The PI Property in Algebras of Polynomial Type

Autores:: Gomez, James
Gallego, Claudia

Tipo de recurso:: Article of journal

Fecha de publicación:: 2026

Institución:: Universidad Pedagógica y Tecnológica de Colombia

Repositorio:: RiUPTC: Repositorio Institucional UPTC

Idioma:: eng

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network_name_str	RiUPTC: Repositorio Institucional UPTC
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dc.title.spa.fl_str_mv	The PI Property in Algebras of Polynomial Type
title	The PI Property in Algebras of Polynomial Type
spellingShingle	The PI Property in Algebras of Polynomial Type PI property, quantum Heisenberg algebras, double Ore extensions, down-up algebras, quantum matrix algebras.
title_short	The PI Property in Algebras of Polynomial Type
title_full	The PI Property in Algebras of Polynomial Type
title_fullStr	The PI Property in Algebras of Polynomial Type
title_full_unstemmed	The PI Property in Algebras of Polynomial Type
title_sort	The PI Property in Algebras of Polynomial Type
dc.creator.fl_str_mv	Gomez, James Gallego, Claudia
dc.contributor.author.spa.fl_str_mv	Gomez, James Gallego, Claudia
dc.subject.none.fl_str_mv	PI property, quantum Heisenberg algebras, double Ore extensions, down-up algebras, quantum matrix algebras.
topic	PI property, quantum Heisenberg algebras, double Ore extensions, down-up algebras, quantum matrix algebras.
publishDate	2026
dc.date.accessioned.none.fl_str_mv	2026-10-17 23:15:06
dc.date.available.none.fl_str_mv	2026-10-17 23:15:06
dc.date.issued.none.fl_str_mv	2026-10-17
dc.type.spa.fl_str_mv	Artículo de revista
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dc.language.iso.eng.fl_str_mv	eng
language	eng
dc.relation.references.eng.fl_str_mv	M. Akhavizadegan and David A. Jordan. Prime ideals of quantized weyl algebras. Glasgow Mathematical Journal, 38(3):283–297, 1996. [2] Nicolás Andruskiewitsch and Frédéric Dumas. On the au- tomorphisms of U+ q (g). In Quantum Groups, volume 12 of IRMA Lectures in Mathematics and Theoretical Physics, pages 107–133. European Mathematical Society, Zürich, 2008. [3] Michael Artin, William Schelter, and John Tate. Quantum deformations of GLn. Communications on Pure and Ap- plied Mathematics, 44(8-9):879–895, 1991. [4] V. V. Bavula. Description of bi-quadratic algebras on 3 generators with PBW basis. Journal of Algebra, 631:695– 730, 2023. [5] V. V. Bavula. The 3-cyclic quantum weyl algebras, their prime spectra and a classification of simple modules (q is not a root of unity). Journal of Noncommutative Geometry, 18:1041–1079, 2024. [6] V. V. Bavula and T. H. Lenagan. Generalized weyl algebras are tensor krull minimal. Journal of Algebra, 239(1):93– 111, 2001. [7] Allen D. Bell. Notes on localization in noncommutative noetherian rings. Manuscript/notes, 2000. [8] Georgia Benkart and Tom Roby. Down–up algebras. Jour- nal of Algebra, 209(1):305–344, 1998. [9] Sanu Bera, Sugata Mandal, Snehasis Mukherjee, and Soumendu Nandy. Simple modules over 3-cyclic quantum weyl algebra at roots of unity, 2024. [10] Sanu Bera and Snehashis Mukherjee. U+ q (B2) and its representations. Communications in Algebra, 0(0):1–18, 2025. [11] Sanu Bera and Snehasis Mukherjee. Pi degree of quan- tized weyl algebras. Proceedings of the Indian Academy of Sciences (Mathematical Sciences), 133:18, 2023. [12] Sanu Bera and Snehasis Mukherjee. On simple modules over the quantum matrix algebra at roots of unity, 2025. [13] Paula A. A. B. Carvalho. Some Noetherian Rings. PhD thesis, University of Glasgow, Glasgow, Scotland, 1997. Available at University of Glasgow Theses Service. [14] Paula A. A. B. Carvalho, Christian Lomp, and Dilek Pusat- Yilmaz. Injective modules over down-up algebras. Glas- gow Mathematical Journal, 52:53–59, 2010. [15] Paula A. A. B. Carvalho and Ian M. Musson. Down–up al- gebras and their representation theory. Journal of Algebra, 228(1):286–310, 2000. [16] Paula A. A. B. Carvalho and Ian M. Musson. Monolithic modules over noetherian rings. Glasgow Mathematical Journal, 53:483–491, 2011. [17] G. Cauchon. Les T-anneaux et les anneaux à identité poly- nomiale noethériens. Thèse, Université de Paris-Sud, Or- say, 1977. [18] K. Chan, J. Gaddis, R. Won, and James J. Zhang. Ozone groups of artin–schelter regular algebras satisfying a poly- nomial identity. Mathematische Zeitschrift, 310:69, 2025. [19] Sergio Chouhy, Estanislao Herscovich, and Andrea Solotar. Hochschild homology and cohomology of down– up algebras. Journal of Algebra, 498:102–128, 2018. [20] R. Daniano and J. Shapiro. Twisted polynomial rings sat- isfying a polynomial identity. Journal of Algebra, 92:116– 127, 1985. [21] Corrado De Concini and Claudio Procesi. Quantum groups. In Giuseppe Zampieri and Andrea D’Agnolo, ed- itors, D-modules, Representation Theory, and Quantum Groups, volume 1565 of Lecture Notes in Mathematics, pages 31–140. Springer, Berlin, 1993. [22] M. Dehn. über die Grundlagen der projektiven Geometrie und allgemeine Zahlsysteme. Math. Ann., 85(1):184–194, 1922. [23] V. Drensky and E. Formanek. Polynomial Identity Rings. Advanced Courses in Mathematics – CRM Barcelona. Birkhäuser, Basel, 2004. [24] Vladimir G. Drinfeld. Quantum groups. In Proceedings of the International Congress of Mathematicians, pages 798– 820, Berkeley, California, 1986. American Mathematical Society. [25] Pavel Etingof. Lie groups and lie algebras ii. MIT Open- CourseWare, Course 18.755, 2024. Full lecture notes, Spring 2024. [26] Jason Gaddis. Two-parameter analogs of the heisen- berg enveloping algebra. Communications in Algebra, 44(11):4637–4653, 2016. [27] Jason Gaddis and Daniel Yee. A family of trivial ozone algebras, 2025. [28] Claudia Gallego and Andrea Solotar. Stable rank of down- up algebras. Journal of Algebra, 526:266–282, 2019. [29] Takahiro Hayashi. q-analogues of clifford and weyl algebras—spinor and oscillator representations of quan- tum enveloping algebras. Communications in Mathemati- cal Physics, 127(1):129–144, 1990. [30] Nathan Jacobson. Collected mathematical papers. Vol. 2. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1989. (1947–1965). [31] Michio Jimbo. A q-difference analogue of u(g) and the Yang–Baxter equation. Letters in Mathematical Physics, 10(1):63–69, 1985. [32] David A. Jordan. A simple localization of the quantized weyl algebra. Journal of Algebra, 174(1):267–281, 1995. 22 J. Gómez and C. Gallego [33] David A. Jordan. Down–up algebras and ambiskew poly- nomial rings. Journal of Algebra, 228(1):311–346, 2000. [34] Irving Kaplansky. Rings with a polynomial identity. Bull. Amer. Math. Soc., 54:575–580, 1948. [35] E. E. Kirkman and L. W. Small. q-analogs of harmonic os- cillators and related rings. Israel Journal of Mathematics, 81(1-2):111–127, 1993. [36] Ellen Kirkman, Ian M. Musson, and D. S. Passman. Noetherian down-up algebras. Proceedings of the Ameri- can Mathematical Society, 127(11):3161–3167, 1999. JS- TOR. [37] Rajesh S. Kulkarni. Down–up algebras and their represen- tations. Journal of Algebra, 245(2):431–462, 2001. [38] Stéphane Launois. Primitive ideals and automorphism group of u+ q (b2). Journal of Algebra and its Applications, 6(1):21–47, 2007. [39] A. Leroy and J. Matczuk. Ore extensions satisfying a poly- nomial identity. Journal of Algebra and Its Applications, 2(3):257–273, 2007. [40] G. Maltsiniotis. Groupes quantiques et structures différen- tielles. Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, 311:831–834, 1990. [41] J. C. McConnell, J. C. Robson, and L. W. Small. Non- commutative Noetherian Rings, volume 30 of Graduate Studies in Mathematics. American Mathematical Society, 2001. [42] J. Pascaud and J. Valette. Polynômes tordus à identité polynomiale. Communications in Algebra, 16:2415–2425, 1988. [43] S. Paul Smith and James J. Zhang. A remark on Gelfand– Kirillov dimension. Proceedings of the American Mathe- matical Society, 126(2):349–352, 1998. [44] M. Takeuchi. A two-parameter quantization of GLn. Pro- ceedings of the Japan Academy, Series A, Mathematical Sciences, 66(5):112–114, 1990. [45] Walter Wagner. über die Grundlagen der projektiven Geometrie und allgemeine Zahlensysteme. Math. Ann., 113(1):528–567, 1937. [46] James J. Zhang and Jun Zhang. Double ore extensions. Journal of Pure and Applied Algebra, 212(12):2668–2690, 2008. [47] James J. Zhang and Jun Zhang. Double extension regular algebras of type (14641). Journal of Algebra, 322(2):373– 409, 2009. 23
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dc.relation.ispartofjournal.spa.fl_str_mv	Ciencia en Desarrollo
dc.relation.citationedition.spa.fl_str_mv	Núm. 3E , Año 2025 : Vol. 16 Núm. 3E (2025): VIII Encuentro Internacional de la Facultad de Ciencias (IMFaS 2025)
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spelling	Gomez, JamesGallego, Claudia2026-10-17 23:15:062026-10-17 23:15:062026-10-170121-748810.19053/uptc.01217488.v16.n3E.2025.211532462-7658https://doi.org/10.19053/uptc.01217488.v16.n3E.2025.21153engUniversidad Pedagógica y Tecnológica de Colombiahttps://creativecommons.org/licenses/by-nc-sa/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2https://revistas.uptc.edu.co/index.php/ciencia_en_desarrollo/article/view/21153PI property, quantum Heisenberg algebras, double Ore extensions, down-up algebras, quantum matrix algebras.The PI Property in Algebras of Polynomial TypeArtículo de revistahttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1info:eu-repo/semantics/articleJournal articleinfo:eu-repo/semantics/publishedVersionTexthttp://purl.org/coar/version/c_970fb48d4fbd8a85M. Akhavizadegan and David A. Jordan. Prime ideals of quantized weyl algebras. Glasgow Mathematical Journal, 38(3):283–297, 1996. [2] Nicolás Andruskiewitsch and Frédéric Dumas. On the au- tomorphisms of U+ q (g). In Quantum Groups, volume 12 of IRMA Lectures in Mathematics and Theoretical Physics, pages 107–133. European Mathematical Society, Zürich, 2008. [3] Michael Artin, William Schelter, and John Tate. Quantum deformations of GLn. Communications on Pure and Ap- plied Mathematics, 44(8-9):879–895, 1991. [4] V. V. Bavula. Description of bi-quadratic algebras on 3 generators with PBW basis. Journal of Algebra, 631:695– 730, 2023. [5] V. V. Bavula. The 3-cyclic quantum weyl algebras, their prime spectra and a classification of simple modules (q is not a root of unity). Journal of Noncommutative Geometry, 18:1041–1079, 2024. [6] V. V. Bavula and T. H. Lenagan. Generalized weyl algebras are tensor krull minimal. Journal of Algebra, 239(1):93– 111, 2001. [7] Allen D. Bell. Notes on localization in noncommutative noetherian rings. Manuscript/notes, 2000. [8] Georgia Benkart and Tom Roby. Down–up algebras. Jour- nal of Algebra, 209(1):305–344, 1998. [9] Sanu Bera, Sugata Mandal, Snehasis Mukherjee, and Soumendu Nandy. Simple modules over 3-cyclic quantum weyl algebra at roots of unity, 2024. [10] Sanu Bera and Snehashis Mukherjee. U+ q (B2) and its representations. Communications in Algebra, 0(0):1–18, 2025. [11] Sanu Bera and Snehasis Mukherjee. Pi degree of quan- tized weyl algebras. Proceedings of the Indian Academy of Sciences (Mathematical Sciences), 133:18, 2023. [12] Sanu Bera and Snehasis Mukherjee. On simple modules over the quantum matrix algebra at roots of unity, 2025. [13] Paula A. A. B. Carvalho. Some Noetherian Rings. PhD thesis, University of Glasgow, Glasgow, Scotland, 1997. Available at University of Glasgow Theses Service. [14] Paula A. A. B. Carvalho, Christian Lomp, and Dilek Pusat- Yilmaz. Injective modules over down-up algebras. Glas- gow Mathematical Journal, 52:53–59, 2010. [15] Paula A. A. B. Carvalho and Ian M. Musson. Down–up al- gebras and their representation theory. Journal of Algebra, 228(1):286–310, 2000. [16] Paula A. A. B. Carvalho and Ian M. Musson. Monolithic modules over noetherian rings. Glasgow Mathematical Journal, 53:483–491, 2011. [17] G. Cauchon. Les T-anneaux et les anneaux à identité poly- nomiale noethériens. Thèse, Université de Paris-Sud, Or- say, 1977. [18] K. Chan, J. Gaddis, R. Won, and James J. Zhang. Ozone groups of artin–schelter regular algebras satisfying a poly- nomial identity. Mathematische Zeitschrift, 310:69, 2025. [19] Sergio Chouhy, Estanislao Herscovich, and Andrea Solotar. Hochschild homology and cohomology of down– up algebras. Journal of Algebra, 498:102–128, 2018. [20] R. Daniano and J. Shapiro. Twisted polynomial rings sat- isfying a polynomial identity. Journal of Algebra, 92:116– 127, 1985. [21] Corrado De Concini and Claudio Procesi. Quantum groups. In Giuseppe Zampieri and Andrea D’Agnolo, ed- itors, D-modules, Representation Theory, and Quantum Groups, volume 1565 of Lecture Notes in Mathematics, pages 31–140. Springer, Berlin, 1993. [22] M. Dehn. über die Grundlagen der projektiven Geometrie und allgemeine Zahlsysteme. Math. Ann., 85(1):184–194, 1922. [23] V. Drensky and E. Formanek. Polynomial Identity Rings. Advanced Courses in Mathematics – CRM Barcelona. Birkhäuser, Basel, 2004. [24] Vladimir G. Drinfeld. Quantum groups. In Proceedings of the International Congress of Mathematicians, pages 798– 820, Berkeley, California, 1986. American Mathematical Society. [25] Pavel Etingof. Lie groups and lie algebras ii. MIT Open- CourseWare, Course 18.755, 2024. Full lecture notes, Spring 2024. [26] Jason Gaddis. Two-parameter analogs of the heisen- berg enveloping algebra. Communications in Algebra, 44(11):4637–4653, 2016. [27] Jason Gaddis and Daniel Yee. A family of trivial ozone algebras, 2025. [28] Claudia Gallego and Andrea Solotar. Stable rank of down- up algebras. Journal of Algebra, 526:266–282, 2019. [29] Takahiro Hayashi. q-analogues of clifford and weyl algebras—spinor and oscillator representations of quan- tum enveloping algebras. Communications in Mathemati- cal Physics, 127(1):129–144, 1990. [30] Nathan Jacobson. Collected mathematical papers. Vol. 2. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1989. (1947–1965). [31] Michio Jimbo. A q-difference analogue of u(g) and the Yang–Baxter equation. Letters in Mathematical Physics, 10(1):63–69, 1985. [32] David A. Jordan. A simple localization of the quantized weyl algebra. Journal of Algebra, 174(1):267–281, 1995. 22 J. Gómez and C. Gallego [33] David A. Jordan. Down–up algebras and ambiskew poly- nomial rings. Journal of Algebra, 228(1):311–346, 2000. [34] Irving Kaplansky. Rings with a polynomial identity. Bull. Amer. Math. Soc., 54:575–580, 1948. [35] E. E. Kirkman and L. W. Small. q-analogs of harmonic os- cillators and related rings. Israel Journal of Mathematics, 81(1-2):111–127, 1993. [36] Ellen Kirkman, Ian M. Musson, and D. S. Passman. Noetherian down-up algebras. Proceedings of the Ameri- can Mathematical Society, 127(11):3161–3167, 1999. JS- TOR. [37] Rajesh S. Kulkarni. Down–up algebras and their represen- tations. Journal of Algebra, 245(2):431–462, 2001. [38] Stéphane Launois. Primitive ideals and automorphism group of u+ q (b2). Journal of Algebra and its Applications, 6(1):21–47, 2007. [39] A. Leroy and J. Matczuk. Ore extensions satisfying a poly- nomial identity. Journal of Algebra and Its Applications, 2(3):257–273, 2007. [40] G. Maltsiniotis. Groupes quantiques et structures différen- tielles. Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, 311:831–834, 1990. [41] J. C. McConnell, J. C. Robson, and L. W. Small. Non- commutative Noetherian Rings, volume 30 of Graduate Studies in Mathematics. American Mathematical Society, 2001. [42] J. Pascaud and J. Valette. Polynômes tordus à identité polynomiale. Communications in Algebra, 16:2415–2425, 1988. [43] S. Paul Smith and James J. Zhang. A remark on Gelfand– Kirillov dimension. Proceedings of the American Mathe- matical Society, 126(2):349–352, 1998. [44] M. Takeuchi. A two-parameter quantization of GLn. Pro- ceedings of the Japan Academy, Series A, Mathematical Sciences, 66(5):112–114, 1990. [45] Walter Wagner. über die Grundlagen der projektiven Geometrie und allgemeine Zahlensysteme. Math. Ann., 113(1):528–567, 1937. [46] James J. Zhang and Jun Zhang. Double ore extensions. Journal of Pure and Applied Algebra, 212(12):2668–2690, 2008. [47] James J. Zhang and Jun Zhang. Double extension regular algebras of type (14641). Journal of Algebra, 322(2):373– 409, 2009. 233E16Ciencia en DesarrolloNúm. 3E , Año 2025 : Vol. 16 Núm. 3E (2025): VIII Encuentro Internacional de la Facultad de Ciencias (IMFaS 2025)PublicationOREORE.xmltext/xml1967https://repositorio.uptc.edu.co/bitstreams/ed782c88-c857-4690-9a77-a9847239f4a9/download9fc32f3c9b1a81603f293e90ccc8ecedMD51falseAnonymousREAD001/23145oai:repositorio.uptc.edu.co:001/231452026-10-22 01:15:18.587https://creativecommons.org/licenses/by-nc-sa/4.0/metadata.onlyhttps://repositorio.uptc.edu.coRepositorio Institucional UPTCbdigital@metabiblioteca.com

The PI Property in Algebras of Polynomial Type

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