On the Structure of Weyl-Type, Witt-Type, and Non-Associative Algebras over Expolynomial Rings

Abstract

This paper introduces a generalized class of Weyl-type, Witt-type, and non-associative algebras constructed over an exponential–polynomial (expolynomial) framework. For fixed scalars ${\iota }_1,\dots ,{\iota }_r\in \mathcal{A}$ and for fixed integers $\mathbf{p}=(p_1,\dots ,p_n)\in {\mathbb{N}}^n$, we define the $\mathbb{F}$-algebra $\mathbb{F}\left[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}\right],$ an expolynomial ring over a field $\mathbb{F}$ of characteristic zero, where $\mathcal{A}$ is an additive subgroup of $\mathbb{F}$ containing $\mathbb{Z}$. This formulation extends the classical Weyl algebra through the integer power parameter $\mathbf{p}$, which generates a family of non-isomorphic simple algebras. The corresponding Weyl-type algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right),$ the Witt-type Lie algebra $W\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right),$ and their non-associative variants are examined in detail. The simplicity, grading, and automorphism structures of these algebras are established, and the dependence of these properties on the deformation parameter $\mathbf{p}$ is analyzed. All the constructed Weyl-type algebras, the corresponding Witt-type Lie algebras, and the non-associative algebras are shown to be simple under derivation structures. Many naturally occurring subalgebras, such as the integer-coefficient subalgebra $A\left(\mathbb{Z}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right)$, are also proven to be simple. Our analysis reveals that different choices of $\mathbf{p}$ result in non-isomorphic algebraic structures while retaining non-commutativity. The results obtained generalize several existing constructions of Weyl-type algebras and lay the theoretical foundation for further developments in transcendental and non-commutative algebraic frameworks.

1. Introduction

The theory of Weyl algebras has played an important role in the development of modern algebra. Originating from the study of differential operators, these algebras form the structural foundation for several branches of mathematics and physics, including quantum mechanics, algebraic geometry, and the theory of partial differential equations [1,2]. They provide a unified algebraic framework describing how differentiation and multiplication interact within operator theory. In its simplest form, the first Weyl algebra, introduced by Hermann Weyl, is defined over a field $\mathbb{F}$ of characteristic zero by

$$A_1\left(\mathbb{F}\right)=\mathbb{F}\langle x,{\partial }_x\mid \langle {\partial }_x,x\rangle =1\rangle ,$$

while the nth Weyl algebra is given by

$$A_n\left(\mathbb{F}\right)=\mathbb{F}\langle x_1,\dots ,x_n,{\partial }_1,\dots ,{\partial }_n\mid \langle {\partial }_i,x_j\rangle ={\delta }_{ij},\langle x_i,x_j\rangle =\langle {\partial }_i,{\partial }_j\rangle =0\rangle ,$$

where $\langle u,v\rangle :=uv-vu$ for all $1\le i,j\le n$. These relations express the essential non-commutativity between differentiation and multiplication operators.

The classical Weyl algebras were introduced in 1927 in connection with the canonical commutation relations of quantum mechanics [3]. Subsequent developments by Dixmier and Alev [4] established their structural properties, including simplicity and automorphism groups, which later inspired the construction of several generalizations. Bavula [5] extended these ideas by defining Generalized Weyl Algebras, while Jing and Zhang [6] introduced quantum deformations linking Weyl structures to quantum groups and q-algebras. Hartwig, Larsson, and Silvestrov [7] subsequently developed twisted generalized Weyl algebras, expanding the theory to encompass more complex non-commutative behaviors. Because of their inherent non-commutativity and absence of zero divisors, Weyl-type algebras continue to serve as a natural testing ground for algebraic deformation and operator theory.

However, observe that the function $y=lnx$ is a fundamental transcendental function, and the real solution of the equation $x^{lnx}=50$ is an irrational number. Similarly, the function $y=e^{e^{x^2}}$ does not satisfy any differential equation over the classical polynomial ring $\mathbb{F}\left[x\right]$, while it does satisfy differential relations inside the larger exponential–polynomial (expolynomial) ring $\mathbb{F}\left[e^{\pm x}\right]$. These examples illustrate a natural limitation of classical Weyl algebras defined over polynomial or exponential rings. Although such algebras are well suited for functions satisfying algebraic or differential relations over $\mathbb{F}\left[x\right]$, they do not adequately capture operators acting on more general transcendental expressions, such as exponential–polynomial functions.

Expolynomial rings therefore provide a natural enlargement of the coefficient algebra that allows one to incorporate these functions in a structurally consistent way, while still admitting a controlled grading structure and a well-defined action of derivations suitable for algebraic analysis. Motivated by this perspective, we enlarge the polynomial ring to an expolynomial ring and introduce a generalized family of Weyl-type, Witt-type, and non-associative algebras constructed over exponential–polynomial extensions of a field $\mathbb{F}$ of characteristic zero. The proposed framework introduces an integer parameter $\mathbf{p}$ that controls the deformation of the algebra and generates a family of algebras of the form

$$\mathbb{F}\left[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}\right],$$

where $\mathcal{A}$ denotes an additive subgroup of $\mathbb{F}$. This formulation generalizes the classical Weyl algebra through the incorporation of the power parameter $\mathbf{p}$, leading to a family of structurally distinct yet related algebras.

Moreover, the inclusion of exponential and transcendental elements extends the framework beyond purely polynomial or Laurent structures, providing a richer algebraic setting for the study of derivations, operator actions, and non-associative extensions. The dependence of the resulting Weyl-type and Witt-type algebras on the deformation parameter $\mathbf{p}$ leads to families of mutually non-isomorphic simple algebras. These properties are established formally in the subsequent sections, where the simplicity and structural relations are derived and analyzed.

The significance of this construction Lies in its generalization of several known algebraic models, unifying polynomial, exponential, and transcendental behaviors within a single non-commutative framework. Although such structures may have potential applications in various analytical and computational domains, the present paper focuses exclusively on their algebraic formulation and theoretical properties. A detailed investigation of the algorithmic and applied aspects of these algebras will be presented in a separate study.

The remainder of paper is organized as follows. Section 2 presents the necessary preliminaries and mathematical framework required for the construction of Weyl-type algebras. In Section 3, we develop the Weyl-type simple algebras and establish their structural properties, including simplicity and automorphism groups. Section 4 provides several algebraic formulas and operator relations that are frequently used in subsequent sections. The construction is further extended in Section 5, where more generalized Weyl-type algebras are introduced and analyzed. In Section 6, we define the corresponding Witt-type Lie algebras and non-associative algebras, highlighting their simplicity and inter relations. Section 7 discusses the conceptual applications and potential future directions, particularly emphasizing the algebraic relevance to cryptographic frameworks. Finally, Section 8 summarizes the main contributions and outlines directions for further research.

2. Preliminaries

This section introduces the essential definitions, notation, and algebraic background that will be used throughout the paper. Unless stated otherwise, all algebras are defined over a field $\mathbb{F}$ of characteristic zero.

2.1. Algebraic Notation and Conventions

Let $\mathbb{Z}$ and $\mathbb{N}$ denote the sets of all integers and all non-negative integers, respectively. Let $\mathbb{F}$ be a field of characteristic zero containing the field $\mathbb{Q}$ of rational numbers. We denote by ${\mathbb{F}}^{*}$ and ${\mathbb{Q}}^{*}$ the sets of all nonzero elements of $\mathbb{F}$ and $\mathbb{Q}$, respectively [8]. Throughout this work, the symbol $\mathbf{p}=(p_1,\dots ,p_n)$ represents an n-tuple of positive integers that specify the powers associated with the variables $x_1,x_2,\dots ,x_n$ in the algebraic term $x^{\mathbf{p}}$ and its exponential extensions. For any additive subgroup $\mathcal{A}\subseteq \mathbb{F}$ containing $\mathbb{Z}$, the notation

$$x^{\mathcal{A}}=\{x^{\alpha }:\alpha \in \mathcal{A}\}$$

denotes the set of all powers of x parameterized by $\mathcal{A}$. Let $\mathcal{T}$ denote the set of all transcendental numbers [9]. Two real numbers ${\iota }_1$ and ${\iota }_2$ are called proportional if either ${\displaystyle \frac{{\iota }_1}{{\iota }_2}}$ or ${\displaystyle \frac{{\iota }_2}{{\iota }_1}}$ is an integer. A positive rational number ${\displaystyle \frac{a}{b}}$ is said to be purely rational if $(a,b)=1$ and $b\ne 1$. A polynomial function $f\left(x\right)\in \mathbb{Z}\left[x\right]$ is purely rational if, for every nonzero $r\in \mathbb{Q}$, the value $f\left(r\right)$ is purely rational. Let $\mathbb{F}[x_1,\dots ,x_n]$ denote the polynomial ring in n commuting variables over $\mathbb{F}$. Its Laurent extension, allowing both positive and negative exponents, is written as

$$\mathbb{F}[x_1^{\pm 1},\dots ,x_n^{\pm 1}].$$

Such extensions are essential when variable inversion is required [10,11]. Similarly, the exponential polynomial algebra is defined by

$$\mathbb{F}[e^{\pm x_1},\dots ,e^{\pm x_n}],$$

and it is known that $\mathbb{F}[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ and $\mathbb{F}[e^{\pm x_1},\dots ,e^{\pm x_n}]$ are isomorphic as $\mathbb{F}$-algebras [12]. These exponential extensions provide a natural algebraic setting for constructing operator-based and transcendental structures. A generalization of the exponential polynomial algebra is the expolynomial algebra, which unifies polynomial and exponential behaviors within a single algebraic framework. Throughout this paper, we consider algebras of the general form

$$\mathbb{F}\left[e^{\pm x_i^{p_i}{e}^{{\iota }_{ij}{x}_i}},e^{{\mathcal{A}}_i{x}_i},x_i^{{\mathcal{A}}_i}\mid 1\le i\le n,1\le j\le r\right],$$

where $\mathbf{p}=(p_1,\dots ,p_n)\in {\mathbb{N}}^n$ and the parameters ${\iota }_{ij}$ are fixed scalars in the additive subgroups ${\mathcal{A}}_i$. This algebra summarizes both polynomial growth and exponential scaling, providing the foundational structure for the Weyl-type, Witt-type, and non-associative algebras developed in the subsequent sections.

2.2. Basic Algebraic Definitions

To make the paper self-contained, we briefly recall a few standard definitions used in the subsequent sections.

Definition 1.

A Lie algebra [13] over a field $\mathbb{F}$ is a vector space V equipped with a bilinear operation $[·,·]:V\times V\to V$, called the Lie bracket, satisfying

1. 

Antisymmetry: $[x,y]=-[y,x]$,

2. 

Jacobi identity: $[x,[y,z\left]\right]+[y,[z,x\left]\right]+[z,[x,y\left]\right]=0$,

for all $x,y,z\in V$.

Definition 2.

The Witt algebra [14] over a field $\mathbb{F}$ is the infinite-dimensional Lie algebra

$$W=\mathrm{Der}\left(\mathbb{F}\left[x^{\pm 1}\right]\right),$$

consisting of all derivations of the Laurent polynomial algebra $\mathbb{F}\left[x^{\pm 1}\right]$. It has a basis $\{x^{k+1}\frac{d}{dx}\mid k\in \mathbb{Z}\}$, and its Lie bracket is given by

$$[x^{m+1}\frac{d}{dx},x^{n+1}\frac{d}{dx}]=(n-m)x^{m+n+1}\frac{d}{dx}.$$

Definition 3.

A non-associative algebra [15] over a field $\mathbb{F}$ is a vector space V equipped with a bilinear multiplication $\mu :V\times V\to V,(x,y)\mapsto xy.$

• No associativity condition is imposed on the multiplication unless stated explicitly. Non-associative algebras include, as important subclasses, Lie, Jordan, and alternative algebras, and they naturally arise in generalizations of Weyl-type and Witt-type constructions.

Definition 4.

An algebra A over a field $\mathbb{F}$ is called simple if it contains no nontrivial two-sided ideals, i.e., the only two-sided ideals of A are $\left\{0\right\}$ and A itself.

Definition 5.

An isomorphism between two $\mathbb{F}$-algebras A and B is a bijective $\mathbb{F}$-linear map $\varphi :A\to B$ that preserves multiplication, that is,

$$\varphi (a+b)=\varphi \left(a\right)+\varphi \left(b\right),\varphi \left(\lambda a\right)=\lambda \varphi \left(a\right),\varphi \left(ab\right)=\varphi \left(a\right)\varphi \left(b\right),$$

for all $a,b\in A$ and $\lambda \in \mathbb{F}$. If such a map exists, the algebras A and B are said to be isomorphic, denoted by $A\cong B$.

Definition 6.

Let A be an $\mathbb{F}$-algebra. An automorphism of A is an $\mathbb{F}$-algebra isomorphism $\varphi :A\to A$, that is, a bijective $\mathbb{F}$-linear map satisfying

$$\varphi \left(ab\right)=\varphi \left(a\right)\varphi \left(b\right)\mathit{for}\mathit{all}a,b\in A.$$

• The set of all automorphisms of A forms a group under composition, denoted by $\mathrm{Aut}\left(A\right)$.

3. Weyl-Type Simple Algebras

To facilitate further discussion of Weyl algebras and their generalizations, we recall the general form of a Weyl-type algebra defined over a commutative F-algebra A. An F-linear mapping $\partial :A\to A$ is called a derivation if it satisfies the Leibniz rule

$$\partial \left(ab\right)=\partial \left(a\right)b+a\partial \left(b\right)\mathrm{for}\mathrm{all}a,b\in A.$$

It follows directly that $\partial \left(\alpha \right)=0$ for every scalar $\alpha \in \mathbb{F}$. Moreover, if $a\in A$ and ∂ is a derivation of A, then the product $a\partial$ defines another derivation on A. We denote by D the collection of all mutually commuting derivations of A. For each element $a\in A$, we associate a linear map $\widehat{a}:A\to A$ given by

$$\widehat{a}\left(b\right)=ab,\mathrm{for}\mathrm{all}b\in A.$$

This map represents the action of left multiplication by a within the algebra A.

Lemma 1.

For any $\partial \in D$ and $a\in A$, the following relation holds:

$$\begin{array}{c}\partial \circ \widehat{a}-\widehat{a}\circ \partial =\widehat{\partial \left(a\right)}\hfill \end{array}$$

(1)

Proof. 

For $b\in A$, $(\partial \circ \widehat{a}-\widehat{a}\circ \partial )\left(b\right)=\partial \left(ab\right)-a\partial \left(b\right)=\partial \left(a\right)b=\widehat{\partial \left(a\right)}\left(b\right)$. □

To proceed with the generalization of Weyl-type constructions, we first recall a key relation that extends the classical Weyl commutation rule. Consider the polynomial algebra $\mathbb{F}[x_1,\dots ,x_n]$. Let ${\partial }_i:={\displaystyle \frac{\partial }{\partial x_i}}$ denote the derivation with respect to $x_i$, and let $\widehat{x_j}$ represent the multiplication operator defined analogously to $\widehat{a}$, such that $\widehat{x_j}\left(b\right)=x_{j}b$ for all $b\in \mathbb{F}[x_1,\dots ,x_n]$. Then, for all $1\le i,j\le n$, the following identity holds:

$${\partial }_i\circ \widehat{x_j}-\widehat{x_j}\circ {\partial }_i={\widehat{\delta }}_{i,j},$$

where ${\delta }_{i,j}$ is the Kronecker delta and ∘ denotes composition of maps (see [1,2,4,16,17]). While some authors denote derivations by $\widehat{\partial }$, we adopt the simpler notation ∂ for brevity. If A is an associative and commutative $\mathbb{F}$-algebra with identity 1, and there exists an element $a_1\in A$ such that $\partial \left(a_1\right)=1$, then the Weyl relation

$$\partial \widehat{a_1}-\widehat{a_1}\partial =\widehat{1},$$

is satisfied, where $\widehat{1}$ denotes the identity operator on A. For simplicity of notation, the composition symbol ∘ will be omitted in subsequent expressions. Hence, the classical Weyl relation

$$\begin{array}{c}\hfill {\partial }_i\widehat{x_j}-\widehat{x_j}{\partial }_i={\widehat{\delta }}_{i,j},\end{array}$$

(2)

appearing in the Weyl algebra $A_n$ can be viewed as a special case of a more general relation of Lemma 1. Moreover, the embedding $a\mapsto \widehat{a}$ preserves multiplication, so that $\widehat{a}\widehat{b}=\widehat{ab}$ for all $a,b\in A$, and hence $\widehat{a}\widehat{b}=\widehat{b}\widehat{a}$. By Lemma 1, we define the Weyl-type algebra over A as

$$A\left(A\right)=\mathbb{F}\langle \widehat{a},\partial \rangle /\mathcal{R},$$

where $a\in A$, $\partial \in D$, and $\mathcal{R}$ denotes the set of relations described in Lemma 1. More explicitly, the ideal $\mathcal{R}$ is generated by the relations

$$\widehat{a}\widehat{b}=\widehat{ab},[{\partial }_i,\widehat{a}]=\widehat{{\partial }_i\left(a\right)},[{\partial }_i,{\partial }_j]=0,$$

for all $a,b\in A$ and $1\le i,j\le n$, where ${\partial }_i\left(a\right)$ denotes the natural action of the derivation ${\partial }_i$ on the element $a\in A$. In particular, when A is an exponential–polynomial algebra, these relations encode the usual Leibniz rule together with the prescribed action of derivations on polynomial and exponential generators.

The algebra $A\left(A\right)$ is clearly a non-commutative $\mathbb{F}$-vector space with multiplication given by composition, and it contains no zero divisors (see [1,2]). We may now define specific Weyl-type algebras, such as

$$A\left(\mathbb{F}[x_1^{\pm 1},\dots ,x_n^{\pm 1}]\right)\mathrm{and}A\left(\mathbb{F}[e^{\pm x_1},\dots ,e^{\pm x_n}]\right),$$

which naturally extend the classical Weyl algebra (see [1,2])

$$A_n:=A\left(\mathbb{F}[x_1,\dots ,x_n]\right).$$

It is well established that the algebras $A_n$, $A\left(\mathbb{F}[x_1^{\pm 1},\dots ,x_n^{\pm 1}]\right)$, and $A\left(\mathbb{F}[e^{\pm x_1},\dots ,e^{\pm x_n}]\right)$ are simple (see [4,5,18]). Specifically, the algebra $A\left(\mathbb{F}[e^{\pm x_1},\dots ,e^{\pm x_n}]\right)$ can be expressed as

$$A\left(\mathbb{F}[e^{\pm x_1},\dots ,e^{\pm x_n}]\right):=\mathbb{F}[\widehat{e^{\pm x_1}},\dots ,\widehat{e^{\pm x_n}},{\partial }_1,\dots ,{\partial }_n]/\mathcal{R},$$

where the relations $\mathcal{R}$ are given by

$$\begin{array}{c}\widehat{e^{x_i}}\widehat{e^{x_j}}=\widehat{e^{x_j}}\widehat{e^{x_i}},\widehat{e^{-x_i}}\widehat{e^{x_j}}=\widehat{e^{x_j}}\widehat{e^{-x_i}},\widehat{e^{-x_i}}\widehat{e^{-x_j}}=\widehat{e^{-x_j}}\widehat{e^{-x_i}},1\le i,j\le n,\hfill \\ {\partial }_u{\partial }_v={\partial }_v{\partial }_u,1\le u,v\le n,\hfill \\ {\partial }_u\widehat{e^{x_u}}-\widehat{e^{x_u}}{\partial }_u=\widehat{e^{x_u}},\mathrm{i}.\mathrm{e}.,\widehat{e^{-x_u}}{\partial }_u\widehat{e^{x_u}}-{\partial }_u=1,1\le u\le n.\hfill \end{array}$$

(3)

We now consider an extended $\mathbb{F}$-algebra that will serve as the base for our Weyl-type construction. Let

$$\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]:=\mathbb{F}[e^{\pm x_1^{p_1}{e}^{x_1}},\dots ,e^{\pm x_n^{p_n}{e}^{x_n}},e^{\pm x_1},\dots ,e^{\pm x_n},x_1,\dots ,x_n],$$

where $\mathbf{p}=(p_1,\dots ,p_n)\in {\mathbb{N}}^n$, and n denotes the number of variables involved. This algebra represents an exponential–polynomial extension of $\mathbb{F}$ obtained by adjoining the exponential elements $e^{\pm x_i^{p_i}{e}^{x_i}}$ and $e^{\pm x_i}$ together with the variables $x_i$$(1\le i\le n)$. Over this extended algebra, we define the corresponding Weyl-type algebra

$$A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]\right)$$

on $\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]$, where the variables $x_1,\dots ,x_n$ and their corresponding derivations ${\partial }_1,\dots ,{\partial }_n$ satisfy relations analogous to those in $A_n$ (see [1,2,19]). This algebra satisfies the following identity:

$$\begin{array}{c}\hfill {\partial }_u\widehat{e^{x_j^{p_j}{e}^{x_j}}{e}^{x_j}}-\widehat{e^{x_j^{p_j}{e}^{x_j}}{e}^{x_j}}{\partial }_u={\delta }_{uj}\widehat{{\partial }_u(e^{x_j^{p_j}{e}^{x_j}}{e}^{x_j})}.\end{array}$$

(4)

along with other similar relations inherited from $A_n$. Furthermore, the algebra is ${\mathbb{Z}}^n$-graded:

$$\begin{array}{c}\hfill A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]\right)=\underset{(a_1,\dots ,a_n)\in {\mathbb{Z}}^n}{⨁}{A}_{(a_1,\dots ,a_n)},\end{array}$$

(5)

where each homogeneous component $A_{(a_1,\dots ,a_n)}$ is the $\mathbb{F}$-subspace spanned by elements of the form

$$\widehat{e^{a_1{x}_1^{p_1}{e}^{x_1}}}\dots \widehat{e^{a_n{x}_n^{p_n}{e}^{x_n}}}\widehat{e^{b_1{x}_1}}\dots \widehat{e^{b_n{x}_n}}\widehat{x_1^{j_1}}\dots \widehat{x_n^{j_n}}\widehat{{\partial }_1^{i_1}}\dots \widehat{{\partial }_n^{i_n}},$$

with $b_k\in \mathbb{Z}$ and $i_k,j_k\in N$ for $1\le k\le n$. A lexicographic order ${>}_{exp}$ can be defined on the elements of $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]\right)$ as follows:

$$\begin{array}{c}\widehat{e^{a_1{x}_1^{p_1}{e}^{x_1}}}\dots \widehat{e^{a_n{x}_n^{p_n}{e}^{x_n}}}\widehat{e^{b_1{x}_1}}\dots \widehat{e^{b_n{x}_n}}\widehat{x_1^{j_1}}\dots \widehat{x_n^{j_n}}{\partial }_1^{i_1}\dots {\partial }_n^{i_n}{>}_{exp}\hfill \\ \widehat{e^{a_1^{\prime }{x}_1^{p_1}{e}^{x_1}}}\dots \widehat{e^{a_n^{\prime }{x}_n^{p_n}{e}^{x_n}}}\widehat{e^{b_1^{\prime }{x}_1}}\dots \widehat{e^{b_n^{\prime }{x}_n}}\widehat{x_1^{j_1^{\prime }}}\dots \widehat{x_n^{j_n^{\prime }}}{\partial }_1^{i_1^{\prime }}\dots {\partial }_n^{i_n^{\prime }},\hfill \end{array}$$

(6)

if $a_1>a_1^{\prime }$, $a_1=a_1^{\prime }$ and $a_2>a_2^{\prime }$, …, or $a_1=a_1^{\prime },\dots ,a_{n-1}=a_{n-1}^{\prime }$, and $a_n>a_n^{\prime }$, that is, according to the standard lexicographic ordering of indices.

Finally, the subalgebra $A\left(\mathbb{F}[e^{0e^x},e^{\pm x},x]\right)$ of $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]\right)$ is the simple algebra $A\left(\mathbb{F}[e^{\pm x},x]\right)$ (see [17]), which corresponds to the $(0,\dots ,0)$-homogeneous component in the above grading. For fixed integers ${\iota }_{11},\dots ,{\iota }_{1r},\dots ,{\iota }_{n1},\dots ,{\iota }_{nm}$, we define the $\mathbb{F}$-algebra

$$\begin{array}{c}\hfill \mathbb{F}\left[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x\right]=\mathbb{F}\left[e^{\pm x_1^{p_1}{e}^{{\iota }_{11}{x}_1}},\dots ,e^{\pm x_n^{p_n}{e}^{{\iota }_{nm}{x}_n}},e^{\pm x_1},\dots ,e^{\pm x_n},x_1,\dots ,x_n\right].\end{array}$$

(7)

The corresponding Weyl-type algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\right)$ is then constructed analogously on this base algebra.

Theorem 1.

The algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]\right)$ is simple.

Proof. 

Let I be a nonzero ideal of $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]\right)$, and let l be a nonzero element of I. We proceed by induction on $\#\left(l\right)$, where $\#\left(l\right)$ denotes the number of distinct homogeneous components (with respect to the ${\mathbb{Z}}^n$-grading) of l that contain nonzero terms. If $\#\left(l\right)=1$, then l is homogeneous. If l Lies in the $(0,\dots ,0)$-homogeneous component, the claim is immediate. Otherwise, suppose l belongs to the $(b_1,\dots ,b_n)$-homogeneous component with at least one $b_i\ne 0$. Then the element

$$\widehat{e^{-b_1{x}_1^{p_1}{e}^{x_1}}}\dots \widehat{e^{-b_n{x}_n^{p_n}{e}^{x_n}}}l$$

is a nonzero element of $A_{(0,\dots ,0)}$. Hence the theorem holds for $\#\left(l\right)=1$. Assume the statement is true for all elements with $\#\left(l\right)=q$, and consider l with $\#\left(l\right)=q+1$. Decompose $l=l_0+l_1$, where $l_0$ is the sum of all elements in the $(0,\dots ,0)$-homogeneous component, and $l_1$ is the sum of all remaining terms. Since the $(0,\dots ,0)$-component

$$A\left(\mathbb{F}[e^{\pm x_1},\dots ,e^{\pm x_n},x_1,\dots ,x_n]\right)$$

is simple, the ideal $\langle l_0\rangle$ generated by $l_0$ contains 1. Consequently, there exists an element $l_1^{\prime }+1\in I$ with $l_1^{\prime }\in I$. If $\#\left(l_1^{\prime }\right)<q$, the induction hypothesis applies. Otherwise, $\#(l_1^{\prime }+1)=q+1$, and $l_1^{\prime }+1$ has a nonzero term in the $(b_1,\dots ,b_n)$-homogeneous component. Consider the commutator

$$e^{-b_1{x}_1^{p_1}{e}^{x_1}}\dots e^{-b_n{x}_n^{p_n}{e}^{x_n}}(l_1^{\prime }+1)-(l_1^{\prime }+1)e^{-b_1{x}_1^{p_1}{e}^{x_1}}\dots e^{-b_n{x}_n^{p_n}{e}^{x_n}}.$$

This element is nonzero and satisfies

$$\#\left(e^{-b_1{x}_1^{p_1}{e}^{x_1}}\dots e^{-b_n{x}_n^{p_n}{e}^{x_n}}(l_1^{\prime }+1)-(l_1^{\prime }+1)e^{-b_1{x}_1^{p_1}{e}^{x_1}}\dots e^{-b_n{x}_n^{p_n}{e}^{x_n}}\right)\le q.$$

By the induction hypothesis, this implies $1\in I$, and hence $I=A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]\right)$. Therefore, the algebra is simple. □

Theorem 2.

The Weyl-type algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\right)$ is simple.

Proof. 

Since the algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\right)=A\left(\mathbb{F}\left[e^{\pm x_1^{p_1}{e}^{{\iota }_{11}{x}_1}},\dots ,e^{\pm x_1^{p_1}{e}^{{\iota }_{1r}{x}_1}},\dots ,e^{\pm x_n^{p_n}{e}^{{\iota }_{n1}{x}_n}},\dots ,e^{\pm x_n^{p_n}{e}^{{\iota }_{nm}{x}_n}},e^{\pm x_1},\dots ,e^{\pm x_n},x_1,\dots ,x_n\right]\right)$ is ${\mathbb{Z}}^{n(r+m)}$-graded, the proof is similar to that of Theorem 1, and hence is omitted. □

Theorem 3.

The Weyl-type algebra $A\left(\mathbb{F}\left[e^{\pm x_1^{p_1}{e}^{{\iota }_{11}{x}_1}},\dots ,e^{\pm x_1^{p_1}{e}^{{\iota }_{1r}{x}_1}},\dots ,e^{\pm x_n^{p_n}{e}^{{\iota }_{n1}{x}_n}},\dots ,e^{\pm x_n^{p_n}{e}^{{\iota }_{nm}{x}_n}},e^{\pm x_1},\dots ,e^{\pm x_n},x_1,\dots ,x_n\right]\right)$ and its subalgebra $A\left(\mathbb{F}\left[e^{\pm x_1^{p_1}{e}^{\iota x_1}},\dots ,e^{\pm x_n^{p_n}{e}^{\iota x_n}},e^{\pm x_1},\dots ,e^{\pm x_n},x_1,\dots ,x_n\right]\right)$ are simple.

Proof. 

Since the algebras are graded algebra, by Theorem 1, the proof of the theorem is straightforward. So let us omit them. □

• For an application of an algebra, it is very important to know the automorphism group of the algebra or to find an isomorphism between those algebras (see [16]).

Proposition 1.

If positive integers $p_1$ and $p_2$ are not equal, or ${\iota }_1$ and ${\iota }_2$ are not proportional, then the algebras $A\left(\mathbb{F}[e^{\pm x^{p_1}{e}^{{\iota }_{1}x}},e^{\pm x},x]\right)$ and $A\left(\mathbb{F}[e^{\pm x^{p_2}{e}^{{\iota }_{2}x}},e^{\pm x},x]\right)$ are not isomorphic.

Proof. 

We can assume that $p_1\le p_2.$ Let us assume that there is an algebra isomorphism $\theta$ from $A\left(\mathbb{F}[e^{\pm x^{p_1}{e}^{{\iota }_{1}x}},e^{\pm x},x]\right)$ to $A\left(\mathbb{F}[e^{\pm x^{p_2}{e}^{{\iota }_{2}x}},e^{\pm x},x]\right)$. We have that $\theta \left(e^x\right)=c_1{e}^{a_1{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{b_{1}x}$ and $\theta \left(e^{x^{p_1}{e}^{{\iota }_{1}x}}\right)=c_2{e}^{a_2{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{b_{2}x}$ with appropriate coefficients and indices. Let us put $\theta (\partial )$ as follows:

$$\begin{array}{c}\theta (\partial )=c{e}^{a{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{bx}{x}^d{\partial }^i+\dots \hfill \end{array}$$

(8)

where $e^{a{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{bx}{x}^d{\partial }^i$ is the maximal term of $\theta (\partial )$ with respect to the order ${>}_{exp}$ and … is the sum of the remaining terms of $\theta (\partial )$ with appropriate coefficients and indices. By $\theta (\partial )\theta \left(e^x\right)-\theta \left(e^x\right)\theta (\partial )=\theta \left(e^x\right),$ we have that

$$\begin{array}{c}c{c}_1{e}^{a{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{bx}{x}^d{\partial }^i{e}^{a_1{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{b_{1}x}+\dots -c{c}_1{e}^{(a+a_1)x^{p_2}{e}^{{\iota }_{2}x}}{e}^{(b+b_1)x}{x}^d{\partial }^i-\dots \hfill \\ =c_1{e}^{a_1{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{b_{1}x}.\hfill \end{array}$$

(9)

This implies that $a=b=d=0$ and $i=1,$ i.e., $\theta (\partial )=c\partial +\dots$. So by (9), these conditions give that $a_1=0$ and $c{b}_1=1,$ i.e., $\theta (\partial )=c\partial +\dots$ and $\theta \left(e^x\right)=c_1{e}^{b_{1}x}.$ By $(c\partial +\dots )\theta \left(x\right)-\theta \left(x\right)(c\partial +\dots )=1,$ we have that $\theta \left(x\right)=c^{-1}x+c^{\prime }$ where $c^{\prime }\in \mathbb{F}.$ By

$$\begin{array}{c}\theta (\partial )\theta \left(e^{x^{p_1}{e}^{{\iota }_{1}x}}\right)-\theta \left(e^{x^{p_1}{e}^{{\iota }_{1}x}}\right)\theta (\partial )=\theta \left(e^{x^{p_1}{e}^{{\iota }_{1}x}}{e}^{{\iota }_{1}x}(p_1{x}^{p_1-1}+{\iota }_1{x}^{p_1})\right)\hfill \\ =\theta \left(e^{x^{p_1}{e}^{{\iota }_{1}x}}\right)\theta \left(e^{{\iota }_{1}x}\right)\theta (p_1{x}^{p_1-1}+{\iota }_1{x}^{p_1}),\hfill \end{array}$$

(10)

we have that

$$\begin{array}{c}c{c}_2\partial e^{a_2{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{b_{2}x}-c{c}_2{c}_2{e}^{a_2{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{b_{2}x}\partial \hfill \\ =c_1^{{\iota }_1}{c}_2{e}^{a_2{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{b_{2}x}{e}^{b_1{\iota }_{1}x}(p_1{(c^{-1}x+c^{\prime })}^{p_1-1}+{\iota }_1{(c^{-1}x+c^{\prime })}^{p_1}).\hfill \end{array}$$

This implies that

$$\begin{array}{c}{a}_{2}c{c}_2{p}_2{e}^{a_2{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{(b_2+{\iota }_2)x}{x}^{p_2-1}+c{c}_2{\iota }_2{e}^{a_2{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{(b_2+{\iota }_2)x}{x}^{p_2}+b_{2}c{c}_2{e}^{a_2{x}^p{e}^{{\iota }_{2}x}}{e}^{b_{2}x}\hfill \\ =c_1^{{\iota }_1}{c}_2{e}^{a_2{x}^{p_2}{e}^{{\iota }_{2}x}}{e}^{(b_2+b_1{\iota }_1)x}(p_1{(c^{-1}x+c^{\prime })}^{p_1-1}+{\iota }_1{(c^{-1}x+c^{\prime })}^{p_1})\hfill \end{array}$$

(11)

If $p_1\ne p_2,$ then the equality (11) does not hold. If $p_1=p_2,$ then ${\iota }_2=b_1{\iota }_1$ implies that ${\iota }_1$ and ${\iota }_2$ are proportional. We have a contradiction for both cases. So the algebras $A\left(\mathbb{F}[e^{\pm x^{p_1}{e}^{{\iota }_1}},e^{\pm x},x]\right)$ and $A\left(\mathbb{F}[e^{\pm x^{p_2}{e}^{{\iota }_2}},e^{\pm x},x]\right)$ are not isomorphic. □

Corollary 1.

If ${\iota }_1$ and ${\iota }_2$ are different prime numbers, then the algebras $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{{\iota }_{1}x}},e^{\pm x},x]\right)$ and $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{{\iota }_{2}x}},e^{\pm x},x]\right)$ are not isomorphic.

Proof. 

The proof of the corollary is straightforward by the proof of Proposition 1. □

Note 1.

For $c_1,c_2\in {\mathbb{F}}^{*},$$k\in \mathbb{N}$ and $d_{k,\partial }\in \mathbb{F},$ if we define $\mathbb{F}$-linear map ${\theta }_{1,c_1,c_2,d(k,\partial )}$ from $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota }},e^{\pm x},x]\right)$ to itself as follows:

$$\begin{array}{c}{\theta }_{1,c_1,c_2,d(k,\partial )}(\partial )=\partial +d_{k,\partial }{x}^k,{\theta }_{1,c_1,c_2,d(k,\partial )}\left(x\right)=x,{\theta }_{1,c_1,c_2,d(k,\partial )}\left(e^x\right)=c_1{e}^x,\hfill \\ \mathrm{and}{\theta }_{1,c_1,c_2,d(k,\partial )}\left(e^{x^{\mathbf{p}}{e}^{\iota x}}\right)=c_2{e}^{x^{\mathbf{p}}{e}^{\iota x}},\hfill \end{array}$$

(12)

then ${\theta }_{1,c_1,c_2,d(k,\partial )}$ can be linearly extended to an automorphism of the algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota }},e^{\pm x},x]\right)$ such that $c_1^{\iota }=1$. For $c_1,c_2\in {\mathbb{F}}^{*},$$k\in \mathbb{N}$ and $d_{k,\partial }\in \mathbb{F},$ if we define F-linear map ${\theta }_{-1,c_1,c_2,d(k,\partial )}$ from $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota }},e^{\pm x},x]\right)$ to itself as follows:

$$\begin{array}{c}{\theta }_{-1,c_1,c_2,d(k,\partial )}(\partial )=-\partial +d_{k,\partial }{x}^k,{\theta }_{-1,c_1,c_2,d(k,\partial )}\left(x\right)=-x,{\theta }_{-1,c_1,c_2,d(k,\partial )}\left(e^x\right)=c_1{e}^x,\hfill \\ \mathrm{and}{\theta }_{-1,c_1,c_2,d(k,\partial )}\left(e^{x^{\mathbf{p}}{e}^{\iota x}}\right)=c_2{e}^{x^{\mathbf{p}}{e}^{\iota x}},\hfill \end{array}$$

(13)

then ${\theta }_{-1,c_1,c_2,d(k,\partial )}$ can be linearly extended to an automorphism of the algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota }},e^{\pm x},x]\right)$ such that $c_1^{\iota }={(-1)}^{p+1}$. The conditions $c_1^{\iota }=1$ and $c_1^{\iota }={(-1)}^{p+1}$ ensure that the images of $e^x$ and $e^{x^{\mathbf{p}}{e}^{\iota x}}$ under ${\theta }_{\pm 1,c_1,c_2,d(k,\partial )}$ are compatible with the defining relations of the algebra, so that the maps extend to well-defined automorphisms.

Proposition 2.

For a positive integer p, the automorphism group $\mathrm{Aut}\left(A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\right)\right)$ of the algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\right)$ is generated by the automorphisms ${\theta }_{1,c_1,c_2,d(k,\partial )}$ and ${\theta }_{-1,c_1,c_2,d(k,\partial )}$ defined in Note 1.

Proof. 

Let $\theta$ be an automorphism of $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\right)$. Let us put $\theta \left(e^x\right)=c_1{e}^{a_1{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{b_{1}x}$ and $\theta \left(e^{x^{\mathbf{p}}{e}^{\iota x}}\right)=c_2{e}^{a_2{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{b_{2}x}$ with appropriate coefficients and indices. Let us put $\theta (\partial )$ as follows:

$$\begin{array}{c}\theta (\partial )=c{e}^{a{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{bx}{x}^d{\partial }^i+\dots \hfill \end{array}$$

(14)

where $e^{a{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{bx}{x}^d{\partial }^i$ is the maximal term of $\theta (\partial )$ with respect to the order ${>}_{exp}$ and … is the sum of the remaining terms of $\theta (\partial )$ with appropriate coefficients and indices. This implies that by similar calculations of (9), we have that $a=b=d=0$ and $i=1,$ i.e., $\theta (\partial )=c\partial +\dots$. So by (9), these conditions give that $a_1=0$ and $c{b}_1=1,$ i.e., $\theta (\partial )=c\partial +\dots$ and $\theta \left(e^x\right)=c_1{e}^{b_{1}x}.$ By $(c\partial +\dots )\theta \left(x\right)-\theta \left(x\right)(c\partial +\dots )=1,$ we have that $\theta \left(x\right)=c^{-1}x+c^{\prime }$ where $c^{\prime }\in \mathbb{F}.$ By

$$\begin{array}{c}\theta (\partial )\theta \left(e^{x^{\mathbf{p}}{e}^{\iota x}}\right)-\theta \left(e^{x^{\mathbf{p}}{e}^{\iota x}}\right)\theta (\partial )=\theta \left(e^{x^{\mathbf{p}}{e}^{\iota x}}{e}^{\iota x}(\mathbf{p}{x}^{\mathbf{p}-1}+\iota x^p)\right)\hfill \\ =\theta \left(e^{x^{\mathbf{p}}{e}^{\iota x}}\right)\theta \left(e^{\iota x}\right)\theta (\mathbf{p}{x}^{\mathbf{p}-1}+\iota x^{\mathbf{p}}),\hfill \end{array}$$

(15)

we have that

$$\begin{array}{c}c{c}_2\partial e^{a_2{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{b_{2}x}-c{c}_2{c}_2{e}^{a_2{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{b_{2}x}\partial \hfill \\ =c_1^{\iota }{c}_2{e}^{a_2{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{b_{2}x}{e}^{b_1\iota x}(\mathbf{p}{(c^{-1}x+c^{\prime })}^{\mathbf{p}-1}+\iota {(c^{-1}x+c^{\prime })}^{\mathbf{p}}).\hfill \end{array}$$

This implies that

$$\begin{array}{c}{a}_{2}c{c}_2\mathbf{p}{e}^{a_2{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{(b_2+\iota )x}{x}^{\mathbf{p}-1}+c{c}_2\iota e^{a_2{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{(b_2+\iota )x}{x}^{\mathbf{p}}+b_{2}c{c}_2{e}^{a_2{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{b_{2}x}\hfill \\ =c_1^{\iota }{c}_2{e}^{a_2{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{(b_2+b_1\iota )x}(\mathbf{p}{(c^{-1}x+c^{\prime })}^{\mathbf{p}-1}+\iota {(c^{-1}x+c^{\prime })}^{\mathbf{p}}).\hfill \end{array}$$

(16)

This implies that $b_1=1$, $b_2=c^{\prime }=0,$$a_2=c_1^l{c}^{-\mathbf{p}}$ and $a_2=c.$ Since the algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota }},e^{\pm x},x]\right)$ is generated by $\partial ,$$x^{\pm 1}$, $\theta \left(e^{\pm x}\right)$, and $e^{\pm x^{\mathbf{p}}{e}^{\iota x}}$, we have that $a_2=1$ or $a_2=-1.$ If $a_2=1$ holds, then $c=1,$ i.e., $c_1^{\iota }=1.$ So the map $\theta$ extends linearly to the automorphism ${\theta }_{1,c_1,c_2,d(k,\partial )}$ of Note 1. If $a_2=-1$, then $c=-1,$ i.e., $c_1^{\iota }={(-1)}^{\mathbf{p}+1}.$ So the map $\theta$ can be linearly extended to the automorphism ${\theta }_{-1,c_1,c_2,d(k,\partial )}$ of Note 1. Since the algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota }},e^{\pm x},x]\right)$ is generated by $\partial ,$$x^{\pm 1}$, $\theta \left(e^{\pm x}\right)$, and $e^{\pm x^{\mathbf{p}}{e}^{\iota x}}$, the automorphism group $\mathrm{Aut}\left(A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota }},e^{\pm x},x]\right)\right)$ is generated by ${\theta }_{1,c_1,c_2,d(k,\partial )}$ and ${\theta }_{-1,c_1,c_2,d(k,\partial )}$ of Note 1. □

• The constructions in this section establish a broad algebraic foundation for exponential polynomial Weyl structures. Subsequent sections extend these results to generalized families and related non-associative and Witt-type Lie algebras.

4. Useful Identities and Operator Relations

A subalgebra $A\subset \mathbb{F}\left[\left[x^{\pm 1}\right]\right]$ is said to be stable if it is invariant under the derivation $\partial ={\displaystyle \frac{d}{dx}}$, that is, $\partial \left(A\right)\subseteq A$.

Lemma 2.

For any $f\left(x\right)$ of a stable subalgebra A of the formal power series ring $\mathbb{F}\left[\left[x^{\pm 1}\right]\right]$,

$$\begin{array}{c}{\partial }^i{x}^a\left(f\left(x\right)\right)=(\left(\genfrac{}{}{0ex}{}{a}{0}\right)x^a{\partial }^i+i\left(\genfrac{}{}{0ex}{}{a}{1}\right)x^{a-1}{\partial }^{i-1}+i(i-1)\left(\genfrac{}{}{0ex}{}{a}{2}\right)x^{a-2}{\partial }^{i-2}+\dots +\hfill \\ i(i-1)\dots (i-b+1)\left(\genfrac{}{}{0ex}{}{a}{b}\right)x^{a-b}{\partial }^{i-b}+\dots +\hfill \\ i(i-1)\dots (i-a+1)\left(\genfrac{}{}{0ex}{}{a}{a}\right){\partial }^{i-a})\left(f\left(x\right)\right)\hfill \end{array}$$

(17)

holds where $\left(\begin{array}{c}a\\ b\end{array}\right)={\displaystyle \frac{a!}{(a-b)!b!}}$ for $a,b,i\in N$. Furthermore, for $g\in A$, the following identities

$$\begin{array}{c}{\partial }^{i}g=g{\partial }^i+\left(\genfrac{}{}{0ex}{}{i}{1}\right)g^{\prime }{\partial }^{i-1}+\dots +\left(\genfrac{}{}{0ex}{}{i}{r}\right)g^{\left(r\right)}{\partial }^{i-r}+\dots +\left(\genfrac{}{}{0ex}{}{i}{i-1}\right)g^{(i-1)}\partial +\left(\genfrac{}{}{0ex}{}{i}{i}\right)g^{\left(i\right)},\hfill \end{array}$$

(18)

$$\begin{array}{c}\partial x^k=x^k\partial +k{x}^{k-1},{\partial }^{i}x=x{\partial }^i+i{\partial }^{i-1},\partial x^{-s}=x^{-s}\partial -s{x}^{-s-1},\hfill \end{array}$$

(19)

$$\begin{array}{c}{\partial }^i{x}^{-k}=x^{-k}{\partial }^i-\left(\genfrac{}{}{0ex}{}{i}{1}\right)k{x}^{-k-1}{\partial }^{i-1}-\left(\genfrac{}{}{0ex}{}{i}{2}\right)k(-k-1)x^{-k-2}{\partial }^{i-2}-\dots -\hfill \\ \left(\genfrac{}{}{0ex}{}{i}{t}\right)k(-k-1)\dots (-k-t+1)x^{-k-t}{\partial }^{i-t}-\dots -\left(\genfrac{}{}{0ex}{}{i}{i}\right)k(-k-1)\dots (-k-i+1)x^{-k-i}\hfill \end{array}$$

(20)

hold where we denote $g\left(x\right)$ as g which is really an operator on A and $k,i,s\in \mathbb{N}.$

Lemma 3.

For the algebra $A\left(\mathbb{F}[e^{\pm x_1},\dots ,e^{\pm x_n}]\right)$, the following

$$\begin{array}{c}{\partial }_u^s\widehat{e^{k{x}_u}}=\widehat{e^{k{x}_u}}{\partial }_u^s+\left(\begin{array}{c}s\\ 1\end{array}\right)k^{s-(s-1)}\widehat{e^{k{x}_u}}{\partial }_u^{s-1}+\dots +\hfill \\ \left(\genfrac{}{}{0ex}{}{s}{t}\right)k^{s-(s-t)}\widehat{e^{k{x}_u}}{\partial }_u^{s-m}+\dots +\left(\genfrac{}{}{0ex}{}{s}{s-1}\right)k^{s-1}\widehat{e^{k{x}_u}}{\partial }_u+k^s\widehat{e^{k{x}_u}},\hfill \end{array}$$

(21)

$$\begin{array}{c}{\partial }_u\widehat{e^{k{x}_u}}=\widehat{e^{k{x}_u}}{\partial }_u+k\widehat{e^{k{x}_u}},\mathrm{i}.\mathrm{e}.,\widehat{e^{-k{x}_u}}{\partial }_u\widehat{e^{k{x}_u}}={\partial }_u+k\hfill \end{array}$$

(22)

hold for $1\le u\le n,$$k\in \mathbb{Z},$ and $t\le s\in \mathbb{N}.$

Proof. 

The statement follows by induction on s, using the basic commutation relation ${\partial }_u\widehat{e^{k{x}_u}}=\widehat{e^{k{x}_u}}{\partial }_u+k\widehat{e^{k{x}_u}}.$ □

• The operator identities derived here provide essential tools for manipulating exponential and polynomial components within Weyl-type constructions. They will be repeatedly employed in the study of generalized Weyl-type, Witt-type, and non-associative algebras presented in the following sections.

5. More Generalized Weyl-Type Simple Algebras

Similar to the $\mathbb{F}$-algebra $\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]$, for fixed nonzero scalars ${\iota }_1,\dots ,{\iota }_n$ and for additive subgroups ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_n$ of $\mathbb{F}$, by Lemma 1, we define the $\mathbb{F}$-algebra $\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x]$$:=\mathbb{F}[e^{\pm x_1^{p_1}{e}^{{\iota }_1{x}_1}},\dots ,e^{\pm x_n^{p_n}{e}^{{\iota }_n{x}_n}},$$e^{{\mathcal{A}}_1{x}_1},\dots ,e^{{\mathcal{A}}_n{x}_n},x_1,\dots ,x_n]$ where ${\mathcal{A}}_u$ contains $\mathbb{Z}$ and ${\iota }_u$, $1\le u\le n$. So we define the Weyl-type algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x]\right)$ on $\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x]$ with the obvious elements and similar appropriate relations of $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x]\right)$ as $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^x},e^{\pm x},x]\right)$. The algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x]\right)$ is a non-commutative algebra and it does not have a zero divisor. Similar to the $\mathbb{F}$-algebra $\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x]$, for fixed scalars ${\iota }_{11},\dots ,{\iota }_{1r}\in {\mathcal{A}}_1,\dots ,{\iota }_{n1},\dots ,$${\iota }_{nm}\in {\mathcal{A}}_n,$ we define the $\mathbb{F}$-algebra $\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]$ $=\mathbb{F}[e^{\pm x_1^{p_1}{e}^{{\iota }_{11}{x}_1}},\dots ,$$e^{\pm x_1^{p_1}{e}^{{\iota }_{1r}{x}_1}},\dots ,$ $e^{\pm x_n^{p_n}{e}^{{\iota }_{n1}{x}_n}},\dots ,$$e^{\pm x_n^{p_n}{e}^{{\iota }_{nm}{x}_n}},$ $e^{{\mathcal{A}}_1{x}_1},\dots ,$$e^{{\mathcal{A}}_n{x}_n},$$x_1^{{\mathcal{A}}_1},\dots ,x_n^{{\mathcal{A}}_n}]$. So we can define the Weyl-type algebra $A\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$$e^{\mathcal{A}x},x^{\mathcal{A}}\left]\right).$ The Weyl-type algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\pm 1}]\right)$ is defined on the $\mathbb{F}$-algebra $\mathbb{F}[e^{\pm x_1^{p_1}{e}_1^{{\iota }_1}},\dots ,$$e^{\pm x_n^{p_n}{e}_n^{{\iota }_n}},e^{{\mathcal{A}}_1},\dots ,e^{{\mathcal{A}}_n},x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ as well. These algebras generalizes the constructions presented in [17,19]. From now on, an additive subgroup $\mathcal{A}$ of $\mathbb{F}$ has an appropriate order which is compatible with the lexicographic order appropriately and it is finitely generated as well. For example, $\mathcal{A}$ can be the additive group $\mathbb{Q}[{\iota }_1,{\iota }_2]=\{a+b{\iota }_1+c{\iota }_2|a,b,c\in \mathbb{Q}\}$ where ${\iota }_1$ and ${\iota }_2$ are given transcendental numbers.

Theorem 4.

The Weyl-type algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right)$ and its subalgebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x]\right)$ are simple.

Proof. 

Both algebras under consideration admit a natural grading by a free abelian group, induced by the degrees of the generators $e^{\pm x^{\mathbf{p}}{e}^{\iota x}}$, $e^{\mathcal{A}x}$, and $x^{\mathcal{A}}$. The proof follows the same approach as that of Theorem 1. In particular, any nonzero ideal contains a nonzero homogeneous element of minimal degree. Using the defining Weyl-type relations, this element can be reduced to an element of the base algebra, which forces the ideal to contain the identity. Hence the ideal coincides with the whole algebra. The same argument applies to the subalgebra, since it inherits the grading and derivation structure from the larger algebra. Therefore, both algebras are simple. □

Corollary 2.

The Weyl-type algebra $A\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$$e^{\mathcal{A}x},x^{\pm 1}\left]\right)$ is simple.

Proof. 

The algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\pm 1}]\right)$ inherits the same Weyl-type relations and grading framework as in Theorem 1. The localization with respect to x does not introduce any nontrivial ideals. Consequently, the simplicity argument of Theorem 1 applies without modification, and the algebra is simple. □

Theorem 5.

If either $p_1$ and $p_2$ are different integers, or transcendental numbers ${\tau }_1$ and ${\tau }_2$ are not equivalent, then the algebras $A\left(\mathbb{F}\right[$$e^{\pm x^{p_1}{e}^{{\tau }_{1}x}},e^{\mathcal{A}x},x\left]\right)$ and $A\left(\mathbb{F}[e^{\pm x^{p_2}{e}^{{\tau }_{2}x}},e^{\mathcal{A}x},x]\right)$ are not isomorphic where $\mathcal{A}$ is an additive group which contains ${\tau }_1$ and ${\tau }_2.$

Proof. 

If $p_1\ne p_2,$ then the proof of the theorem is similar to the proof of Proposition 1. Assume that $p_1=p_2$ and let ${\tau }_1$ and ${\tau }_2$ are not equivalent. Put $\theta \left(e^x\right)=c_1{e}^{a_1{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{1}x}$, $\theta \left(e^{{\tau }_{1}x}\right)=c_2{e}^{a_2{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{2}x}$ and $\theta \left(e^{x^{p_1}{e}^{{\tau }_{1}x}}\right)=c_3{e}^{a_3{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{3}x}$ with appropriate coefficients and indices. Let us put $\theta (\partial )$ as follows:

$$\begin{array}{c}\theta (\partial )=c{e}^{a{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{bx}{x}^d{\partial }^i+\dots \hfill \end{array}$$

(23)

where $e^{a{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{bx}{x}^d{\partial }^i$ is the maximal term of $\theta (\partial )$ with respect to the order ${>}_{exp}$ and … is the sum of the remaining terms of $\theta (\partial )$ with appropriate coefficients and indices. By (24) and $\theta (\partial )\theta \left(e^x\right)-\theta \left(e^x\right)\theta (\partial )=\theta \left(e^x\right),$ we have that

$$\begin{array}{c}c{c}_1{e}^{a{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{bx}{x}^d{\partial }^i{e}^{a_1{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{1}x}+\dots -c{c}_1{e}^{(a+a_1)x^{p_1}{e}^{{\tau }_{2}x}}{e}^{(b+b_1)x}{x}^d{\partial }^i-\dots \hfill \\ =c_1{e}^{a_1{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{1}x}.\hfill \end{array}$$

(24)

This implies that $a=b=d=0$ and $i=1,$ i.e., $\theta (\partial )=c\partial +\dots$ hold. So by (24), these conditions imply that $a_1=0$ and $c{b}_1=1,$ i.e., $\theta (\partial )=c\partial +\dots$ and $\theta \left(e^x\right)=c_1{e}^{b_{1}x}.$ By $\theta (\partial )\theta \left(e^{{\tau }_{1}x}\right)-\theta \left(e^{{\tau }_{1}x}\right)\theta (\partial )={\tau }_1\theta \left(e^{{\tau }_{1}x}\right)$, we have that

$$\begin{array}{c}c{c}_2\partial e^{a_2{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{2}x}-c{c}_2{e}^{a_2{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{2}x}\partial =c_2{\tau }_1{e}^{a_2{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{2}x}.\hfill \end{array}$$

This implies that $a_2=0$ and $c{b}_2={\tau }_1,$ i.e., $\theta \left(e^{{\tau }_{1}x}\right)=c_2{e}^{b_2{\tau }_{1}x}$. By $(c\partial +\dots )\theta \left(x\right)-\theta \left(x\right)(c\partial +\dots )=1,$ we have that $\theta \left(x\right)=c^{-1}x+c^{\prime }$ where $c^{\prime }\in \mathbb{F}.$ By

$$\begin{array}{c}\theta (\partial )\theta \left(e^{x^{p_1}{e}^{{\tau }_{1}x}}\right)-\theta \left(e^{x^{p_1}{e}^{{\tau }_{1}x}}\right)\theta (\partial )=\theta \left(e^{x^{p_1}{e}^{{\tau }_{1}x}}{e}^{{\tau }_{1}x}(p_1{x}^{p_1-1}+{\tau }_1{x}^{p_1})\right)\hfill \\ =\theta \left(e^{x^{p_1}{e}^{{\tau }_{1}x}}\right)\theta \left(e^{{\tau }_{1}x}\right)\theta (p_1{x}^{p_1-1}+{\tau }_1{x}^{p_1}).\hfill \end{array}$$

(25)

So we have that

$$\begin{array}{c}c{c}_3\partial e^{a_3{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{3}x}-c{c}_3{e}^{a_3{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{3}x}\partial \hfill \\ =c_2{c}_3{e}^{a_3{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{3}x}{e}^{b_2{\tau }_{1}x}(p_1{(c^{-1}x+c^{\prime })}^{p_1-1}+{\tau }_1{(c^{-1}x+c^{\prime })}^{p_1})\hfill \end{array}$$

(26)

This implies that

$$\begin{array}{c}{a}_{3}c{c}_3{p}_1{e}^{a_3{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{(b_3+{\tau }_2)x}{x}^{p_1-1}+c{c}_2{\tau }_2{e}^{a_3{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{(b_3+{\tau }_2)x}{x}^{p_1}+b_{3}c{c}_3{e}^{a_3{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{b_{3}x}\hfill \\ =c_2{c}_3{e}^{a_3{x}^{p_1}{e}^{{\tau }_{2}x}}{e}^{(b_3+b_1{\tau }_1)x}(p_1{(c^{-1}x+c^{\prime })}^{p_1-1}+{\tau }_1{(c^{-1}x+c^{\prime })}^{p_1})\hfill \end{array}$$

(27)

This gives that ${\tau }_2=b_1{\tau }_1$ and $c^{\prime }\ne 0$. Since the equality (27) does not hold, we have a contradiction. Thus the algebras $A\left(\mathbb{F}\right[$$e^{\pm x^{p_1}{e}^{{\tau }_1}},e^{\mathcal{A}x},x\left]\right)$ and $A\left(\mathbb{F}[e^{\pm x^{p_2}{e}^{{\tau }_2}},e^{\mathcal{A}x},x]\right)$ are not isomorphic. □

Theorem 6.

There are uncountably many pairwise non-isomorphic simple Weyl-type algebras.

Proof. 

Since there are uncountably many non-equivalent transcendental numbers, the proof of the theorem is straightforward by Theorem 2. □

6. Generalized Witt-Type Lie Algebras and Non-Associative Algebras

For the $\mathbb{F}$-algebra $\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]$$=\mathbb{F}[e^{\pm x_1^{p_1}{e}^{{\iota }_{11}{x}_1}},\dots ,$$e^{\pm x_1^{p_1}{e}^{{\iota }_{1r}{x}_1}},\dots ,$ $e^{\pm x_n^{p_n}{e}^{{\iota }_{n1}{x}_n}},\dots ,$$e^{\pm x_n^{p_n}{e}^{{\iota }_{nm}{x}_n}},$ $e^{{\mathcal{A}}_1{x}_1},\dots ,$$e^{{\mathcal{A}}_n{x}_n},$$x_1^{{\mathcal{A}}_1},\dots ,x_n^{{\mathcal{A}}_n}]$, we define the Witt-type Lie algebra $W\left(\mathbb{F}\right[$$e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}\left]\right)$$=\left\{f{\partial }_u\right|f\in$$\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}],$$1\le u\le n\}$ with the following Lie bracket:

$$\begin{array}{c}[f{\partial }_v,g{\partial }_w]=f{\partial }_v\circ g{\partial }_w-g{\partial }_w\circ f{\partial }_v=f\left({\partial }_v\left(g\right)\right){\partial }_w-g\left({\partial }_w\left(f\right)\right){\partial }_v\hfill \end{array}$$

(28)

where $f{\partial }_v,g{\partial }_w\in$$W\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right)$ (see [18,19,20,21,22]). For the $\mathbb{F}$-algebra $\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x]$, we define the Witt-type algebra $W\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x]\right)$ as $W\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$$e^{\mathcal{A}x},$$x^{\mathcal{A}}\left]\right).$ Similarly, we define the Witt-type Lie algebra $W\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\pm 1}]\right)$ as well. Because of the equality (27), it is natural to introduce a non-associative product whose antisymmetrization recovers the Witt-type Lie bracket. In this way, the non-associative product ∗ encodes the derivation action underlying the Witt-type Lie algebra, while discarding the skew-symmetry condition. Accordingly, we define the non-associative algebra $N\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right)$ with the multiplication ∗ given by

$$\begin{array}{c}f{\partial }_u\ast g{\partial }_v=f\left({\partial }_u\left(g\right)\right){\partial }_v\hfill \end{array}$$

(29)

for $f{\partial }_u,g{\partial }_v\in$$N\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right).$ Then the non-associative algebra $N\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$$e^{\mathcal{A}x},x^{\mathcal{A}}\left]\right)$ has the right identity ${\sum }_{1\le w\le n}(x_w+c_w){\partial }_w$ where $c_w\in \mathbb{F},$$1\le w\le n.$ Similarly, for any associative-commutative algebra A, using all the derivations of A and the similar multiplication of (29), we define the non-associative algebra $N\left(A\right)$ on A (see [19]). For the non-associative algebra $N\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$$e^{\mathcal{A}x},x^{\mathcal{A}}\left]\right)$, we define the Lie bracket on the algebra as (28), then we have the anti-symmetrized algebra $N^{-}\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$ $e^{\mathcal{A}x},x^{\mathcal{A}}\left]\right)$ which is the Witt-type Lie algebra $W\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$$e^{\mathcal{A}x},x^{\mathcal{A}}\left]\right)$ (see [23]). Similarly, the following algebras can be defined analogously, and the corresponding theorems hold with evident proofs, which are therefore omitted. We can also define the non-associative algebra $N\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$$e^{\mathcal{A}x},x^{\mathcal{A}}\left]\right)$ on $\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]$ and its antisymmetrized algebra $N\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$$e^{\mathcal{A}x},x^{\mathcal{A}}{\left]\right)}^{-}$ which is the Witt-type Lie algebra $W\left(\mathbb{F}\right[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},$$e^{\mathcal{A}x},x^{\mathcal{A}}\left]\right)$. Similarly, the Witt-type Lie algebras $W\left(\mathbb{F}[x_1^{{\mathcal{A}}_1},\dots ,x_n^{{\mathcal{A}}_n}]\right)$ and $W\left(\mathbb{F}[e^{{\mathcal{A}}_1{x}_1},\dots ,e^{{\mathcal{A}}_n{x}_n}]\right)$ can be defined (see [24]) and we define the non-associative algebras $N\left(\mathbb{F}[x_1^{{\mathcal{A}}_1},\dots ,x_n^{{\mathcal{A}}_n}]\right)$ and $N\left(\mathbb{F}[e^{{\mathcal{A}}_1{x}_1},\dots ,e^{{\mathcal{A}}_n{x}_n}]\right)$. Thus, we conclude that for a given associative algebra A, the algebra $W\left(A\right)$ represents a Witt-type algebra, while $N\left(A\right)$ denotes the corresponding non-associative algebra.

Proposition 3.

The Witt-type Lie algebras $W\left(\mathbb{F}[x_1^{{\mathcal{A}}_1},\dots ,x_n^{{\mathcal{A}}_n}]\right)$ and $W\left(\mathbb{F}[e^{{\mathcal{A}}_1{x}_1},\dots ,e^{{\mathcal{A}}_n{x}_n}]\right)$ are isomorphic.

Proof. 

If we define an F-linear map $\theta$ from the Witt-type Lie algebras from $W\left(\mathbb{F}[x_1^{{\mathcal{A}}_1},\dots ,x_n^{{\mathcal{A}}_n}]\right)$ to $W\left(\mathbb{F}[e^{{\mathcal{A}}_1{x}_1},\dots ,e^{{\mathcal{A}}_n{x}_n}]\right)$ such that for any $x_u^{a_u+1}{\partial }_u$, $1\le u\le n,$ of the Witt-type Lie algebras $W\left(\mathbb{F}[x_1^{{\mathcal{A}}_1},\dots ,x_n^{{\mathcal{A}}_n}]\right)$, $\theta \left(x_u^{a_u+1}{\partial }_u\right)=e^{a_u{x}_u}{\partial }_u,$ then $\theta$ can be linearly extended to a Lie algebra isomorphism between them. □

• The following unified results follow from arguments analogous to those used in Theorem 1. In each case, the non-associative or Witt-type algebra is constructed over a graded commutative base algebra, and the grading is compatible with the defining derivation or Lie bracket. Any nonzero ideal therefore contains a nonzero homogeneous element of minimal degree, which implies that the ideal coincides with the whole algebra. The same reasoning applies to the corresponding subalgebras, as they inherit the relevant grading and structural relations.

Theorem 7.

Let A be one of the following $\mathbb{F}$-algebras:

$$\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}],\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x^{\mathcal{A}}],\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x}],$$

and let $A^{\prime }$ denote any of the corresponding subalgebras

$$\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x],\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x}],\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x}].$$

Then the non-associative algebra $N\left(A\right)$ and each of its subalgebras $N\left(A^{\prime }\right)$ are simple.

Theorem 8.

Let A be one of the $\mathbb{F}$-algebras

$$\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x^{\mathcal{A}}]\mathrm{or}\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x],$$

and let $A^{\prime }$ denote the corresponding subalgebra

$$\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\mathrm{or}\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x}],$$

respectively. Then the Witt-type Lie algebra $W\left(A\right)$ and its Lie subalgebra $W\left(A^{\prime }\right)$ are simple.

Corollary 3.

The Witt-type Lie algebra $W\left(\mathbb{F}[x_1^{{\mathcal{A}}_1},\dots ,x_n^{{\mathcal{A}}_n}]\right)$ and the non-associative algebra $N\left(\mathbb{F}[x_1^{{\mathcal{A}}_1},\dots ,x_n^{{\mathcal{A}}_n}]\right)$ are simple.

Note 2.

The automorphism group $Au{t}_{Lie}\left(W\left(\mathbb{F}[e^{\pm x},x]\right)\right)$ of the Lie algebra $W\left(\mathbb{F}[e^{\pm x},x]\right)$ is generated by ${\varphi }_{\alpha }$, ${\psi }_{\beta }$, and τ such that

$$\begin{array}{c}{\psi }_{\alpha }(e^{mx}{x}^n\partial )={\alpha }^m{e}^{mx}{x}^n\partial ,\hfill \\ {\psi }_{\beta }(e^{mx}{x}^n\partial )=e^{mx}{(x+\beta )}^n\partial ,\hfill \\ \tau (e^{mx}{x}^n\partial )={(-1)}^{n-1}{e}^{-mx}{x}^n\partial \hfill \end{array}$$

(30)

for $e^{mx}{x}^n\partial \in W\left(\mathbb{F}[e^{\pm x},x]\right)$, $\alpha \in {\mathbb{F}}^{*}$ and $\beta \in \mathbb{F}$ and those maps are linearly extended to $W\left(\mathbb{F}[e^{\pm x},x]\right)$ (see [20]).

• The automorphism group $Au{t}_{Lie}\left(W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)\right)$ of $W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)$ is not in the literature, let us find its automorphism group here.

Proposition 4.

The Lie algebra $W\left(\mathbb{F}[e^{\pm x^p{e}^{\iota x}},e^{\pm x},x]\right)$ is self-centralized, i.e., for every non-zero element l of $W\left(\mathbb{F}[e^{\pm x^p{e}^{\iota x}},e^{\pm x},x]\right)$, the dimension of the centralizer of l is one.

Proof. 

The proof of the proposition is standard, so let us omit (see [20]). □

Note 3.

For the Lie algebra $W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)$, if we define F-linear maps ${\varphi }_{\alpha }$ and τ such that

$$\begin{array}{c}{\psi }_{\alpha }(e^{mx}{x}^n\partial )={\alpha }^m{e}^{mx}{x}^n\partial ,\hfill \\ \tau (e^{mx}{x}^n\partial )={(-1)}^{n-1}{e}^{-mx}{x}^n\partial ,\hfill \end{array}$$

(31)

then they can be linearly extended to the Lie algebra $W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)$ for $e^{mx}{x}^n\partial \in W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)$, $\alpha \in {\mathbb{F}}^{*}$. Let H be the subgroup of $Au{t}_{Lie}\left(W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)\right)$ which is generated by $\left\{{\psi }_{\alpha }\right|\alpha \in {\mathbb{F}}^{*}\}$ and N be the subgroup of $Au{t}_{Lie}\left(W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)\right)$ which is generated by $\tau .$ (see [20]).

Theorem 9.

The automorphism group $Au{t}_{Lie}\left(W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)\right)$ of $W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)$ is generated by ${\varphi }_{\alpha }$ and τ of Note 2. Furthermore, $Au{t}_{Lie}\left(W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)\right)=N⋊H$ where H and N are subgroups of $W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)$ which are defined in Note 2.

Proof. 

Note that the Lie algebra $W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)$ is $\mathbb{Z}$-graded as follows:

$$W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)={\oplus }_{a\in \mathbb{Z}}{W}_a$$

where $W_a$ is the vector subspace of $W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)$ which is spanned by $\{e^{ax}{x}^u\partial |u\in \mathbb{Z}\}.$ Let $\theta$ be an automorphism of $W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)$. Let us assume that $\theta (\partial )\notin W_0$ where $W_0$ is a simple Lie algebra which is isomorphic to the centerless Virasoro algebra. Let us put

$$\begin{array}{c}\theta (\partial )=c_{a,i}{e}^{ax}{x}^i\partial +\dots \hfill \end{array}$$

(32)

where $e^{ax}{x}^i\partial$, $a\ne 0,$ is the maximal term of $\theta (\partial )$ with respect to the lexicographic order and with appropriate coefficients. By $\left[\theta \right(\partial ),\theta (x\partial \left)\right]=\theta (\partial ),$ the maximal term of $\theta (x\partial )$ is $e^{ax}{x}^i\partial$. So there is a non-zero scalar c such that $\left[\theta \right(\partial ),\theta (x\partial -c\partial \left)\right]\ne \theta (\partial )$. This contradiction shows that $\theta (\partial )\in W_0$. This argument shows that any nonzero homogeneous component of $\theta (\partial )$ outside $W_0$ leads to a contradiction with the Lie bracket relations, and hence $\theta (\partial )$ must Lie in $W_0$. By changing the roles of $\theta (\partial )$ and $\theta (x\partial ),$ we have that $\theta (x\partial )\in W_0.$ Similarly, by $[\theta (x\partial ),\theta (x^i\partial )]=(i-1)x^i\partial$, we have that $\theta (x^i\partial )\in W_0$ for $i\in \mathbb{Z}.$ Since $W_0$ is a simple Lie algebra and $\theta$ is an automorphism preserving Lie brackets, the image $\theta \left(W_0\right)$ must coincide with $W_0$, i.e., $\theta \left(W_0\right)=W_0.$ It is well-known that the automorphism group $Aut\left(W_0\right)$ is generated by two kinds of automorphisms $\varphi$ and $\psi$ of Note 2 such that for any basis element $\theta (x^{\alpha }\partial )$ of $W_0$,

$$\begin{array}{c}\varphi (x^{\alpha }\partial )=c^{1-\alpha }{x}^{\alpha }\partial \hfill \end{array}$$

(33)

$$\begin{array}{c}\psi (x^{\alpha }\partial )={(-1)}^{\alpha }{c}^{\alpha -1}{x}^{2-\alpha }\partial \hfill \end{array}$$

(34)

respectively, i.e., they can be linearly extended to the Lie algebra automorphisms of $W_0$ where $c\in {\mathbb{F}}^{*}$ and $\alpha \in \mathbb{Z}.$ Let us put

$$\begin{array}{c}\theta (e^x\partial )=c_{b,k}{e}^{bx}{x}^k\partial +\dots \hfill \end{array}$$

(35)

where $e^{bx}{x}^k\partial$ is the maximal term of $\theta (e^x\partial )$ as (32). Let us assume that $\theta$ holds (34), i.e., $\theta (\partial )=c{x}^2\partial$. This implies that by (35), the equality $[c{x}^2\partial ,\theta (e^x\partial )]=\theta (e^x\partial )$ does not hold. This contradiction shows that (33) holds. This implies that by (30), we have that $c=-1$. By (30), we have that $\theta (e^{mx}{x}^n\partial )={\alpha }^m{e}^{mx}{x}^n\partial$ and $\theta (e^{mx}{x}^n\partial )={(-1)}^{n-1}{e}^{-mx}{x}^n\partial$ for $\alpha \in {\mathbb{F}}^{*}$ and for any basis element $e^{mx}{x}^n\partial$. Since N is a normal subgroup of $Au{t}_{Lie}\left(W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)\right)$ and $H\cap N=\left\{id\right\}$, every automorphism can be uniquely expressed as a product of elements from N and H, which yields the semidirect product decomposition $Au{t}_{Lie}\left(W\left(\mathbb{F}[e^{\pm x},x^{\pm 1}]\right)\right)=N⋊H$. □

Lemma 4.

The matrix algebra $M_m\left(\mathbb{F}\right)$ is isomorphic to the subalgebra $\mathcal{G}$ of $N\left(\mathbb{F}\right[e^{\pm x^p{e}^{\iota x}},$$e^{\pm x},x^{\pm 1}\left]\right)$ which is spanned by $\left\{(x_u+c_u){\partial }_v\right|c_u\in \mathbb{F},1\le u,v\le m\}.$

Proof. 

If we define an F-linear map from $M_m\left(\mathbb{F}\right)$ to $\mathcal{G}$ as follows:

$$\theta \left({ϵ}_{uv}\right)=(x_u+c_u){\partial }_v,1\le u,v\le m,$$

where $\mathcal{G}$ is a subring of $Ho{m}_{\mathbb{F}}(V_d,V_d)$ and $c_u$ is a fixed scalar for $1\le u\le m.$ Then $\theta$ can be linearly extended to the algebra isomorphism from $M_m\left(\mathbb{F}\right)$ to $\mathcal{𝒢}$. □

7. Applications and Future Directions

The algebraic structures developed in this work, including the Weyl-type, Witt-type, and related non-associative algebras over expolynomial rings, are primarily studied here from a structural and theoretical perspective. In this context, their intrinsic non-commutativity, graded composition, and operator-based formulation suggest possible directions for future applications, particularly in areas where algebraic complexity plays a significant role. For instance, elements of the Weyl-type algebra $A\left(\mathbb{Z}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\right)$ can be expressed as finite sums of the form

$$l=\sum _j{c}_{a_j,b_j,d_j,i_j}{e}^{a_j{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{b_{j}x}{x}^{d_j}{\partial }^{i_j},$$

where the coefficients and exponents are determined by the underlying algebraic structure. Powers of such elements,

$$l^h={\left(\sum _j{c}_{a_j,b_j,d_j,i_j}{e}^{a_j{x}^{\mathbf{p}}{e}^{\iota x}}{e}^{b_{j}x}{x}^{d_j}{\partial }^{i_j}\right)}^h,$$

lead to increasingly intricate combinations governed by the non-commutative multiplication rules and the grading of the algebra. From a theoretical point of view, this behavior illustrates how relatively simple generators can give rise to highly structured and nonlinear expressions.

Such algebraic complexity may be of interest in future investigations related to cryptography and information security, where non-commutative algebraic systems have previously been explored as potential sources of computational hardness. In particular, the presence of rich grading structures, large families of non-isomorphic algebras, and non-associative variants $N\left(\mathbb{Z}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\right)$ suggests possible relevance to algebraic constructions aimed at increasing resistance to structural or algebraic attacks. At present, the focus remains on establishing the underlying algebraic framework, which provides a foundation for the development of cryptographic constructions. A systematic study of how these algebraic structures could be incorporated into practical cryptographic primitives—such as key exchange protocols, pseudorandom sequence generation, or algebra-based encryption models—will be explored in future studies. This offers a promising direction building upon the algebraic foundations established in this paper.

8. Conclusions

In this work, we have constructed and analyzed several classes of generalized Weyl-type algebras $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right)$, their corresponding Witt-type Lie algebras $W\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right)$, and the associated non-associative algebras $N\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right)$. Each of these algebras has been shown to be simple under natural gradation and derivation structures, with the simplicity following from graded ideal arguments developed in the main body of the paper. In addition, the automorphism groups of several of these algebras were explicitly described, revealing semidirect product structures that generalize classical results for Weyl and Witt algebras.

Furthermore, by varying transcendental parameters $\iota$, we obtain uncountably many pairwise non-isomorphic algebras, thereby extending the known family of Weyl-type constructions in a significant way. Subalgebras such as $A\left(\mathbb{Z}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\pm x},x]\right)$ possess desirable algebraic properties, including non-commutativity, absence of zero divisors, and the inclusion of transcendental coefficients. All conclusions stated here follow directly from the structural results established in the preceding sections. These characteristics suggest potential applications in algebraic cryptography, where non-commutative and structurally rich algebras can enhance computational security. Future investigations will focus on exploring such applications and establishing explicit connections between the constructed algebras and cryptographic frameworks, particularly those involving recursive key exchange and algebraic encoding schemes.