{"title":"自由Rota-Baxter族代数和自由(三)树形族代数","authors":"Yuanyuan Zhang, Xing Gao, Dominique Manchon","doi":"10.1007/s10468-022-10198-3","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we first construct the free Rota-Baxter family algebra generated by some set <i>X</i> in terms of typed angularly <i>X</i>-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"26 6","pages":"2967 - 3002"},"PeriodicalIF":0.5000,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Free Rota-Baxter Family Algebras and Free (tri)dendriform Family Algebras\",\"authors\":\"Yuanyuan Zhang, Xing Gao, Dominique Manchon\",\"doi\":\"10.1007/s10468-022-10198-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we first construct the free Rota-Baxter family algebra generated by some set <i>X</i> in terms of typed angularly <i>X</i>-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.</p></div>\",\"PeriodicalId\":50825,\"journal\":{\"name\":\"Algebras and Representation Theory\",\"volume\":\"26 6\",\"pages\":\"2967 - 3002\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-02-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebras and Representation Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10468-022-10198-3\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebras and Representation Theory","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-022-10198-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
在本文中，我们首先用类型化的角X装饰平面有根树来构造由某个集合X生成的自由罗塔-巴克斯特族代数。作为一个应用，我们只用角装饰平面有根树（而不是森林）就得到了自由罗塔-巴克斯特代数的新构造，这与 K. Ebrahimi-Fard 和 L. Guo 通过角装饰平面有根林的已知构造截然不同。然后，我们将自由树枝状（或三树枝状）族代数嵌入权重为零（或一）的自由罗塔-巴克斯特族代数中。最后，我们证明自由 Rota-Baxter 族代数是自由（三）树枝状族代数的普遍包络代数。
Free Rota-Baxter Family Algebras and Free (tri)dendriform Family Algebras
In this paper, we first construct the free Rota-Baxter family algebra generated by some set X in terms of typed angularly X-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.
期刊介绍：
Algebras and Representation Theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including Lie algebras and superalgebras, rings of differential operators, group rings and algebras, C*-algebras and Hopf algebras, with particular emphasis on quantum groups.
The journal contains high level, significant and original research papers, as well as expository survey papers written by specialists who present the state-of-the-art of well-defined subjects or subdomains. Occasionally, special issues on specific subjects are published as well, the latter allowing specialists and non-specialists to quickly get acquainted with new developments and topics within the field of rings, algebras and their applications.
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