. studied local superderivations of a simple Lie superalgebra. They proved that every local superderivation is a superderivation for basic classical Lie superalgebras (except $\mathrm{A}(1, 1)$), the strange Lie superalgebra $\mathrm{q}_{n}$, and Cartan-type Lie superalgebras over the complex field. In ^[3,11], Camacho and Wu et al. reached a similar conclusion for a particular class of solvable Lie superalgebras and the super-Virasoro algebras over the complex field. One can also consider local superderivations of a Lie superalgebra in their modules. When the simple modules of a Lie superalgebra are completely clear, it is possible to determine the local superderivations of a Lie superalgebra for all simple modules. In ^[10], Wang et al. determined the simple modules of the orthogonal symplectic Lie superalgebra $\mathrm{osp}(1, 2)$ over a field of prime characteristic. In ^[8,9], Wang et al. studied the $2$-local derivations of Lie algebra $\mathrm{sl}(2)$ for all simple modules and the first cohomology of $\mathrm{osp}(1, 2)$ with coefficients in simple modules over a field of prime characteristic.

In this paper, we are interested in determining all local superderivations of the Lie superalgebra $\mathrm{osp}(1, 2)$ for all simple modules over a field of prime characteristic. The paper is structured as follows: In Section 2, we recall the basic concepts and establish several lemmas. In Lemma 2.1, we show the connection between the bases of the simple module and the bases of the inner superderivation space. We introduce the notion of local superderivations for a Lie superalgebra to any finite-dimensional module (see Definition 2.1). By ^[10], any simple module of $\mathrm{osp}(1, 2)$ is isomorphic to some simple module $L_{\chi}(\lambda)$ for highest weight $\lambda$ and $p$-character $\chi$, and $\chi$ is either regular nilpotent, regular semisimple, or restricted. The first cohomology of $\mathrm{osp}(1, 2)$ with coefficients in $L_{\chi}(\lambda)$ was described in ^[9], from which we obtain the bases of the vector space of superderivations. We introduce the method to determine the local superderivations of $\mathrm{osp}(1, 2)$ to $L_{\chi}(\lambda)$ of parity $\alpha$ in Lemma 2.2. In Section 3 (resp. Section 4), we show that every local superderivation of $\mathrm{osp}(1, 2)$ to $L_{\chi}(\lambda)$ with $\chi$ being regular nilpotent or regular semisimple (resp. $\chi$ is restricted) is a superderivation.

2.   Preliminaries

In this paper, the underlying field $\mathbb{F}$ is algebraically closed and of prime characteristic $p > 2$, and $\mathbb{Z}_{2} = \{\overline{0}, \overline{1}\}$ is the additive group of order two with addition, in which $\overline{1}+\overline{1} = \overline{0}$. Recall that a $\mathbb{Z}_{2}$-graded vector space $V = V_{\overline{0}}\oplus V_{\overline{1}}$ is also called a superspace, where the elements of $V_{\overline{0}}$ (resp. $V_{\overline{1}}$) are said to be even (resp. odd). For $\alpha\in \mathbb{Z}_{2}$, any element $v$ of $V_{\alpha}$ is said to be homogeneous of parity $\alpha$, denoted by $|v| = \alpha$. Write $\{x_{1}, \ldots, x_{p}\mid y_{1}, \ldots, y_{q}\}$ implying that $x_{i}$ is even and $y_{j}$ is odd in a superspace. If $\{x_{1}, \ldots, x_{p}\mid y_{1}, \ldots, y_{q}\}$ is a $\mathbb{Z}_{2}$-homogeneous basis of a $\mathbb{Z}_{2}$-graded vector space $V$, we write $V = \langle x_{1}, \ldots, x_{p}\mid y_{1}, \ldots, y_{q}\rangle$. Denote by $\mathrm{Hom}(V, W)$ the set consisting of all the $\mathbb{F}$-linear maps from $V$ to $W$, where $V$ and $W$ are $\mathbb{Z}_{2}$-graded vector spaces. We define the $\mathbb{Z}_{2}$-gradation on $\mathrm{Hom}(V, W)$ by $\mathrm{Hom}(V, W)_{\alpha} = \{\phi\in \mathrm{Hom}(V, W)\mid\phi(V_{\beta})\subset W_{\alpha+\beta}, \; \beta\in \mathbb{Z}_{2} \}$.

Let $L$ be a Lie superalgebra and $M$ an $L$-module. Recall that a $\mathbb{Z}_{2}$-homogeneous linear map of parity $\alpha$, $\phi : L \rightarrow M$, is called a superderivation of parity $\alpha$ if

Write $\mathrm{Der}(L, M)_{\alpha}$ for the set of all superderivations of $L$ to $M$ of parity $\alpha$. It is easy to verify that $\mathrm{Der}(L, M)_{\alpha}$ is a vector space. Denote

For a $\mathbb{Z}_{2}$-homogeneous element $m \in M$, define the linear map $D_{m}$ of $L$ to $M$ by $D_{m}(x) = (-1)^{|x||m|}x. m$, where $x \in L$. Then $D_{m}$ is a superderivation of parity $|m|$. Let $\mathrm{Ider}(L, M)$ be the vector space spanned by all $D_{m}$ with $\mathbb{Z}_{2}$-homogeneous elements $m \in M$. Then every element in $\mathrm{Ider}(L, M)$ is called an inner superderivation. It is easy to check that

is an even linear map. Then we have the following lemma, which is simple and useful.

Lemma 2.1. Let $\mathrm{H^{0}}(L, M) = 0$. Then the linear map $\mathfrak{D}$ (defined by Eq (2.1)) is a linear isomorphism. In particular, $\{D_{m_{1}}, D_{m_{2}}, \ldots, D_{m_{k}}\}$ is a basis of $\mathrm{Ider}(L, M)$ if and only if $\{m_{1}, m_{2}, \ldots, m_{k}\}$ is a basis of $M$.

Recall the well-known fact that the first cohomology of $L$ with coefficients in $L$-module $M$ is

Obviously, $\mathrm{H^{1}}(L, M) = 0$ is equivalent to $\mathrm{Der}(L, M) = \mathrm{Ider}(L, M)$.

Definition 2.1. A $\mathbb{Z}_{2}$-homogeneous linear map $\phi_{\alpha}$ of a Lie superalgbra $L$ to $L$-mod $M$ of parity $\alpha$ is called a local superderivation if, for any $x\in L$, there exists a superderivation $D_{x}\in \mathrm{Der}(L, M)_{\alpha}$ (depending on $x$) such that $\phi_{\alpha}(x) = D_{x}(x)$.

Let $B_{\alpha} = \{D_{1}, D_{2}, \ldots, D_{m}\}$ be a basis of $\mathrm{Der}(L, M)_{\alpha}$ and $T_{\alpha}\in \mathrm{Hom}(L, M)_{\alpha}$. For $x\in L$, we write $M(B_{\alpha}; x)$ for the matrix $(D_{1}x\; D_{2}x\; \ldots\; D_{m}x)$ and $M(B_{\alpha}, T_{\alpha}; x)$ for the matrix $(M(B_{\alpha}; x)\; T_{\alpha}x)$, where $\alpha\in \mathbb{Z}_{2}$. The following lemma can be easily verified by Definition 2.1.

Lemma 2.2. Let $T_{\alpha}$ be a homogeneous linear map of a Lie superalgbra $L$ to $L$-mod $M$ of parity $\alpha$. Then $T_{\alpha}$ is a local superderivation of parity $\alpha$ if and only if the rank of $M(B_{\alpha}; x)$ is equal to the rank of $M(B_{\alpha}, T_{\alpha}; x)$ for any $x\in L$ and $\alpha\in \mathbb{Z}_{2}$.

Set $h: = E_{22}-E_{33}, \; e: = E_{23}, \; f: = E_{32}, \; E: = E_{13}+E_{21}, \; F: = E_{12}-E_{31}$, where $E_{ij}$ is the $3\times3$ matrix unit. Recall that $\{h, e, f, \mid E, F \}$ is the standard $\mathbb{Z}_{2}$-homogeneous basis of the Lie superalgebra $\mathrm{osp}(1, 2)$. Hereafter, we write $L$ for $\mathrm{osp}(1, 2)$ over $\mathbb{F}$ and $L_{\chi}(\lambda)$ for the simple module of $L$ with the highest weight $\lambda$ and $p$-character $\chi$. Recall the basic properties of $L_{\chi}(\lambda)$, which we discuss in this paper (see ^[10], Section 6). There are three orbits of $\chi\in L_{\overline{0}}^{\ast}$:

(1) regular nilpotent: $\chi(e) = \chi(h) = 0$ and $\chi(f) = 1$;

(2) regular semisimple: $\chi(e) = \chi(f) = 0$ and $\chi(h) = a^{p}$ for some $a\in \mathbb{F} \setminus \{0\}$;

(3) restricted: $\chi(e) = \chi(f) = \chi(h) = 0$.

That is, the $p$-character $\chi$ is either regular nilpotent, regular semisimple, or 0. We have the following standard basis for $L_{\chi}(\lambda)$. For $\lambda < p$, we have $L_{0}(\lambda) = \langle v_{0}, v_{2}, \ldots, v_{2\lambda-2}\mid v_{1}, v_{3}, \ldots, v_{2\lambda-1}\rangle$. For $\chi\neq 0$, we have $L_{\chi}(\lambda) = \langle v_{0}, v_{2}, \ldots, v_{2p-2}\mid v_{1}, v_{3}, \ldots, v_{2p-1}\rangle$. The $L$-action is given by

By [9, Theorem 1.2], we have $\mathrm{H}^{1}(L, L_{\chi}(\lambda)) = \langle 0\mid \psi_{1}, \; \psi_{2}\rangle$ for $(\lambda, \chi) = (p-1, 0)$, where

Otherwise, $\mathrm{H}^{1}(L, L_{\chi}(\lambda)) = \langle 0\mid 0\rangle$. By Lemma 2.1, we have

3.   The regular nilpotent or semisimple case

In this section, we shall characterize local superderivations of $L$ to the simple module $L_{\chi}(\lambda)$, where $\chi\neq 0$. We have (see Section 2)

Let $D_{i}$ be the matrix of $D_{v_{i}}$ under the standard ordered bases of $L$ and $L_{\chi}(\lambda)$. That is,

By the definition of innner superderivations, we have

Hereafter, write $\varepsilon_{i}$ for the $5$-dimensional column vector in which $i$ entry is $1$ and the other entries are $0$ as well as $E_{i, j}$ (resp. $\tilde{E}_{i, j}$) for the $2p\times 5$ (resp. $p\times p$) matrix in which $(i, j)$ entry is $1$ and the other entries are $0$. Then for $t\in\{0, 1, \ldots, p-2\}$, we have

For convenience, put $I = \{1, 2, 3, 4\}$ and $Y = \{y_{1}, y_{2}, \ldots, y_{9}\}$, where $y_{i} = \varepsilon_{i+1}$, $y_{4+j} = \varepsilon_{j}+\varepsilon_{4}$, $y_{6+m} = \varepsilon_{m}+\varepsilon_{5}$ for $i\in I, \; j\in I\setminus \{3, 4\}$ and $m\in I\setminus \{4\}$. We introduce the following symbols for $k\in\{1, 2, \ldots, p\}$:

Proposition 3.1. Suppose that $p$-character $\chi\neq0$. Let $T_{\alpha}$ be a homogeneous linear map of $L$ to $L_{\chi}(\lambda)$ of parity $\alpha$. Then the following statements hold:

(1) Suppose that $\chi$ is regular nilpotent. The matrices $M(B_{\alpha}, T_{\alpha}; x)$ and $M(B_{\alpha}; x)$ have the same rank for any $x\in L$ if and only if $M(B_{\alpha}, T_{\alpha}; y_{i})$ and $M(B_{\alpha}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{2}, y_{4}, y_{6}\}$ if $\alpha = \bar{0}$ and $y_{i}\in Y\setminus\{y_{3}, y_{4}, y_{6}\}$ if $\alpha = \bar{1}$.

(2) Suppose that $\chi$ is regular semisimple. The matrices $M(B_{\alpha}, T_{\alpha}; x)$ and $M(B_{\alpha}; x)$ have the same rank for any $x\in L$ if and only if $M(B_{\alpha}, T_{\alpha}; y_{i})$ and $M(B_{\alpha}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{4}, y_{6}\}$ if $\alpha = \bar{0}$ and $y_{i}\in Y\setminus\{y_{3}, y_{8}\}$ if $\alpha = \bar{1}$.

Proof. Set $T_{\bar{0}} = \begin{pmatrix} A & 0 \\ 0 & B \\ \end{pmatrix}$ and $T_{\bar{1}} = \begin{pmatrix} 0 & C \\ D & 0 \\ \end{pmatrix}$, where $A, \; D\in M_{p, 3}$ and $B, \; C\in M_{p, 2}$. Write $a_{ij}$, $b_{ql}$, $c_{il}$ and $d_{qj}$ for the elements of matrix blocks $A$, $B$, $C$, and $D$, respectively, where $i, j\in\{1, 2, 3\}$, $q, l\in\{1, 2\}$. Let $X = (x_{1}, x_{2}, \ldots, x_{5})^{\mathrm{T}}$ be the coordinate of any element $x\in L$ under the standard basis of $L$. In this proof, we write $l\in I$, $k\in I\setminus\{p\}$, $m\in I\setminus\{p-1, p\}$, $t\in I\setminus \{1\}$, where $I = \{1, 2, \ldots, p\}$.

(1) If $\chi$ is regular nilpotent, that is, $\chi(f) = 1$, $\chi(e) = \chi(h) = 0$. Then we have

Then, $M(B_{\bar{0}}, T_{\bar{0}}; x) = \begin{pmatrix} M(B_{\bar{0}}; x) & T_{\bar{0}}x \\ \end{pmatrix}$ and $M(B_{\bar{1}}, T_{\bar{1}}; x) = \begin{pmatrix} M(B_{\bar{1}}; x) & T_{\bar{1}}x \\ \end{pmatrix}$.

Since the matrices $M(B_{\bar{0}}, T_{\bar{0}}; y_{i})$ and $M(B_{\bar{0}}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{2}, y_{4}, y_{6}\}$, we have

It follows that for any $x\in L$,

That is, $T_{\bar{0}}x$ is a linear combination of $\{D_{0}x, D_{2}x, \ldots, D_{2p-2}x\}$. Hence, $M(B_{\bar{0}}, T_{\bar{0}}; x)$ and $M(B_{\bar{0}}; x)$ have the same rank for any $x\in L$.

Since the matrices $M(B_{\bar{1}}, T_{\bar{1}}; y_{i})$ and $M(B_{\bar{1}}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{3}, y_{4}, y_{6}\}$, we have

It follows that for any $x\in L$,

That is, $T_{\bar{1}}x$ is a linear combination of $\{D_{1}x, D_{3}x, \ldots, D_{2p-1}x\}$. Therefore, $M(B_{\bar{1}}, T_{\bar{1}}; x)$ and $M(B_{\bar{1}}; x)$ have the same rank for any $x\in L$.

(2) If $\chi$ is regular semisimple, that is, $\chi(e) = \chi(f) = 0$, $\chi(h) = a^{p}$, where $a\in \mathbb{F} \setminus \{0\}$. Then, for any $\alpha\in \mathbb{Z}_{2}$, we have

Therefore, $M(B_{\bar{0}}, T_{\bar{0}}; x) = \begin{pmatrix} M(B_{\bar{0}}; x) & T_{\bar{0}}x \\ \end{pmatrix}$ and $M(B_{\bar{1}}, T_{\bar{1}}; x) = \begin{pmatrix} M(B_{\bar{1}}; x) & T_{\bar{1}}x \\ \end{pmatrix}$.

A similar calculation, as in the case of regular nilpotent, shows that

It follows that for any $x\in L$,

That is, $T_{\bar{0}}x$ is a linear combination of $\{D_{0}x, D_{2}x, \ldots, D_{2p-2}x\}$. Hence, $M(B_{\bar{0}}, T_{\bar{0}}; x)$ and $M(B_{\bar{0}}; x)$ have the same rank for any $x\in L$.

Since $M(B_{\bar{1}}, T_{\bar{1}}; y_{i})$ and $M(B_{\bar{1}}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{3}, y_{8}\}$, we have

It follows that for any $x\in L$,

That is, $T_{\bar{1}}x$ is a linear combination of $\{D_{1}x, D_{3}x, \ldots, D_{2p-1}x\}$. Therefore, $M(B_{\bar{1}}, T_{\bar{1}}; x)$ and $M(B_{\bar{1}}; x)$ have the same rank for any $x\in L$. □

By Lemma 2.2, as a direct consequence of Proposition 3.1, we have the following theorem:

Theorem 3.1. Let $L_{\chi}(\lambda)$ be the simple module of $\mathrm{osp}(1, 2)$ with the highest weight $\lambda$ and $p$-character $\chi$. Suppose that $p$-character $\chi$ is regular nilpotent or regular semisimple. Then every local superderivation of $\mathrm{osp}(1, 2)$ to $L_{\chi}(\lambda)$ is a superderivation.

4.   The restricted case

In this section, we shall characterize local superderivations of $\mathrm{osp}(1, 2)$ to the simple module $L_{0}(\lambda)$. We have (see Section 2)

Let $D_{i}$ be the matrix of $D_{v_{i}}$ under the standard ordered bases of $L$ and $L_{0}(\lambda)$. That is,

By the definition of inner superderivations, we have

Write $\tilde{\varepsilon}_{i}$ for the $\lambda$-dimensional column vector in which $i$ entry is $1$ and the other entries are $0$ as well as $\hat{E}_{i, j}$ for the $(2\lambda+1)\times 5$ matrix in which $(i, j)$ entry is $1$ and the other entries are $0$. Then for $m\in\{0, 1, \ldots, \lambda-1\}$, $n\in\{0, 1, \ldots, \lambda-2\}$, we have

Proposition 4.1. Let $T_{\bar{0}}$ be a homogeneous linear map of L to $L_{0}(\lambda)$ of parity $\bar{0}$, where $\lambda\in\{0, 1, \ldots, p-1\}$. Then the matrices $M(B_{\bar{0}}, T_{\bar{0}}; x)$ and $M(B_{\bar{0}}; x)$ have the same rank for any $x\in L$ if and only if $M(B_{\bar{0}}, T_{\bar{0}}; y_{i})$ and $M(B_{\bar{0}}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{3}, y_{4}, y_{8}\}$.

Proof. Set $T_{\bar{0}} = \begin{pmatrix} A & 0 \\ 0 & B \\ \end{pmatrix}$, where $A\in M_{\lambda+1, 3}$ and $B\in M_{\lambda, 2}$. Denote by $a_{ij}$ and $b_{ql}$ the elements of matrix blocks $A$ and $B$, respectively, where $i, j\in\{1, 2, 3\}$, $q, l\in\{1, 2\}$. Let $X = (x_{1}, x_{2}, \ldots, x_{5})^{\mathrm{T}}$ be the coordinate of any element $x\in L$ under the standard basis of $L$. In this proof, we write $k\in \{1, 2, \ldots, \lambda\}$, $t\in \{1, 2, \ldots, \lambda-1\}$.

It is obviously true that the proposition holds for $\lambda = 0$. In the following, we assume that $\lambda$ is not equal to 0. Denote

Then we have

Therefore, $M(B_{\bar{0}}, T_{\bar{0}}; x) = \begin{pmatrix} M(B_{\bar{0}}; x) & T_{\bar{0}}x \\ \end{pmatrix}$. Since the matrices $M(B_{\bar{0}}, T_{\bar{0}}; y_{i})$ and $M(B_{\bar{0}}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{3}, y_{4}, y_{8}\}$, we have

Therefore, for any $x\in L$, we have

That is, $T_{\bar{0}}x$ is a linear combination of $\{D_{0}x, D_{2}x, \ldots, D_{2\lambda}x\}$. Therefore, $M(B_{\bar{0}}, T_{\bar{0}}; x)$ and $M(B_{\bar{0}}; x)$ have the same rank for any $x\in L$. □

Proposition 4.2. Let $T_{\bar{1}}$ be a homogeneous linear map of L to $L_{0}(\lambda)$ of parity $\bar{1}$, where $\lambda\in\{0, 1, \ldots, p-1\}$. Then the following statements hold:

(1) Suppose that $\lambda\neq p-1$. The matrices $M(B_{\bar{1}}, T_{\bar{1}}; x)$ and $M(B_{\bar{1}}; x)$ have the same rank for any $x\in L$ if and only if $M(B_{\bar{1}}, T_{\bar{1}}; y_{i})$ and $M(B_{\bar{1}}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{8}\}$.

(2) Suppose that $\lambda = p-1$. The matrices $M(B_{\bar{1}}, T_{\bar{1}}; x)$ and $M(B_{\bar{1}}; x)$ have the same rank for any $x\in L$ if and only if $M(B_{\bar{1}}, T_{\bar{1}}; y_{i})$ and $M(B_{\bar{1}}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{1}, y_{2}, y_{3}, y_{4}, y_{8}\}$.

Proof. Let $T_{\bar{1}} = \begin{pmatrix} 0 & C \\ D & 0 \\ \end{pmatrix}$, where $C\in M_{\lambda+1, 2}$ and $D\in M_{\lambda, 3}$. Denote by $c_{il}$ and $d_{kj}$ the elements of matrix blocks $C$ and $D$, respectively, where $i, j\in\{1, 2, 3\}$, $k, l\in\{1, 2\}$. Let $X = (x_{1}, x_{2}, \ldots, x_{5})^{\mathrm{T}}$ be the coordinate of any element $x\in L$ under the standard basis of $L$. In this proof, we write $k\in J$, $m\in J\setminus\{1\}$, $t\in J\setminus\{\lambda\}$, where $J = \{1, 2, \ldots, \lambda\}$.

(1) Let $\lambda\neq p-1$. Denote

Then we have

Therefore, $M(B_{\bar{1}}, T_{\bar{1}}; x) = \begin{pmatrix} M(B_{\bar{1}}; x) & T_{\bar{1}}x \\ \end{pmatrix}$.

Since the matrices $M(B_{\bar{1}}, T_{\bar{1}}; y_{i})$ and $M(B_{\bar{1}}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{8}\}$, we have

Then, for any $x\in L$, we have

That is, $T_{\bar{1}}x$ is a linear combination of $\{D_{1}x, D_{3}x, \ldots, D_{2\lambda-1}x\}$. Therefore, $M(B_{\bar{1}}, T_{\bar{1}}; x)$ and $M(B_{\bar{1}}; x)$ have the same rank for any $x\in L$.

(2) Let $\lambda = p-1$. Using the fact that $M(B_{\bar{1}}, T_{\bar{1}}; y_{i})$ and $M(B_{\bar{1}}; y_{i})$ have the same rank for any $y_{i}\in Y\setminus\{y_{1}, y_{2}, y_{3}, y_{4}, y_{8}\}$, we have

Therefore, for any $x\in L$, we have

That is, $T_{\bar{1}}x$ is a linear combination of $\{D_{1}x, D_{3}x, \ldots, D_{2p-1}x, \psi_{1}x, \psi_{2}x\}$. Therefore, $M(B_{\bar{1}}, T_{\bar{1}}; x)$ and $M(B_{\bar{1}}; x)$ have the same rank for any $x\in L$. □

By Lemma 2.2, as a direct consequence of Propositions 4.1 and 4.2, we have the following result:

Theorem 4.1. Let $L_{\chi}(\lambda)$ be the simple module of $\mathrm{osp}(1, 2)$ with the highest weight $\lambda$ and $p$-character $\chi$. Suppose that $p$-character $\chi$ is restricted. Then every local superderivation of $\mathrm{osp}(1, 2)$ to $L_{\chi}(\lambda)$ is a superderivation.

5.   Conclusions

Let $L$ be the orthogonal symplectic Lie superalgebra $\mathrm{osp}(1, 2)$ over an algebraically closed field of prime characteristic $p > 2$. By ^[10], any simple module of $L$ is isomorphic to some simple module $L_{\chi}(\lambda)$ for highest weight $\lambda$ and $p$-character $\chi$, and $\chi$ is either regular nilpotent, regular semisimple, or restricted. According to Theorems 3.1 and 4.1, the following conclusion can be summarized: Every local superderivation of $L$ to any simple module is a superderivation over an algebraically closed field of prime characteristic $p > 2$.

We give an example. Every local superderivation of $L$ to a 1-dimensional trivial module of $L$ is a superderivation. In fact, according to the definition of superderivations, we can obtain that every superderivation of $L$ to a 1-dimensional trivial module is equal to $0$. According to Definition 2.1, we know that in this case, every local superderivation is equal to $0$.

Author contributions

Shiqi Zhao: Writing-original draft, Editing; Wende Liu and Shujuan Wang: Supervision, Writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The research was supported by the Postgraduate Innovation and Scientific Research Topic of the School of Mathematical Statistics of Hainan Normal University (No. styc202201), the NSF of Hainan Province (No. 121MS0784) and the NSF of China (No. 12061029).

Conflict of interest

The authors declare no conflicts of interest.