Structural Properties of The Clifford–Weyl Algebra 𝒜 q ±

Jia Zhang and Gulshadam Yunus ()
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Jia Zhang: College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China
Gulshadam Yunus: College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China

Mathematics, 2025, vol. 13, issue 17, 1-15

Abstract: 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.

Keywords: Clifford–Weyl algebra; iterated skew polynomial algebra; Gröbner–Shirshov basis; PBW basis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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