{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T05:19:31Z","timestamp":1782364771590,"version":"3.54.5"},"reference-count":14,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,20]],"date-time":"2023-07-20T00:00:00Z","timestamp":1689811200000},"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":["11771135"],"award-info":[{"award-number":["11771135"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["21C0498"],"award-info":[{"award-number":["21C0498"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2019-TD-15"],"award-info":[{"award-number":["2019-TD-15"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Hunan Provincial Department of Education","award":["11771135"],"award-info":[{"award-number":["11771135"]}]},{"name":"Hunan Provincial Department of Education","award":["21C0498"],"award-info":[{"award-number":["21C0498"]}]},{"name":"Hunan Provincial Department of Education","award":["2019-TD-15"],"award-info":[{"award-number":["2019-TD-15"]}]},{"name":"Hunnan Institute of Science and Technology","award":["11771135"],"award-info":[{"award-number":["11771135"]}]},{"name":"Hunnan Institute of Science and Technology","award":["21C0498"],"award-info":[{"award-number":["21C0498"]}]},{"name":"Hunnan Institute of Science and Technology","award":["2019-TD-15"],"award-info":[{"award-number":["2019-TD-15"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The centroid of Lie algebra is a basic concept and a necessary tool for studying the structure of Lie algebraic structure. The extended Heisenberg algebra is an important class of solvable Lie algebras. In any Lie algebra, the anti symmetry of Lie operations is an important property of Lie algebra. This article investigates the centroids and structures of 2n+2 dimensional extended Heisenberg algebras, where all invertible elements form a group and all elements form a ring. Then, its main research results are extended to infinite dimensional extended Heisenberg algebras.<\/jats:p>","DOI":"10.3390\/sym15071451","type":"journal-article","created":{"date-parts":[[2023,7,21]],"date-time":"2023-07-21T01:52:35Z","timestamp":1689904355000},"page":"1451","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Study on the Centroid of a Class of Solvable Lie Algebras"],"prefix":"10.3390","volume":"15","author":[{"given":"Demin","family":"Yu","sequence":"first","affiliation":[{"name":"School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chan","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiejing","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.jpaa.2005.06.007","article-title":"The centroid of extended affine and root graded Lie algebras","volume":"205","author":"Benkart","year":"2005","journal-title":"J. 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Symmetry, 15.","DOI":"10.3390\/sym15051115"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1451\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:15:42Z","timestamp":1760127342000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1451"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,20]]},"references-count":14,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["sym15071451"],"URL":"https:\/\/doi.org\/10.3390\/sym15071451","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,20]]}}}