{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:43:00Z","timestamp":1760146980275,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,26]],"date-time":"2024-12-26T00:00:00Z","timestamp":1735171200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we analyze the Lie bialgebra (LB) and quantize the generalized loop planar-Galilean conformal algebra (GLPGCA) W(\u0393). Additionally, we prove that all LB structures on W(\u0393) possess a triangular coboundary. We also quantize W(\u0393) using the Drinfeld-twist quantization technique and identify a group of noncommutative algebras and noncocommutative Hopf algebras.<\/jats:p>","DOI":"10.3390\/axioms14010007","type":"journal-article","created":{"date-parts":[[2024,12,26]],"date-time":"2024-12-26T21:13:18Z","timestamp":1735247598000},"page":"7","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lie Bialgebra Structures and Quantization of Generalized Loop Planar Galilean Conformal Algebra"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0230-684X","authenticated-orcid":false,"given":"Yu","family":"Yang","sequence":"first","affiliation":[{"name":"School of Electronics and Information Engineering, Taizhou University, Taizhou 318000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7968-7188","authenticated-orcid":false,"given":"Xingtao","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,26]]},"reference":[{"key":"ref_1","first-page":"667","article-title":"Constant quasiclassical solutions of the Yang\u2013Baxter quantum equation","volume":"273","year":"1983","journal-title":"Sov. Math. Dokl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s11464-019-0761-0","article-title":"Lie bialgebra structures on generalized loop Schr\u00f6dinger\u2013Virasoro algebras","volume":"14","author":"Chen","year":"2019","journal-title":"Front. Math. China"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1685","DOI":"10.4134\/BKMS.b150884","article-title":"Lie super-bialgebras on generalized loop super-virasoro algebras","volume":"53","author":"Dai","year":"2016","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1063\/1.4935652","article-title":"Lie bialgebra structures on the deformative Schr\u00f6dinger\u2013Virasoro algebras","volume":"56","author":"Fa","year":"2015","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1063\/1.3187784","article-title":"Lie bialgebra structures on the Schr\u00f6dinger\u2013Virasoro Lie algebra","volume":"50","author":"Han","year":"2009","journal-title":"J. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/S0252-9602(10)60040-9","article-title":"Lie super-bialgebra structures on generalized super-Virasoro algebras","volume":"30","author":"Yang","year":"2010","journal-title":"Acta Math. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1007\/s11401-015-0904-x","article-title":"Lie bialgebras of Generalized loop Virasoro algebras","volume":"36","author":"Wu","year":"2013","journal-title":"Chin. Ann. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1063\/1.4952701","article-title":"Lie super-bialgebra and quantization of the super Virasoro algebra","volume":"57","author":"Yuan","year":"2016","journal-title":"J. Math. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01587938","article-title":"Quantization of Lie bialgebras I","volume":"2","author":"Etingof","year":"1996","journal-title":"Sel. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1007\/s10114-011-8080-8","article-title":"Quantizations of the W Algebra W(2, 2)","volume":"27","author":"Bo","year":"2011","journal-title":"Acta. Math. Sin. (Engl. Ser.)"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"98","DOI":"10.4172\/1736-4337.1000172","article-title":"Quantization of the q-analog Virasoro-like algebras","volume":"4","author":"Cheng","year":"2010","journal-title":"J. Gen. Lie Theory Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.aim.2014.10.022","article-title":"Deformation quantization of Leibniz algebras","volume":"270","author":"Dherin","year":"2015","journal-title":"Adv. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.jalgebra.2004.04.016","article-title":"Quantizations of the Witt algebra and of simple Lie algebras in characteristic p","volume":"280","author":"Grunspan","year":"2003","journal-title":"J. Algebra"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"902","DOI":"10.1016\/j.jalgebra.2007.02.019","article-title":"Quantizations of generalized-Witt algebra and of Jacobson-Witt algebra in the modular case","volume":"312","author":"Hu","year":"2007","journal-title":"J. Algebra"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2888","DOI":"10.1016\/j.laa.2008.01.020","article-title":"Quantization of generalized Virasoro-like algebras","volume":"428","author":"Song","year":"2008","journal-title":"Linear Algebra Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1142\/S100538670900042X","article-title":"Quantization of Lie Algebras of Generalized Weyl Type","volume":"16","author":"Yue","year":"2009","journal-title":"Algebra Colloq."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1662","DOI":"10.1063\/1.531629","article-title":"Quantization of a loop extended SU(2) affine Kac\u2013Moody algebra","volume":"37","year":"1996","journal-title":"J. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"037","DOI":"10.1088\/1126-6708\/2009\/07\/037","article-title":"Galilean conformal algebras and AdS\/CFT","volume":"2009","author":"Bagchi","year":"2009","journal-title":"J. High Energy Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"052307","DOI":"10.1063\/1.3371191","article-title":"Affine Extension of Galilean Conformal Algebra in 2 + 1 Dimensions","volume":"51","author":"Hosseiny","year":"2010","journal-title":"J. Math. Phys."},{"key":"ref_20","first-page":"1","article-title":"Comments on Galilean conformal field theories and their geometric realization","volume":"5","author":"Martelli","year":"2010","journal-title":"J. High Energy Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/978-4-431-54270-4_21","article-title":"Some properties of planar Galilean conformal algebras","volume":"36","author":"Aizawa","year":"2013","journal-title":"Lie Theory Its Appl. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1606","DOI":"10.1080\/03081087.2017.1364340","article-title":"Biderivations and linear commuting maps on the planar Galilean conformal algebra","volume":"66","author":"Chi","year":"2018","journal-title":"Linear Multilinear Algebra"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1142\/S1005386719000221","article-title":"Left-Symmetric Algebra Structures on the planar Galilean conformal algebra","volume":"26","author":"Chi","year":"2019","journal-title":"Algebra Colloq."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0034-4877(16)30052-0","article-title":"Structure of The planar Galilean Conformal algebra","volume":"78","author":"Gao","year":"2016","journal-title":"Rep. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/s00220-021-04302-9","article-title":"Representations of the planar Galilean Conformal algebra","volume":"391","author":"Gao","year":"2022","journal-title":"Comm. Math. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1080\/03081087.2018.1429380","article-title":"Biderivations of the planar Galilean conformal algebra and their applications","volume":"67","author":"Tang","year":"2019","journal-title":"Linear Multilinear Algebra"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s00009-018-1162-4","article-title":"Structures of Generalized Loop Schr\u00f6dinger\u2013Virasoro algebras","volume":"15","author":"Chen","year":"2018","journal-title":"Mediterr. J. Math."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kac, V.G. (1990). Infinite-Dimensional Lie Algebras, Cambridge University Press. [3rd ed.].","DOI":"10.1017\/CBO9780511626234"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"071707","DOI":"10.1063\/1.5100918","article-title":"Non-weight modules over the affine-Virasoro algebra of type A1","volume":"60","author":"Chen","year":"2019","journal-title":"J. Math. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1550041","DOI":"10.1142\/S0129167X1550041X","article-title":"Structures of generalized loop super-Virasoro algebras","volume":"26","author":"Dai","year":"2015","journal-title":"Int. J. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1545","DOI":"10.1080\/00927872.2012.744029","article-title":"Structures of generalized loop Virasoro algebras","volume":"42","author":"Wu","year":"2014","journal-title":"Comm. Algebra"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2517","DOI":"10.1080\/00927872.2021.2009492","article-title":"Structures of generalized loop planar Galilean conformal algebras","volume":"50","author":"Yang","year":"2021","journal-title":"Comm. Algebra"},{"key":"ref_33","first-page":"798","article-title":"Quantum groups","volume":"Volumes 1 and 2","year":"1987","journal-title":"Proceedings of the International Congress of Mathematicians"},{"key":"ref_34","unstructured":"Wang, P. (2020). Lie Bialgebra Structures and Classification of Simple Weight Modules with Finite-Dimensional Weight Spaces over the Planar Galilean Conformal Algebra, Henan University."},{"key":"ref_35","first-page":"222","article-title":"Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang\u2013Baxter equations","volume":"27","year":"1983","journal-title":"Sov. Math. Dokl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:00:57Z","timestamp":1760115657000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,26]]},"references-count":35,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010007"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010007","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,12,26]]}}}