On the Structure and Homological Regularity of the q-Heisenberg Algebra

The q-Heisenberg algebra hn(q) is a significant class of solvable polynomial algebras, and it unifies the canonical commutation relations of Heisenberg algebras and the deformation theory of quantum groups. In this paper, we employ Gröbner-Shirshov basis theory and PBW (Poincare´-Birkhoff-Witt) basis techniques to systematically investigate hn(q). Our main results establish that: hn(q) possesses an iterated skew-polynomial algebra structure, and it satisfies the important homological regularity properties of being Auslander regular, Artin-Schelter regular, and Cohen-Macaulay. These findings provide deep insights into the algebraic structure of hn(q), while simultaneously bridging the gap between noncommutative algebra and quantum representation theory. Furthermore, our constructive approach yields computable methods for studying modules over hn(q), opening new avenues for further research in deformation quantization and quantum algebra.