{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:27:25Z","timestamp":1760232445100,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T00:00:00Z","timestamp":1667952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11901240","12001174"],"award-info":[{"award-number":["11901240","12001174"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Hopf algebras, as a crucial generalization of groups, have a very symmetric structure and have been playing a prominent role in mathematical physics. In this paper, let H be a dual quasi-Hopf algebra which is a more general Hopf algebra structure. A. Balan firstly introduced the notion of right-right Yetter-Drinfeld modules over H and studied its Galois extension. As a continuation, the aim of this paper is to introduce more properties of Yetter-Drinfeld modules. First, we will describe all the other three kinds of Yetter-Drinfeld modules over H, and the monoidal and braided structure of the categories of Yetter-Drinfeld modules explicitly. Furthermore, we will prove that the category HHYDfd of finite dimensional left-left Yetter-Drinfeld modules is rigid. Then we will compute explicitly the canonical isomorphisms in HHYDfd. Finally, as an application, we will rewrite the isomorphisms in the case of coquasitriangular dual quasi-Hopf algebra.<\/jats:p>","DOI":"10.3390\/sym14112358","type":"journal-article","created":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T02:44:32Z","timestamp":1667961872000},"page":"2358","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Canonical Isomorphisms in the Yetter-Drinfeld Categories for Dual Quasi-Hopf Algebras"],"prefix":"10.3390","volume":"14","author":[{"given":"Yan","family":"Ning","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Application Technology, Jining University, Qufu 273155, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daowei","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Application Technology, Jining University, Qufu 273155, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaofan","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,9]]},"reference":[{"key":"ref_1","first-page":"1419","article-title":"Quasi-Hopf algebras","volume":"1","author":"Drinfeld","year":"1990","journal-title":"Leningrad Math. 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Graduate Texts in Mathematics 155.","DOI":"10.1007\/978-1-4612-0783-2"},{"key":"ref_14","first-page":"45","article-title":"Braided Hopf algebras over Abelian finite groups","volume":"63","author":"Andruskiewitsch","year":"1999","journal-title":"Bol. Acad. Ciencias (C\u00f2rdoba)"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2358\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:12:55Z","timestamp":1760145175000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2358"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,9]]},"references-count":14,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112358"],"URL":"https:\/\/doi.org\/10.3390\/sym14112358","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,11,9]]}}}