Rota-Baxter簇系统与Gr\"obner-Shirshov基

Rota-Baxter Family Systems and Gr\"{o}bner-Shirshov Bases

中文关键词:  Rota-Baxter簇系统  Rota-Baxter簇代数  Gr\"obner-Shirshov基

英文关键词:Rota-Baxter family systems  Rota-Baxter family algebras  Gr\"obner-Shirshov bases

中文摘要:

      从Yang-Baxter簇方程和Volterra积分方程得到的Rota-Baxter簇代数的概念出发,我们引入Rota-Baxter簇系统的概念,推广了Brzezinski提出的Rota-Baxter系统.我们证明这个概念也与结合Yang-Baxter簇对和pre-Lie簇代数有关.此外,作为Rota-Baxter簇系统的一个类比,我们引入平均簇系统的概念,并证明平均簇系统会得到dialgebra簇结构.我们还研究dendriform代数上的Rota-Baxter簇系统,并展示它们如何诱导quadri簇代数结构.最后,我们用Gr\"obner-Shirshov基的方法给出Rota-Baxter簇系统的一个线性基.

英文摘要:

      Motivated by the concept of Rota-Baxter family algebras arising from associative Yang-Baxter family equations and Volterra integral equations, we introduce the notion of a Rota-Baxter family system which generalizes the Rota-Baxter system proposed by Brzezi$\acute{\mathrm{n}}$ski. We show that this notion is also related to an associative Yang-Baxter family pair and the pre-Lie family algebras. Furthermore, as an analogue of Rota-Baxter family system, we introduce a notion of averaging family system and prove that an averaging family system induces a dialgebra family structure. We also study Rota-Baxter family systems on a dendriform algebra and show how they induce quadri family algebra structures. Finally, we give a linear basis of the Rota-Baxter family system by the methods of Gr\"obner-Shirshov bases.