{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T09:36:49Z","timestamp":1768383409937,"version":"3.49.0"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T00:00:00Z","timestamp":1768176000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundations of China","doi-asserted-by":"crossref","award":["12061068"],"award-info":[{"award-number":["12061068"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"crossref","award":["2024D01C37"],"award-info":[{"award-number":["2024D01C37"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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 Gr\u00f6bner-Shirshov basis theory and PBW (Poincare\u00b4-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.<\/jats:p>","DOI":"10.3390\/axioms15010054","type":"journal-article","created":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T12:44:44Z","timestamp":1768221884000},"page":"54","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Structure and Homological Regularity of the q-Heisenberg Algebra"],"prefix":"10.3390","volume":"15","author":[{"given":"Yabiao","family":"Wang","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gulshadam","family":"Yunus","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1007\/BF01328377","article-title":"\u00dcber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen","volume":"33","author":"Heisenberg","year":"1925","journal-title":"Z. Phys."},{"key":"ref_2","unstructured":"Zhang, H.S. (2003). The Automorphism Groups of Heisenberg Lie Algebras and the Standard Kac-Moody Algebras and the Completely Reducible of Integrable Modules. [Ph.D. Thesis, Capital Normal University]."},{"key":"ref_3","unstructured":"Ji, G.Z. (2019). Algebra Rota-Baxter Operators of Heisenberg Lie Algebra. [Master\u2019s Thesis, Harbin University of Science and Technology]."},{"key":"ref_4","unstructured":"Zhou, C.Y. (2021). The Quasi-Automorphism and Automorphism of Heisenberg Lie Algebras. [Master\u2019s Thesis, Suzhou University of Science and Technology]."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/BF02099007","article-title":"The quantum Poincar\u00e9-Birkhoff-Witt theorem","volume":"143","author":"Berger","year":"1992","journal-title":"Comm. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Rosenberg, A.L. (1995). Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-8430-2"},{"key":"ref_7","unstructured":"Wess, J. (1999, January 9\u201316). q-Deformed Heisenberg Algebras. Geometry and Quantum Physics. Proceedings of the 38. Internationale Universit\u00e4tswochen f\u00fcr Kern-und Teilchenphysik, Schladming, Austria."},{"key":"ref_8","first-page":"550","article-title":"Gr\u00f6bner-Shirshov bases and structural properties of q-Heisenberg algebras","volume":"41","author":"Zhang","year":"2024","journal-title":"J. Xinjiang Univ. Natural Sci. Ed."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2550318","DOI":"10.1142\/S0219498824500920","article-title":"A note on regularities of the standard quantized matrix algebra Mq(n)","volume":"23","author":"Tuniyaz","year":"2024","journal-title":"J. Algebra Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Zhang, J., and Yunus, G. (2025). Structural Properties of The Clifford-Weyl Algebra Aq\u00b1. Mathematics, 13.","DOI":"10.3390\/math13172823"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Li, H. (2011). Gr\u00f6bner Bases in Ring Theory, World Scientific Publishing.","DOI":"10.1142\/8223"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Goodearl, K.R., and Warfield, R.B. (2004). An Introduction to Noncommutative Noetherian Rings, Cambridge University Press.","DOI":"10.1017\/CBO9780511841699"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"McConnell, J.C., and Robson, J.C. (2001). Noncommutative Noetherian Rings, American Mathematical Society.","DOI":"10.1090\/gsm\/030"},{"key":"ref_14","first-page":"107288","article-title":"Dualizing complexes and perverse modules over certain singular algebras","volume":"372","author":"Yekutieli","year":"2020","journal-title":"Adv. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1017\/S0017089500008843","article-title":"Some properties of non-commutative regular rings","volume":"34","author":"Levasseur","year":"1992","journal-title":"Glasg. Math. J."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Weibel, C.A. (1994). An Introduction to Homological Algebra, Cambridge University Press.","DOI":"10.1017\/CBO9781139644136"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1112\/jlms\/50.2.276","article-title":"Auslander-regular algebras and maximal orders","volume":"50","author":"Stafford","year":"1994","journal-title":"J. Lond. Math. Soc."},{"key":"ref_18","unstructured":"Li, H., and Van Oystaeyen, F. (1996). Zariskian Filtrations, Kluwer Academic Publishers. K-Monographs in Mathematics."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Li, H. (2021). Noncommutative Polynomial Algebras of Solvable Type and Their Modules: Basic Constructive-Computational Theory and Methods, Chapman and Hall\/CRC Press.","DOI":"10.1201\/9781003213192"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Levandovskyy, V., and Sch\u00f6nemann, H. (2003, January 3\u20136). Plural: Acomputeralgebrasystem for noncommutative polynomial algebras. Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, Philadelphia, PA, USA.","DOI":"10.1145\/860854.860895"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Li, H. (2002). Noncommutative Gr\u00f6bner Bases and Filtered-graded Transfer, Springer. Lecture Notes in Mathematics.","DOI":"10.1007\/b84211"},{"key":"ref_22","unstructured":"Krause, G.R., and Lenagan, T.H. (1991). Growth of Algebras and Gelfand-Kirillov Dimension, American Mathematical Society. Graduate Studies in Mathematics."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3520","DOI":"10.1080\/00927872.2018.1424863","article-title":"An elimination lemma for algebras with PBW bases","volume":"46","author":"Li","year":"2018","journal-title":"Commun. Algebr."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/54\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T05:17:45Z","timestamp":1768367865000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/54"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,12]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["axioms15010054"],"URL":"https:\/\/doi.org\/10.3390\/axioms15010054","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,12]]}}}