Structural Properties of The Clifford–Weyl Algebra ${𝒜}_q^{\pm }$

Abstract

The Clifford–Weyl algebra ${𝒜}_q^{\pm }$, as a class of solvable polynomial algebras, combines the anti-commutation relations of Clifford algebras ${𝒜}_q^{+}$ with the differential operator structure of Weyl algebras ${𝒜}_q^{-}$. It exhibits rich algebraic and geometric properties. This paper employs Gröbner–Shirshov basis principles in concert with Poincaré–Birkhoff–Witt (PBW) basis methodology to delineate the iterated skew polynomial structures within ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$. By constructing explicit PBW generators, we analyze the structural properties of both algebras and their modules using constructive methods. Furthermore, we prove that ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$ are Auslander regular, Cohen–Macaulay, and Artin–Schelter regular. These results provide new tools for the representation theory in noncommutative geometry.

1. Introduction

The classical Clifford algebra was introduced in the 19th century by William Kingdon Clifford. In the 1980s, Drinfeld and Jimbo introduced quantum enveloping algebras (universally known as quantum groups), initiating a paradigm shift in algebraic deformation theory. Their foundational work established the systematic study of quantum groups as quantized analogues of classical algebraic structures. These quantum groups are obtained through q-deformations (parameterized by $q\in {\mathbb{C}}^{\times }$, $q\ne \pm 1$) of Lie algebras $\mathfrak{g}$. Their representation theory $\mathrm{Rep}\left({𝒰}_q\left(\mathfrak{g}\right)\right)$ reveals deep connections with integrable systems [1] and quantum field theory. Notably, quantum Weyl algebras form fundamental q-deformed structures, whose systematic theory was developed by Guianquanto and Zhang [2]. These algebras serve as essential algebraic tools for deformation quantization.

Hayashi introduced a q-deformation of Clifford algebras by defining generators ${\left\{e_i\right\}}_{i=1}^n$ satisfying the q-anticommutation relations and established their correspondence with representations of Drinfeld–Jimbo algebras [3]. This q-deformation preserves the essential geometric properties of the classical algebra while introducing quantum symmetries through the deformation parameter q. The classical limit occurs at $q=1$, recovering the standard Clifford algebra $𝒞\mathsf{\ell }\left(V\right)$, whereas the regime $q\ne 1$ manifests quantum effects through noncommutative geometry and entanglement operators. The proposed Clifford–Weyl algebra ${𝒜}_q^{\pm }$ synthesizes algebraic structures via Fermionic generators and Bosonic generators. This framework provides a categorical equivalence between the Drinfeld–Jimbo quantum algebras ${𝒰}_q\left(\mathfrak{g}\right)$ and the Clifford–Weyl algebra through the following canonical isomorphism:

$${𝒜}_q^{\pm }\cong {𝒰}_q\left({\mathfrak{a}}_n\right)\otimes {𝒰}_q\left({\mathfrak{c}}_n\right),$$

where the q-deformation mechanism intertwines the canonical (anti)commutation relations to systematically generate quantum groups of types $A_n$ and $C_n$.

Building upon Hayashi’s foundational work, researchers have extended the unification of q-Clifford and q-Weyl algebras to higher-dimensional and non-Abelian settings. This is achieved through the introduction of multi-index generators $\left\{e_{m,n}\right\}$ and q-parameters dependent on structure constants, defining non-Abelian q-anticommutation relations, written as follows:

$$e_{m,n}{e}_{k,l}+q^{B(m,n,k,l)}{e}_{k,l}{e}_{m,n}={\delta }_{mk}{\delta }_{nl},$$

where $B(m,n,k,l)$ is determined by the structure coefficients of the Lie algebra $\mathfrak{g}$. Such generalizations play pivotal roles in topological quantum field theories (TQFTs) and noncommutative geometry. The combined algebra admits a canonical ${\mathbb{Z}}_2$-graded superalgebra structure, where q-Clifford generators $e_i$ (odd grading), q-Weyl generators $x_j$ (even grading) interact through cross-commutation relations, written as follows:

$$e_i{x}_j=q{x}_j{e}_i\left(\mathrm{odd-even cross-commutation}\right).$$

The Clifford–Weyl algebra ${𝒜}_q^{\pm }$—which unifies Clifford-type anticommutation relations and Weyl-type commutation relations through a q-deformation—has seen deepened investigations in quantum group representation theory, supersymmetric models and noncommutative geometric frameworks.

Since the algebra ${𝒜}_q^{\pm }$ is not a solvable polynomial algebra, the analysis of its homological properties (including Artin–Schelter, Auslander, and Cohen–Macaulay regularities) presents significant challenges. We therefore initiate the first systematic study of its substructures by focusing on the subalgebras ${𝒜}_q^{+}$ (q-Clifford algebra) and ${𝒜}_q^{-}$ (q-Weyl algebra).

The paper is organized as follows. In Section 2, we compute compositions between defining relations for the algebras ${𝒜}_{\mathrm{q}}^{+}$ and ${𝒜}_{\mathrm{q}}^{-}$, and derive their Gröbner–Shirshov bases. Using irreducible elements, we construct a PBW basis, demonstrating that both are solvable polynomial algebras. In Section 3, we derive explicit skew-polynomial structures for each algebra. In Section 4, we analyze the structural properties of ${𝒜}_q^{+}$, ${𝒜}_q^{-}$ and their modules. In Section 5, we synthesize the main conclusions and propose potential extensions to other related areas.

2. Preliminaries

We first recall some notations and definitions from [4,5,6,7].

Let K be a field with multiplicative group $K^{*}=K\setminus \left\{0\right\}$ and A a non-empty set of symbols. Let $\langle A\rangle$ be the free monoid on A(with identity 1) and $K\langle A\rangle$ the free associative K-algebra generated by A. To define the leading term $\overline{h}$ for each element $h\in K\langle A\rangle$, we fix a monomial order ≺ on $\langle A\rangle$, which then induces an order on $K\langle A\rangle$. An element $h\in K\langle A\rangle$ is called monic if the coefficient of its leading term $\overline{h}$ equals 1. For two monic elements $h,j\in K\langle A\rangle$ with leading terms $\overline{h}$ and $\overline{j}$, their composition is defined as follows:

(1)

If there exist $c,d\in \langle A\rangle$ such that $\overline{h}c=d\overline{j}=\omega$, and $\left|\overline{c}\right|+\left|\overline{d}\right|>\left|w\right|$ (where $\left|w\right|$ denotes the length of w), then the intersection composition of h and j relative to w is defined as ${(h,j)}_{\omega }=hc-dj$;

(2)

If there exist $c,d\in \langle A\rangle$ such that $\overline{h}=c\overline{j}d=\omega$, then their inclusion composition relative to w is defined as ${(h,j)}_{\omega }=h-cjd$.

Remark 1. 

For both intersection and inclusion compositions defined above, the leading term satisfies $\overline{{(h,j)}_{\omega }}\prec \omega$.

Let $S\subseteq K\langle A\rangle$ be monic. For $h,j\in K\langle A\rangle$ and $\omega \in \langle A\rangle$, we write $h\equiv jmod(S;\omega )$ if $h-j={\sum }_i{\alpha }_i{c}_i{s}_i{d}_i$ with ${\alpha }_i\in K$, $c_i,d_i\in \langle A\rangle$, $s_i\in S$, and $\overline{c_i{s}_i{d}_i}\prec \omega$. A composition $(h,j)\omega$ is trivial modulo S if it satisfies $(h,j)\omega \equiv 0mod(S,\omega )$, meaning ${(h,j)}_{\omega }={\sum }_i{\alpha }_i{c}_i{s}_i{d}_i$ with $\overline{c_i{s}_i{d}_i}\prec \omega$.

Definition 1. 

If all compositions among the polynomials in S are trivial modulo $S,$ then S is called a Gröbner–Shirshov basis in $K\langle A\rangle$ for the ideal $\left(S\right).$ If, in addition, no composition of inclusion exists in S, then S is called a minimal Gröbner–Shirshov basis.

Suppose $D=K[d_1,\dots ,d_n]$ is a finitely generated K-algebra admitting a PBW basis, written as follows:

$$𝓑=\left\{d^k=d_1^{k_1}{d}_2^{k_2}\cdots d_n^{k_n}\mid k=(k_1,k_2,\dots ,k_n)\in {\mathbb{N}}^n\right\}.$$

Let ≺ be a total ordering on the PBW basis $𝓑$. For any nonzero element $g\in D$, there exists a unique representation, written as follows:

$$g={\lambda }_1{d}^{k\left(1\right)}+{\lambda }_2{d}^{k\left(2\right)}+\cdots +{\lambda }_j{d}^{k\left(j\right)},$$

where $d^{k\left(1\right)}\prec d^{k\left(2\right)}\prec \cdots \prec d^{k\left(j\right)}$, with ${\lambda }_i\in K^{*}$, $d^{k\left(i\right)}=d_1^{k_{1i}}{d}_2^{k_{2i}}\cdots d_n^{k_{ni}}\in 𝓑$, and $1⩽i⩽j$.

The elements of PBW basis $𝓑$ are called monomials. For any nonzero element $g\in D$, its leading monomial, denoted $\mathbf{LM}\left(g\right)$, is defined as the unique maximal monomial $d^{k\left(j\right)}\in 𝓑$ appearing in the canonical representation of g with respect to the total order ≺. Correspondingly: the scalar coefficient ${\lambda }_j\in K^{*}$ associated with the leading monomial $\mathbf{LM}\left(g\right)$ of the element g is referred to as the leading coefficient of g, and it is denoted by $\mathbf{LC}\left(g\right)$; the $\mathbf{LT}\left(g\right)={\lambda }_j{d}^{k\left(j\right)}$ is defined as the leading term of g.

Definition 2. 

Let $D=K[d_1,\dots ,d_n]$ be a finitely generated K-algebra with a PBW basis $𝓑$. A monomial order is defined as the total ordering ≺ specified on the set $𝓑$, if it satisfies the following conditions:

(1) 

≺ is a well-order;

(2) 

For $d^{\gamma },d^{\alpha },d^{\beta },d^{\eta }\in 𝓑$, if $d^{\gamma }\ne 1$, $d^{\beta }\ne d^{\gamma }$, and $d^{\gamma }=\mathbf{LM}\left(d^{\alpha }{d}^{\beta }{d}^{\eta }\right)$, then $d^{\beta }\prec d^{\gamma }$ (hence $1\prec d^{\gamma }$ for all $d^{\gamma }\ne 1$);

(3) 

For $d^{\gamma },d^{\alpha },d^{\beta },d^{\eta }\in 𝓑$, if $d^{\alpha }\prec d^{\beta }$, $\mathbf{LM}\left(d^{\gamma }{d}^{\alpha }{d}^{\eta }\right)\ne 0$, $\mathbf{LM}\left(d^{\gamma }{d}^{\beta }{d}^{\eta }\right)\notin \{0,1\}$, then $\mathbf{LM}\left(d^{\gamma }{d}^{\alpha }{d}^{\eta }\right)\prec \mathbf{LM}\left(d^{\gamma }{d}^{\beta }{d}^{\eta }\right).$

Definition 3. 

The finitely generated K-algebra $D=K[d_1,\dots ,d_n]$ is called a solvable polynomial algebra if the following conditions are satisfied:

(1) 

D has a PBW basis, written as follows:

$$𝓑=\left\{d^k=d_1^{k_1}{d}_2^{k_2}\cdots d_n^{k_n}\mid k=(k_1,k_2,\dots ,k_n)\in {\mathbb{N}}^n\right\};$$

(2) 

There exists a monomial order ≺ on $𝓑$, and for each pair $(i,j)$ with $1⩽i<j⩽n$, there exist scalars ${\lambda }_{ji}\in K^{*}$ and elements $g_{ji}\in D$, such that the following relations:

$$d_j{d}_i={\lambda }_{ji}{d}_i{d}_j+g_{ji},$$

satisfy $\mathbf{LM}\left(g_{ji}\right)\prec d_i{d}_j$ whenever $g_{ji}\ne 0$.

Definition 4. 

Let q be a nonzero complex number with $q^4\ne 1$, n be a positive integer, and K be a field of characteristic 0. The algebra ${𝒜}_q^{\pm }$ over $K=\mathbb{C}$ is generated by the elements ${\omega }_s,{\omega }_s^{-1},{\psi }_s,{\psi }_s^{*}(1⩽s⩽n)$, subject to the following relations:

$$\begin{array}{cc}{\omega }_t{\omega }_s={\omega }_s{\omega }_t,{\omega }_s{\omega }_s^{-1}={\omega }_s^{-1}{\omega }_s=1,{\omega }_t^{-1}{\omega }_s^{-1}={\omega }_s^{-1}{\omega }_t^{-1}\hfill & 1⩽s<t⩽n,\hfill \\ {\psi }_t{\psi }_s={\psi }_s{\psi }_t,{\psi }_t^{*}{\psi }_s^{*}={\psi }_s^{*}{\psi }_t^{*}\hfill & 1⩽s<t⩽n,\hfill \\ {\omega }_t{\psi }_s{\omega }_t^{-1}-q^{\pm {\delta }_{st}}{\psi }_s={\omega }_s{\psi }_t^{*}{\omega }_s^{-1}-q^{\mp {\delta }_{st}}{\psi }_t^{*}=0\hfill & 1⩽s,t⩽n,\hfill \\ {\psi }_t^{*}{\psi }_s={\psi }_s{\psi }_t^{*}\hfill & s\ne t,\hfill \\ {\psi }_s{\psi }_s^{*}\pm q^{-2}{\psi }_s^{*}{\psi }_s={\omega }_s^2\hfill & 1⩽s⩽n,\hfill \\ {\psi }_s{\psi }_s^{*}\pm q^2{\psi }_s^{*}{\psi }_s={\omega }_s^{-2}\hfill & 1⩽s⩽n.\hfill \end{array}$$

We will consider two subalgebras ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$ of ${𝒜}_q^{\pm }$, which are called the q-Clifford algebra and q-Weyl algebra, respectively.

Now we explicitly define the q-Clifford algebra ${𝒜}_q^{+}$. Let $W=\{{\omega }_s,{\psi }_s,{\psi }_s^{*}\mid s\in I\left(n\right)\}$, where $I\left(n\right)=\{s\in \mathbb{N}\mid 1⩽s⩽n\}$. This set W serves as the set of generators for ${𝒜}_q^{+}$. In the free associative K-algebra $K\langle W\rangle$ generated by W, the defining relations S of ${𝒜}_q^{+}$ in $K\langle W\rangle$ are the following:

$$\begin{array}{ccc}\hfill & \left(a\right).f_1={\omega }_t{\omega }_s-{\omega }_s{\omega }_t\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & \left(b\right).f_2={\psi }_t{\psi }_s-{\psi }_s{\psi }_t\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & \left(c\right).f_3={\psi }_t^{*}{\psi }_s^{*}-{\psi }_s^{*}{\psi }_t^{*}\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & \left(d\right).f_4={\psi }_s{\omega }_t-q^{{\delta }_{st}}{\omega }_t{\psi }_s\hfill & 1⩽s,t⩽n,\hfill \\ \hfill & \left(e\right).f_5={\psi }_s^{*}{\omega }_t-q^{-{\delta }_{st}}{\omega }_t{\psi }_s^{*}\hfill & 1⩽s,t⩽n,\hfill \\ \hfill & \left(g\right).f_6={\psi }_s^{*}{\psi }_t-{\psi }_t{\psi }_s^{*}\hfill & s\ne t,\hfill \\ \hfill & \left(h\right).f_7={\psi }_s^{*}{\psi }_s-q^2{\psi }_s{\psi }_s^{*}+q^2{\omega }_s^2\hfill & 1⩽s⩽n.\hfill \end{array}$$

Then ${𝒜}_q^{+}\cong K\langle W\rangle /\langle S\rangle ,$ where $\langle S\rangle$ denotes the two-sided ideal of $K\langle W\rangle$ generated by the set S.

Let $W^{*}$ is a set of all finite-length words (including the empty word) formed by elements of W. First, we define the lexicographic order ${\prec }_{\mathrm{lex}}$ on W: for ${\omega }_s,{\psi }_s,{\psi }_s^{*}\in W$, $1⩽s⩽n$, written as follows:

$$\begin{array}{ccc}\hfill & {\omega }_s{\prec }_{lex}{\omega }_t\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & {\psi }_s{\prec }_{lex}{\psi }_t\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & {\psi }_s^{*}{\prec }_{lex}{\psi }_t^{*}\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & {\omega }_l{\prec }_{lex}{\psi }_j{\prec }_{lex}{\psi }_k^{*}\hfill & 1⩽l,j,k⩽n.\hfill \end{array}$$

To each ${\omega }_s,{\psi }_s,{\psi }_s^{*}$ ($1⩽s⩽n$), we assign a degree of 1. Let $\left|e\right|$ denote the degree of e, for $e,f\in W^{*}$, written as follows:

$$e{\prec }_{d-lex}f\iff \left\{\begin{array}{c}\left|e\right|<\left|f\right|\hfill \\ or\left|e\right|=\left|f\right|ande{\prec }_{lex}f.\hfill \end{array}\right.$$

It can be directly verified that ${\prec }_{d-lex}$ defines a monomial order on $W^{*}$. Specifically, ${\prec }_{d-lex}$ is a well-ordering and satisfies the following compatibility condition:

$$e{\prec }_{d-lex}f\Rightarrow wer{\prec }_{d-lex}wfr,\forall e,f,w,r\in W^{*}.$$

Theorem 1. 

Let $q\in \mathbb{C}\setminus \left\{0\right\}$ satisfy $q^4\ne 1$, and let n be a positive integer. Let $M=\langle S\rangle$ is the ideal of the algebra ${𝒜}_q^{+}$ generated by a set S. Equip $K\langle W\rangle$ with the degree-lexicographic monomial order ${\prec }_{d-lex}$. Then the set S of defining relations forms a Gröbner–Shirshov basis of the ideal M.

Proof. 

Let $(a\wedge b)$ denote the composition of relations $\left(a\right),\left(b\right)\in S$. Since the computation of compositions follows analogous procedures, we provide detailed calculations for one illustrative example below:

$$g=f_6={\psi }_s^{*}{\psi }_t-{\psi }_t{\psi }_s^{*},b=f_2={\psi }_t{\psi }_s-{\psi }_s{\psi }_t,\omega =\overline{f_6}{\psi }_s={\psi }_s^{*}\overline{f_2},1⩽s<t⩽n.$$

$$\begin{array}{cc}\hfill {(g\wedge b)}_{\omega }={(f_6\wedge f_2)}_{\omega }& =f_6{\psi }_s-{\psi }_s^{*}{f}_2;\hfill \\ \hfill & ={\psi }_s^{*}{\psi }_t{\psi }_s-{\psi }_t{\psi }_s^{*}{\psi }_s-{\psi }_s^{*}{\psi }_t{\psi }_s+{\psi }_s^{*}{\psi }_s{\psi }_t;\hfill \\ \hfill & =-{\psi }_t{\psi }_s^{*}{\psi }_s+{\psi }_s^{*}{\psi }_s{\psi }_t;\hfill \\ \hfill & \equiv -{\psi }_t{\psi }_s{\psi }_s^{*}+{\psi }_s{\psi }_s^{*}{\psi }_t;\hfill \\ \hfill & \equiv -{\psi }_s{\psi }_t{\psi }_s^{*}+{\psi }_s{\psi }_t{\psi }_s^{*};\hfill \\ \hfill & \equiv 0mod(S,\omega ).\hfill \end{array}$$

Applying analogous computations to all possible compositions shows that each resolves trivially (reduces to zero), which completes the proof of the theorem. □

Corollary 1. 

Let $q\in \mathbb{C}\setminus \left\{0\right\}$ satisfy $q^4\ne 1$, n be a positive integer, then the q-Clifford algebra ${𝒜}_q^{+}\cong K\langle W\rangle /M$ has the PBW basis given by the following:

$$𝓑=\{{\omega }_1^{k_1}{\omega }_2^{k_2}\cdots {\omega }_n^{k_n}{\psi }_1^{l_1}{\psi }_2^{l_2}\cdots {\psi }_n^{l_n}{\psi }_1^{{*}^{t_1}}{\psi }_2^{{*}^{t_2}}\cdots {\psi }_n^{{*}^{t_n}}\mid k_i,l_i,t_i\in \mathbb{N},i\in I\left(n\right)\}.$$

Now we define the q-Weyl algebra ${𝒜}_q^{-}$. Let $W^{\prime }=\{{\omega }_s^{-1},{\psi }_s,{\psi }_s^{*}\mid s\in I\left(n\right)\}$ (where $I\left(n\right)=\{s\in \mathbb{N}\mid 1⩽s⩽n\}$) be the generators of q-Weyl algebra ${𝒜}_q^{-}$, and the defining relations $S^{\prime }$ of ${𝒜}_q^{-}$ in the free associative K-algebra $K\langle W^{\prime }\rangle$ are as follows:

$$\begin{array}{ccc}\hfill & \left(a^{\prime }\right).g_1={\omega }_t^{-1}{\omega }_s^{-1}-{\omega }_s^{-1}{\omega }_t^{-1}\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & \left(b^{\prime }\right).g_2={\psi }_t{\psi }_s-{\psi }_s{\psi }_t\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & \left(c^{\prime }\right).g_3={\psi }_t^{*}{\psi }_s^{*}-{\psi }_s^{*}{\psi }_t^{*}\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & \left(d^{\prime }\right).g_4={\psi }_s{\omega }_t^{-1}-q^{{\delta }_{st}}{\omega }_t^{-1}{\psi }_s\hfill & 1⩽s,t⩽n,\hfill \\ \hfill & \left(e^{\prime }\right).g_5={\psi }_s^{*}{\omega }_t^{-1}-q^{-{\delta }_{st}}{\omega }_t^{-1}{\psi }_s^{*}\hfill & 1⩽s,t⩽n,\hfill \\ \hfill & \left(g^{\prime }\right).g_6={\psi }_s^{*}{\psi }_t-{\psi }_t{\psi }_s^{*}\hfill & s\ne t,\hfill \\ \hfill & \left(h^{\prime }\right).g_7={\psi }_s^{*}{\psi }_s-q^2{\psi }_s{\psi }_s^{*}+q^2{\omega }_s^{-1}{\omega }_s^{-1}\hfill & 1⩽s⩽n.\hfill \end{array}$$

Then ${𝒜}_q^{-}\cong K\langle W^{\prime }\rangle /\langle S^{\prime }\rangle$, where two-sided ideal of $K\langle W^{\prime }\rangle$ generated by the set $S^{\prime }$ is denoted by $\langle S^{\prime }\rangle$.

$W^{{\prime }^{*}}$ is a set of all finite-length words (including the empty word) formed by elements of $W^{\prime }$. Similarly, we define the lexicographic order ${\prec }_{\mathrm{lex}}$ on $W^{\prime }$: for ${\omega }_s^{-1},{\psi }_s,{\psi }_s^{*}\in W^{\prime }$ ($1⩽s⩽n$), written as follows:

$$\begin{array}{ccc}\hfill & {\omega }_s^{-1}{\prec }_{lex}{\omega }_t^{-1}\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & {\psi }_s{\prec }_{lex}{\psi }_t\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & {\psi }_s^{*}{\prec }_{lex}{\psi }_t^{*}\hfill & 1⩽s<t⩽n,\hfill \\ \hfill & {\omega }_l^{-1}{\prec }_{lex}{\psi }_j{\prec }_{lex}{\psi }_k^{*}\hfill & 1⩽l,j,k⩽n.\hfill \end{array}$$

And to each ${\omega }_s^{-1},{\psi }_s,{\psi }_s^{*}$ ($1⩽s⩽n$), assign a degree of 1. Let $|e^{\prime }|$ denote the degree of $e^{\prime }$, for $e^{\prime },f^{\prime }\in W^{{\prime }^{*}}$, written as follows:

$$e^{\prime }{\prec }_{d-lex}{f}^{\prime }\iff \left\{\begin{array}{c}|e^{\prime }|<|f^{\prime }|\hfill \\ or|e^{\prime }|=|f^{\prime }|and{e}^{\prime }{\prec }_{lex}{f}^{\prime }.\hfill \end{array}\right.$$

It can be directly verified that ${\prec }_{d-\mathrm{lex}}$ is a monomial order on $W^{{\prime }^{*}}$, i.e., ${\prec }_{d-\mathrm{lex}}$ is a well-order and satisfies the following: $\forall e^{\prime },f^{\prime },w,r\in W^{{\prime }^{*}},$$e^{\prime }{\prec }_{d-lex}{f}^{\prime }$ implies $w{e}^{\prime }r{\prec }_{d-lex}w{f}^{\prime }r$.

Theorem 2. 

Let $q\in \mathbb{C}\setminus \left\{0\right\}$ satisfy $q^4\ne 1$, and let n be a positive integer. Assume that $M^{\prime }=\langle S^{\prime }\rangle$ is an ideal of ${𝒜}_q^{-}$ generated by the set $S^{\prime }$. When $K\langle W^{\prime }\rangle$ is endowed with the degree lexicographic order ${\prec }_{d-lex}$, the defining relations $S^{\prime }$ of the q-Weyl algebra ${𝒜}_q^{-}$ form the Gröbner–Shirshov basis for $M^{\prime }$.

Proof. 

Analogously to the proof for ${𝒜}_q^{+}$, the set $S^{\prime }$ can be shown to form Gröbner–Shirshov basis in free associative algebra $K\langle W^{\prime }\rangle$ under the monomial order ${\prec }_{d-lex}$. □

Corollary 2. 

Let $q\in \mathbb{C}\setminus \left\{0\right\}$ satisfy $q^4\ne 1$, and let n be a positive integer, then the q-Weyl algebra ${𝒜}_q^{-}\cong K\langle W^{\prime }\rangle /M^{\prime }$ has the following PBW basis:

$$𝓑=\{{\omega }_1^{-1^{k_1}}{\omega }_2^{-1^{k_2}}\cdots {\omega }_n^{-1^{k_n}}{\psi }_1^{l_1}{\psi }_2^{l_2}\cdots {\psi }_n^{l_n}{\psi }_1^{{*}^{t_1}}{\psi }_2^{{*}^{t_2}}\cdots {\psi }_n^{{*}^{t_n}}\mid k_i,l_i,t_i\in \mathbb{N},i\in I\left(n\right)\}.$$

Theorem 3. 

Let $q\in \mathbb{C}\setminus \left\{0\right\}$ satisfy $q^4\ne 1$, n be positive integer. Then the q-Clifford algebra ${𝒜}_q^{+}$ is a solvable polynomial algebra.

Proof. 

The monomial order ${\prec }_{d-lex}$ on $𝓑$ is calculated as follows:

$${\psi }_1^{*}{\prec }_{d-lex}{\psi }_2^{*}{\prec }_{d-lex}\cdots {\prec }_{d-lex}{\psi }_n^{*}{\prec }_{d-lex}{\psi }_1{\prec }_{d-lex}\cdots {\prec }_{d-lex}{\psi }_n{\prec }_{d-lex}{\omega }_1{\prec }_{d-lex}\cdots {\prec }_{d-lex}{\omega }_n,$$

where $1{\prec }_{d-lex}x$ for all $x\in 𝓑$. Since the defining relations of ${𝒜}_q^{+}$ satisfy Definitions 2 and 3 under ${\prec }_{d-lex}$, it follows that ${𝒜}_q^{+}$ is a solvable polynomial algebra. □

Similarly, we have the following:

Theorem 4. 

Let $q\in \mathbb{C}\setminus \left\{0\right\}$ satisfy $q^4\ne 1$, n be positive integer. Then the q-Weyl algebra ${𝒜}_q^{-}$ is a solvable polynomial algebra.

3. Iterated Skew Polynomial Structure of the $\mathit{q}$-Clifford Algebra ${\mathbf{𝒜}}_{\mathit{q}}^{\mathbf{+}}$ and the $\mathit{q}$-Weyl Algebra ${\mathbf{𝒜}}_{\mathit{q}}^{\mathbf{-}}$

As established in Section 2, defining relations of the q-Clifford algebra ${𝒜}_q^{+}$ and the q-Weyl algebra ${𝒜}_q^{-}$ form Gröbner–Shirshov bases. From irreducible elements of these Gröbner–Shirshov bases, the PBW bases of ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$ can be constructed. In this section, we explicitly and iteratively demonstrate how the skew polynomial structures of ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$ depend on their PBW bases. Through this iterated skew polynomial structure, we derive additional structural properties of these algebras.

Theorem 5. 

The q-Clifford algebra ${𝒜}_q^{+}$ and the q-Weyl algebra ${𝒜}_q^{-}$ are Noetherian domains.

Proof. 

${𝒜}_q^{+}$ has no zero divisors, and for all nonzero elements $g,h\in {𝒜}_q^{+}$, $\mathbf{LM}\left(gh\right)=\mathbf{LM}\left(g\right)\mathbf{LM}\left(h\right)$. Moreover, since every nonzero one-sided ideal admits a finite Gröbner–Shirshov basis, it follows from the definition of zero divisors and the characterization of Noetherian domains that ${𝒜}_q^{+}$ is a Noetherian domain. The proof for ${𝒜}_q^{-}$ follows similarly.

Based on these findings, we now derive the skew polynomial structures of ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$. To this end, we first recall the definition of skew polynomial rings.

Definition 5. 

The skew polynomial ring $S=R[t;\sigma ,\delta ]$ is generated by a ring R and a variable t, whose elements are polynomials of the following form:

$$g=\sum _{i=1}^n{r}_i{t}^i(r_i\in R),$$

where t does not commute with elements of $R,$ but satisfies the following:

• Every polynomial has a unique representation as $g={\sum }_{i=1}^n{r}_i{t}^i$ with $r_i\in R$. Equivalently, S is a free left R-module with basis $\{1,t,t^2,\dots ,t^k,\dots$}.
• For all $r\in R$, $tr\in Rt+R$, $tr=\sigma \left(r\right)t+\delta \left(r\right)$, where $\sigma \left(r\right),\delta \left(r\right)\in R$.

The conditions imply that $\sigma :R\to R$ is a ring endomorphism satisfying $\sigma \left(1\right)=1$, and $\delta :R\to R$ is a $\sigma$-derivation (i.e., $\delta \in {End}_{\mathbb{Z}}\left(R\right)$) such that for all $x,y\in R$, the following is obtained:

$$\delta (x+y)=\delta \left(x\right)+\delta \left(y\right),\delta \left(xy\right)=\sigma \left(x\right)\delta \left(y\right)+\delta \left(x\right)y.$$

To elucidate the skew polynomial structure of the q-Clifford algebra ${𝒜}_q^{+}$ and the q-Weyl algebra ${𝒜}_q^{-}$, we begin by analyzing the concrete case of ${𝒜}_q^{+}\left(2\right)$, the specialization of ${𝒜}_q^{+}$ to $n=2$. Its structure is completely characterized by the given relations, which are direct instantiations of the general relations of ${𝒜}_q^{+}$ for two generators ($s,t\in \{1,2\}$).

The defining relations of ${𝒜}_q^{+}\left(2\right)$ are given as follows:

$$\begin{array}{cc}\hfill & {\omega }_2{\omega }_1={\omega }_1{\omega }_2,{\psi }_2{\psi }_1={\psi }_1{\psi }_2,{\psi }_2^{*}{\psi }_1^{*}={\psi }_1^{*}{\psi }_2^{*},\hfill \\ \hfill & {\psi }_1{\omega }_2={\omega }_2{\psi }_1,{\psi }_2{\omega }_1={\omega }_1{\psi }_2,{\psi }_1{\omega }_1=q{\omega }_1{\psi }_1,{\psi }_2{\omega }_2=q{\omega }_2{\psi }_2,\hfill \\ \hfill & {\psi }_1^{*}{\omega }_2={\omega }_2{\psi }_1^{*},{\psi }_2^{*}{\omega }_1={\omega }_1{\psi }_2^{*},{\psi }_1^{*}{\omega }_1=q^{-1}{\omega }_1{\psi }_1^{*},{\psi }_2^{*}{\omega }_2=q^{-1}{\omega }_2{\psi }_2^{*},\hfill \\ \hfill & {\psi }_1^{*}{\psi }_2={\psi }_2{\psi }_1^{*},{\psi }_2^{*}{\psi }_1={\psi }_1{\psi }_2^{*},\hfill \\ \hfill & {\psi }_1^{*}{\psi }_1=q^2{\psi }_1{\psi }_1^{*}-q^2{\omega }_1^2,\hfill \\ \hfill & {\psi }_2^{*}{\psi }_2=q^2{\psi }_2{\psi }_2^{*}-q^2{\omega }_2^2.\hfill \end{array}$$

By Theorem 5, ${𝒜}_q^{+}\left(2\right)$ is a domain and possesses a PBW basis, written as follows:

$$𝓑=\left\{{\omega }_1^{k_1}{\omega }_2^{k_2}{\psi }_1^{l_1}{\psi }_2^{l_2}{\psi }_1^{{*}^{t_1}}{\psi }_2^{{*}^{t_2}}\mid k_i,l_i,t_i\in \mathbb{N},i=1,2\right\}.$$

Starting from the commutative polynomial ring $R_2=K\left[{\omega }_1\right]$, it is straightforward to verify the following:

$${𝒜}_2^{+}=R_2[{\omega }_2;{\alpha }_2][{\psi }_1;{\theta }_1][{\psi }_2;{\theta }_2][{\psi }_1^{*};{\sigma }_1,{\delta }_1][{\psi }_2^{*};{\sigma }_2,{\delta }_2],$$

where the following are defined:

• the algebra automorphism ${\alpha }_2:R_2\to R_2$ (with $R_2=K\left[{\omega }_1\right]$) is defined by the following:

  $${\alpha }_2\left(g\left({\omega }_1\right)\right)=g\left({\omega }_1\right),$$
• the algebra automorphism ${\theta }_1:K[{\omega }_1,{\omega }_2]\to K[{\omega }_1,{\omega }_2]$ is given by the following:

  $${\theta }_1\left(g({\omega }_1,{\omega }_2)\right)=g(q{\omega }_1,{\omega }_2),$$
• the algebra automorphism ${\theta }_2:K[{\omega }_1,{\omega }_2,{\psi }_1]\to K[{\omega }_1,{\omega }_2,{\psi }_1]$, such that the following is obtained:

  $${\theta }_2\left(g({\omega }_1,{\omega }_2,{\psi }_1)\right)=g({\omega }_1,q{\omega }_2,{\psi }_1),$$
• the algebra automorphism ${\sigma }_1:K[{\omega }_1,{\omega }_2,{\psi }_1,{\psi }_2]\to K[{\omega }_1,{\omega }_2,{\psi }_1,{\psi }_2]$, such that the following is obtained:

  $${\sigma }_1\left(g({\omega }_1,{\omega }_2,{\psi }_1,{\psi }_2)\right)=g(q^{-1}{\omega }_1,{\omega }_2,q^2{\psi }_1,{\psi }_2),$$
• ${\delta }_1$ is the ${\sigma }_1$-derivation on $K[{\omega }_1,{\omega }_2,{\psi }_1,{\psi }_2]$ such that the following is obtained:

  $${\delta }_1\left(g({\omega }_1,{\omega }_2,{\psi }_1,{\psi }_2)\right)=g({\omega }_1,{\omega }_2,-q^2{\omega }_1^2,{\psi }_2),$$
• ${\sigma }_2$ is the algebra automorphism of $K[{\omega }_1,{\omega }_2,{\psi }_1,{\psi }_2,{\psi }_1^{*}]$ such that the following is obtained:

  $${\sigma }_2\left(g({\omega }_1,{\omega }_2,{\psi }_1,{\psi }_2,{\psi }_1^{*})\right)=g({\omega }_1,q^{-1}{\omega }_2,{\psi }_1,q^2{\psi }_2,{\psi }_1^{*}),$$
• ${\delta }_2$ is the ${\sigma }_2$-derivation on $K[{\omega }_1,{\omega }_2,{\psi }_1,{\psi }_2,{\psi }_1^{*}]$ such that the following is obtained:

  $${\delta }_2\left(g({\omega }_1,{\omega }_2,{\psi }_1,{\psi }_2,{\psi }_1^{*})\right)=g({\omega }_1,{\omega }_2,{\psi }_1,-q^2{\omega }_2^2,{\psi }_1^{*}).$$

Proposition 1. 

The q-Clifford algebra ${𝒜}_q^{+}$ is an iterated skew polynomial ring of the form, written as follows:

$${𝒜}_q^{+}=K[{\omega }_1,\dots ,{\omega }_n][{\psi }_1;{\theta }_1]\cdots [{\psi }_n;{\theta }_n][{\psi }_1^{*};{\sigma }_1,{\delta }_1]\cdots [{\psi }_n^{*};{\sigma }_n,{\delta }_n],$$

and the q-Weyl algebra ${𝒜}_q^{-}$ is an iterated skew polynomial ring of the following form:

$${𝒜}_q^{-}=K[{\omega }_1^{-1},\dots ,{\omega }_n^{-1}][{\psi }_1;{\theta }_1]\cdots [{\psi }_n;{\theta }_n][{\psi }_1^{*};{\sigma }_1,{\delta }_1]\cdots [{\psi }_n^{*};{\sigma }_n,{\delta }_n].$$

Proof. 

By Corollary 1 and Theorem 5, ${𝒜}_q^{+}$ is a domain with a PBW basis, written as follows:

$$𝓑=\left\{{\omega }_1^{k_1}{\omega }_2^{k_2}\cdots {\omega }_n^{k_n}{\psi }_1^{l_1}{\psi }_2^{l_2}\cdots {\psi }_n^{l_n}{\psi }_1^{{*}^{t_1}}{\psi }_2^{{*}^{t_2}}\cdots {\psi }_n^{{*}^{t_n}}\mid k_i,l_i,t_i\in \mathbb{N},i\in I\left(n\right)\right\}.$$

The skew polynomial structure of ${𝒜}_q^{+}$ is realized through an iterated extension of skew polynomial subalgebras.

First, we construct the following:

$$A_1=K\left[{\omega }_1\right][{\omega }_2;{\alpha }_2]\cdots [{\omega }_n;{\alpha }_n],$$

where for each $1<k⩽n$, ${\alpha }_k$ is the automorphism of $K[{\omega }_1,{\omega }_2,\dots ,{\omega }_{k-1}]$ defined by the following:

$${\alpha }_k\left(g({\omega }_1,{\omega }_2,\dots ,{\omega }_{k-1})\right)=g({\omega }_1,{\omega }_2,\dots ,{\omega }_{k-1}).$$

Next, construct the skew polynomial subalgebra as follows:

$$A_2=A_1[{\psi }_1;{\theta }_1]\cdots [{\psi }_n;{\theta }_n]=K[{\omega }_1,\dots ,{\omega }_n][{\psi }_1;{\theta }_1]\cdots [{\psi }_n;{\theta }_n],$$

where the algebra automorphism ${\theta }_1:A_1=K[{\omega }_1,\dots ,{\omega }_n]\to K[{\omega }_1,\dots ,{\omega }_n]$, such that the following is obtained:

$${\theta }_1\left(g({\omega }_1,{\omega }_2,\dots ,{\omega }_n)\right)=g(q{\omega }_1,{\omega }_2,\dots ,{\omega }_n),$$

the algebra automorphism ${\theta }_2:K[{\omega }_1,\dots ,{\omega }_n,{\psi }_1]\to K[{\omega }_1,\dots ,{\omega }_n,{\psi }_1]$, such that the following is obtained:

$${\theta }_2\left(g({\omega }_1,\dots ,{\omega }_n,{\psi }_1)\right)=g({\omega }_1,q{\omega }_2,{\omega }_3,\dots ,{\omega }_n,{\psi }_1),$$

for any $k(1<k⩽n)$, ${\theta }_k$ is algebra automorphism of $K[{\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_{k-1}]$, such that the following is obtained:

$${\theta }_k\left(g({\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_{k-1})\right)=g({\omega }_1,{\omega }_{k-1},q{\omega }_k,{\omega }_{k+1},\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_{k-1}).$$

And construct the skew polynomial algebra, written as follows:

$$A_3=A_2[{\psi }_1^{*};{\sigma }_1,{\delta }_1]\cdots [{\psi }_n^{*};{\sigma }_n,{\delta }_n]=K[{\omega }_1,\dots ,{\omega }_n][{\psi }_1;{\theta }_1]\cdots [{\psi }_n;{\theta }_n][{\psi }_1^{*};{\sigma }_1,{\delta }_1]\cdots [{\psi }_n^{*};{\sigma }_n,{\delta }_n],$$

where ${\sigma }_1$ is the algebra automorphism of $K[{\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_n]$, such that the following is obtained:

$${\sigma }_1\left(g({\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_n)\right)=g(q^{-1}{\omega }_1,{\omega }_2,\dots ,{\omega }_n,q^2{\psi }_1,{\psi }_2,\dots ,{\psi }_n),$$

${\delta }_1$ is the ${\sigma }_1$-derivation on $K[{\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_n]$, such that the following is obtained:

$${\delta }_1\left(g({\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_n)\right)=g({\omega }_1,\dots ,{\omega }_n,-q^2{\omega }_1^2,{\psi }_2,\dots ,{\psi }_n),$$

about any $k(1<k⩽n)$, ${\sigma }_k$ is algebra automorphism of $K[{\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_n,{\psi }_1^{*},\dots ,{\psi }_{k-1}^{*}]$, such that the following is obtained:

$$\begin{array}{cc}\hfill {\sigma }_k& \left(g\left({\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_n,{\psi }_1^{*},\dots ,{\psi }_{k-1}^{*}\right)\right)\hfill \\ & =g({\omega }_1,\dots ,{\omega }_{k-1},q^{-1}{\omega }_k,{\omega }_{k+1},\dots ,{\omega }_n,\hfill \\ \hfill & {\psi }_1,\dots ,{\psi }_{k-1},q^2{\psi }_k,{\psi }_{k+1},\dots ,{\psi }_n,{\psi }_1^{*},\dots ,{\psi }_{k-1}^{*}),\hfill \end{array}$$

${\delta }_k$ is the ${\sigma }_k$-derivation on $K[{\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_n,{\psi }_1^{*},\dots ,{\psi }_{k-1}^{*}]$, such that the following is obtained:

$$\begin{array}{cc}\hfill {\delta }_k(& g\left({\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_n,{\psi }_1^{*},\dots ,{\psi }_{k-1}^{*}\right))\hfill \\ & =g\left({\omega }_1,\dots ,{\omega }_n,{\psi }_1,\dots ,{\psi }_{k-1},-q^2{\omega }_k^2,{\psi }_{k+1},\dots ,{\psi }_n,{\psi }_1^{*},\dots ,{\psi }_{k-1}^{*}\right).\hfill \end{array}$$

Through this iterative construction, the skew polynomial structure of the q-Clifford algebra ${𝒜}_q^{+}$ is explicitly realized. An analogous procedure yields the corresponding structure for the q-Weyl algebra ${𝒜}_q^{-}$. □

4. Structural Properties of the $\mathit{q}$-Clifford Algebra ${\mathbf{𝒜}}_{\mathit{q}}^{\mathbf{+}}$ and the $\mathit{q}$-Weyl Algebra ${\mathbf{𝒜}}_{\mathit{q}}^{\mathbf{-}}$

This section characterizes the structure of both the q-Clifford algebra ${𝒜}_q^{+}$ and the q-Weyl algebra ${𝒜}_q^{-}$ through application of results from Section 2 and Section 3.

Proposition 2 

([8,9]). Let $D=K[d_1,\dots ,d_n]$ be a solvable polynomial algebra equipped with an admissible system $(B,\prec )$. Then D is isomorphic to $K\langle X\rangle /\langle 𝒢\rangle$, where $K\langle X\rangle =K\langle X_1,\dots ,X_n\rangle$ is the free associative K-algebra on n generators $X=\{X_1,\dots ,X_n\}$. Denote by $𝒢$ the Gröbner basis of the ideal $\langle 𝒢\rangle$ in $K\langle X\rangle$ about the monomial order ≺, satisfying one of the following two conditions:

$$\mathrm{LM}\left(𝒢\right)=\{X_m{X}_l\mid 1⩽l<m⩽n\},$$

or

$$\mathrm{LM}\left(𝒢\right)=\{X_l{X}_m\mid 1⩽l<m⩽n\}.$$

Under these conditions, the following properties hold:

• Gelfand–Kirillov dimension: $\mathrm{GK}.\dim D=n$.
• When $𝒢$ consists of $\mathbb{N}$-graded homogeneous elements in the free associative K-algebra $K\langle X\rangle$, then the global homological dimension $\mathrm{GL}.\dim D=n$. If $𝒢$ does not consist of homogeneous elements, then $\mathrm{GL}.\dim D⩽n$.
• Suppose that the free associative K-algebra $K\langle X\rangle$ is $\mathbb{N}$-graded, with each generator $X_l$ having degree 1. If $𝒢$ consists of quadratic homogeneous elements forming a homogeneous Gröbner basis in $K\langle X\rangle$, then the algebra D is a homogeneous quadratic Koszul algebra. In contrast, if these conditions are not met, D is not a homogeneous quadratic Koszul algebra.

Based on Proposition 2, the following conclusions hold.

Theorem 6. 

With the notation fixed above, the q-Clifford algebra ${𝒜}_q^{+}$ and the q-Weyl algebra ${𝒜}_q^{-}$ have the following properties:

• Gelfand–Kirillov dimension: $GK.dim{𝒜}_q^{+}=3n$, $GK.dim{𝒜}_q^{-}=3n$.
• Global homological dimension: $GL.dim{𝒜}_q^{+}=3n$, $GL.dim{𝒜}_q^{-}=3n$.
• Both ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$ are homogeneous quadratic Koszul algebras.

Proof. 

Regarding the monomial order ${\prec }_{d-\mathrm{lex}}$ defined on $W^{*}$, where $W^{*}$ represents all finite-length words (including the empty word) generated by elements of W, written as follows:

$${\omega }_1{\prec }_{d-lex}\cdots {\prec }_{d-lex}{\omega }_n{\prec }_{d-lex}{\psi }_1{\prec }_{d-lex}\cdots {\prec }_{d-lex}{\psi }_n{\prec }_{d-lex}{\psi }_1^{*}{\prec }_{d-lex}{\psi }_2^{*}{\prec }_{d-lex}\cdots {\prec }_{d-lex}{\psi }_n^{*}.$$

The set S of defining relations forms a Gröbner–Shirshov basis for the ideal $J=\langle S\rangle$, with all leading terms in S explicitly determined as follows:

$$\begin{array}{cccc}\hfill & {\omega }_t{\omega }_{s}with{\omega }_s{\prec }_{d-lex}{\omega }_t\hfill & & where 1⩽s⩽t⩽n,\hfill \\ \hfill & {\psi }_t{\psi }_{s}with{\psi }_s{\prec }_{d-lex}{\psi }_t\hfill & & where 1⩽s<t⩽n,\hfill \\ \hfill & {\psi }_t^{*}{\psi }_s^{*}with{\psi }_s^{*}{\prec }_{d-lex}{\psi }_t^{*}\hfill & & where 1⩽s<t⩽n,\hfill \\ \hfill & {\omega }_t{\psi }_{s}with{\psi }_s{\prec }_{d-lex}{\omega }_t\hfill & & where 1⩽s,t⩽n,\hfill \\ \hfill & {\omega }_s{\psi }_t^{*}with{\psi }_t^{*}{\prec }_{d-lex}{\omega }_s\hfill & & where 1⩽s,t⩽n,\hfill \\ \hfill & {\psi }_s{\psi }_t^{*}with{\psi }_t^{*}{\prec }_{d-lex}{\psi }_s\hfill & & where 1⩽s\ne t⩽n.\hfill \end{array}$$

The algebra ${𝒜}_q^{+}$ satisfies the hypotheses of Proposition 2; analogously, ${𝒜}_q^{-}$ satisfies them as well.

(1)–(3) By Theorem 3, the solvable polynomial algebra ${𝒜}_q^{+}$ satisfies the conditions of Proposition 2 and is an $\mathbb{N}$-graded algebra defined by a homogeneous quadratic Gröbner basis, each generator ${\omega }_s$, ${\psi }_s$, ${\psi }_s^{*}$ is assigned a degree of 1. Therefore, according to Proposition 2, (1)–(3) hold. The proof for ${𝒜}_q^{-}$ follows analogously. □

Theorem 7. 

For both the q-Clifford algebra ${𝒜}_q^{+}$ and q-Weyl algebra ${𝒜}_q^{-}$, the following statements hold:

• The Hilbert series is ${\displaystyle \frac{1}{{(1-t)}^{3n}}}$.
• For any nonzero left ideal $M\subset {𝒜}_q^{+}$ and $M^{\prime }\subset {𝒜}_q^{-}$ we have the following:

  $$GK.dim({𝒜}_q^{+}/M)<GK.dim{𝒜}_q^{+}=3n,GK.dim({𝒜}_q^{-}/M^{\prime })<GK.dim{𝒜}_q^{-}=3n,$$

  with explicit algorithms to compute these dimensions.
• For finitely generated modules N over ${𝒜}_q^{+}$ and $N^{\prime }$ over ${𝒜}_q^{-}$, both the finite free resolutions and the projective dimensions $pdim\left(N\right)$, $pdim\left(N^{\prime }\right)$ are algorithmically computable.
• For finitely generated graded modules N, $N^{\prime }$ over $\mathbb{N}$-graded ${𝒜}_q^{+}$, ${𝒜}_q^{-}$ (with generators in degree 1), both minimal homogeneous generating sets and minimal finite graded free resolutions are algorithmically constructible.

Proof. 

(1) By Corollary 1, ${𝒜}_q^{+}$ has a PBW basis written as follows:

$$𝓑=\{{\omega }_1^{k_1}{\omega }_2^{k_2}\cdots {\omega }_n^{k_n}{\psi }_1^{l_1}{\psi }_2^{l_2}\cdots {\psi }_n^{l_n}{\psi }_1^{{*}^{t_1}}{\psi }_2^{{*}^{t_2}}\cdots {\psi }_n^{{*}^{t_n}}\mid k_i,l_i,t_i\in \mathbb{N},i\in I\left(n\right)\}.$$

And ${𝒜}_q^{-}$ has a PBW basis written as follows:

$$𝓑=\{{\omega }_1^{-1^{k_1}}{\omega }_2^{-1^{k_2}}\cdots {\omega }_n^{-1^{k_n}}{\psi }_1^{l_1}{\psi }_2^{l_2}\cdots {\psi }_n^{l_n}{\psi }_1^{{*}^{t_1}}{\psi }_2^{{*}^{t_2}}\cdots {\psi }_n^{{*}^{t_n}}\mid k_i,l_i,t_i\in \mathbb{N},i\in I\left(n\right)\}.$$

Thus, the Hilbert series of ${𝒜}_q^{+}$ is ${\displaystyle \frac{1}{{(1-t)}^{3n}}}$ and that of ${𝒜}_q^{-}$ is ${\displaystyle \frac{1}{{(1-t)}^{3n}}}$, where t is a formal variable.

(2) By Theorem 6 (1), the Gelfand–Kirillov dimensions satisfy the following:

$$GK.dim{𝒜}_q^{+}=3n.$$

Since Theorem 3 establishes that ${𝒜}_q^{+}$ are quadratic solvable polynomial algebras, we deduce from [8] that for any proper left ideal $M\subset {𝒜}_q^{+}$, written as follows:

$$GK.dim{𝒜}_q^{+}/M<3n.$$

Alternatively, this inequality follows from the classical Gelfand–Kirillov dimension theorem [10] via the Noetherian domain property of ${𝒜}_q^{+}$. Furthermore, there exists an explicit algorithm to compute these dimensions. The proof for ${𝒜}_q^{-}$ follows analogously.

(3) By [8], for a solvable polynomial algebra D under its monomial ordering, every finitely generated D-module N admits a finite free resolution. Consequently, algorithms exist to compute such resolutions as well as their projective dimensions.

(4) Let N be a finitely generated graded ${𝒜}_q^{+}$-module. Then $N={\oplus }_{q\in \mathbb{N}}{N}_q$, where each $N_q$ is a K-subspace of N. Nonzero elements in $N_q$ are termed homogeneous of degree q. Consequently, minimal homogeneous generating sets and minimal finite graded resolutions for N are algorithmically computable. The proof for ${𝒜}_q^{-}$ follows analogously.

For both the q-Clifford algebra ${𝒜}_q^{+}$ and the q-Weyl algebra ${𝒜}_q^{-}$, every one-sided ideal possesses the elimination property. For a more precise statement, we recall the Elimination Lemma. □

Lemma 1 

([7]). Let $D=K[d_1,d_2,\dots ,d_n]$ be the free K-algebra, and assume that D admits a PBW basis $𝓑=\{d^k=d_1^{k_1}{d}_2^{k_2}\cdots d_n^{k_n}\mid k=(k_1,k_2,\dots ,k_n)\in {\mathbb{N}}^n\}$.

For a subset $E=\{d_{j_1},d_{j_2},\dots ,d_{j_t}\}\subset \{d_1,d_2,\dots ,d_n\}$, $j_1<j_2<\cdots <j_t$, let the following be valid:

$$F=\left\{d_{j_1}^{k_1}{d}_{j_2}^{k_2}\cdots d_{j_t}^{k_t}\mid (k_1,k_2,\dots ,k_t)\in {\mathbb{N}}^t\right\},V\left(F\right)=K-spanF.$$

Let H be a nonzero left ideal of D, defining the left D-module $D/H$. If there exists a subset $E=\{d_{j_1},d_{j_2},\dots ,d_{j_t}\}\subset \{d_1,d_2,\dots ,d_n\}$ with $j_1<j_2<\cdots <j_t$ such that $V\left(F\right)\cap H=\left\{0\right\}$, then write the following:

$$\mathrm{GK}.\dim (D/H)⩾t.$$

Therefore, suppose that the GK dimension of quotient algebra $D/H$ is finite, specifically $\mathrm{GK}.\dim (D/H)=m<n$, where n denotes the number of generators of the algebra D. Then, for any subset $E=\{d_{j_1},d_{j_2},\dots ,d_{j_{m+1}}\}$ of the generator set $\{d_1,d_2,\dots ,d_n\}$ with $1⩽j_1<j_2<\cdots <j_{m+1}⩽n$, the intersection of the subspace $V\left(F\right)$ and the ideal H is non-trivial, i.e., $V\left(F\right)\cap H\ne \left\{0\right\}$. In particular, for any subset $E=\{d_1,d_2,\dots ,d_s\}$ of the generator set with $m+1⩽s⩽n-1$, the non-triviality of the intersection $V\left(F\right)\cap H\ne \left\{0\right\}$ still holds.

To state elimination property for ${𝒜}_q^{+}$, denote the set of generators of ${𝒜}_q^{+}$ by the following:

$$W=\{{\omega }_1,\cdots ,{\omega }_n,{\psi }_1,\cdots ,{\psi }_n,{\psi }_1^{*},\cdots ,{\psi }_n^{*}\},{𝒜}_q^{+}=K[{\omega }_1,\cdots ,{\omega }_n,{\psi }_1,\cdots ,{\psi }_n,{\psi }_1^{*},\cdots ,{\psi }_n^{*}].$$

Then for a subset, write the following:

$S=\{{\omega }_{i_1},\cdots ,{\omega }_{i_r},{\psi }_{i_1},\cdots ,{\psi }_{i_t},{\psi }_{i_1}^{*},\cdots ,{\psi }_{i_k}^{*}\}\subset \{{\omega }_1,\cdots ,{\omega }_n,{\psi }_1,\cdots ,{\psi }_n,{\psi }_1^{*},\cdots ,{\psi }_n^{*}\},$$i_1<i_2<\cdots <i_r,i_1<i_2<\cdots <i_t,i_1<i_2<\cdots <i_k$, denote the following:

$$F=\left\{{\omega }_{i_1}^{{\alpha }_1}\cdots {\omega }_{i_r}^{{\alpha }_r}{\psi }_{i_1}^{{\beta }_1}\cdots {\psi }_{i_t}^{{\beta }_t}{\psi }_{i_1}^{{*}^{{\gamma }_1}}\cdots {\psi }_{i_k}^{{*}^{{\gamma }_k}}\right\},$$

where $({\alpha }_1,\dots ,{\alpha }_r)\in {\mathbb{N}}^r,({\beta }_1,\dots ,{\beta }_t)\in {\mathbb{N}}^t,({\gamma }_1,\dots ,{\gamma }_k)\in {\mathbb{N}}^k,$

$$V\left(F\right)=K-spanF.$$

Theorem 8. 

Fix the notation above, and let H be a left ideal of ${𝒜}_q^{+}$. In this context, the following two assertions hold:

• GK.dim${𝒜}_q^{+}/H<3n=$ GK.dim${𝒜}_q^{+}$. If GK.dim${𝒜}_q^{+}/H=m$, then

  $$V\left(F\right)\cap H\ne \left\{0\right\},$$

  for any subset, written as follows:

  $$S=\{{\omega }_{i_1},\cdots ,{\omega }_{i_r},{\psi }_{i_1},\cdots ,{\psi }_{i_t},{\psi }_{i_1}^{*},\cdots ,{\psi }_{i_k}^{*}\}\subset \{{\omega }_1,\cdots ,{\omega }_n,{\psi }_1,\cdots ,{\psi }_n,{\psi }_1^{*},\cdots ,{\psi }_n^{*}\},$$

  where $i_1<i_2<\cdots <i_r$, $i_1<i_2<\cdots <i_t$, $i_1<i_2<\cdots <i_k$, and $r+t+k=m+1$. In particular, for any $S=\{{\omega }_1,\cdots ,{\omega }_s,{\psi }_1,\cdots ,{\psi }_t,{\psi }_1^{*},\cdots ,{\psi }_l^{*}\}$, we have $m+1⩽s+t+l⩽3n-1$,

  $$V\left(F\right)\cap H\ne \left\{0\right\}.$$
• Without knowing the exact value of GK.dim${𝒜}_q^{+}/H$, the elimination property of the left ideal $H={\sum }_{i=1}^m{𝒜}_q^{+}{\xi }_i$ of ${𝒜}_q^{+}$ can be achieved computationally.

A similar result holds true for ${𝒜}_q^{-}$.

Proof. 

(1) Adopting the fixed notation above, ${𝒜}_q^{+}$ has a PBW basis $𝓑$ by Corollary 1. From Theorem 7 (2), we have GK.dim${𝒜}_q^{+}/H<3n$, and the elimination property can be derived from Lemma 1.

(2) Let ${\prec }_{\mathrm{d}-\mathrm{lex}}$ be a monomial order on PBW basis $𝓑$ of ${𝒜}_q^{+}$, and assume $V\left(F\right)$ is defined as in (1). By employing an elimination order ⋖ relative to F (constructible via [8] if ${\prec }_{\mathrm{d}-\mathrm{lex}}$ is not an elimination order), noncommutative Buchberger algorithm for solvable polynomial algebras produces a Gröbner basis $𝒢$ for H, satisfying the following:

$$H\cap V\left(F\right)\ne \left\{0\right\}\iff 𝒢\cap V\left(F\right)\ne \varnothing .$$

The analogous result holds for ${𝒜}_q^{-}$, inheriting the elimination property. □

We recall, the definitions of Auslander regularity, Cohen–Macaulay property, Artin–Schelter regularity, and maximal orders in quotient algebras before stating the next result.

Definition 6 

([11]). (1) Auslander regular: A ring R is called Auslander regular if, for every finitely generated left R-module M and any integer $k⩾0$, the module ${Ext}_R^k(M,R)$ satisfies $j_R\left(N\right)⩾k$ for all right R-submodules N, where $j_R\left(N\right)$ is the minimal integer such that ${Ext}_R^{j_R\left(N\right)}(N,R)\ne 0$ and it has finite global homological dimension.

(2) Cohen–Macaulay property: A ring R is said to possess the Cohen–Macaulay property if, for all finitely generated left R-modules M, the equation $GK.dim\left(M\right)+j_R\left(M\right)=GK.dim\left(R\right)$ is satisfied.

(3) Artin–Schelter regular: Consider the connected $\mathbb{N}$-graded K-algebra X, suppose X is generated in degree 1, where $X={⨁}_{i\in \mathbb{N}}{X}_i$, $X_0=K$ is a central subalgebra, $dim\left(X_i\right)<\infty$ for every non-negative integer i, and the algebra X is generated by the degree 1 component $X_1$. Let $X^{+}={⨁}_{i⩾1}{X}_i$ denote the graded radical of X. An algebra X is called an Artin–Schelter regular algebra of dimension d if it satisfies the following three conditions: the global homological dimension of X is $gl.dim\left(X\right)=d$, the Gelfand–Kirillov dimension of X is finite, ${Ext}_X^n(K_X,X)={\delta }_{nd}\left(K_X\right)$, where K is identified with the quotient algebra $X/X^{+}$.

(4) Stably free: A K-algebra X is stably free if for every finitely generated projective X-module P, there exist integers a, b such that $P\oplus X^a=X^b$.

(5) Maximal order: Let X be a domain and $Q\left(X\right)$ be its quotient division ring. If, for some K-algebra S such that $X\subseteq S$, the condition that there exist $a,b\in X-\left\{0\right\}$ with $aSb\subseteq X$ implies $X=S$, then X is called a maximal order in $Q\left(X\right)$.

Proposition 3 

([12]). Suppose that X is a Noetherian K-algebra which satisfies both the Cohen–Macaulay property and Auslander regularity. If X is stably free, then X is a domain and a maximal order in its quotient division ring $Q\left(X\right)$.

Proposition 4 

([13]). If a ring R is left-right Noetherian, Auslander regular, then the following hold:

• Any skew polynomial ring $R[t;\sigma ]$ is left-right Noetherian and Auslander regular, where σ is an automorphism of R.
• Any skew polynomial ring $R[t;\sigma ,\delta ]$ is left-right Noetherian and Auslander regular, where σ is an automorphism of R and δ is a σ-derivation.

Theorem 9. 

The q-Clifford algebra ${𝒜}_q^{+}$ and the q-Weyl algebra ${𝒜}_q^{-}$ satisfy the following properties:

• The algebras ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$ are Auslander regular.
• The algebras ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$ satisfy the Cohen–Macaulay property.
• The algebras ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$ are Artin–Schelter regular.
• Both ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$ are maximal orders in their respective quotient division algebras $Q\left({𝒜}_q^{+}\right)$ and $Q\left({𝒜}_q^{-}\right)$.

Proof. 

(1) Since ${𝒜}_q^{+}$ is left-right Noetherian and has global homological dimension $3n$, by the proof of Proposition 1, the subalgebra $A_1=K[{\omega }_1,\dots ,{\omega }_n]$ is an iterated skew polynomial algebra obtained via a sequence of automorphisms ${\alpha }_k$ ($1<k⩽n$) over the commutative polynomial ring $K\left[{\omega }_1\right]$. The algebra ${𝒜}_q^{+}$ is constructed iteratively from the subalgebras $A_1$, $A_2$, and $A_3$ through a series of algebra automorphisms ${\sigma }_k$ and ${\sigma }_k$-derivations ($1⩽k⩽n$). By Proposition 4, ${𝒜}_q^{+}$ is an Auslander regular algebra.

(2) By Theorem 3, ${𝒜}_q^{+}$ is a solvable polynomial algebra. According to Theorems 5 and 6 (1) (2), ${𝒜}_q^{+}$ is Noetherian with $gl.dim{𝒜}_q^{+}=3n=GK.dim{𝒜}_q^{+}$. Since ${𝒜}_q^{+}$ has an iterated skew polynomial structure, it follows from [14] that ${𝒜}_q^{+}$ satisfies the Cohen–Macaulay property.

(3) As a connected $\mathbb{N}$-graded K-algebra with degree 1 generators, ${𝒜}_q^{+}$ satisfies Auslander regularity. Consequently, applying [15] fundamental result establishes its Artin–Schelter regularity.

(4) By Theorem 5, ${𝒜}_q^{+}$ is Noetherian. According to [8], ${𝒜}_q^{+}$ is stably free. By (1) and (2), ${𝒜}_q^{+}$ is Cohen–Macaulay and Auslander regular. Then, Proposition 3 implies that ${𝒜}_q^{+}$ is a maximal order in its quotient division algebra $Q\left({𝒜}_q^{+}\right)$.

One can prove the statement for ${𝒜}_q^{-}$ by following an analogous argument. □

Remark 2. 

For the mixed algebra ${𝒜}_q^{\pm }={𝒜}_q^{+}\otimes {𝒜}_q^{-}$ (where q is not a root of unity), its homological properties are established through the regularity of its subalgebras and tensor product theorems. Specifically: Auslander regularity transfers via Levasseur’s tensor product theorem [16]; The Cohen–Macaulay property follows from the dimension formulas; Artin–Schelter regularity is preserved under tensor products of connected $\mathbb{N}$-graded algebras. Complete proofs require consistency of filtration structures in the subalgebras and application of graded tensor product theory, with detailed exposition reserved for subsequent work.

5. Conclusions

This work provides the first systematic analysis of the Clifford–Weyl algebra ${𝒜}_q^{\pm }$ via its subalgebras ${𝒜}_q^{+}$ (q-Clifford algebra) and ${𝒜}_q^{-}$ (q-Weyl algebra). Through an innovative synthesis of Gröbner–Shirshov basis theory and PBW basis methods, we establish the following three fundamental advances:

• Structural Clarity: An explicit construction of both subalgebras as iterated skew polynomial rings that resolves the non-solvability obstruction of ${𝒜}_q^{\pm }$.
• Homological Characterization: A rigorous proof that both ${𝒜}_q^{+}$ and ${𝒜}_q^{-}$ are Artin–Schelter-regular, Auslander-regular, and Cohen–Macaulay-regular Noetherian domains satisfying the following:

  $$\mathrm{gk}.\dim \left({𝒜}_q^{+}\right)=\mathrm{gk}.\dim \left({𝒜}_q^{-}\right)=3n.$$
• Algorithmic Foundation: We develop a computational framework for module theory that enables the following:

  (1)

  Dimension bounds for quotient modules (<$3n$);

  (2)

  Algorithmic construction of free resolutions;

  (3)

  Verifiable elimination properties for left ideals.

These results not only elucidate the algebraic architecture of ${𝒜}_q^{\pm }$ but also provide practical tools for further exploration in noncommutative geometry and representation theory. The homological regularity provides rigorous homological invariants for singularity classification in the quantum coordinate ring $A_q\left(X\right)={𝒜}_q^{\pm }{\otimes }_{\mathbb{C}}\mathcal{𝒪}\left(X\right)$; the computational framework supports algorithmic implementation of noncommutative connections on ${𝒜}_q^{\pm }$-principal bundles; the elimination property opens new pathways for constructing Lorentz-covariant representations of q-deformed quantum field operators. The established iterated skew polynomial structures deepen the understanding of Clifford–Weyl algebras and lay a methodological foundation for integrating quantum algebra with geometry through their regularity and computability.