A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay

Maysaa Al Qurashi; Saima Rashid; Fahd Jarad; Maysaa Al Qurashi; Saima Rashid; Fahd Jarad

doi:10.3934/mbe.2022605

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 12950-12980. doi: 10.3934/mbe.2022605

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Research article Special Issues

A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay

1.
Department of Mathematics, King Saud University, P. O. Box 22452, Riyadh 11495, Saudi Arabia
2.
Department of Mathematics, Saudi Electronic University, Riyadh, Saudi Arabia
3.
Department of Mathematics, Government College University, Faisalabad 38000, Pakistan
4.
Department of Physics, Government College University, Faisalabad 38000, Pakistan
5.
Department of Mathmatics, Cankaya University, Ankara, Turkey
6.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
7.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

Received: 20 July 2022 Revised: 31 July 2022 Accepted: 02 August 2022 Published: 05 September 2022

Recently, researchers have become interested in modelling, monitoring, and treatment of hepatitis B virus infection. Understanding the various connections between pathogens, immune systems, and general liver function is crucial. In this study, we propose a higher-order stochastically modified delay differential model for the evolution of hepatitis B virus transmission involving defensive cells. Taking into account environmental stimuli and ambiguities, we presented numerical solutions of the fractal-fractional hepatitis B virus model based on the exponential decay kernel that reviewed the hepatitis B virus immune system involving cytotoxic T lymphocyte immunological mechanisms. Furthermore, qualitative aspects of the system are analyzed such as the existence-uniqueness of the non-negative solution, where the infection endures stochastically as a result of the solution evolving within the predetermined system's equilibrium state. In certain settings, infection-free can be determined, where the illness settles down tremendously with unit probability. To predict the viability of the fractal-fractional derivative outcomes, a novel numerical approach is used, resulting in several remarkable modelling results, including a change in fractional-order $\delta$ with constant fractal-dimension $\varpi$ , $\delta$ with changing $\varpi$ , and $\delta$ with changing both $\delta$ and $\varpi$ . White noise concentration has a significant impact on how bacterial infections are treated.

Keywords:

HBV model,
Fractal-fractional Caputo-Fabrizio differential operators,
existence and uniqueness,
qualitative analysis,
numerical solution

Citation: Maysaa Al Qurashi, Saima Rashid, Fahd Jarad. A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 12950-12980. doi: 10.3934/mbe.2022605

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Abstract

1. Introduction

The study of algebraic automorphisms on different algebraic systems is a classic direction in algebra. Usually, it is very difficult to determine the automorphisms of an algebra. How to describe the automorphisms in an algebra is still an open problem. A well-studied example is the automorphism group of an incidence algebra ^[1,2]. Andruskiewitsch and Dumas studied the algebra automorphisms and Hopf algebra automorphisms of the positive part of the quantum enveloping algebra of simple complex finite-dimensional Lie algebras in ^[3]. For more works on the algebra automorphisms of other algebras, please refer to ^{[4,5,6,7,8,9,10]}.

The purpose of this paper is to find an effective method to determine the automorphisms of finite-dimensional algebras. Using the method, we not only describe all automorphisms of low-dimensional algebras, but also identify some good automorphisms of high-dimensional algebras.

The paper is organized as follows:

In Section 2, we establish a criterion for automorphisms of finite-dimensional algebras. In Section 3, as an application, we give all automorphisms of the single-parameter generalized quaternion algebra. As special cases of the single-parameter generalized quaternion algebra, all automorphisms on the semi-quaternion algebra and the split semi-quaternion algebra are given. Since Sweedler's 4-dimensional Hopf algebra, as an algebra, is a split semi-quaternion algebra, all algebraic automorphisms in Sweedler's 4-dimensional Hopf algebra are described.

2. The criteria for automorphisms on finite-dimensional algebras

Throughout the paper, $\mathbb{R}$ denotes the real number field. All algebras are over $\mathbb{R}$ , and linear refers to $\mathbb{R}$ -linear. Given a matrix $M$ , $M^{T}$ denotes the transpose of $M$ .

Let

$\eta_{1} = \left( \begin{array}{c} 1\\ 0 \\ \vdots \\ 0\\ \end{array} \right),\eta_{2} = \left( \begin{array}{c} 0\\ 1 \\ \vdots \\ 0\\ \end{array} \right),\cdots, \eta_{n} = \left( \begin{array}{c} 0\\ 0 \\ \vdots \\ 1\\ \end{array} \right)$

be the standard basis of $\mathbb{R}^{n}$ .

Let $A$ be a finite-dimensional algebra with unit $1$ and generators $g_{1}, g_{2}, \cdots, g_{s}$ which are subject to certain relationships. Assume that $\{\alpha_{1} = 1, \alpha_{2}, \cdots, \alpha_{n}\}$ is a basis for $A$ . Using the relationships among the $g_{i}$ , we have

$\begin{equation} \left( \begin{array}{c} \alpha_{1}\\ \alpha_{2} \\ \vdots \\ \alpha_{n}\\ \end{array} \right)(\alpha_{1},\alpha_{2},\cdots, \alpha_{n}) = \mathbf{U}_{1}\alpha_{1}+\mathbf{U}_{2}\alpha_{2}+\cdots+\mathbf{U}_{n}\alpha_{n}, \end{equation}$

(2.1)

where each $\mathbf{U_{i}}$ is a $n\times n$ digital matrix. By dividing $\mathbf{U}_{i}$ into block matrices by the columns, we obtain

$\mathbf{U}_{i} = (_{i}\gamma_{1},_{i}\gamma_{2},\cdots, _{i}\gamma_{n}), \ _{i}\gamma_{j} = \left( \begin{array}{c} _{i}u_{1j} \\ _{i}u_{2j} \\ \vdots \\ _{i}u_{nj} \\ \end{array} \right).$

Construct the following matrices

$\mathbf{W}_{i} = (_{1}\gamma_{i}, _{2}\gamma_{i}, \cdots, _{n}\gamma_{i}), i = 1,2,\cdots, n.$

Definition 2.1. With the matrices $\mathbf{U}_{i}(i = 1, 2, \cdots, n)$ as above. We call $\mathbf{W}_{i}(i = 1, 2, \cdots, n)$ the matrix induced from the $i$ -th columns of $\{U_{j}\}_{j = 1}^{n}$ .

Lemma 2.2. Let $\mathcal{P}$ be a linear transformation on $A$ , and

$P = \left( \begin{array}{cccc} a_{11} & a_{12}& \cdots & a_{1n}\\ a_{21} & a_{22}& \cdots & a_{2n} \\ \vdots &\vdots& \vdots & \vdots \\ a_{n1} & a_{n2}& \cdots & a_{nn} \\ \end{array} \right) = (\xi_{1},\xi_{2},\cdots,\xi_{n}),$

the matrix of $\mathcal{P}$ with respect to the basis $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}$ . Then we have

$\begin{eqnarray} \mathcal{P}(\alpha_{i})\mathcal{P}(\alpha_{j}) = (\alpha_{1}, \alpha_{2},\cdots, \alpha_{n})\left( \begin{array}{cccc} \xi_{j}& & & \\ & \xi_{j} & & \\ & & \ddots& \\ & & &\xi_{j}\\ \end{array} \right)^{T}\mathbf{C}^{T}\xi_{i}, \end{eqnarray}$

(2.2)

for all $i, j = 1, 2, \cdots, n$ , where

$\mathbf{C} = (\mathbf{U}_{1}, \mathbf{U}_{2}, \cdots , \mathbf{U}_{n}).$

Proof. For all $i, j$ , since

$\begin{eqnarray*} \mathcal{P}(\alpha_{i})\mathcal{P}(\alpha_{j}) & = &\xi_{i}^{T} \left( \begin{array}{c} \alpha_{1} \\ \alpha_{2} \\ \vdots \\ \alpha_{n} \\ \end{array} \right) (\alpha_{1},\alpha_{2},\cdots, \alpha_{n})\xi_{j}\\ & = &\xi_{i}^{T} \left( \mathbf{U}_{1}\alpha_{1}+\mathbf{U}_{2}\alpha_{2}+\cdots+\mathbf{U}_{n}\alpha_{n} \right)\xi_{j}\\ & = &\xi_{i}^{T} \mathbf{U}_{1} \xi_{j}\alpha_{1} +\xi_{i}^{T} \mathbf{U}_{2}\xi_{j}\alpha_{2}+\cdots +\xi_{i}^{T} \mathbf{U}_{n} \xi_{j}\alpha_{n}\\ & = &\xi_{i} ^{T}\mathbf{C}\left( \begin{array}{cccc} \xi_{j}& & & \\ & \xi_{j} & & \\ & & \ddots& \\ & & &\xi_{j}\\ \end{array} \right) \left( \begin{array}{c} \alpha_{1} \\ \alpha_{2} \\ \vdots \\ \alpha_{n} \\ \end{array} \right), \end{eqnarray*}$

it follows that (2.2) holds. □

Example 2.3. Recall that a single-parameter quaternion $q$ is an expression of the form

$q = a_{0}+a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3},$

where $a_{0}, a_{1}, a_{2}, a_{3}$ are real numbers and $e_{1}, e_{2}, e_{3}$ satisfy the following equalities:

$e_{1}^{2} = -\mu, e_{2}^{2} = 0, e_{3}^{2} = 0, e_{1}e_{2} = e_{3} = -e_{2}e_{1}, e_{2}e_{3} = 0 = -e_{3}e_{2}, e_{3}e_{1} = \mu e_{2} = -e_{1}e_{3},$

where $0\neq\mu\in \mathbb{R}$ . The set of single-parameter quaternions is denoted by $\mathbb{H}_{\mu}$ ^[8], and $\mathbb{H}_{\mu}$ is an associative algebra. We call $\mathbb{H}_{\mu}$ an algebra of single-parameter quaternions (or single-parameter quaternion algebra).

Observe that $\mathbb{H}_{\mu}$ is a 4-dimensional algebra with basis $1, e_{1}, e_{2}, e_{3}$ . By computing

$\begin{eqnarray*} \left( \begin{array}{c} 1\\ e_{1} \\ e_{2} \\ e_{3}\\ \end{array} \right)(1,e_{1},e_{2}, e_{3})& = & \left( \begin{array}{cccc} 1 & e_{1}& e_{2} & e_{3} \\ e_{1} &-\mu & e_{3} & -\mu e_{2}\\ e_{2} & -e_{3} & 0 & 0\\ e_{3} & \mu e_{2} & 0 & 0 \\ \end{array} \right)\\ & = & \left( \begin{array}{cccc} 1 & 0& 0 & 0 \\ 0 & -\mu & 0 & 0\\ 0 &0& 0 & 0\\ 0 & 0 & 0 & 0 \\ \end{array} \right)1+\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)e_{1}\\ &&+\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -\mu \\ 1 & 0 & 0 & 0 \\ 0 & \mu & 0 & 0 \\ \end{array} \right)e_{2}+\left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right)e_{3}. \end{eqnarray*}$

We have the corresponding $\mathbf{U}_{i} (i = 1, 2, 3, 4)$ as follows:

$\mathbf{U}_{1} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -\mu & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right), \mathbf{U}_{2} = \left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right),\mathbf{U}_{3} = \left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -\mu \\ 1 & 0 & 0 & 0 \\ 0 & \mu & 0 & 0 \\ \end{array} \right),$

$\mathbf{U}_{4} = \left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right).$

Thus we have

$\mathbf{C} = \left( \begin{array}{cccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & -\mu & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -\mu & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \mu & 0 & 0 & 1 & 0 & 0 & 0 \\ \end{array} \right).$

Let $\mathcal{P}$ be a linear transformation on $\mathbb{H}_{\mu}$ , and

$P = \left( \begin{array}{cccc} a_{11} & a_{12}& a_{13} & a_{14}\\ a_{21} & a_{22}& a_{23} & a_{24} \\ a_{31} & a_{32}& a_{33} & a_{34} \\ a_{41} & a_{42}& a_{43} & a_{44} \\ \end{array} \right) = (\xi_{1},\xi_{2},\xi_{3},\xi_{4}),$

the matrix of $\mathcal{P}$ with respect to the basis $\alpha_{1} = 1, \alpha_{2} = e_{1}, \alpha_{3} = e_{2}, \alpha_{4} = e_{3}$ . For instance, we aim to compute $\mathcal{P}(e_{1})\mathcal{P}(e_{2})$ , i.e., $\mathcal{P}(\alpha_{2})\mathcal{P}(\alpha_{3})$ . Since

$\left( \begin{array}{cccc} a_{13} & 0 & 0 & 0 \\ a_{23} & 0 & 0 & 0 \\ a_{33} & 0 & 0 & 0 \\ a_{43} & 0 & 0 & 0 \\ 0 & a_{13} & 0 & 0 \\ 0 & a_{23} & 0 & 0 \\ 0 & a_{33} & 0 & 0 \\ 0 & a_{43} & 0 & 0 \\ 0 & 0 & a_{13} & 0 \\ 0 & 0 & a_{23} & 0 \\ 0 & 0 & a_{33} & 0 \\ 0 & 0 & a_{43} & 0 \\ 0 & 0 & 0 & a_{13} \\ 0 & 0 & 0 & a_{23} \\ 0 & 0 & 0 & a_{33} \\ 0 & 0 & 0 & a_{43} \\ \end{array} \right)^{T}\mathbf{C}^{T}\left( \begin{array}{c} a_{12} \\ a_{22} \\ a_{32} \\ a_{42} \\ \end{array} \right) = \left( \begin{array}{c} a_{12} a_{13}-a_{22} a_{23} \mu \\ a_{13} a_{22}+a_{12} a_{23} \\ -a_{22} a_{43} \mu +a_{23} a_{42} \mu +a_{13} a_{32}+a_{12} a_{33} \\ -a_{23} a_{32}+a_{22} a_{33}+a_{13} a_{42}+a_{12} a_{43} \\ \end{array} \right)$

by Lemma 2.2, we have

$\mathcal{P}(e_{1})\mathcal{P}(e_{2}) = (1,e_{1},e_{2},e_{3})\left( \begin{array}{c} a_{12} a_{13}-a_{22} a_{23} \mu \\ a_{13} a_{22}+a_{12} a_{23} \\ -a_{22} a_{43} \mu +a_{23} a_{42} \mu +a_{13} a_{32}+a_{12} a_{33} \\ -a_{23} a_{32}+a_{22} a_{33}+a_{13} a_{42}+a_{12} a_{43} \\ \end{array} \right).$

Theorem 2.4. With notation as shown in Lemma 2.2. Then $\mathcal{P}$ is an automorphism if and only if $\xi_{1} = \eta_{1}$ and the matrix $P$ is an invertible matrix and satisfies the following equation:

$\begin{equation} (\mathbf{W}_{1},\mathbf{W}_{2},\cdots, \mathbf{W}_{n})\left( \begin{array}{cccc} P & 0 &0 & 0 \\ 0 & P & 0 & 0 \\ 0 & 0 & \ddots& 0 \\ 0 & 0 & 0 & P \\ \end{array} \right)^{T} = P^{T}\mathbf{C}\left( \begin{array}{cccc} P & 0 &0 & 0 \\ 0 & P & 0 & 0 \\ 0 & 0 & \ddots& 0 \\ 0 & 0 & 0 & P \\ \end{array} \right)\mathbf{K}, \end{equation}$

(2.3)

where $\mathbf{C}$ is shown in Lemma 2.2 and

$\mathbf{K} = \left( \begin{array}{ccccccccccccc} \eta_1 & 0 & \cdots & 0 & \eta_2 & 0 & \cdots & 0 & \cdots & \eta_n & 0 & \cdots & 0 \\ 0 & \eta_1 & \cdots & 0 & 0 & \eta_2 & \cdots & 0 & \cdots & 0 & \eta_n & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \cdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \eta_1 & 0 & 0 & \cdots & \eta_2 & \cdots & 0 & 0 & \cdots & \eta_n \\ \end{array} \right).$

Proof. From (2.1), it follows that

$\begin{equation} \alpha_{i} \alpha_{j} = (\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n})\left( \begin{array}{c} _{1}u_{ij} \\ _{2}u_{ij} \\ \vdots \\ _{n}u_{ij} \\ \end{array} \right), \forall i,j = 1,2,\cdots, n. \end{equation}$

(2.4)

For a fixed $j$ , by using (2.4), we have

$\mathcal{P}(\alpha_{i}\alpha_{j}) = (\alpha_{1},\alpha_{2},\cdots,\alpha_{n})P\left( \begin{array}{c} _{1}u_{ij} \\ _{2}u_{ij} \\ \vdots \\ _{n}u_{ij} \\ \end{array} \right).$

Since $\mathcal{P}(\alpha_{i}\alpha_{j}) = \mathcal{P}(\alpha_{i})\mathcal{P}(\alpha_{j})$ , for all $i$ , it follows that

$\left( \begin{array}{cccc} \xi_{j}& & & \\ & \xi_{j} & & \\ & & \ddots& \\ & & &\xi_{j}\\ \end{array} \right)^{T}\mathbf{C}^{T}\xi_{i} = P\left( \begin{array}{c} _{1}u_{ij} \\ _{2}u_{ij} \\ \vdots \\ _{n}u_{ij} \\ \end{array} \right),$

which is equivalent to

$\begin{eqnarray*} \left( \begin{array}{cccc} \xi_{j}& & & \\ & \xi_{j} & & \\ & & \ddots& \\ & & &\xi_{j}\\ \end{array} \right)^{T}\mathbf{C}^{T}P = P \left( \begin{array}{cccc} _{1}u_{1j} & _{1}u_{2j} & \cdots & _{1}u_{nj} \\ _{2}u_{1j}& _{2}u_{2j} & \cdots & _{2}u_{nj} \\ \vdots & \vdots &\vdots & \vdots \\ _{n}u_{1j} & _{n}u_{2j} & \cdots & _{n}u_{nj} \\ \end{array} \right) = P \mathbf{W}^{T}_{j}. \end{eqnarray*}$

Thus

$\begin{eqnarray*} &&(\mathbf{W}_{1},\mathbf{W}_{2},\cdots, \mathbf{W}_{n})\left( \begin{array}{cccc} P & 0 &0 & 0 \\ 0 & P & 0 & 0 \\ 0 & 0 & \ddots& 0 \\ 0 & 0 & 0 & P \\ \end{array} \right)^{T}\\ & = &P^{T}\mathbf{C}\left( \begin{array}{ccccccccccccc} \xi_{1} & 0 & \cdots & 0 & \xi_{2} & 0 & \cdots & 0 & \cdots & \xi_{n} & 0 & \cdots & 0 \\ 0 & \xi_{1} & \cdots & 0 & 0 & \xi_{2} & \cdots & 0 & \cdots & 0 & \xi_{n} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \cdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \xi_{1} & 0 & 0 & \cdots & \xi_{2} & \cdots & 0 & 0 & \cdots & \xi_{n} \\ \end{array} \right)\\ & = &P^{T}\mathbf{C}\left( \begin{array}{cccc} P & 0 &0 & 0 \\ 0 & P & 0 & 0 \\ 0 & 0 & \ddots& 0 \\ 0 & 0 & 0 & P \\ \end{array} \right)\mathbf{K}. \end{eqnarray*}$

From $\mathcal{P}(1) = 1$ , we obtain $\xi_{1} = \eta_{1}$ . The proof is completed.

Example 2.5. Let $\mathbf{U}_{i} (i = 1, 2, 3, 4)$ and $\mathbf{C}$ be given in Example 2.3. By Definition 2.1, we can obtain the desired $\mathbf{W}_{i}(i = 1, 2, 3, 4)$ as follows:

$\mathbf{W}_{1} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right), \mathbf{W}_{2} = \left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ -\mu & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & \mu & 0 \\ \end{array} \right), \mathbf{W}_{3} = \left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right),$

$\mathbf{W}_{4} = \left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & -\mu & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right).$

Thus, we have

$(\mathbf{W}_{1},\mathbf{W}_{2},\mathbf{W}_{3},\mathbf{W}_{4}) = \left( \begin{array}{cccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & -\mu & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -\mu & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)$

and

$\mathbf{K} = \left( \begin{array}{cccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right).$

3. Application to $\mathbb{H}_{\mu}$

In this section, we consider the application of Theorem 2.4 to $\mathbb{H}_{\mu}$ . To describe all automorphisms on $\mathbb{H}_{\mu}$ , we need to determine the matrices $P$ that satisfy the condition (2.3). Let

$P = \left( \begin{array}{cccc} 1 & a_{12} & a_{13} & a_{14} \\ 0 & a_{22} & a_{23} & a_{24} \\ 0 & a_{32} & a_{33} & a_{34} \\ 0 & a_{42} & a_{43} & a_{44} \\ \end{array} \right).$

Since

$\begin{eqnarray*} &&\text{LHS of (2.3)} \\& = & (\mathbf{W}_{1},\mathbf{W}_{2},\mathbf{W}_{3}, \mathbf{W}_{4})\left( \begin{array}{cccc} P & 0 &0 & 0 \\ 0 & P & 0 & 0 \\ 0 & 0 & P& 0 \\ 0 & 0 & 0 & P \\ \end{array} \right)^{T}\\ & = &\left( \begin{array}{cccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & -\mu & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -\mu & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\\ &&\left( \begin{array}{cccccccccccccccc} 1 & a_{12} & a_{13} & a_{14} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a_{22} & a_{23} & a_{24} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a_{32} & a_{33} & a_{34} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a_{42} & a_{43} & a_{44} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & a_{12} & a_{13} & a_{14} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a_{22} & a_{23} & a_{24} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a_{32} & a_{33} & a_{34} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a_{42} & a_{43} & a_{44} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & a_{12} & a_{13} & a_{14} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{22} & a_{23} & a_{24} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{32} & a_{33} & a_{34} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{42} & a_{43} & a_{44} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & a_{12} & a_{13} & a_{14} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{22} & a_{23} & a_{24} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{32} & a_{33} & a_{34} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{42} & a_{43} & a_{44} \\ \end{array} \right)^{T}\\ & = &(\mathbf{M}_{1}, \mathbf{M}_{2},\mathbf{M}_{3},\mathbf{M}_{4}), \end{eqnarray*}$

where

$\mathbf{M}_{1} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ a_{12} & a_{22} & a_{32} & a_{42} \\ a_{13} & a_{23} & a_{33} & a_{43} \\ a_{14} & a_{24} & a_{34} & a_{44} \\ \end{array} \right), \mathbf{M}_{2} = \left( \begin{array}{cccc} a_{12} & a_{22} & a_{32} & a_{42} \\ -\mu & 0 & 0 & 0 \\ -a_{14} & -a_{24} & -a_{34} & -a_{44} \\ a_{13} \mu & a_{23} \mu & a_{33} \mu & a_{43} \mu \\ \end{array} \right),$

$\mathbf{M}_{3} = \left( \begin{array}{cccc} a_{13} & a_{23} & a_{33} & a_{43} \\ a_{14} & a_{24} & a_{34} & a_{44} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right), \mathbf{M}_{4} = \left( \begin{array}{cccc} a_{14} & a_{24} & a_{34} & a_{44} \\ a_{13} (-\mu ) & a_{23} (-\mu ) & a_{33} (-\mu ) & a_{43} (-\mu ) \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right),$

and

$\begin{eqnarray*} &&\text{RHS of (2.3)} \\& = & P^{T}\mathbf{C}\left( \begin{array}{cccc} P & 0 &0 & 0 \\ 0 & P & 0 & 0 \\ 0 & 0 & \ddots& 0 \\ 0 & 0 & 0 & P \\ \end{array} \right)\mathbf{K}\\ & = &\left( \begin{array}{cccc} 1 & a_{12} & a_{13} & a_{14} \\ 0 & a_{22} & a_{23} & a_{24} \\ 0 & a_{32} & a_{33} & a_{34} \\ 0 & a_{42} & a_{43} & a_{44} \\ \end{array} \right)^{T}\left( \begin{array}{cccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & -\mu & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -\mu & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \mu & 0 & 1 & 0 & 0 & 0 \\ \end{array} \right)\\ &&\left( \begin{array}{cccccccccccccccc} 1 & a_{12} & a_{13} & a_{14} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a_{22} & a_{23} & a_{24} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a_{32} & a_{33} & a_{34} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a_{42} & a_{43} & a_{44} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & a_{12} & a_{13} & a_{14} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a_{22} & a_{23} & a_{24} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a_{32} & a_{33} & a_{34} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a_{42} & a_{43} & a_{44} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & a_{12} & a_{13} & a_{14} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{22} & a_{23} & a_{24} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{32} & a_{33} & a_{34} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{42} & a_{43} & a_{44} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & a_{12} & a_{13} & a_{14} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{22} & a_{23} & a_{24} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{32} & a_{33} & a_{34} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{42} & a_{43} & a_{44} \\ \end{array} \right)\\ &&\left( \begin{array}{cccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)\\ & = &(\mathbf{N}_{1}, \mathbf{N}_{2},\mathbf{N}_{3},\mathbf{N}_{4}), \end{eqnarray*}$

where

$\mathbf{N}_{1} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ a_{12} & a_{22} & a_{32} & a_{42} \\ a_{13} & a_{23} & a_{33} & a_{43} \\ a_{14} & a_{24} & a_{34} & a_{44} \\ \end{array} \right),$

$\mathbf{N}_{2} = \left( \begin{array}{cccc} a_{12} & a_{22} & a_{32} & a_{42} \\ a_{12}^2-a_{22}^2 \mu & 2 a_{12} a_{22} & 2 a_{12} a_{32} & 2 a_{12} a_{42} \\ a_{12} a_{13}-a_{22} a_{23} \mu & a_{13} a_{22}+a_{12} a_{23} & -a_{23} a_{42} \mu +a_{22} a_{43} \mu +a_{13} a_{32}+a_{12} a_{33} & a_{23} a_{32}-a_{22} a_{33}+a_{13} a_{42}+a_{12} a_{43} \\ a_{12} a_{14}-a_{22} a_{24} \mu & a_{14} a_{22}+a_{12} a_{24} & -a_{24} a_{42} \mu +a_{22} a_{44} \mu +a_{14} a_{32}+a_{12} a_{34} & a_{24} a_{32}-a_{22} a_{34}+a_{14} a_{42}+a_{12} a_{44} \\ \end{array} \right) ,$

$\mathbf{N}_{3} = \left( \begin{array}{cccc} a_{13} & a_{23} & a_{33} & a_{43} \\ a_{12} a_{13}-a_{22} a_{23} \mu & a_{13} a_{22}+a_{12} a_{23} & -a_{22} a_{43} \mu +a_{23} a_{42} \mu +a_{13} a_{32}+a_{12} a_{33} & -a_{23} a_{32}+a_{22} a_{33}+a_{13} a_{42}+a_{12} a_{43} \\ a_{13}^2-a_{23}^2 \mu & 2 a_{13} a_{23} & 2 a_{13} a_{33} & 2 a_{13} a_{43} \\ a_{13} a_{14}-a_{23} a_{24} \mu & a_{14} a_{23}+a_{13} a_{24} & -a_{24} a_{43} \mu +a_{23} a_{44} \mu +a_{14} a_{33}+a_{13} a_{34} & a_{24} a_{33}-a_{23} a_{34}+a_{14} a_{43}+a_{13} a_{44} \\ \end{array} \right) ,$

$\mathbf{N}_{4} = \left( \begin{array}{cccc} a_{14} & a_{24} & a_{34} & a_{44} \\ a_{12} a_{14}-a_{22} a_{24} \mu & a_{14} a_{22}+a_{12} a_{24} & -a_{22} a_{44} \mu +a_{24} a_{42} \mu +a_{14} a_{32}+a_{12} a_{34} & -a_{24} a_{32}+a_{22} a_{34}+a_{14} a_{42}+a_{12} a_{44} \\ a_{13} a_{14}-a_{23} a_{24} \mu & a_{14} a_{23}+a_{13} a_{24} & -a_{23} a_{44} \mu +a_{24} a_{43} \mu +a_{14} a_{33}+a_{13} a_{34} & -a_{24} a_{33}+a_{23} a_{34}+a_{14} a_{43}+a_{13} a_{44} \\ a_{14}^2-a_{24}^2 \mu & 2 a_{14} a_{24} & 2 a_{14} a_{34} & 2 a_{14} a_{44} \\ \end{array} \right) ,$

we have $\mathbf{M}_{i} = \mathbf{N}_{i}(i = 1, 2, 3, 4)$ , and obtain the following system of equations:

$a_{22}^2 \mu -a_{12}^2-\mu = 0,$

(a1)

$-2 a_{12} a_{22} = 0,$

(a2)

$-2 a_{12} a_{32} = 0,$

(a3)

$-2 a_{12} a_{42} = 0,$

(a4)

$a_{22} a_{23} \mu -a_{12} a_{13}+a_{14} = 0,$

(a5)

$-a_{13} a_{22}-a_{12} a_{23}+a_{24} = 0,$

(a6)

$-a_{23} a_{42} \mu +a_{22} a_{43} \mu -a_{13} a_{32}-a_{12} a_{33}+a_{34} = 0,$

(a7)

$a_{23} a_{32}-a_{22} a_{33}-a_{13} a_{42}-a_{12} a_{43}+a_{44} = 0,$

(a8)

$a_{13} (-\mu )+a_{22} a_{24} \mu -a_{12} a_{14} = 0,$

(a9)

$-a_{23} \mu -a_{14} a_{22}-a_{12} a_{24} = 0,$

(a10)

$-a_{33} \mu -a_{24} a_{42} \mu +a_{22} a_{44} \mu -a_{14} a_{32}-a_{12} a_{34} = 0,$

(a11)

$-a_{43} \mu +a_{24} a_{32}-a_{22} a_{34}-a_{14} a_{42}-a_{12} a_{44} = 0,$

(a12)

$a_{22} a_{23} \mu -a_{12} a_{13}-a_{14} = 0,$

(a13)

$-a_{13} a_{22}-a_{12} a_{23}-a_{24} = 0,$

(a14)

$a_{23} a_{42} \mu -a_{22} a_{43} \mu -a_{13} a_{32}-a_{12} a_{33}-a_{34} = 0,$

(a15)

$-a_{23} a_{32}+a_{22} a_{33}-a_{13} a_{42}-a_{12} a_{43}-a_{44} = 0,$

(a16)

$a_{23}^2 \mu -a_{13}^2 = 0,$

(a17)

$-2 a_{13} a_{23} = 0,$

(a18)

$-2 a_{13} a_{33} = 0,$

(a19)

$-2 a_{13} a_{43} = 0,$

(a20)

$a_{23} a_{24} \mu -a_{13} a_{14} = 0,$

(a21)

$-a_{14} a_{23}-a_{13} a_{24} = 0,$

(22)

$-a_{24} a_{43} \mu +a_{23} a_{44} \mu -a_{14} a_{33}-a_{13} a_{34} = 0,$

(a23)

$a_{24} a_{33}-a_{23} a_{34}-a_{14} a_{43}-a_{13} a_{44} = 0,$

(a24)

$a_{13} \mu +a_{22} a_{24} \mu -a_{12} a_{14} = 0,$

(a25)

$a_{23} \mu -a_{14} a_{22}-a_{12} a_{24} = 0,$

(a26)

$a_{33} \mu +a_{24} a_{42} \mu -a_{22} a_{44} \mu -a_{14} a_{32}-a_{12} a_{34} = 0,$

(a27)

$a_{43} \mu -a_{24} a_{32}+a_{22} a_{34}-a_{14} a_{42}-a_{12} a_{44} = 0,$

(a28)

$a_{23} a_{24} \mu -a_{13} a_{14} = 0,$

(a29)

$-a_{14} a_{23}-a_{13} a_{24} = 0,$

(a30)

$a_{24} a_{43} \mu -a_{23} a_{44} \mu -a_{14} a_{33}-a_{13} a_{34} = 0,$

(a31)

$-a_{24} a_{33}+a_{23} a_{34}-a_{14} a_{43}-a_{13} a_{44} = 0,$

(a32)

$a_{24}^2 \mu -a_{14}^2 = 0,$

(a33)

$-2 a_{14} a_{24} = 0,$

(a34)

$-2 a_{14} a_{34} = 0,$

(a35)

$-2 a_{14} a_{44} = 0.$

(a36)

All automorphisms on $\mathbb{H}_{\mu}$ can be described as follows:

Theorem 3.1. Each automorphism $\mathcal{P}$ on $\mathbb{H}_{\mu}$ has one of the following forms:

(i) $\mathcal{P}(1) = 1$ , $\mathcal{P}(e_{1}) = - e_{1}+ a e_{2} +d e_{3}$ , $\mathcal{P}(e_{2}) = b e_{2}+\frac{c}{\mu} e_{3}$ , $\mathcal{P}(e_{3}) = c e_{2}-b e_{3},$

(ii) $\mathcal{P}(1) = 1$ , $\mathcal{P}(e_{1}) = e_{1}+ a e_{2} +d e_{3}$ , $\mathcal{P}(e_{2}) = b e_{2}-\frac{c}{\mu} e_{3}$ , $\mathcal{P}(e_{3}) = c e_{2}+b e_{3},$

where $a, b, c, d$ are parameters and $\frac{\mu b^2+c^2}{\mu }\neq 0$ .

Proof. From (a6) and (a14), we obtain $a_{24} = 0$ . Therefore, from (a33), we have $a_{14} = 0$ . Taking $a_{24} = 0$ and $a_{14} = 0$ in (a25) and (a26) yields $a_{13} = a_{23} = 0$ .

If $a_{12} = 0$ , then, from (a1), we have $a_{22} = 1$ or $-1$ . If $a_{22} = 1$ , then the system of Eqs (a1)–(a36) is equivalent to the following system of equations

$a_{43} \mu +a_{34} = 0, a_{44}-a_{33} = 0.$

Thus, we obtain the desired $P$ as follows:

$P = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & a_{32} & a_{33} & a_{34} \\ 0 & a_{42} & -\frac{a_{34}}{\mu } & a_{33} \\ \end{array} \right),$

which is just the (ⅱ) of Theorem 3.1. If $a_{22} = -1$ , then we get the (ⅰ) of Theorem 3.1.

If $a_{12}\neq 0$ , from (a34)–(a36), one has $a_{34} = a_{44} = 0$ , which makes $P$ degenerate.

If $\mu = 1$ , then $\mathbb{H}_{\mu}$ is the algebra of semi-quaternions. If $\mu = -1$ , then $\mathbb{H}_{\mu}$ is the algebra of split semi-quaternions. By applying Theorem 3.1 to these special cases, we have the following:

Corollary 3.2. Each automorphism $\mathcal{P}$ on the algebra of semi-quaternions has one of the following forms:

(i) $\mathcal{P}(1) = 1$ , $\mathcal{P}(e_{1}) = - e_{1}+ a e_{2} +d e_{3}$ , $\mathcal{P}(e_{2}) = b e_{2}+c e_{3}$ , $\mathcal{P}(e_{3}) = c e_{2}-b e_{3},$

(ii) $\mathcal{P}(1) = 1$ , $\mathcal{P}(e_{1}) = e_{1}+ a e_{2} +d e_{3}$ , $\mathcal{P}(e_{2}) = b e_{2}-c e_{3}$ , $\mathcal{P}(e_{3}) = c e_{2}+b e_{3},$

where $a, b, c, d$ are parameters and $b^2+c^2\neq 0$ .

Corollary 3.3. Each automorphism $\mathcal{P}$ on the algebra of split semi-quaternions has one of the following forms:

(i) $\mathcal{P}(1) = 1$ , $\mathcal{P}(e_{1}) = - e_{1}+ a e_{2} +d e_{3}$ , $\mathcal{P}(e_{2}) = b e_{2}-c e_{3}$ , $\mathcal{P}(e_{3}) = c e_{2}-b e_{3},$

(ii) $\mathcal{P}(1) = 1$ , $\mathcal{P}(e_{1}) = e_{1}+ a e_{2} +d e_{3}$ , $\mathcal{P}(e_{2}) = b e_{2}+c e_{3}$ , $\mathcal{P}(e_{3}) = c e_{2}+b e_{3},$

where $a, b, c, d$ are parameters and $c^2-b^2 \neq 0$ .

Corollary 3.3 gives us an extra surprise. Using Corollary 3.3, we can determine all automorphisms on Sweedler's 4-dimensional Hopf algebra. First, recall that Sweedler algebra $\mathbb{H}_{4}$ is generated by two elements $g$ and $\nu$ subject to

$g^{2} = 1, \nu^{2} = 0, g\nu+\nu g = 0.$

The comultiplication, antipode, and counit of $\mathbb{H}_{4}$ are given by

$\Delta(g) = g\otimes g, \Delta(\nu) = g\otimes\nu+\nu\otimes 1, \varepsilon(g) = 1, \varepsilon(\nu) = 0, S(g) = g, S(\nu) = -g\nu.$

Note that the dimension of $\mathbb{H}_{4}$ is four with $1, g, \nu, g\nu$ forming a basis for $\mathbb{H}_{4}$ . By setting $e_{1} = g, e_{2} = \nu, e_{3} = gv$ , we see that $\mathbb{H}_{4}$ as an algebra is a split semi-quaternion algebra. Thus, by Corollary 3.3, we have the following result.

Corollary 3.4. Each automorphism $\mathcal{P}$ on Sweedler's 4-dimensional Hopf algebra has one of the following forms:

(i) $\mathcal{P}(1) = 1$ , $\mathcal{P}(g) = - g+ a \nu +d g\nu$ , $\mathcal{P}(\nu) = b \nu-c g\nu$ , $\mathcal{P}(g\nu) = c \nu-b g\nu,$

(ii) $\mathcal{P}(1) = 1$ , $\mathcal{P}(g) = g+ a \nu +d g\nu$ , $\mathcal{P}(\nu) = b \nu+c g\nu$ , $\mathcal{P}(g\nu) = c \nu+b g\nu,$

where $a, b, c, d$ are parameters and $c^2-b^2 \neq 0$ .

Use of AI tools declaration

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

Acknowledgments

The authors sincerely thank the referees for numerous very valuable comments and suggestions on this article. This work was supported by the National Natural Science Foundation of China (No. 12271292), the Natural Science Foundation of Shandong Province (No. ZR2022MA002) and the Natural Science Foundation of Xinjiang Uygur Autonomous Region (No. 2019D01B04).

Conflict of interest

The authors declare there are no conflicts of interest.

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