{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:46:19Z","timestamp":1760147179969,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,15]],"date-time":"2023-01-15T00:00:00Z","timestamp":1673740800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11801477","2017J05016"],"award-info":[{"award-number":["11801477","2017J05016"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["11801477","2017J05016"],"award-info":[{"award-number":["11801477","2017J05016"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A Lie bialgebra is a vector space endowed simultaneously with the structure of a Lie algebra and the structure of a Lie coalgebra, and some compatibility condition. Moreover, Lie brackets have skew symmetry. Because of the close relation between Lie bialgebras and quantum groups, it is interesting to consider the Lie bialgebra structures on the Lie algebra L related to the Virasoro algebra. In this paper, the Lie bialgebras on L are investigated by computing Der(L,\u00a0L\u2297L). It is proved that all such Lie bialgebras are triangular coboundary, and the first cohomology group H1(L,\u00a0L\u2297L) is trivial.<\/jats:p>","DOI":"10.3390\/sym15010239","type":"journal-article","created":{"date-parts":[[2023,1,16]],"date-time":"2023-01-16T03:10:34Z","timestamp":1673838634000},"page":"239","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lie Bialgebra Structures on the Lie Algebra L Related to the Virasoro Algebra"],"prefix":"10.3390","volume":"15","author":[{"given":"Xue","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Xiamen University of Technology, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yihong","family":"Su","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xiamen University of Technology, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jia","family":"Zheng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xiamen University of Technology, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kac, V.G. (1990). Infinite-Dimensional Lie Algebras, Cambridge University Press. [3rd ed.].","DOI":"10.1017\/CBO9780511626234"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Dong, C.Y., and Lepowsky, J. (1993). Generalized Vertex Algebras and Relative Vertex Operators, Birkhauser. Progress in Math.","DOI":"10.1007\/978-1-4612-0353-7"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Allison, B., Azam, S., Berman, S., Gao, Y., and Pianzola, A. (1997). Extended Affine Lie Algebras and Their Root Systems, Memoirs of the American Mathematical Society.","DOI":"10.1090\/memo\/0603"},{"key":"ref_4","first-page":"91","article-title":"Structures of the Lie algebra L related to Virasoro algebra","volume":"22","author":"Chen","year":"2014","journal-title":"J. Xiamen Univ. Technol."},{"key":"ref_5","unstructured":"Zhao, Q. (2014). On Structures of Two Classes of Infinite Dimensional Lie Algebras. [Master\u2019s Thesis, Xiamen University]. (In Chinese)."},{"key":"ref_6","first-page":"667","article-title":"Constant quasiclassical solutions of the Yang-Baxter quantum equation","volume":"28","author":"Drinfeld","year":"1983","journal-title":"Soviet Math. Dokl."},{"key":"ref_7","first-page":"798","article-title":"Quantum Groups","volume":"Volumes 1 and 2","author":"Drinfeld","year":"1987","journal-title":"Proceedings of the International Congress of Mathematicians"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0022-4049(93)90116-B","article-title":"Witt and Virasoro algebras as Lie bialgebras","volume":"87","author":"Taft","year":"1993","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1006\/aima.1994.1062","article-title":"A class of infinite-dimensional Lie bialgebras containing the Virasoro algebras","volume":"107","author":"Michaelis","year":"1994","journal-title":"Adv. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/S0022-4049(99)00045-6","article-title":"Classification of the Lie bialgebra structures on the Witt and Virasoro algebras","volume":"151","author":"Ng","year":"2000","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1007\/s11425-006-0533-7","article-title":"Lie bialgebras of generalized Witt type","volume":"49","author":"Song","year":"2006","journal-title":"Sci. China Ser. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1915","DOI":"10.1007\/s10114-005-0742-y","article-title":"Lie bialgebras of generalized Virasoro-like type","volume":"22","author":"Wu","year":"2006","journal-title":"Acta Math. Sin. Engl. Ser."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1080\/00927870701776649","article-title":"Lie bialgebra structures on Lie algebras of generalized Weyl type","volume":"36","author":"Yue","year":"2008","journal-title":"Comm. Algebra"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"083504","DOI":"10.1063\/1.3187784","article-title":"Lie bialgebra structures on the Schr\u00f6dinger-Virasoro Lie algebra","volume":"50","author":"Han","year":"2009","journal-title":"J. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1142\/S1005386710000192","article-title":"Lie bialgebra structures on the W-algebra W (2,2)","volume":"17","author":"Li","year":"2010","journal-title":"Algebra Colloq."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.jalgebra.2012.03.009","article-title":"Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra","volume":"359","author":"Liu","year":"2012","journal-title":"J. Algebra"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"111706","DOI":"10.1063\/1.4935652","article-title":"Lie bialgebra structures on the deformative Schr\u00f6dinger-Virasoro algebras","volume":"56","author":"Fa","year":"2015","journal-title":"J. Math. Phys."},{"key":"ref_18","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":"2015","journal-title":"Chin. Ann. Math. Ser. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s11464-019-0761-0","article-title":"Lie bialgebra structures on generalized loop Schr\u00f6dinger-Virasoro algebras","volume":"14","author":"Chen","year":"2019","journal-title":"Front. Math. China"},{"key":"ref_20","first-page":"1039","article-title":"Dual Lie bialgebras of Witt and Virasoro types","volume":"43","author":"Song","year":"2013","journal-title":"Sci. Sin. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/239\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:06:35Z","timestamp":1760119595000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/239"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,15]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010239"],"URL":"https:\/\/doi.org\/10.3390\/sym15010239","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,1,15]]}}}