Cocycle Hopf 代数和 Reynolds 算子的研究

Both Rota-Baxter operator and Rota-Baxter algebra are the algebraization of the integral operator and have been studied intensively in recent years. Furthermore, In Gian-Carlo Rota's view, Reynolds operators are infinitesimal analogs of the Rota-Baxter operators. So far, there were only a few papers about Reynolds operators from analytic point of view. This program will study further around the following two problems associated with Rota-Baxter operators and Reynolds operators. The first one is to study the cocycle Hopf algebras on free Rota-Baxter algebras. We plan to carry out this issue as following aspects : combinatorial description of the coproduct on cocycle Hopf algebras will be determined, the dual cocycle Hopf algebra will be studied, the Milnor-Moore theorem for cocycle Hopf algebras will be explored; The second one is to study the Reynolds operators from algebraic point of view. We proceed in the following aspects: the free Reynolds algebra will be constructed, the enumeration of Reynolds words will be studied, Reynolds operators in some algebras (such as polynomial algebras and so on) will be determined. The research of this program will promote the development of the theories of Rota-Baxter operators and Reynolds operators respectively, which has not only important theoretical significance, but also very good application prospects.

Rota-Baxter算子和Rota-Baxter代数是积分算子的代数化，对它们的研究方兴未艾。另外，G.-C.Rota认为Reynolds算子是“无穷小的Rota-Baxter算子”。目前Reynolds算子的研究几乎都在分析学范畴中进行。本项目将围绕与Rota-Baxter算子和Reynolds算子相关的如下两个问题展开：(1)自由Rota-Baxter代数上cocycle Hopf代数的研究，包括cocycle Hopf代数余乘的组合解释，对偶cocycle Hopf代数的研究，cocycle Hopf代数上Milnor-Moore定理的刻画。(2)Reynolds算子的代数化研究，包括自由Reynolds代数的构造，Reynolds字的记数，Reynolds算子在一些代数（多项式代数等）上的构造及其性质。此项工作将推动这两类算子的研究，具有重要的理论意义以及很好的应用前景。

Reynolds算子最早出现在流体力学的研究之中，是一种对群作用取平均的线性算子。自上世纪六、七十年代在分析学中得到广泛应用和发展之后，对Reynolds算子的研究停滞在分析学的领域之中。我们最先开始了Reynolds算子的代数化研究，主要包括：1. Reynolds代数的基本性质，譬如探究了其Rota-Baxter算子的关系、“可复制”性等；2. 构造了集合上交换和非交换自由对象以及多项式代数上Reynolds算子；3. 给出了Reynolds代数基的组合解释。最近，A. Das给出了Reynolds算子的李代数版本，至此出现了Reynolds结合代数和Reynolds李代数两个新的代数结构，我们认为开展对Reynolds代数相关的研究是有意义的。今后我们对Reynolds 代数的研究将从结合和Lie两个方向开展，包括自由Reynolds李代数和Reynolds余代数构造，Reynolds family代数和量子 Reynolds 代数的研究，Reynolds代数的形变理论和Operad理论等。

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