Structural Properties of The Clifford–Weyl Algebra 𝒜 q ±

The Clifford–Weyl algebra 𝒜 q ± , 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 + 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 + and 𝒜 q − are Auslander regular, Cohen–Macaulay, and Artin–Schelter regular. These results provide new tools for the representation theory in noncommutative geometry.