{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-022-10198-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.
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。
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