Rota-Baxter TD Algebra and Quinquedendriform Algebra

• 来源期刊: 代数集刊（英文版）
• 关键词: dendriform algebra Rota-Baxter algebra RBTD algebra free objects operads rooted trees quinquedendriform algebra
• 年，卷（期）: 2017,（1）
• 页码范围: 53-74
• 页数: 22页
• 语种: 英文
• DOI: 10.1142/S1005386717000037

摘要

A dendriform algebra defined by Loday has two binary operations that give a two-part splitting of the associativity in the sense that their sum is associative.Similar dendriform type algebras with three-part and four-part splitting of the associativity were later obtained.These structures can also be derived from actions of suitable linear operators,such as a Rota-Baxter operator or TD operator,on an associative algebra.Motivated by finding a five-part splitting of the associativity,we consider the Rota-Baxter TD (RBTD) operator,an operator combining the Rota-Baxter operator and TD operator,and coming from a recent study of Rota's problem concerning linear operators on associative algebras.Free RBTD algebras on rooted forests are constructed.We then introduce the concept of a quinquedendriform algebra and show that its defining relations are characterized by the action of an RBTD operator,similar to the cases of dendriform and tridendriform algebras.