{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T15:40:53Z","timestamp":1767109253049,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,16]],"date-time":"2021-07-16T00:00:00Z","timestamp":1626393600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"publisher","award":["LQ16A010011"],"award-info":[{"award-number":["LQ16A010011"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871421"],"award-info":[{"award-number":["11871421"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Consider a Lie color algebra, denoted by L. Our aim in this paper is to study the Lie triple derivations TDer(L) and generalized Lie triple derivations GTDer(L) of Lie color algebras. We discuss the centroids, quasi centroids and central triple derivations of Lie color algebras, where we show the relationship of triple centroids, triple quasi centroids and central triple derivation with Lie triple derivations and generalized Lie triple derivations of Lie color algebras L. A classification of Lie triple derivations algebra of all perfect Lie color algebras is given, where we prove that for a perfect and centerless Lie color algebra, TDer(L)=Der(L) and TDer(Der(L))=Inn(Der(L)).<\/jats:p>","DOI":"10.3390\/sym13071280","type":"journal-article","created":{"date-parts":[[2021,7,18]],"date-time":"2021-07-18T21:18:52Z","timestamp":1626643132000},"page":"1280","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Generalized Lie Triple Derivations of Lie Color Algebras and Their Subalgebras"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1922-9430","authenticated-orcid":false,"given":"Sania","family":"Asif","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8114-3542","authenticated-orcid":false,"given":"Zhixiang","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"493","DOI":"10.4153\/CJM-1960-044-x","article-title":"Generalized Lie elements","volume":"12","author":"Ree","year":"1960","journal-title":"Can. J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1063\/1.1628837","article-title":"Classification of derivation-simple color algebras related to locally finite derivations","volume":"45","author":"Su","year":"2004","journal-title":"J. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/BF02785957","article-title":"Simple color algebras of Weyl type","volume":"137","author":"Su","year":"2003","journal-title":"Israel J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1006\/jabr.1997.7050","article-title":"Constructing simple Lie superalgebras from associative graded algebras","volume":"195","author":"Montgomery","year":"1997","journal-title":"J. Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1063\/1.524113","article-title":"Generalized Lie algebras","volume":"20","author":"Scheunert","year":"1979","journal-title":"J. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/s40840-017-0486-8","article-title":"Generalized derivations of multiplicative n-ary Hom-\u03c9 color algebras","volume":"42","author":"Beites","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"740","DOI":"10.1006\/jabr.1995.1327","article-title":"Delta ideal of Lie color algebras","volume":"177","author":"Bergen","year":"1995","journal-title":"J. Algebra"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1006\/aima.2000.1942","article-title":"Representations of Lie color algebras","volume":"157","author":"Feldvoss","year":"2001","journal-title":"Adv. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1086","DOI":"10.1080\/03081087.2015.1072492","article-title":"Generalized derivations of (color) n-ary algebras","volume":"64","author":"Kaygorodov","year":"2016","journal-title":"Linear Multilinear Algebra"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1006\/jabr.1995.1207","article-title":"Delta methods in enveloping algebras of Lie color algebras","volume":"75","author":"Wilson","year":"1995","journal-title":"J. Algebra"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/BF02784122","article-title":"On Lie derivations of Lie ideals of prime algebras","volume":"123","author":"Beidar","year":"2001","journal-title":"Israel J. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1006\/jabr.1996.0281","article-title":"Lie derivations of the skew elements of prime rings with involution","volume":"184","author":"Swain","year":"1996","journal-title":"J. Algebra"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3737","DOI":"10.1080\/00927870903236228","article-title":"Generalized derivations of Lie superalgebras","volume":"38","author":"Zhang","year":"2010","journal-title":"Comm. Algebra"},{"key":"ref_14","first-page":"260","article-title":"Generalized derivations of Lie triple systems","volume":"41","author":"Zhou","year":"2016","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_15","first-page":"65","article-title":"Isometries of bi-invariant pseudo-Riemannian metrices on Lie groups","volume":"29","year":"1989","journal-title":"Geom. Dedicata"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1841","DOI":"10.1080\/00927872.2018.1523422","article-title":"Lie triple derivations of incidence algebras","volume":"47","author":"Wang","year":"2019","journal-title":"Comm. Algebra"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1016\/j.laa.2005.12.003","article-title":"Lie triple derivations of nest algebras","volume":"416","author":"Zhang","year":"2006","journal-title":"Linear Algebra Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1016\/j.laa.2012.04.015","article-title":"Lie triple derivations of triangular algebras","volume":"437","author":"Xiao","year":"2012","journal-title":"Linear Algebra Appl."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Qi, X. (2014). Characterization of Multiplicative Lie Triple Derivations on Rings. Abstr. Appl. Anal.","DOI":"10.1155\/2014\/739730"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1647","DOI":"10.1080\/00927872.2011.649224","article-title":"Triple derivations of perfect lie algebras","volume":"41","author":"Zhou","year":"2013","journal-title":"Comm. Algebra"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1016\/j.indag.2016.11.012","article-title":"Triple derivations and triple homomorphisms of perfect Lie superalgebras","volume":"28","author":"Zhou","year":"2015","journal-title":"Indag. Math."},{"key":"ref_22","first-page":"29","article-title":"The centroid of n-Lie algebras","volume":"25","author":"Bai","year":"2004","journal-title":"Algebr. Groups Geom."},{"key":"ref_23","first-page":"9","article-title":"The centroid of a Lie triple Algebra","volume":"404219","author":"Liu","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_24","first-page":"933","article-title":"Derivations and Extensions of Lie Color Algebra","volume":"28B","author":"Zhang","year":"2008","journal-title":"Acta Math. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1080\/03081080903350153","article-title":"Generalized Lie triple derivations","volume":"59","author":"Li","year":"2011","journal-title":"Linear Multilinear Algebra"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1007\/s00025-012-0241-2","article-title":"Generalized Derivations of Lie Color Algebras","volume":"63","author":"Chen","year":"2013","journal-title":"Results Math."},{"key":"ref_27","first-page":"203","article-title":"Centroids and quasi-centroids of finite dimensional Leibniz algebras","volume":"113","author":"Almutari","year":"2017","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Fiidow, M.A., Rakhimov, I.S., and Hussain, S.S. (2015, January 25\u201327). Derivations and Centroids of Associative Algebras. Proceedings of the IEEE Proceedings of International Conference on Research and Education in Mathematics (ICREM7), Kuala Lumpur, Malaysia.","DOI":"10.1109\/ICREM.2015.7357059"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1280\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:30:44Z","timestamp":1760164244000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1280"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,16]]},"references-count":28,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071280"],"URL":"https:\/\/doi.org\/10.3390\/sym13071280","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,7,16]]}}}